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 muchimproved
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 NeuFinder4 (NF4) deflectionload 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:
 Shafts with more than normal curvature would curve
upwards on the transformed graph.
 Shafts with less than normal curvature would curve
downwards on the transformed graph.
 Shafts with more than normal slope would slope upper
left to lower right on the transformed graph.
 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 ad above are very
worthwhile goals. They take the shafttoshaft profile differences
and turn them into an easilyvisible 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.
JAM13
USA
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
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 handwaving 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 ad. A few things that
could be better:
 The red curve (Mercury Savage) and the yellow curve
(EI70) 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
rawdata plots.
Is there a way to improve Messner's formula as it applies to NF4
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
NF4. 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 NF4, 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 NF4
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 NF4 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 NF4 beam length.
Whether it is 7" or some other number, this difference represents an
"offset" to
be subtracted from the NF4 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 NF4 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:
 Perhaps the set of shafts is not totally representative. It
may be biased by having tipstiff shafts and not tipflexible 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.
 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 NF4 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 ad 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 VistaPro (light blue) with a flexible
midsection and stiff butt and tip.
 The TrueTemper EI70 (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. 
Conclusion
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 offsetadjusted 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
