MOI Matching  But Which Axis?
Dave Tutelman  November 18, 2015
Let's challenge the usual assumption
that MOI matching should be done using the butt of the club as the axis
for moment of inertia. Here are the likely axes one might use, why the
butt of the club is the one we do
use, and a formula so you can transform the axis if you want to try a
different approach.
In the past few days, I had several inquiries why we do MOI matching
using a moment of inertia around an axis at the butt of the club. It is
for certain that we do. The commercial moment of inertia instruments
(the pendulum and the springbased SpeedMatch) measure throught the
butt axis. So
does my technique
using a swingweight scale. But why?
A brief physics digression about moment
of inertia before we start. (Note that I also have an introductory article
on moment of inertia.)
Moment of inertia is not a
property of an object, at least not a single number that attaches to
the object. Mass is a property of an object; give me an object and I
can measure its mass. On the other hand, the vibrational frequency of a
golf club is not just a property of the shaft stiffness; it depends on
other things as well. Change the clubhead weight, and the frequency
changes. Same for the unclamped length of the club, and even the
clamp's pressure and design.
Moment of inertia is a property of the
object plus
some
other condition. Moment of inertia is a body's
resistance to rotating about an axis. But which axis? That is the key
question! Moment of inertia will vary depending on which axis you
rotate the object about. Here's
an experiment you can do to see for yourself.
 Take a heavy wedge and grip it at the balance point
of the
club. Twist it back and forth; your forearm is now the axis for the
moment of
inertia of the club. Feel how much effort it takes to twist it.
 Now
grip the handle near the butt, and twist back and forth again. It takes
a lot more effort, doesn't it? That is because the moment of inertia
through an axis at the butt is much greater than the moment of inertia
through the center of gravity.
In fact, the lowest moment of inertia
for any body is through the center of gravity; that's just physics. The
MOI increases as the
axis is moved away from the CG.
With a little thought, we can come up with three different axes that
might make
sense for a momentofinertia approach to heft matching golf clubs.
Let's review what they are, and why we use the axis at the butt.

Axis at the
butt
Commercially
available MOI meters for golf clubs measure through an
axis at the butt of the club. That goes for the Wishon pendulum and the
GolfMechanix SpeedMatch.
Rationale:
The major hinge between the arms and the club is the left wrist. The
left wrist is on a level with the butt of the club, for most golfers'
grips. For swings that rely mostly on hinging rather than applying
torque to the club, this is probably a reasonable assumption. 
Axis at
the
midhands point
Most
biomechanics studies these days assume that the club should be measured
from the "midhands point". That is where the two hands meet or the
middle of the overlap if there is an overlap. This is typically about 4
inches below the butt.
Rationale:
If both hands are active in applying forces and torques to the club,
then this is a reasonable assumption. Forces and torques are applied by
each hand individually, in fact each finger and other point of contact
between the hands and the club. For purposes of analysis (what
biomechanists do), all the forces and torques can and should be
combined into a single force and a single torque. For the best analysis
results, these should be applied at about the midpoint of the
components they represent. Hence the importance of the midhands point. 
Axis at the
club's
center of rotation
One
could make the argument that the important moment of inertia is at the
axis representing the center of the club's rotation in space. For most
of the swing, the center of rotation of the entire club is not within
the body of the club itself. It is closer to the golfer, and varies all
over the place during the swing. So I'm not sure where in the swing you
would want to take the axis to measure the moment of inertia. But, just
for argument's sake, let's assume just before impact. The diagram at
the right shows where that axis would be. Next we will see why it would
be there.
Here is a picture of a good golf swing at impact. The clubhead speed
comes from two sources:
 The rotation of the club about the hands. That speed
is the angular velocity of the club times the length of the club.
 The
speed of the hands moving forward. This simply adds to #1. (If the two
velocities are not in the same direction, we would use vector addition
of the velocities instead of just adding the speeds. But we'll keep it
simple here.)
We
can find the center of rotation from the two speeds and the length of
the club. To see how, let's look at a rotating disk. (Some of you are
old enough to see right away that it's a 45rpm phonograph record.) Any
point on the disk has some speed at which it is moving. That speed is
directly proportional to the distance from the center of rotation.
Let's apply this to the golf swing above.
Proportionality means:
 The clubhead speed divided by the distance from
clubhead to center of rotation is
equal to...
 The hand speed divided by the distance from the hands
to the center of rotation.
It
is a matter of simple algebra to find the center of rotation, given
both speeds and the length of the club. (Maybe you can't do it now. But
when you were in high school, you had to solve problems like this
in order to pass math.)
Let's run through an example. A good
swing with a driver results in a hand speed of 1920% of the clubhead
speed. I have looked at this in videos, mathematical
simulations, and technical papers on biomechanics; 19% or 20% is
remarkably constant. So we can say hand speed is 1/5 of clubhead speed.
That means that the length of the club is 4/5 of the distance from the
clubhead to the center of rotation. Therefore, with a typical 45"
driver, the center of rotation is about 11" outside the butt of the
club, toward the golfer.
But wait! It's not that simple. The
1920% is for good golf swings. If you are fitting a mid or
highhandicap golfer (or even some low'cappers), the center of
rotation could vary drastically. Let's look at two fairly common swing
faults, which
move the center of rotation in opposite directions.
 If the
golfer casts the club, then the center of rotation at impact is closer
to the golfer's body. In the extreme case, where all lag is
expended well before impact, the club and arms are moving as if they
were part of the same phonograph disk. So the center of rotation would
be somewhere between the golfer's left shoulder and neck, the axis
about which the arms are rotating.
 If the golfer decelerates
instead of keeping the hands moving through impact, then the hand speed
is less than 19% of the clubhead speed. In the extreme case, where the
hand speed goes to zero at impact, the center of rotation is the butt
of the club, because the hands are not moving  all clubhead speed is
due to the rotation of the club.
We learn from this that, if the center of rotation is a good axis for
club matching, it has to be done in a very custom way.
You need to know the golfer's swing in detail to even know where the
center of rotation is. And note that we arbitrarily picked the center
of rotation just before impact. More relevant (though more difficult to
determine) might be the point in the downswing of maximum acceleration,
or maximum angular acceleration, or... well, I don't really know.
Rationale:
I don't know of any serious rationale for using this moment of inertia
for club matching. But from time to time it is proposed by people
learning physics but not yet experienced with analysis. They see an
article on golf clubs or baseball bats, showing where the actual center
of rotation is, and they say, "Hmmm..." Their next step is to email me
 which is my personal rationale for writing this article. I've
answered two of those
emails in the past week, so I want something I can point to
instead of explaining again from scratch.

Which one
should we
use?
I'm
not going to get into a deep discussion of the physics of the golf
swing here. Instead, I'm going to rely on empirical data  experience
 from the real world of clubfitting. The data is anecdotal, so I
can't say this is the scientific last word. But there is quite a bit of
the data from varied sources and different kinds of experiments, and it
all points in the same direction. So there is a good likelihood that it
is valid.
The bottom line is: matching
moment of inertia at the butt seems to work well.
Nobody has reported results for any of the other axes, so we don't have
comparative data. But we have confirmation that MOI matching at the
butt works better for
most golfers than, say, swingweight matching.
Let's review where some of this data is coming from.
 Tom
Wishon claims to have a lot of data supporting this. I have not seen
the data, but I have heard from quite a few happy Wishon disciples who
use the technique. Together, they have lots of fittings  and
report happy and successful customers after delivering sets
made to
this specification.
 Golfmechanix
has built two different instruments over the past decade to match
moment of inertia. The first was a pendulum; their current product is
based on horizontal oscillation using a spring. Both measure moment of
inertia at the butt.
The next two points involve inferring MOI from a
swingweight pattern. I developed this approach, with some early help
(in 1994) from TJ Field. You can see more about it in another article.
 I have received email from two people who fitted
themselves clubbyclub in an adhoc manner. These were not experienced
clubfitters, just serious golfers messing with their own clubs' hefts
until
they felt right. Both
reported to me that each club had a different ideal
swingweight; they found that surprising, so it isn't a case of
confirmation bias. The swingweights they reported were almost exactly
the 1.3
points per inch slope that signifies an MOI match at the butt axis.
 ...Which brings us to my own fitting experiences. My approach
is to fit a long iron and a short iron separately, and build the set on
whatever swingweight slope they turn out to be. The process is to take
the clubs and a roll of lead tape to the range, and add and subtract
lead tape to each of the clubs until it gives the best consistency.
Then I take the tapedup clubs home and measure their swingweight; no
room for confirmation bias there. I haven't done a lot of
these fittings  I am not really in the custom club business  but
those I have done have come out close to 1.3 points per inch.
So I have quite a bit of confidence that MOI matching at the butt axis
works well.

Does it
even matter?
Here's a crucial question that we haven't
even asked yet. How
much difference does it make which axis you use? For
instance, how different is a set that is MOI matched at the butt from a
set that is MOI matched at the midhands point? Will the difference
exceed the normal build tolerances for a custom clubmaker? Put simply, does it even matter which axis you choose, as long as you
match moment of inertia?
(For
reference purposes, the tolerances I apply to my own work assume that
most golfers will not be affected by a single swingweight point, and no
golfers by a half swingweight point. Over the years, I have
hearsay evidence of only one counterexample; British clubfitter Richard
Kempton mentioned to me a golfer who seemed to be affected by
a change of less than a quarter swingweight point for one particular
club in his
set.)
To answer
this question, I measured two MOImatched sets of irons.
Both sets were matched with the axis at the butt of the clubs. One set
was graphiteshafted and the other steelshafted; I believe this will
span most interesting data points for the question. Based on these
measurements, I computed
how different each club's MOI would be if it
were taken four inches down the grip, at the midhands axis. (The
formula for computing the change of MOI as we change the axis is
presented in the next section of this article, and the derivation of
the formula in the following section.) Here is a table of the
change in MOI when the axis goes from the butt to a 4inch midhands
point. Bear in mind when you look at the table:
 The
entries are not the total MOI of the club. They are the amount by which
the MOI is reduced when you go from butt MOI to midhands MOI.
 The total variation (the bottom row of the table) is the difference
in reduction between the longest club and the shortest. If the
difference were zero  that is, if the MOI reduction was the same for
every club  then the set would still be MOImatched, just at a lower
MOI. So this number is a measure of how much the match changes as you move the axis. Not the MOI itself, but the match across the set.
Club 
Reduction
in moment of inertia
(in kgcm^{2}) 
Graphite
shafts 
Steel
shafts 
3 
— 
588 
4 
563 
— 
5 
571 
598 
6 
576 
603 
7 
581 
609 
8 
591 
615 
9 
590 
615 
PW 
594 
616 
Total
variation
of MOI 
31
(1.6 points
equiv) 
28
(1.4 points
equiv) 
What does this tell us? Well, we have a set that is MOI matched at the
butt. The MOI at the midhand point is of course considerably lower
(because the
midhand axis is closer to the center of gravity than is the butt
axis).
The important thing is that the reduction of MOI is not the same across
the set. There is more reduction in the short irons than the long
irons. The short irons are about 30 kgcm^{2}
"lighter" in MOI than the long irons. So these clubs, which were
matched to a butt axis, are not
matched to a midhands axis.
Is this enough to matter? Yes. One swingweight point is about the same
heft difference as 20 kgcm^{2}. We have here a
30 kgcm^{2} difference across the set,
equivalent to 1.5 swingweight points. That's not a lot, but enough
where a significant minority of golfers can feel the difference or see
a performance
difference.
What would we have to do to the set in order to MOImatch it at the
midhand axis? The chart above tells us that the short irons are
getting heavier but not fast enough; we don't have enough "swingweight
slope". For the
butt match, the slope gave us a total variation across the set of about
4 swingweight points. It should be 1.5 points more if we want a
midhands match, or 5.5 points over the set. Doing the arithmetic, a
midhands match would call for
a swingweight slope of 1.8 points per inch. Remember, the rule of thumb
for a butt match is a slope of 1.3 points per inch.
So:
 There is a difference between buttmatching a set of
irons and midhandsmatching them.
 The difference is that a midhands match needs a
swingweight slope of almost 40% more than a butt match. (1.8 instead of
1.3)
It is worth repeating here that several independent clubbyclub
fittings have just turned out to have a slope of 1.3. Once could be an
accident, but but it came within a few percent of 1.3 in every case
that I know of. The fact that they all are close to 1.3  and not
close to 1.8 
suggests
that the butt axis is
the right place to measure MOI in order to match a set of
irons.
Just an interesting side note: A flat slope  i.e., a perfect
swingweight match  seems to correspond to an MOI match 7.5 inches
beyond the end of the shaft. That's 7.5" above the butt. It is within
¼" of the center of rotation for a 6iron (the middle of the set you're
matching) from the mid1900s. Coincidence? Probably. But perhaps Robert
Adams knew more than we give him credit for when he invented
swingweight. Perhaps he was deliberately matching MOI, but mistakenly
picked the axis as the center of rotation at impact. Hmmmm.

Transforming
axes
In the previous section, I computed the change in moment of inertia
when going from the butt axis to the midhands axis. Here is the
formula I used, and a derivation of the formula for those interested.
The distance from
butt to balance point is G. The balance point is where the force of gravity acts.
The target axis is distance A
closer to the balance point. (If you want the axis outside the club,
like the center of rotation, A
is negative.)
The total weight of the club is M.
The moment of inertia at the butt is I_{b},
and at the target axis is I_{a}.
Then:
I_{a
= }I_{b}
 M A (2GA)
If you want to experiment with club matching using an axis other
than the butt, you can use the formula to transform what you can
measure into the axis you want to use for matching.
Derivation of the formula
The formula is based on the parallel axis theorem that every physicist
and engineer learned as a college freshman or sophomore. In case you
aren't familiar with it, here is an easytounderstand reference.
That and high school algebra is enough to derive the formula.
Let's call the moment of inertia at the center of mass I_{o}.
From the parallel axis theorem, we know:
I_{b} =
I_{o} + MG^{2}
We will solve for I_{o},
because we will need it later.
I_{o}
= I_{b}  MG^{2}
(1)
Similarly, we can use the parallel axis theorem to get I_{a}.
I_{a}
= I_{o} + M
(GA)^{2}
(2)
But we already know I_{o};
we found it in (1) above. Plugging that back into (2), we get:
I_{a}
= I_{b}
 MG^{2}
+ M (GA)^{2 }
(3)
Now it is just a matter of algebra to simplify (3). We'll show a few
key steps.
I_{a}
= I_{b}
 MG^{2} +
MG^{2}  2MAG
+ MA^{2}
I_{a}
= I_{b}
 M A (2GA)
This is our formula for finding moment of inertia about any axis
parallel to the measurement we already have at the butt axis.

Conclusions
When we match clubs by moment of inertia, we
usually do so using an axis through the butt of the club. An argument
could be made for an axis at the midhands point, or perhaps the center
of rotation for the club as a whole. But it appears that the butt axis
gives the best overall match of clubs.
Last
updated  Sept 27, 2016
