Shaft Deflection Profiling:

Plotting the data

Dave Tutelman -- October 11, 2008

Abstract - It is difficult to learn much about a shaft profile from a raw plot of deflection vs beam length. In a previous article, I introduced the notion of transforming the raw deflection load data to make the graph easier to read. Here is a much-improved transform. It is easier to understand, as well as no longer depending measuring some "standard" shaft. Instead, the data is subjected to a simple calculation not based on comparison with any other shaft. Using this model, the plot itself is:

Load * (BeamLength - Offset)    vs    (BeamLength - Offset)

Good values for Offset are 7" or 8.4".

The Problem

Let's review why transforming the data is important. The figure shows a plot of the raw NeuFinder-4 (NF-4) deflection-load data for four different shafts. I chose these particular shafts because they had the biggest profile differences among the shafts I had measured. The differences should be readily apparent on any profile graph we plot. In fact...

A graph is useful if -- and only if -- you can learn something or get some insight from eyeballing it. There are good and bad things about this graph, as far as eyeball nourishment is concerned:
  • It is easy to see which shafts have the stiffest tips, and by how much; the curves are quite separated from each other at the left -- the tip end of the curve.
  • It is a little harder to assess the shafts' relative butt stiffness. That is because the butt loads are much lower. It is hard to plot the tips and butts on the same scale and still preserve the separation of butt loads. (This could perhaps be accomplished by plotting it on logarithmic graph paper, but that would not solve the next problem.)
  • It is really difficult to look at the graph and gain much intuition about the profile. All the curves have roughly the same shape. They are all sloped from upper left to lower right, and are concave upward. The amount of concavity differs. For instance, it isn't hard to see that the light blue curve (the Fuji shaft) is more concave than the others. The red shaft (Mercury Savage)probably has the least concavity and the greatest slope.
But these differences are easy to see mostly because the plots are done together on the same graph. Suppose we had to look at each plot separately? We see this below.

It isn't nearly as easy to tell whether a particular shaft has more or less curvature, more or less slope. And we still have the other plots on the same page -- even if they aren't plotted on the same graph -- to help a little with comparison. Looking at a single curve without any others visible would tell us almost nothing about the profile.

This is why we need some transformation on the raw data to emphasize differences in the shape of the profile.

My first attempts at a transform depended on comparing the measured shaft with some ideal or "standard" shaft profile, which would show up as a horizontal line on the graph. Using this model:
  1. Shafts with more than normal curvature would curve upwards on the transformed graph.
  2. Shafts with less than normal curvature would curve downwards on the transformed graph.
  3. Shafts with more than normal slope would slope upper left to lower right on the transformed graph.
  4. Shafts with less than normal slope would slope lower left to upper right on the transformed graph.
Dependence on some arbitrary ideal shaft was a fundamental weakness of the scheme. But points a-d above are very worthwhile goals. They take the shaft-to-shaft profile differences and turn them into an easily-visible shape. I now have a transform that accomplishes the shape goals without requiring measurement of a real shaft as a standard.

Messner's Suggestion

The key hint came from Jay Messner, who posted the following note to the Tom Wishon online forum on January 2, 2005.
Posted - Jan 02 2005 :  10:02:45 PM

I also use Excel for my data when doing zone profiling. One problem with just using raw frequencies for graphing is that when plotted the differences between shaft profiles is difficult to distinguish because of the large range of the frequency across all the zones on the graph. One method I've used is to multiply the frequency by the extended beam length for each zone as this produces, just by luck, a fairly constant result. I've had some shafts that are almost a perfect flat line using this method. The real benefit is when you compare shafts you can really see the differences in zone stiffness.

I've seen other methods described that basically graph deviation from a "standard" shaft.

Jay writes it off to "just by luck", but there is more than luck at work here. The red curve in this graph is a plot of  y=1/x. It is the same basic shape as the plots of load vs beam length, at least as far as the naked eye can see.

Note that, if we were to use Messner's formula of load times length on a 1/x plot like this, it would give a perfectly horizontal line -- just as he says.

So perhaps a "neutral" (let's use this word instead of "standard") shaft has a load profile of 1/BeamLength.

I could even provide a hand-waving sort of argument that says a neutral shaft should give a horizontal line when we multiply load by beam length. Load is a force, so load times length is a torque. What we are saying is that, if the shaft has a uniform "stiffness" (whatever that means) all along its length, then deflecting the tip a constant amount should produce a uniform torque on the support. Yes, it's pretty vague, but gives us a bit of intuition about why Messner's formula might work -- and not just be a lucky coincidence.

Let's try it for deflection profiles.

Here is the result of plotting load times length for the same four shafts as above.

It does compress the "dynamic range" of the graph we look at, so it is easier to compare the magnitudes for the different shafts along the length. But it is still not ideal; it still does not meet all of criteria a-d. A few things that could be better:
  • The red curve (Mercury Savage) and the yellow curve (EI-70) are still roughly a sloped line. The slope is rather different, so that is an improvement. But, without the other for comparison, it would not be easy to see the difference.
  • Every single shaft has a net slope downward to the right. We should want shaft differences to show up as a slope difference in kind, not just degree.
  • No shaft is concave downward. On the positive side, the light blue curve (Fujikura) is the only one that is still concave upward; that's good because it had the strongest concavity of the raw-data plots.
Is there a way to improve Messner's formula as it applies to NF-4 deflection profiling? Yes there is.

Beam Length Offset

Beam length is beam length, right? Not hardly. Beam length turns out to be quite different for a frequency meter and an NF-4. Here is the difference.

When a shaft is measured for frequency profile, the end closer to the butt is clamped in a firm cantilever kind of support. Typically this is a 5" clamp. The "beam length" is the unsupported length of the shaft, from the front of the clamp to the tip of the shaft.

In an NF-4, there is no clamp. Instead, the shaft is bent between three sets of bearings. (The left, middle, and right bearing sets are designated L, M, and R in the diagram.) The left bearing block presses downward on the shaft, and the middle bearing block presses upward. The distance between them is 9.4". (Don't ask, it's historical. It's called the NF-4 because it has predecessors.) Then the shaft is deflected by the right bearing block pressing down on the tip. The "beam length" is the total length between the left and right bearings.

So the beam lengths are different. Jay Messner says that frequency times beam length is a good transformation for frequency profiles. So... Is there some length measure on the NF-4 that is equivalent to the beam length of a cantilever clamp? Turns out there is.

Almost all the shaft bend occurs between the middle bearings and the right bearings. The shaft between the middle and left bearings is stiffer, and there is a shorter section of it. If the shaft had no bend at all between the two leftmost bearing blocks, then it would be acting as if those two bearing sets constituted a clamp. In other words, the equivalent beam length would be exactly the distance from the middle bearings to the right bearings.

But there is some small amount of bend to the left of the middle bearing. So the equivalent beam length extends a little to the left of the middle bearing. I have done some analysis, and it seems that the equivalent beam length is about 7" less than the conventional NF-4 beam length. Whether it is 7" or some other number, this difference represents an "offset" to be subtracted from the NF-4 beam length when we compute the transform.

So let us see what happens when we plot load times equivalent length, where the equivalent length is the beam length minus some offset. Here are the graphs for offsets of 4", 6", 8", 10", and 12".

Just by eye, we can see that the offset of 8 inches is the least biased, the one where positive and negative slopes balance best. That is remarkably close to the theoretical difference of 7", the difference between the NF-4 beam length and the equivalent beam length. Some experimentation shows that an offset of 8.4" gives the most balanced display for this set of shafts.

The discrepancy between 8.4" and the theoretical 7" might be explained by one of:
  1. Perhaps the set of shafts is not totally representative. It may be biased by having tip-stiff shafts and not tip-flexible shafts. That does not square with other studies I have made with this transform, using a pretty wide variety of shafts. If anything, the selection in the graph is biased to soft tips, not stiff. But still it may be the case.
  2. Perhaps the 1/x curve that justifies multiplying load by length is close to representing a neutral shaft, but not exact. There is some evidence for this. I plotted the same shafts' profiles using actual measured frequencies. I got fairly similar curves, with the least bias at 1.25 inches of offset. So the NF-4 and frequency curves differ by 7.15" (that's 8.4"-1.25") -- almost exactly the expected 7".
Here are the four shafts we started with, plotted using an offset of 8.4". The transform satisfies our earlier criteria a-d for a good visual display. We can easily see four very different shaft profiles with the naked eye:
  • The Mercury Performance (dark blue) with a fairly stiff butt and very flexible tip.
  • The Mercury Savage (red) with the most flexible butt and the stiffest tip.
  • The Fujikura Vista-Pro (light blue) with a flexible midsection and stiff butt and tip.
  • The TrueTemper EI-70 (yellow) with a stiff midsection and flexible butt and tip.

And all of this is visually very apparent. Even without the other curves for comparison, the shapes are unambiguous in what they tell us. That is the value we look for in a profile graph.


It is difficult to learn much about a shaft profile from a raw plot of deflection vs beam length. You can infer much more about a shaft's profile if it is plotted vs the offset-adjusted beam length as:
Load * (BeamLength - Offset)
An offset of 8.4" works best visually. An offset of 7" matches the theory. Either one is vastly superior to plotting the raw data.

Last modified -- Oct 12, 2008