EI Machine to Profile Golf Shafts
Dave Tutelman  November 14, 2008
The engineering description of a flex profile is the
variation of EI
along the shaft. EI is a structural term, the abbreviation of "E times
I", where:
 E stands for
"modulus
of elasticity", a measure of the stiffness of the material. In
particular, it says how much force it takes to stretch the material a
given amount. Steel would have a very high E, and rubber a very low E.
(Do not confuse elasticity with strength. Strength is the force it
takes to damage the material, not to stretch it.)
 I stands for
"area
moment of inertia", a measure of stiffness of the cross sectional geometry of the
shaft. Thicker walls or a bigger diameter make for a higher I.
This is different from the way clubfitters think of shaft profiles.
When you use a frequency meter or a NeuFinder to measure a profile,
each measurement is a weighted average of stiffness over a
section of the shaft from the tip to some some point X, and you vary X
to make a
graph of the profile. EI measures the local stiffness at X, not an average from X to the tip.
How it works
If you
support a beam (say, a golf shaft) at two points and apply a
force halfway between those points, you will deflect (bend) the beam.
It is pretty easy to calculate how much the
beam flexes. The wellknown formula for deflection at the middle, where the force is applied,
is
where:
 y is the deflection
 F is the force
 L is the length between the supports
So the way to measure the EI along a shaft is to support part
of it between two points and do one of two things:
 Either apply a force in the middle between the
supports, and
measure the deflection, or
 Deflect the middle a known amount, and measure the
force required to do it.
Then
you compute EI by solving the equation above:
What is an EI profile? The
stiffness of a golf shaft varies along its length; the
EI can easily be 34
times higher at the butt than at the
tip. An EI profile is a graph of the EI over the length of the shaft.
The way you measure it is to go through the steps above (apply force,
measure deflection, and compute EI) for a series of points along the
length of the shaft, then make a graph of those measurements.
The
formula works well if the stiffness doesn't vary much over the length
of
shaft between the
supports. But the EI may change quickly enough that it can vary
significantly even in the short distance between the supports. If EI
varies too much between the supports, the formula loses accuracy. In
the diagram, the closetogether (green) set of supports does not see
too much change in EI, while the fartherapart (blue) pair of supports
sees a substantial change. If we consider the measured EI as the EI
halfway between the supports, the green is obviously more accurate.
So the trick is to find a
combination of a force and a distance between supports so that:
 You are able to deflect the shaft enough to measure
the
deflection and force with some precision (implies a big force
or a big distance).
 The distance is small enough so that the EI is fairly
constant between the supports (implies a small distance).
 You don't damage the
shaft (implies a small force).
Obviously these criteria are in conflict. The art of designing an EI
machine is finding a good tradeoff of distance and force. 
Description of my EI machine
Here are a couple of pictures of a machine I made to measure the EI
profile of golf shafts.
The first picture is an overview of the machine. What you see is an
orange shaft resting on two supports 11" apart. The shaft is preloaded
with a small weight hanging on it in the middle of the 11"
span. There is also a 15pound weight hooked to a storage loop on the machine;
in
this picture the weight is not loading the shaft, but just waiting to be used.
The
second picture is a
closeup of the business end of the measurement. It shows:
 The hook of the preload weight assembly draped over
the
shaft, in the middle of the 11" span.
 The 15pound weight now hooked to the bottom of the
preload
weight. The shaft is now being bent by 15 more pounds than it was
before.
 A dial indicator measuring exactly how much the shaft
is
deflected by the additional 15 pounds. The probe of the dial indicator
rests on a flat spot machined into the top of the hook.
 A ruler to indicate the position on the shaft
represented
by the
reading. The measured distance is that from the weight (at the middle
of the span) to the tip of the shaft.
Version 2
Since the
pictures were taken, I have replaced the dial indicator with a digital
indicator. Advantages:
 Better precision. The dial
indicator
reads to .001", and
the digital reads to .0005". Precision of this magnitude is indeed an
issue, because 15 pounds only flexes the butt of the shaft .015" to
.020" with an
11" span.
 No arithmetic. The digital
indicator has
a zero/tare button.

The precision issue
As noted back in the first section, the trick of designing a good EI
meter is to balance the need for:
 A small distance between the supports (to
keep EI constant over the span).
 A large distance between the supports (for
measurement precision).
 A small force (to avoid damaging the shaft).
 A large force (for measurement precision).
I
don't have a neat solution to the problem. This machine represents a
tradeoff, with all parameters in a very workable range but certainly
not ideal. A distance of 11" allows an EI variation of over 50% in many
shafts. A force of 15 pounds is not really close to the damage point.
It was chosen because a machine that applied force with removable
weights was easy to build and use; a much larger force would have made
the
machine difficult to use. But, even if there were a different mechanism
to apply the force, I would not want to use a force many times larger,
because damage to the shaft would be a definite possibility.
Let's
look at what sort of precision we can expect from the machine. Here is
a sample EI profile measured using the EI machine. In this graph EI is
in units of pound inches squared. (No, not pounds per square inch; it's
multiplication, not division.)
The points on the graph are the result of solving the original equation. The solution
is:
EI = 
FL^{3}
48 y 
= 
15 * 11^{3}
48 y 
= 
415.9
y 
We measured y at 5" intervals, and
computed EI using the formula. Simple! But perhaps
not very precise.
The values of y that we measured ranged from .0183 (near the butt we
get very small deflections, because the butt is stiff) to .0562 (near
the tip). But bear in mind that the smallest distance we can measure is
only .0005, and that is with the digital indicator; the dial indicator
has a resolution of .001.
That means that our measurements cannot be more precise than the
resolution, and that is .0005. So the precision of the measurements
ranged from
to
One percent is a pretty good resolution. Three percent is probably good
enough for profiling, but not good enough for shaft matching nor
quality control of shafts. And stiffer shafts will show even smaller
deflections, meaning that the precision can be as coarse as 5% or 6%.
Again, it will demonstrate the general shape of a profile, but
you would not want to use the measurement for anything else.
I have several ways in mind that the precision can be increased. But
the existing instrument gives profile shapes, and that is all I intend
to use it for. My frequency meter and NF4 are quite sufficient to do
matching, and are more convenient for profiling. I am exploring a
computer algorithm from M. Brillouette ("On Measuring the Flexural
Rigidity Distribution of Golf Shafts", Science and Golf IV, 2002) to
convert cantilever measurements (like frequency or NF4 measurements)
to EI profiles. If I can mechanize it with an Excel spreadsheet, and if
my measurements prove its value, then I can use my NF4 to yield EI profiles as well. If so, I will probably abandon my EI
machine as not being worth the lab space.

Precision vs Accuracy
In another article, I
point out the difference between precision and accuracy. The EI machine
is an extreme example, because there is actually a tradeoff between
precision and accuracy. Most of the simple things you can do to increase accuracy will decrease precision.
We have seen that there is an inherent inaccuracy due to the 11"
spacing between the supports. The EI value can change by as much as 50%
over that distance. So the measurement is not really the EI at the
center of measurement, but rather a weighted average IE over the 11"
span. You can increase the accuracy by decreasing the distance between
the supports.
Let's do that and see what happens. Let's look at the measurement at
31" from the tip, where we measured the deflection at .0183", which
computed to an EI of 22700 pound inches squared. We'll cut the span in
half, from 11" to 5½". According to the equation, the new deflection
is:
y = 
FL^{3}
48 EI 
= 
15 * 5.5^{3}
48 * 22700 
= .0022" 
This is eight times smaller than it was with a span of 11". (Not
surprising. Deflection depends on the cube of length. We divided the
length by 2, and 2 cubed is 8.) So what happened to the precision? It
is now
This is really bad precision! The result of improving
accuracy by a factor of 2 was that precision went to hell in a
handbasket. 
SummaryAn
EI machine shows a shaft's flex profile as the stiffness at the point
where the measurement is taken. That may or may not be the best way for
custom clubmakers to look at a profile, but it definitely is the way
shaft design engineers see profiles.
A simple EI machine is
easy to build and easy to understand. It is much harder to build one
that combines precision and accuracy. Precision demands larger
deflections, while accuracy demands a smaller span between the supports
 which are in conflict with each other..Acknowledgement: I'd
like to thank Don Johnson of BTM Clubs, who shared photos and details
of the EI machine that he built. Knowing what he did and what gave him
problems saved me a lot of time and resulted in a simpler design from
the outset. Thanks, Don.
This article is based on the "description" section, which was originally published in April 2006. Last modified  Nov 15, 2008
