Mathematical Basis of Rollie's Dad's Method

I had been working on this and posting about it in September 2010 or so,
and was getting confused. I retreated to the shop and took a bunch of
measurements on lathe and mill, and tried to fit data to the then math
model, and was getting odd results. Then my day job got too busy, and
all else drifted to a stop. Now I'm back on the task, prompted by a
passing mention on RCM, and may have figured it out. At least
mathematically, as I have not done any added shop work, but it may be a
while, so I'll post what I have.
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The original purpose of ³Rollie's Dad's Method² (RDM) was to align the
spindle rotation axis of a metalworking lathe parallel to its bedway,
such that the lathe would machine perfect cylinders. Many people found
the instructions somewhat difficult to use, even though they seemed
simple enough. It was not clear what RDM's abilities and limitations
might be, and it seemed likely that some unknown limitation was causing
the difficulties of application. It also appeared that RDM could have
applications to other machine tools. For these reasons, a mathematical
analysis was undertaken, and is summarized here.
The physical setup is that a quite smooth and round and reasonably
straight bar is clamped very firmly in the machine spindle. Nearby
there is a dial indicator mounted firmly to a machine component that is
designed to move parallel to the spindle rotation axis. The tip of the
dial indicator's probe rides on and is perpendicular to the side of the
bar. The bar is manually rotated by rotating the spindle, and the
following things are measured and recorded at various places along the
bar: the maximum and minimum indicator readings, the distance from the
spindle face, and the diameter of the bar.
The maximum dial indicator readings are those where the probe tip is at
the greatest distance from the spindle rotation axis, while the minimum
dial indicator readings are those closest to the rotation axis. Runout
is the absolute difference between maximum and minimum readings.
A rotation axis is by definition always an ideal straight line. If by
some misadventure the spindle is bent, the rotation axis may no longer
coincide with the spindle centerline, or be correctly aligned with other
parts of the machine, but the rotation axis is still a straight line.
The spindle with firmly clamped bar together form a rigid body. By
definition, the distances between the various points within a rigid body
do not change when the body is rotated and translated in any way. This
implies that all the points of the combined rotating bar and spindle
assembly independently describe circles centered on and perpendicular to
the spindle rotation axis. This is always true, regardless of the
details, however complex.
For various reasons, the actual spindle rotation axis may not be quite
parallel to the ways upon which that mobile machine component moves, and
our intent is to measure this deviation from parallelism despite use of
test bars that are not exactly straight.
A simple math model is introduced here. We define two functions that
together describe the test bar, being the centerline function and the
radius function. The centerline function gives the location of the
local centerline of the bar as a function of distance along the spindle
rotation axis. The radius function gives the radius of the bar as a
function of distance along the same spindle rotation axis.
Note the subtle shift from bar centerline to spindle rotation axis just
above. This simplifying approximation works because these two lines are
necessarily very close to one another, so the error thus committed isn't
large.
The bar is assumed to be everywhere round, when measured in a plane
perpendicular to the spindle rotation axis. There is a small error
caused by using planes perpendicular to the rotation axis, versus the
bar axis, as the cross section of a round bar measured at an angle to
the bar axis is an ellipse. However, such errors are insignificant if
the angle between spindle axis and bar centerline is small, the usual
case here, so we will ignore this error and use the measured bar
diameter without correction.
The maximum shown by a dial indicator is the radius function plus the
centerline function, while the minimum is the radius function minus the
centerline function; this is true everywhere along the rotation axis.
This implies that given the maximum and minimum functions, one can
reconstruct the radius and centerline functions. Specifically, the sum
of the max and min functions yields twice the radius function, while the
max function minus the min function yields twice the centerline
function. We will call these the reconstructed radius and centerline
functions respectively.
Expressed as equations, we get:
ReconstructedRadiusFn = (MaxFn+MinFn)/2, and
ReconstructedCenterlineFn = (MaxFn-MinFn)/2.
The reconstructed radius function is some combination of the actual
radius function of the bar and the deviation of the actual rotation axis
from the ideal rotation axis. Measurement of the actual diameter of the
bar at the test points along the axis allows the bar radius function to
be subtracted, yielding an indication of the deviations of the actual
rotation axis from ideal.
One worry is that a bar with multiple bends, such that it cannot rest
flat upon a surface plate, would cause problems. Also, the spindle axis
and bar centerline are almost never coplanar, instead passing at a small
angle to one another and yet never intersecting. Another worry is that
if the bar centerline axis crosses the rotation axis, which often
happens in practice, the method may fail. These worries are related, as
discussed below.
Because the bar and spindle form a rigid body, and because the bar is
everywhere round and large enough to completely contain the spindle
rotation axis, one will always get a reliable dial indicator reading,
and taking the average of max and min readings will always yield the
reconstructed radius function. But there is a twist. If the centerline
function crosses (or closely approaches) the spindle rotation axis, the
reconstructed radius function will have a minimum at the point of
closest approach. If the two lines cross closely or at a large angle,
the minimum will be quite sharp and dramatic. If the lines are almost
parallel and/or distant, the minimum will be quite broad, and the
reconstructed radius function will not be linear even if the test rod is
perfectly straight and cylindrical.
A simple test to determine how close the rod centerline approaches the
spindle axis is to also measure the clock angle of the spindle (with
respect to the headstock) at each measurement location. The indication
of passage nearby to the rotation axis is that the clock angle of the
maximum indicator reading will vary along the test bar, being on
opposite sides of the bar on opposite sides of the point of closest
approach. In other words, the side of the bar that's maximum (minimum)
on one side of crossover becomes minimum (maximum) on the other side of
crossover.
This is most likely the root cause of the confusing results from RDM,
which implicitly assumes that the test bar centerline is essentially
coplanar with the rotation axis, and that there is no crossover or near
passage. In the more common case that one or both of these implicit
assumptions is violated, the results can be equivocal or confusing.
Specifically, if one measures at only two locations, the apparent
orientation of the spindle rotation axis will vary with the test
locations relative to the location of closest approach (~crossover).
What to do? The simplest thing to do is to measure at no less than five
locations along the bar, and plot the computed distance to the rotation
axis (correcting for the actual bar diameter) as a function of location
along the bar. If this plot forms a reasonably straight line, standard
RDM will work well enough. If instead the plot forms a curve or V with
a minimum somewhere in the middle, there is a crossover. If one is
lucky, the minimum will be near one end or the other, often the spindle
end, and one can use the reasonably straight portion of the plot. If
not, one can try various bars and bar orientations in the spindle and
see if some combination is OK.
While it is possible to handle the crossover situation mathematically,
it's a bit more bookkeeping and math than is comfortable. The basic
approach is to measure and record the clock angle of the spindle at the
five or more maxima, and fit a line to the data, in cylindrical
coordinates.
The reconstructed centerline function is an unknown compound of the
imperfections of both bar and spindle. This compound cannot be resolved
into separate bar and spindle functions without making multiple
measurements with the bar repositioned to various clock angles with
respect to the spindle. However, repositioning the bar involves
unclamping the bar, which may shift a bit every time the rod is
repositioned, so it may take much effort to get statistically stable
separation of spindle and bar imperfections, unless the imperfections
are gross.
In theory, one could use a collection of test bars and machines, and
mathematically tease it all apart. But it's far too much work for the
limited accuracy it will yield, and anyone with enough machines to use
such a method should just get a real precision test bar.
Some practical details:
The bar must be quite round. This implies that raw stock will not work
as received, and one must instead use rods that are machined to be quite
round. Precision shafting is round enough, and the slight deviations
from straight will be compensated for by the RDM algorithm. In RDM, it
is suggested that one use the piston rods from old Pearson struts and
shock absorbers. Alternately, raw stock rods may be turned round in
each of the places that will be measured.
The bar must not permanently bend under test stresses, instead remaining
elastic. The bar must not droop due to gravity. The bar must be large
enough that the rotation axis is always within the test bar, at least at
the measurement points. All this implies that the bar diameter cannot
be too small, although it could be hollow to reduce weight, and that
hardening is helpful.
The bar must be held very firmly, so it does not shift in the collet or
chuck when the free end of the test bar is bumped sideways. The key is
to hold the rod so firmly that the rod will flex elastically before it
will shift, despite the considerable leverage of a long rod held in a
short collet. One way to tell if the grip is sufficiently firm is to
rap the free end sideways with a mallet and see how quickly the
oscillation damps down. If there is no oscillation, the clamp is not
nearly firm enough.
For the record, spring collets are usually not nearly firm enough,
because unless one is so lucky that the rod diameter exactly matches the
collet's actual inside diameter, the rod is held only by a circle of
contact, versus true area contact.
Application to typical machine tools:
In a lathe, the application is direct, the ³moving machine component²
being the carriage, and one performs the algorithm twice, once with the
probe axis of the dial indicator horizontal, and once again with the
probe axis vertical.
In a lathe, a 3-jaw scroll chuck ought to be firm enough, but a 5C
collet chuck may or may not be firm enough.
In a vertical mill, the ³moving machine component² is the knee, and the
relevant bedway is the vertical dovetail. What is measured is how well
the mill is trammed, where perfect tramming is defined as spindle
rotation axis being parallel to the vertical dovetails upon which the
knee moves. Note that this definition is not quite the same as the
usual definition of tramming, which makes the spindle axis perpendicular
to the table surface. In a new machine, the definitions are essentially
equivalent, but with wear on the table and the table ways, these two
definitions will drift apart.
Alternately, one may move only the quill, but this will tell us nothing
about tram.
As a practical matter, it's best to clamp all axes when making a
measurement on a vertical mill, and to use for instance a R8 arbor with
integral ER25 (or the like) head to clamp the rod. R8 spring collets
are not really firm enough, unless one is lucky.
Joe Gwinn
Ref: ³Rollie's Dad's Method of Lathe Alignment²,
, originally
"
formatting link
" (which no
longer works), Copyright 1997 by the New England Model Engineering
Society (NEMES).
Reply to
Joseph Gwinn
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I believe the part about the rotation axis still being a straight line, even with a bent spindle, is incorrect.
One may get lucky with using an inaccurate center in the spindle to achieve a close approximation of the workpiece end nearest the headstock rotating around an axis, but not the spindle's centerline axis.
With a bent spindle, a collet or chuck won't have a centerline axis, as the spindle's centerline axis will be an orbiting line, extending from the bend in the spindle.
A misaligned headstock with a straight spindle will exhibit rotation around a straight line, although not aligned to the bed.
Reply to
Wild_Bill
The spindle maybe, but the bar certainly isn't. It deflects from gravity and indicator contact pressure. Test it with your finger.
I just chucked an aluminum rod 1" dia x 12" long and touched an indicator that reads to 0.0001" to the outer end. The contact pressure of a second new indicator changes the reading by a few tenths. Light finger pressure moves it a thousandth easily. If the indicator spring compression force increases with the reading you have an error.
On my old tenth-reading indicators the drag from dirt and wear changes with the reading.
The light force of the new indicator deflects an 8mm printer rod 0.003" at 1 foot from the chuck.
jsw
Reply to
Jim Wilkins
The mechanical centerline of a bent spindle would rotate in a cone, but only if the spindle and its bearing aren't worn out of round or damaged during chuck removal. My Sears AA lathe had both a bent spindle and an egg-shaped bearing and the rotational axis varied with cutting pressure. Add in about 0.02" of bed wear, lower in front so the carriage rocked, and Rollie Gaucher's Dad's Method wouldn't guarantee a straight cut on it.
My South Bend's problem is that the tailstock spindle shifts with extension and clamping so a workpiece run between centers needs to be checked for taper every time, by measuring the diameters of trial cuts and changing the tailstock's position on the ways.
jsw
Reply to
Jim Wilkins
Wow, that's an eye opener...
Jon
Reply to
Jon Anderson
If the spindle is bent, and the chuck face is pointing off at a 45 degree angle, the test bar you install in it will still rotate around a straight line that is the spindle's axis of rotation. The text bar's centerline certainly isn't parallel with the spindle's axis of rotation, but the test bar's rotation axis is.
Reply to
Pete C.
In article , Jim Wilk> > > ...>
How do you know that the rod is not shifting in the chuck?
Steel is three times stiffer than aluminum, and also will cause less drag on the dial indicator.
An 8mm diameter bar is far too slender.
Joe Gwinn
Reply to
Joseph Gwinn
No, the rotation axis is always a straight line: .
Not necessarily. If the spindle and headstock are good, the rotation axis and spindle axis will coincide, but will be at some fixed angle to the bedway.
Joe Gwinn
Reply to
Joseph Gwinn
The chuck was stationary, I only measured static lateral deflection of the rod with a few representative forces. The printer rod is a typical diameter for them. IIRC Rollie's Dad used a shock absorber rod, but I no longer attend NEMES meetings to ask.
My point was to show that the spindle and rod are NOT necessarily a mathematical rigid body revolving symmetrically on the spindle bearings' central axis, not to suggest either material for use. Personally I'd go to the place out on 102 that makes hydraulic cylinders and pick out some rod cutoffs from his scrap bin.
You can easily check your choice of test bar for deflection caused by the indicator by pushing on it with a second similar one.
The deflection is a unidirectional offset error in the computed axis position that changes with carriage (indicator) position, and its magnitude can be significant with the wrong but tempting choice of rod. To magnify and clarify the error think about using dry spaghetti for the rod.
jsw, lab tech, gotta watch out for this kind of experimental error.
Reply to
Jim Wilkins
In article , Jim Wilk> > > ...
This does not answer my question about the rod shifting in the chuck.
The rods used by Rollie's Dad (RD) had to be at least 1" in diameter, made of some pretty good steel. Somehow, I don't think RD had much trouble with deflection due to dial indicators.
Stiffness varies as the cube of rod diameter, so it doesn't take all that large a rod to eliminate deflection as a practical problem.
In theory, yes, everything deflects to some degree. In practice, one can make the deflection pretty small.
Yep.
I did say this, well imply this, but perhaps with insufficient clarity and force.
Joe Gwinn > jsw, lab tech, gotta watch out for this kind of experimental error.
Reply to
Joseph Gwinn
Snip
Snip
I can't agree with the concept that you have named as "there is a twist"
I believe that the nonlinearty of the radius function that you are discussing is a complexity that doesn't exist . The case described is that of a test bar both slightly displaced and mounted at an angle so that it may cross the spindle axis somewhere near the middle of the rod. This means tha the bar is displaced in opposite directions at each end. This results in large TIRs near the ends and little or no TIR near the middlle.
Although it may look odd there is nothing wrong with this. RDM's method of using the halfway point of the TIRs STILL correctly indicates the surface of a cylinder that is truly parallel to the spindle axis. There is nothing nonlinear in the cylinder defined by this process
Provided the bed is straight, ANY pair of traverse readings will correctly show the same angular deviation from spindle axis.
The angular accuracy of the measurement is mainly determined by the traverse distance between measurement points and the reading accuracy of the dial indicator. It doesn't matter whether there's a small or large eccentricity at the measurement points. A secondary problem is bar deflection resulting from the force required to move the dial indicator pointer. The best solution is of course adequate bar diameter to length ratio. A useful cross check is to observe the dial indicator movement resulting from say a ten thou/1/4mm cross slide movement close up to the chuck compared with the result of the same movement applied at the far unsupported end of the bar.
Gravity droop of the bar is much less important because it occurs at right angles to the measurement plane. Simple pythagoras shows that even a gravity droop as large as 5 thou on a 1/2" bar would only result in 1/4 thou radius error.
Jim
Reply to
pentagrid
A straight rotation axis for a bent spindle exists between the headstock bearings.
A spindle could possibly be bent between the bearings or near the spindle nose bearing (but let's not examine 2 bends).
A lathe can't be properly aligned for any useful work in the usual sense which utilizes a chuck with a bent spindle, nor could a mill be trammed properly with a bent spindle.
A lathe with a bent spindle could produce a shaft turned between centers, but using a chuck for workholding isn't practical for performing work or alignment.
I just didn't understand why the bent spindle issue was just skipped over without any precaution.. particularly when the previous and following paragraphs are regarding preparation for the test setups which recommend the use of a chuck.
A bent spindle changes the normal workholding centerline axis of a chuck to a conical path as Jim W mentioned (my use of the term orbiting). The jaws of a chuck mounted on a bent spindle can't be trued by grinding, and actually, the chuck introduces a significant weight imbalance to the spinning spindle.
It's not practical to attempt to machine parts with a bent spindle and a chuck.. regardless of how many measurements (or adjustments) are made. A bored hole will exhibit taper, center-drilling the far end of a workpiece isn't possible since the point of the drill will attempt to inscribe a circle on the end of the workpiece.
A bent spindle won't even provide an accurate rotating point on a properly ground center.. the point of the center will wag/orbit within the workpiece's center-drilled hole. Placing a stationary flat surface (perpendicular to the spindle's axis) against the point of a center which is mounted in a bent spindle.. the point will inscribe a circle on the flat surface, instead of rotating precisely around the center's point.
Reply to
Wild_Bill
The chuck is a low-mileage Bison 3-jaw bought new, and was definitely tight on the test bars. It's a valid concern on an old lathe that needs this test.
In general I think a poor grip on the test rod would show up as unrepeatability, the reading could shift between two values at certain chuck positions.
My ex trade school South Bend has a good headstock and ways but the AA/ Sears I bought first had received considerable use and showed most of the associated wear, plus that model has an easily bent spindle nose, so I got to learn a lot about lathe faults, measurement and restoration, guided by "Machine Tool Reconditioning".
I've also had considerable formal training and practical experience in precision measurement and have learned to carefully examine all the assumptions of mathematical perfection, especially the actual stiffness of kinematic instrument structures, which at the microscopic scale act like rubber. I'm not just picking on you.
jsw
Reply to
Jim Wilkins
Actually the effect of a bent spindle is the same as a worn 3-jaw chuck. As long as the eccentric wear in the spindle bearing has been corrected (with solder and scraping) and the clearance taken up, the work piece still revolves around a single axis even though its OD wobbles. It can be center drilled and turned or bored straight, but not removed and replaced.
My mill had a slightly bent spindle that I didn't notice for years. It simply cut a bit larger than the end mill's diameter. I finally found a reamer and cleaned it up.
jsw
Reply to
Jim Wilkins
Mathematically, I think that it's just a practical application of the method of least squares. It'll wouk to align the headstock, but it won't correct other machining problems produced by ways that are curved from wear.
Reply to
Denis G.
If they are severe enough to swamp misalignment issues you might as well turn short sections to measured diameter. Then the lathe is still useable.
jsw
Reply to
Jim Wilkins
Agreed. Every once in a while I like to go back to an article that I found here called "In Praise of Klunkers." My old link doesn't seem to work anymone, but if I find it again, I'll have to store it on disk.
Reply to
Denis G.
In article , "Denis G." wrote:
Joe Gwinn
Reply to
Joseph Gwinn
In article , Jim Wilk> > >...
Even new chucks can be off, but yes.
I found that smacking the end of the bar and listening to be pretty effective a test, if the bar was thin enough that it would elastically bend with humanly reasonable forces. If there was slipping, one gets a clunk sound, and no tone (or smoothly decreasing oscillation).
I have been ignoring elasticity except to say that there should not be so much droop, mainly because I expect that people will use large enough bars turning slowly enough that elasticity won't cause problems. This allows one to assume an ideal rigid body, even though in practice no such thing exist.
Joe Gwinn
Reply to
Joseph Gwinn
Not quite. Actually, the TIR in the middle may me similar to that at the ends, if the distance at closest approach is no small compared to that at the ends.
Hmm. I think I'll set up a 3D math model of this case, and see. I already have most of such a model, and it may be able to generate understandable pictures.
I would think that even a worn bed would be straight enough.
Hmm? True, but I don't think I claimed otherwise. Are you just making an added observation?
I suppose that the best solution would be to use two dial indicators, one on each side of the bar, so the probe forces mostly cancel out.
Or, use a stiff enough bar that it doesn't matter. To put some perspective on it, dial indicator forces are in the tens of grams, while cutting forces are in the tens of kilograms.
For horizontal-plane adjustment, yes, But what about vertical-place measurements?
Joe Gwinn
Reply to
Joseph Gwinn

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