Mathematical analysis of Rollie's Dad's Method

As I mentioned in "Re: Millrite MVI spindle bearing repair - Second report" posted on 5 September 2010, I'm looking for a way to measure total runout
without use of a $250 precision test bar, and Rollie's Dad's Method of Lathe Alignment <http://www.neme-s.org/Rollie%27s_Dad%27s_Method.pdf was suggested as an approach that could be adapted to the task. We will call this the RDM method, or just RDM.
Reading various postings about attempts to use RDM, people seemed to be having some problems getting it to work. As did I. I may now know why. A simple but fundamental error may have crept in over the years.
Consider a circle rotating about an axis displaced from the center of the circle. (This is all in 2D, and the various axes are perpendicular to the plane of the circle.) Using RDM's nomenclature, the radius of the circle is R, and the distance between rotation axis and circle center is X. In other words, X is the runout.
Using a dial indicator or a dial test indicator, we will rotate the circle about the rotation axis and measure the maximum and minimum values. We have adjusted the indicator so that measurements are all positive (or all negative), with greater absolute values signifying greater distances from the axis of rotation. We will assume positive measurements in the following paragraphs.
Now, by geometry, the maximum reading will be (R+X), and the minimum reading will be (R-X).
By RDM, we compute 0.5*[(R+X)+(R-X)]= 0.5*[2R]= R, which is the radius of the circle, regardless of the runout X. If we measure the diameter D with a micrometer and compute R-D/2 as suggested, what we get is a measure of the departure from roundness of the circle. We do not get the runout, which has already cancelled out.
If we instead compute the difference, subtracting the smaller measurement from the larger measurement, it's the circle radius that cancels out instead, and we now get the runout 0.5*[(R+X)-(R-X)]= 0.5*[2X]= X that we seek, uncontaminated by the radius R of the circle. (Assuming that the "circle" is in fact round enough.)
Wherever we compute this difference, the radius R of the circle at that location will cancel, yielding the total runout X at that location.
Now, I bet that Rollie's Dad knew this and was solving for X and not for R, so a small error crept in as the method was passed along.
If one measures X at different locations along a round rod, it is possible to fit the data to a linear equation, and this equation can be used to predict total runout as a function of position along the rod.
I made the needed measurements on my Millrite, so will use the data in the following example:
Close to collet: Max=0.0020", Min=0.0015", so X=0.00025".
Away from collet by 4.135" (from the DRO):Max=0.0040", Min=0.00145", X=0.001275".
Now, fit these data points to the equation y=a*x+b (X and x are not the same), with all runouts multiplied by 1000 for convenience:
0.25=a*0+b, so b=0.25 mils.
1.275= a*4.135+0.25, so a=0.2479 mils per inch.
The full equation is thus y= 0.2479*x+0.25, yielding mils of total runout as a functionm of distance in inches from where the "close to collet" measurement was taken.
In other words, the total runout is a quarter mil plus a quarter mil per inch along the rod.
As mentioned earlier, the Millrite MVI specs are 0.0005" total runout near the spindle nose, and 0.001" at 8" from the spindle nose, using a test bar. This yields the equation y=0.5+0.0625*x.
Near the spindle nose, we are seeing only 0.00025" total runout, half the allowed 0.0005" runout. This is probably due solely to the lateral runout of the bearing closest to the nose, and cannot be much improved.
At 8", we would see about 0.25+8* 0.2479= 2.2332 mils, or 0.002233" total runout, which exceeds the 8" total limit by a factor of 2.233.
The key problem is the angular error, 0.2479 versus 0.0625 mils per inch.
A machine made in 1965 need not apologize for having only twice the runout it had in its youth. That said, properly orienting the outer races may help a great deal.
I should also list the fundamental assumptions underlying the above methods:
First, while the rod need not be straight, it must be quite round at all places measured, so a piece of raw stock will not work. What will likely work the best is precision ground shafting, which is quite round but may have a few thousandths of curve per foot.
Second, in the above linear fit, we implicitly assumed that the line between the two measurements does not cross the axis of rotation. While this is usually true, it is not guaranteed. A quick test is to measure in at least three places along the rod, and plot the value of X as a function of position along the rod. If they fall in a line, no significant crossover. If they form a V, there is crossover. If necessary, one can keep track of runout directions and fit to the actuals.
Third, we implicitly assume that the measurements all fall on a common line (are colinear), and that this common line and the axis of rotation together define a common plane (in other words, the lines are not skewed with respect to one another). This is never quite true, although is is usually true enough. To detect skew, one measures both runout and clock angle at a minimum of three places along the bar and does some fancy math.
I hasten to add that the machine accuracy specs make the same assumptions.
Joe Gwinn
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Although the article doesn't say that, it may be that RDM is ultimately trying to measure bed twist, but the method as published cannot achieve that. The published method measures rod diameter despite runout, and there is no way to deduce bed twist from rod diameter.
What the article claims to be measuring is the deviation of the spindle rotation axis from parallelism with the ways, in two dimensions, horizontal and vertical. The published method cannot do this, but changing only the math from summing the runout max and min to taking the difference allows this deviation to be measured.
What was done with the information was to cleverly shim the headstock where it rests on the bed, to achieve parallelism.
Actually, with the sum, we don't get the full radius, we get the change from some unknown constant, because we never zero the dial indicator at the unknown center of rotation, we set the dial indicator up at some convenient offset, and go from there.
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On reflection, I think Ade V has put his finger on the answer.
Rollie's Dad (RD) was measuring using a dial indicator on the lathe carriage sensing the spinning rod, the intent being to see how well aligned the headstock was, using a random bit of round rod held in a chuck with random crookedness. So, RD wanted to cancel the runout and crookedness, yielding the diameter of the rod plus some constant.
The easiest way to visualize the runout is to imagine the spindle axis tracing out a cone in space. To align the headstock, the axis of that cone is made parallel to the bedway. The raw measurement is a combination of runout, actual rod diameter, and deviation of cone axis from parallel. Cancelling the runout yields the local apparent rod radius, which is the combination of actual rod diameter and cone axis deviation. If the rod is a perfect cylinder, with constant radius everywhere, then a constant dial indicator reading as the carriage moves implies that the cone axis is parallel to the bedway. If the rod radius varies with location, one must measure the actual rod radius and subtract it to get the distance to the cone axis.
Now, by contrast, I'm currently interested in the runout that RD ignores, and want to ignore the rod radius that RD uses.
So, to summarize (in the context of a vertical mill):
One half the *sum* of the the indicator measurements (corrected for rod radius) yields the deviation from perfect parallelism between Z-Axis ways and/or quill motion, ignoring runout and chuck crookedness. This is RD's Method (RDM).
One half the *difference* of the the indicator measurements yields the runout, ignoring crookedness and imprecise parallelism.
One set of measurements can be used to compute both non-parallelism and runout. In practice, the only part of non-parallelism that is adjustable in most vertical mills is tramming.
One can also deduce crookedness of the chuck by running the test with the same bar rotated into different positions, so bar crookedness can be separated mathematically from chuck crookedness.
A practical note: I've found that R8 spring collets don't hold the rod quite rigidly enough, probably because of a very slight mismatch between actual rod diameter and actual hole diameter of the collet, so the rod is clamped in a ring versus over an extended area. If one tugs on the rod, it will permanently shift by a few tenths, and won't usually return to zero if the rod is plucked and allowed to vibrate down to zero. A Jacobs Superchuck is somewhat more secure in that while it also moves, the bar returns to ~zero when it is plucked. I will next try an Albrecht keyless chuck, which is likely far more precisely made than the superchuck.
What should work far better is a R8 to ER arbor, as ER collets have a far better grip on a rod than a R8 spring collet. And ER collets have many more uses than a test bar.
We may also be seeing the R8 arbor shifting in the spindle, but given the taper it should return to zero when the bar is plucked. A test bar would have the same problem with shifting in the spindle.
Joe Gwinn
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On Mon, 13 Sep 2010 09:12:17 -0400, Joseph Gwinn

From reading your original post.
Rollie Dads Method of measuring bed twist is useful and well established.
For the commonplace case of a reasonable straight and round test bar it gives the correct answer.
As you point out, Rollie's "correction" for diameter is in error. At both the near and the far measurement points, the rotational axis is truly defined by the difference between the highest and lowest clock readings and is independent of diameter and diameter change.
However the detail content of Rollie's description is very useful. I think it might be helpful to post to the drop box an agreed RTM Mk2. Changes needed could be pretty small :-
In each paras 5 and 8 delete the last sentence
"why this method works" - delete or modify
Jim
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wrote:

I think it measures far more than just bed twist. Any misalignment between spindle axis and bedway should show up.

But lots of people seem to have had problems. Now, I was never sure if it was due to inadequate explanation, or some error in the method itself. Or maybe both.
I'll have to try RDM on my lathe, for the experience. I think I recall having tried it, and having gotten nowhere, but don't know why. It may have been that I didn't really understand what I was doing mathematically, and so was doing random things.
I ordered a R8 to ER25 collet chuck today, so I will soon be able to get better runout data.

I'll have to think about this, as it seems to me that the difference yields the pure runout regardless of where the cone axis might be.

I had been thinking along these lines as well. At the very least there is an extension to RDM, and there may be a small correction as well.

I'm not sure I understand. Please quote the sentences to be deleted.

Yes, modify.
Joe Gwinn
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On Wed, 15 Sep 2010 23:46:28 -0400, Joseph Gwinn

It's true that the method also shows up spindle alignment and possible carriage alignment change but in practical terms, adjustment of bed twist is normally the only available method of correction.
RDM is often proposed as an alternative to the use of a precision level which only detects bed twist . It may be worth discussing this method in an RDM revision because the method is often described as a series of level measurements of the bed surface with the lathe as a whole needing to be precisely level.
If you have precision level it, is very much simpler to mount the level on the crossslide and observe the change in reading as the carriage is traversed.
It is unnecessary for the lathe to be precisely level because you are now measuring directly the effect of bed twist or distortion on the cutting tool location. Even if the lathe were large enough and flimsy enough for gravity induced deflections to be significant this would be indicated directly by this method
Both para 5 and 8 amended to only read "Average the high an low readings (add together and divide by two) to get the "near end average distance". Delete " if you suspect .........."
An alternative wording could be " Note as the reference distance,the mid point between the two readings"
The centre line of the bar mounted in the chuck describes a cone whose centre line is coincident with the lathe rotational axis. The reference distances describe a line truly parallel to that axis. The reference distances assume that the lathe bed is straight. Bends or bumps would introduce their own errors.
Jim
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snipped-for-privacy@yahoo.com wrote:

RD also mentioned shimming the headstock where it rests upon the bedway, but you are right that the adjustment options are limited. In practice one might do both in alternation, so the lathe converges to as perfect alignment as can be obtained given only those two "knobs".

This could work, but it would be necessary to separate the effects of headstock misalignment and bed twist, or one could end up turning the wrong knob, and making things progressively worse.

I do have a 6" Starrett model 98-6 precision level (0.005" per foot per division), although I usually slide it around on the tops of the bedway V rails, the method recommended in the Clausing manual. This will not work for all bedway designs, but the ride-the-carriage method should work universally.
The tops of the V rails do not wear in normal use, as the carriage rests on and wears away the flanks of the V rails, so for older machines using the level on the tops of the rails should be more accurate than riding the carriage. But I'll have to think about this - I don't know how important it will be in practice.

Yes. Maybe this also explains how people adjusted lathes in ships at sea.

Ahh. These reference the original RDM description, not my postings.
The original description could be clearer for sure. I would start with the why-this-works explanation of the math, then move on to some specific applications, on the theory that the detailed method is easier to understand if one knows how and why it all fits together.

Not to mention uneven bed wear near the headstock, often an issue with HSM iron. But again, I wonder how important this is in practice.
Joe Gwinn
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On Thu, 16 Sep 2010 09:06:46 -0400, Joseph Gwinn

While it is true that there are many secondary errors that can contribute to the total alignment error I believe that bed twist is both the commonest major error and is also the easiest to correct.
With level measurement, if the sides of the V ways are worn, it is even more important to use the carriagre mount method as this then shows the total effect of both wear and misalignment on the cutting tool location.
Jim
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snipped-for-privacy@yahoo.com wrote:

[snip]
[snip]
Bed twist is certainly easy to remedy, but in an old lathe, many things are worn a bit, and there is a distinct limit to how close one can get to the original factory performance. So, by "important in practice", I'm assuming an old lathe used by a HSMer.
People also shim headstocks, and I've always wondered if they were unknowingly fixing a bed twist the hard way.

It's true that the carriage mounted indicator will show the combined effect of misalignment and wear of bedway (and carriage), but if one adjusts the headstock so the spindle axis parallels the effective bedway near the headstock, it will be misaligned away from the headstock. It's a tradeoff to be sure, and most work is close to the headstock. But shorter workpieces are less sensitive to misalignment in the first place.
I guess my instinct is that it's best to align the spindle axis with the unworn bedway. But this will be a matter of personal preference, at least partly determined by how worn one's lathe really is.
Joe Gwinn
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