3 shafts in 3 holes Tolerance Analysis

[loughnane]

I'm writing some TA code and am having a tough time wrapping my mind around the math for the following.

I've got a plate. On this plate I have 3 holes (.130 dia) in a row, with 1" between each hole.

I've got another plate. On this plate I have 3 shafts (.125 dia) in a row, with 1" between each hole.

Tolerances are .001" on every dimension. I assume a CpK of 1.00 (plus/minus 3sigma)

Nominally, the two plates go together. What I'm working on is, if I make 1,000,000 of these parts, how many will fail?

My first crack at it is below. Note that I used only the hole/shaft radii for Hole 1 and Hole 3 (the outer holes), whereas Hole 2 I used the diameter because it is in the center.




Dim Tolerance
Hole - 1 (radius) 0.065 0.003
Hole - 2 (diameter) 0.130 0.005
Hole - 3 (radius) 0.065 0.003
Shaft - 1 (radius) 0.063 0.003
Shaft - 2 (diameter) 0.125 0.005
Shaft - 3 (radius) 0.063 0.003
Distance 1 - 2 1.000 0.005
Distance 2 - 3 1.000 0.005

Nominal Clearance 0.0100
Worst- case tolerance 0.030
RSS Tolerance 0.01118034
Sigma (assuming RSS = 3sigma) 0.00372678
Failures per million 3645.179046

Here is where I say that I don't feel comfortable with this, as I can't quite think of how I would sketch it out on paper.

Does this sound right?

Chris Loughnane - Product Design

http://www.pdnotebook.com

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[pmarc]

loughnane,

First of all post a sketch how these two parts are dimensioned.

 

[drawoh]

loughnane,

Arbitrarily select one hole and its interfacing shaft. Apply GD&T positional tolerances to both features.

Your part is located by an external feature, possibly the other two holes, possibly a couple of outside edges. It does not matter.

The shaft occupies a space defined by its diameter ØDs plus its positional tolerance ØG.

Ss = Ds + G

To assemble your parts, your hole must clear this space. If your hole is dimensioned with a tolerance ±t and located to a positional tolerance of ØG, it clears a space...

Sh = Dh - t - G

If Sh < Ss, your assembly can fail, although it is not guaranteed to.

On my drawings, I like to dimension holes ØD+t/-0, and I set the position tolerance to zero at MMC. This simplifies the math.

JHG

 

[Belanger]

Is the only thing that aligns the two parts the holes/shafts? Or are the two parts butted up to common edges? I'm just trying to clarify what the datums are....






John-Paul Belanger
Certified Sr. GD&T Professional
Geometric Learning Systems

 

[pmarc]

This is exactly what I wanted to figure out by asking for sketches of the parts.

 

[loughnane]

drawoh,

I agree with your method... the difficulty I am having is in converintg "can fail, although it is not guaranteed to" to a statistical equation.

attached is a basic dimensioning scheme. This sample is not going to be a production part, rather a case study used to prove out some mathematics.

So there is nothing locating the two parts aside from these three shafts.

Thanks for all the quick responses as well... sorry I didn't respond as quickly.

Chris Loughnane - Product Design

http://www.pdnotebook.com

http://www.twitter.com/DesignNotebook

 

[drawoh]

loughnane,

The problem with any sort of statistical distribution is that the bell curves and the various sigma values have nothing whatsoever to do with your tolerances.

The tolerances on your drawing tell your fabricator what is required from the parts you will accept from him. You can apply tolerances that, if met, are 100% guaranteed to work.

What you need to know for your statistics, is the capability of your manufacturing process. You are going to have to talk to your fabricator, and/or you are going to have to inspect hundreds of parts to establish standard deviation. Your inspection of hundreds of parts is valid for the vendor and his current setup, only.

Do you really need to be this fancy?

You can chat with your fabricator, and set tolerances he guarantees he can achieve. The worst case is that you will have to enlarge your holes a bit to make it all work.

JHG

 

[drawoh]

loughnane,

I just looked at your blog, and your article statistical tolerance analysis basics: Root Sum Square (RSS).

I think you missed an expense. You have a million parts, 99.7% of which meet specifications. Your scrap cost is not just the 3000 non-conforming parts. You have to add the inspection process.

Another option would be for workers to toss un-assemblable parts onto a separate pile, which could be searched and sorted, later, for stuff that assembles.

There was a good article on Japanese cars which claimed that they apply sloppy tolerances on their drawings, and provide the assembly workers with spacers to make sure everything assembles properly. My interpretation of all this was that the Japanese were preparing their drawings properly, with achievable tolerances. The suppliers were taking them seriously, and manufacturing could make accurate assumptions about the dimensions of their parts.

You have an interesting blog.

JHG

 

[loughnane]

Drawoh,

I'm assuming that the cpk is 1.000. Ultimately I'm using the math to drive some analysis.

For example, on the front end of a project we usually have to downselect between mechanism concepts. it is nice to be able to punch in dimensions that describe the mechanism. Once the dims are in I have vlookup functions that determine the tolerance based on DIN 16901, Injection Molding Handbook, Machinery Handbook etc.

Once I've got the tolerance stack i assume a CPK of 1.00 (1.33 seems to be more of an average, but better to be safe on the front end rather than tolerancing ourselves into a corner). Then I can see roughly how many failures I will have based on my annual quantities.

The only hole so far is that while positional tolerance stacks are basic bath, when you get into true position of holes and such it gets a little more complicated... like the 3 holes and 3 pins I've got now.

You make a good point on inspection. I spent about a year in a manufacturing environment and one of the big takeaways was the flaws inherent in inspection. I've heard it said that 100% inspection is only 80% accurate, so on the back end I think that is certainly something to keep in mind. I suppose the hope though is that we (with the help of tolerance analysis) select such a robust mechanism concept on the front end that minimal inspection is needed on the back end.

It will probably be a bit until I break down the math of this (things are really busy), but I will be sure to post up my solution once I have it coded.

Do you have a link to that article? Or recall where you saw it?

Chris Loughnane - Product Design

http://www.pdnotebook.com

http://www.twitter.com/DesignNotebook

 

[drawoh]

loughnane,

Here is the article: The Quest for Imperfection.

My interpretation of the article is that the Japanese are producing fabrication drawings with realistic tolerances, and that the suppliers are meeting those tolerances. The relative lack of inspection reflects the confidence of the Japanese that the tolerances are being met.

One of the things I need to do to ensure an assembly work, is a tolerance stack-up analysis. If I have tolerances on the drawings, I can do that analysis. If my vendors meet my tolerances, my analysis is valid, and I can use it to ensure my assembly works.

Your analysis is feasible if I set my tolerances to ± the manufacturer's 3[&sigma;]. We can assume that the variation is a bell curve about the median dimension.

Aside from the fact that much of my tolerancing is not based on 3[&sigma;], you have to consider alternate sources of error. Perhaps someone setting up a machine failed to notice a hex nut sitting next to a stop. Everything now is out by .190". For a lot of processes, the mean measurement will vary from setup to setup.

A very good design for manufacture strategy is to keep all tolerances well outside the ±3[&sigma;] of their manufacturers.

JHG

 

[loughnane]

Thanks for the article and all the insight.... always good to get an informed opinion.

hope to be back up here soon.

Chris Loughnane - Product Design

http://www.pdnotebook.com

http://www.twitter.com/DesignNotebook

 

[Belanger]

Good stuff, Chris!

Also be aware that there is a middle-ground approach -- while a worst-case stack is usually too restrictive, it can be said that the RRS method is "too good to be true."

RSS is great in theory, but in the real world there may be other factors that influence things. So many companies use a fudge factor coefficient in front of the RSS formula. This is called "modified RSS" and was first proposed by a guy named Bender back in the 1960s.

He took the theoretical data and overlayed it with actual lab data, and found that a coefficient of 1.5 aligned the two sets of data.

Similar to that is the "Gilson method," which uses a variable fudge factor, based on the number of dimensions being stacked.

So if you're writing code for real-world use, maybe check into these other variations of the RSS approach.

John-Paul Belanger
Certified Sr. GD&T Professional
Geometric Learning Systems

 

[KENAT]

I asked a question a while back on if anyone had anything on applying RSS (or variations there on) to hole patterns specified with GD&T/Position and didn't get much usefull response.

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[Belanger]

Kenat -- Yes, when I teach tolerance stacks I go through a simple spreadsheet method to calculate stacks, including position and even bonus/shift tolerance. Then there's a column further to the right that simply does RSS or modifed RSS on those results.

The RSS itself is no big deal for position or and GD&T control; simply do the RSS formula on the totals. The hard part is factoring in the position (and bonus/shift) to begin with -- GD&T is often not a bilateral tolerance, so it's different than stacking simple +/- numbers.

John-Paul Belanger
Certified Sr. GD&T Professional
Geometric Learning Systems

 

[loughnane]

Belanger,

I've seen some people do something similar to what you mention. Specifically, they average the difference between the RSS stackup and the worst case stackup.

While that may work, you are going to have an awful hard time defending it mathematically. Aren't you better off just doing an RSS stack and adjusting your assumed CPK?

Chris Loughnane - Product Design

http://www.pdnotebook.com

http://www.twitter.com/DesignNotebook

 

[drawoh]

KENAT,

What decisions are you going to make with all this analysis?

Take the case that I manufacture 10,000 widgets, and I carefully inspect thirty pieces, all on a weekly basis. I establish the mean and the standard deviation.

If [&mu;]±3[&sigma;] are well inside my tolerance range, I have a batch of good parts, and I can claim to be a zero defects manufacturer.

If [&mu;]±3[&sigma;] extends somewhere outside my tolerance range, a significant number of my parts do not conform. I can either say to heck with it, and ship as is, or I can inspect every single piece in the questionable batch.

Probably, I will quarantine the batch. The effort required for inspection will be significant, and it may be cheaper to scrap everything. In effect, if my process fails, my scrap rate is 100%. I fix my process to get everything back in specification.

Another possibility is that I will learn that my tolerance range is less than 6[&sigma;]. In this case, I have to inspect 100% of my components as an integral part of my manufacturing process. The inspection and scrap would have to built into my costing.

In terms of decisions and costs, there is probably not much difference between 0.5% non-conforming, and 5% non-conforming. At some point, scrap becomes a significant part of production costs, and the statistical analysis will help you to predict it.

My basic DFMA strategy would be to try very hard to ensure that my tolerances are way looser than my manufacturer's 6[&sigma;] values.

JHG

 

[KENAT]

Well, my role has changed quite a bit since then so probably nothing!;-)

We do low volume, relatively high precision (some more than others) stuff.

We sometimes have situations where the required functional tolerance approaches (or is tighter than) typical manufacturing capability based on worst case tolerance analysis.

In these cases taking into account that 'worst case' is typically overly conservative can have use.

We often don't have the luxury of tolerancing well inside the manufacturers capability.

However, my question was specifically about hole patterns dimensioned using position tol, not run of the mill +- tol stacks.

As Belanger it's a bit more complex with the impact of the second dimension, MMC etc.

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