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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 spring-based 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 moment-of-inertia 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 mid-hands point

Most biomechanics studies these days assume that the club should be measured from the "mid-hands 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 mid-hands 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 19-20% 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 19-20% is for good golf swings. If you are fitting a mid- or high-handicap 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 club-by-club in an ad-hoc 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 taped-up 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 mid-hands 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 MOI-matched sets of irons. Both sets were matched with the axis at the butt of the clubs. One set was graphite-shafted and the other steel-shafted; 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 mid-hands 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 4-inch mid-hands 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 mid-hands 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 MOI-matched, 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 kg-cm²)
Club	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 mid-hand point is of course considerably lower (because the mid-hand 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 kg-cm² "lighter" in MOI than the long irons. So these clubs, which were matched to a butt axis, are not matched to a mid-hands axis.

Is this enough to matter? Yes. One swingweight point is about the same heft difference as 20 kg-cm². We have here a 30 kg-cm² 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 MOI-match it at the mid-hand 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 mid-hands match, or 5.5 points over the set. Doing the arithmetic, a mid-hands 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 butt-matching a set of irons and mid-hands-matching them.
The difference is that a mid-hands 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 club-by-club 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 6-iron (the middle of the set you're matching) from the mid-1900s. 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 mid-hands 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 (2G-A)

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 easy-to-understand 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 + MG²

We will solve for I_o, because we will need it later.

I_o = I_b - MG² (1)

Similarly, we can use the parallel axis theorem to get I_a.

I_a = I_o + M (G-A)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 - MG² + M (G-A)² (3)

Now it is just a matter of algebra to simplify (3). We'll show a few key steps.

I_a = I_b - MG² + MG² - 2MAG + MA²

I_a = I_b - M A (2G-A)

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 mid-hands 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