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 "