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RESULTS OF MOI-MATCHING EXPERIMENT

Dave Tutelman -- Jan-May 1995

In the first half of 1995, I was experimenting with MOI matched clubs. This was based on math and physics I did in 1994 suggesting this would be a better way to match clubs than swingweight. I posted my experiences to rec.sport.golf, the golfing interest newsgroup on the Internet. Here is an anthology of those postings.

PART 1

I recently posted several articles to this newsgroup arguing that the real measure of a club's "feel" is not swingweight (SW), but moment of inertia (MOI). My thesis was based on analysis, though supported by some small amount of evidence. However, there's nothing like a good experiment to test a theory, so I set as my winter clubmaking project the construction of an MOI-matched (as opposed to the usual SW-matched) set of irons. If my theory were correct, the clubs would all feel and swing the same -- the long like the short.

I have made the clubs, and done a few tests at the driving range (hey, this is the Northeastern US in the winter). I found the results very
surprising, though I shouldn't have been surprised if I had thought through the implications of a feel-matched set of clubs. Of course, not all the results are in. Remember that:

Two large buckets at the range is a very limited trial.
The only point of the tests so far is to see whether the clubs are competent, and whether the long irons feel like the short irons. Thus....
I haven't checked out distance, nor an A/B test against my old clubs.
The experiment wasn't double-blind, as a proper human-based test should be. It wasn't even single-blind. Face it; a hitting test where the hitter is the clubs' designer is the least valid experiment one could imagine.

Nevertheless, I thought the results of the first two buckets were interesting enough to send out now. If/when I get more data, you'll see more in this space.

THE CLUBS

I made the irons from:

Z-Model clubheads, a clone of the Ping Zing. Not that I like them all that much (they're ugly as sin), but they were cheap (experiment, remember) and met my specs. Well, their specs met my specs. When they arrived, the weights were all over the lot and the 9I was a loft double for the 8I. After a lot of grinding (Dremel moto-tool), a little lead tape, and 2.5 degrees of bend in a 17-4 stainless steel head (don't try this at home, kids), I had the right set of heads.
Dynamic Lite shafts. They're my current favorite on woods, and a secondary point of this experiment is to see how I like them on irons.
Line grip, a look-alike of the grips Ping uses. Again, not my favorite, but if I don't like the clubs then a consistent Ping-alike will be easier to sell. If I keep them, the next grips will be Victory cords.

I have written a program that will MOI-match (or, optionally, SW-match) a set of clubs from the component specs. It does this by recommending a length for each club, based on head weight, raw weight and raw length of the shaft, and the trimmed measurements of a "base club" -- the one you want all your clubs to feel like.

Since I was hitting the 7I really well for the last part of 1994, I used my current 7I as the base club, and the computer gave me a set of lengths that would give me all the Z-Models at the same MOI as my current 7I. I also cranked out the measurements that would give me a SW-match, for comparison purposes. Here are the two sets:

Length for Length for Club MOI-match SW-match ---- ---------- ---------- 2I 39.2 39.7 3I 38.6 39.0 4I 38.1 38.4 5I 37.7 37.9 6I 37.2 37.3 7I 36.7 36.7 8I 36.2 36.2 9I 35.9 35.7
Since the 7I was the base club, it's not surprising that the computer recommends the same length for both sets' 7I. But the MOI-matched set has shorter long irons and longer short irons. The range of lengths, from 2I through 9I, is almost 3/4" shorter for the MOI set than the SW set.

What does this say about SW-matched sets if my theory is right that MOI has more to do with feel than SW does? It means that, in a SW-matched set, a long iron will FEEL heavier than a short iron ... by about four swingweight points (2/3 of an inch).

THE FIRST DATA

OK, let's take 'em out to the range!

My first bucket was a big disappointment. Oh, I hit the short irons very well, high straight darts at the target. But I couldn't hit the longer irons at all. Starting with the 5I, I was displaying an alarming tendency to top the ball -- and I seldom top. (My common mishits are fat or quack.) I found it hard to get the clubface on the 5I, and harder still as the clubs got longer. Not a single decent 2I or 3I shot in the whole bucket.

Gloom and dejection on the drive home. But early the next morning (hey, some people sing in the shower, I do design) a really wierd thought struck me. I've been using the Sam Snead model for positioning the ball; play the ball near the middle of the stance up to the the 6I, then move it forward in the stance as the irons get longer. (This is in distinction to the Nicklaus model, where the ball is always in the same place.) Think about this:

Suppose my experiment were perfectly successful, and the short and long irons felt identical. Then I'd swing them the same. If I were swinging my long irons exactly like my short irons, the long ones would touch down near the middle of my stance, and be on the way up again when they reached the ball. Instant top!

Well, I had to wait a couple of weeks before the weather gave me an opportunity for another bucket. That was this afternoon (2/10/95, my youngest son's 20th birthday; Happy Big 2-0, Dan.) Hit a bucket, using both the Snead positioning and the position-everything-like-a-7-iron positioning. Bingo! My solid and straight hits were all from the middle of my stance -- even with the 2I. Remember, unlike SW-matched sets, my MOI-matched set doesn't feel heavier in the long irons, so I had no problem squaring the clubface in time. It felt and reacted exactly the way the short irons do.

I haven't checked to see if I'm losing distance; it's a possibility. But if this helps me to hit a 3I on the sweet spot as often as I do a 7I, I won't miss the yardage. After all, it's only potential yardage if I hit it right only 1/3 of the time.

More news as it breaks.
Dave

PS - I'd like to thank the folks who reviewed my early drafts of the MOI analysis, and especially T.J. Field (who took some initial tries at MOI-matching rules last Fall to get me interested) and Guy Cooper (whose experience with Z-model-2 clubheads warned me to measure and correct them when my components arrived).

PART 2

This is a continuation of a report on a set of clubs that I made to test my theory that clubs should be matched by moment of inertia, not swingweight. The original theoretical work was posted in November 1994, and is available from the clubmaker archives at dunkin.princeton.edu. the first installment of the testing was posted last week. So much for the background....

Who says that hard work is its own reward? I stayed at work late Friday night to finish up what I was doing, and was rewarded much more concretely. I passed another late-stayer in the hall who muttered something about, "Fifties this weekend ..... tee time at Shark River ..... need a fourth." I didn't need to be asked twice. Actually, I didn't need to be asked once; I just volunteered.

I've never played golf in February before, but I knew Shark River was all temporary greens. I didn't care. This was an opportunity for my new MOI-matched irons to see some live combat; don't need real greens for that. To make sure I stayed focused on the performance of the new irons, I left my woods in the trunk of the car. I intended to use the 2-iron from the tee (or the 3 if it turned out I couldn't handle the 2).

The most interesting outcome of the round was a confirmation of what I had observed at the range. The clubs feel so much alike that I must play the ball from the same point in my stance for all the clubs. When I try to play the long irons more forward in my stance than the short irons, I top the long clubs.

It took me most of the front nine to internalize this fact. It was real hard to look at a 2-iron addressing the ball in the middle of my stance, and not swing with something extra in my right hand to keep from pushing the shot way right. But that was a mistake, as I proved repeatedly early in the round. Doing anything extra to prevent a push invariably resulted in a hook. Not a duck-hook, just a long, strong shot that curved further to the left than a sane golfer would want to be. All too frequently the shot entered the trees, albeit well down the fairway.

By late in the front nine, I had adjusted to this phenomenon and started to hit really good long iron shots. The trick was to swing the clubhead without any conscious effort to do anything but make good contact with the ball at the moment of release. My 2-irons off the tee were almost where the other three's metalwoods were, most of the time; and I was generally in better position and out of trouble. I hit some of the best 3I, 4I, and 5I shots of my life. More importantly, I was getting solid and straight hits from my long irons FAR more frequently than I'm used to; the reliability was more like my 7I -- which, OF COURSE, was exactly the point of the MOI-matched set.

So the phenomenon I observed at the range does transport to the course.
The remaining questions to be answered are:

Trajectory - see below.
Distance - see below.
Head-to-head comparison with SW-matched irons - if the weather cooperates next weekend, I'll be at the range checking this out.
Serious, carefully-controlled experiment - I'm currently in no position to do this, and would appreciate any help I can get.

TRAJECTORY:

I'm hitting the ball lower than I did with my old clubs. This is especially true of the long irons, but also true of the shorter irons to
some smaller extent. A couple of possible reasons:

Higher bend point on the long irons. The new clubs have Dynamic Lite shafts (high bend point) compared to the FlexFlow shafts on my older, SW-matched set. The FlexFlows have low bend points on the long irons, varying to high bend points on the short irons.
The clubhead is de-lofted at impact because I'm playing the ball further back in my stance -- at least with the long irons.
Less loft or higher COG of the clubhead. I really doubt this. I measured the lofts, and they're remarkably similar to my old Tour Model IVs. And the Zings (the Z-Model heads are Zing clones) are notorious for a low COG and a high trajectory. So that's not it.

In summary, either the delofting of the clubhead or the shaft bend point -- or both -- is the reason.

DISTANCE:

I was concerned (as were those who posted or Emailed comments on the first "results" post) that the slightly shorter shafts of the long irons would cut distance. Well, actually I wasn't as concerned as the respondents; the analysis that led me to MOI in the first place said that distance would not be heavily affected by shaft length. (Hold your fire on this until you've seen and understood the analysis; then we can argue.)

For whatever reason, loss of distance doesn't seem to be an issue. The lower trajectory seems to carry with it more distance. I was hitting 3I, 4I, and 5I shots that approached (maybe exceeded) anything I hit since I was in my twenties. On a 180-yard par 3, where I normally need a perfect 3I to hit the green, I backed off to my 6I. That's normally my 150-yard club, but I was hitting to a temporary green in front of the normal green. The ball went straight over the winter pin and landed on the front of the summer green, 170 or so from the tee.

One thing leads me to believe the difference was the lower trajectory, and that it was due to delofting of the clubhead. I felt that the most pronounced lowering of trajectory was in the 2I, which was very low even off the tee. I believe my 2I may have lost some distance, and probably didn't gain any. I was probably 180-200 on a decent hit yesterday, and maybe 190 on a good hit normally. But normally a hit like that with a 2I is maybe twice a year; yesterday more than half my 2I tee shots were going 180-200. I probably ought to settle for the reliability. (One other thing to check out: with the increased length of my 3I, is it now ALWAYS longer than the 2I? If so, leave the 2I home and carry the 64-degree wedge that Bob Dietrich recommended last week.)

PART 3

This is a continuation of a report on a set of clubs that I made to test my theory that clubs should be matched by moment of inertia, not swingweight. The original theoretical work was posted in November 1994, and is available from the clubmaker archives at dunkin.princeton.edu. Several installments of the testing have been posted over the last few weeks. So much for the background....

A few more results on the experiment comparing MOI-matching with SW-matching. We had a break in the winter weather, and I took some clubs to the range for the long-awaited A/B test. I took a 3I, 5I, 7I, and 9I from each of two sets. I itemize the differences between the sets below, so you can speculate (as, in all honesty, I do) on which differences are due to the heft-matching differences and which to other factors. Once again, let me admit that this is FAR from a controlled experiment.

OLD SET NEW SET ======= ======= SW-matched MOI-matched Flex-Flow shafts Dynamic Lite shafts R-flex R-flex, 3/4" tipped Golfsmith Tour Model ProPride Z-Model heads IV heads Victory Cord grips Off-brand Line grips
Given all the practice I've been able to get this winter ( :-( ), I was swinging like the proverbial rusty gate. I was not hitting at all consistently, so the only basis for comparison of the clubs themselves have to be ONLY the best hits with each club. In the comparisons below:

The SW-matched set is referred to consistently as the "old set", and the MOI-matched set as the "new set".
All statements about trends refer to good, solid hits with good, solid swings. The generalizations are not about ALL shots, but just the good, solid ones (unless explicitly noted otherwise).

All those caveats out of the way, here are the comparisons between the sets. First, let me state the two major observations by way of summary. Then, for those who are interested in how I reached these conclusions, a description of the tests:

MOI-matching causes the clubs to swing the same across the set, to the extent that the ball must be played from the same point in the stance for all clubs. SW-matching requires the longer clubs to be played more forward in the stance, or a different swing to be used.
A higher-kickpoint shaft gives a lower trajectory. (Traditionally, this has been the conventional wisdom. It has been contradicted by some over the last 2-3 years. After this test, I still believe in the older wisdom.)

DIRECTION:

Playing the ball from the middle of my stance, the new set hit pretty much straight. The 9I was maybe a shade to the right of the longer clubs, but not much.

Playing the ball from the middle of my stance, the old set hit long irons more to the right and short irons more to the left. The effect was not as marked as I expected, but it was decidedly there. The 9I was straight to slightly left, the 7I was straight, the 5I took off straight and faded, and the 3I was definitely pushed and/or faded when hit long.

When I moved the ball forward in my stance for the long irons (5I and 3I), I couldn't hit the new set at all, but the old set straightened out nicely. (Actually, I did manage to hit a couple of good shots with the new set, but I had to use a very exaggerated weight shift where even my head and shoulders come forward in the downswing -- the sort of swing that used to result in a wild push last year with the old set.)

When I gripped down on the long irons in the old set (to approximate a lighter swingweight, closer to MOI-matching), I got results much like the new set in regard to direction.

A brief word about less-than-well-hit shots: I got a lot more curve (slice and, especially, hook) from poor swings with the old irons than the new ones. That's probably due to their being a little too flexible for me. (Dynacraft's DSFI book would refer to this as a "High Risk, High Yield" shaft for my swing.)

DISTANCE:

This driving range wasn't laid out well for distance-marking. Those shots I saw land near some landmark didn't give me enough basis for saying that one set had an advantage over another in distance. There's another range nearby that's better for distance estimation, but no heaters in the stalls. Have to wait another week or two for the weather.

TRAJECTORY:

This was interesting. As I had noted on the range and the golf course in earlier posts, the new clubs seemed to have a lower trajectory -- especially in the long irons -- than I was used to. This was confirmed in the A/B tests, with dramatic differences observed. I was uncertain whether to attribute this to (1) the higher-kickpoint Dynamic Lite shafts or (2) the delofting of the clubface due to playing the ball back further in my stance for long irons.

I was able to devise a test to distinguish between (1) and (2). By gripping down (about 3/8" for the 5I and 3/4" for the 3I), I was able to hit both the old and new sets straight from the middle of my stance. When I did that, the trajectory difference was still quite noticeable, especially with the 3I. I therefore conclude that a low kickpoint gives a higher trajectory than a high kickpoint, in spite of the apparent shift in the "conventional wisdom".

But there's still one more factor that might be at work, even if we've eliminated the delofting of the clubface. The old set's shafts are a little too soft for my swing. The DSFI book points out that it is probably curving forward at the moment of impact. This just might increase the effective loft of the club and provide a higher trajectory. I doubt this is what's doing it, but I'll leave open the possibility. (The reason for my doubt is that I didn't notice much trajectory difference in the short irons, where both shafts have high kickpoints.)

CONSISTENCY:

As noted at the top of this article, I was extremely inconsistent with ALL the clubs (even my beloved seven-irons in both sets). I am therefore reluctant to draw any conclusions about the relative consistency or ease of hitting between the sets. That will have to wait until the season is underway and I have a more repeatable swing.

LIE:

In response to my post on construction and measuring of MOI-matched sets, someone noted that I might be upsetting the match of lies across the set. That's a very real possibility, as I'm changing the length relationship between the longest and shortest clubs, without changing their lie relationship. So I did a dynamic lie test for both sets, checking with a 9I, a 6I, and a 3I.

The results are puzzling. The old set has perfect lie. The new set seems to have a consistent lie error of about 1 degree flat across the set. No explanation for now, unless the Z-Models just plain don't meet their lie specs. (I didn't measure the static lie of each clubhead when they arrived, but one of the lofts was way off, and the weights were all over the lot. So I don't have great confidence in the specs.)

A FOLLOWUP ARTICLE

In response to my posts about my MOI-matching experience, Guy Cooper wrote some questions on stuff he was having trouble with. Here was my response, with his questions embedded in the usual email convention...

In article <950215205141.2020243a@VAX.ETOWN.EDU>, Guy Cooper <MYERSJA@VAX.ETOWN.EDU> wrote: > I follow this "broad-brush" explaination of how > your MOI program figures shaft length...but are more than head > wt., and a shaft's raw wt. & length factors? Well, let's take the possible factors and see why I don't bother with them: 1. Trimmed weight. Actually, the program does this quite precisely. It iterates the length calculation, estimating trimmed weight from the raw weight, raw length, and trimmed length. Then it recomputes the MOI and re-trims, until the difference of trim is under .01". 2. COG of the shaft. It may not be dead center but, except for the deliberately tip weighted shafts, we don't lose much accuracy by the assumption. 3. Clubhead COG. This can move, and affect MOI. We can definitely ignore this in building a matched set from similar heads. It's harder to ignore if the idea is to match a set across heads of radically different design -- but that's not the usual goal of a clubmaker. It's also a problem if we're trying to match precisely the feel of a base club with a radically different head; all I can say is that we may be a little off the MOI of the base club, but the set itself should be internally matched. Matching across the set is more important than a perfect match to the base club -- IMHO, but it's just a HO. 4. Shafting depth. More precisely, how much do we REALLY trim the shaft to get a given length of club? I've taken a hot guess in the program that the shaft tip is 1.5" above the ground, but this number can be set in the input data for the clubheads involved. 5. Other non-MOI "feel" factors. I'll say a little about this below, but the program only deals with MOI. Other factors besides tip-to-ground distance that can be set for a data run include: - MOI fulcrum. I believe in measuring from the butt as a fair approximation of the wrist hinge. But others suggest the middle of the hands' grip. The program allows you to change the value from my default. - Swingweight fulcrum. There are both 12" and 14" scales. - Grip weight and COG. I oppose the use of grip weight to affect "feel". In short, I believe it doesn't. But it WILL affect numerical swingweight, and I've included it in the swingweight calculations. > I can almost grasp what you might be attempting to do given > one shaft, ie TT's Dynamic Lite with CSG's "Zing" head; then, > deciding to swap "Zing" heads for say, "King Snakes" and > attempting to duplicate the "feel" of the "Zing" 7i in the new > set.... > But, am I to understand, as in your example, that > if I like the feel of my current 7i, I could duplicate its feel > even if I moved to a different shaft "profile?" Given two shafts > w/ same wt. but different step configurations, or even > composition, say steel vs. titanium or even hvy. wt. graphite, > are you suggesting that original feel could be duplicated? > Is it simply..."Feel" (mostly) = MOI? Yes and no. Feel (again IMHO -- I'll defend that opinion against idle speculation and individual anecdote, but not against carefully collected data) is a combination of "heft" and "flex". I'm convinced that the "heft" factor is completely MOI _for_practical_purposes_ (more below). The other factors you note are "flex" factors, and I'm not even trying to deal with them in this study. Next year, I hope to have a frequency meter, and I'll turn my attention there. Where I say above that MOI == "heft" factor _for_practical_purposes_, here's what I mean. I'm aware that Jorgensen's book proposes the first three moments (weight, first moment, and moment of inertia) as being the heft factors. Indeed, one can feel all three when one lifts the club, and waggles it. But my analysis indcates that DURING THE DOWNSWING the MOI completely dominates the other two moments, especially in the production of clubhead speed. I believe that this "feel" is what counts in producing performance from a club. In the very limited testing I've done so far, I've seen no contraindications of the theory. > Thought I'd been following this thread enuf not to have > to ask the above question, and I'm not asking that you reveal > any "secret ingredients," but has some general discussion > taken place as to why MOI over Sw. Wt.... No secret ingredient. I posted it all quite publicly in November, and the articles should be in Marcelo's archive. There's a 3-page summary and a separate lengthy analysis, complete with equations and references. > .... or for that matter frequency matching? Separate issue entirely, as I said above. By shaft selection and tip trimming, you can adjust frequency as a "flex" factor, completely independent of the "heft" factors. > > Suppose my experiment were perfectly successful, and the short > > and long irons felt identical. Then I'd swing them the same. > > If I were swinging my long irons EXACTLY like my short irons, > > the long ones would touch down near the middle of my stance, > > and be on the way up again when they reached the ball. Instant > > top! > > It may have less to do w/ feel and more to do w/ shorter length. That had occurred to me. But the difference is that with these clubs I can hit straight with the long irons from the middle of my stance. If I try that with long irons from a SW-matched set, I can't square the clubface in time and I push or slice. > >I haven't checked to see if I'm losing distance; it's a possibility. > > Wouldn't suprise me, but like you if the quality of your > contact is much improved, it's more than a fair trade. Dynacraft > would suggest the shorter length would be a direct result of > shorter clublengths, if my memory serves. "If memory serves" indeed!!! That's pretty sneaky, Guy. Lend me your copy of Dynacraft's shaft fitting book so you can ask me to look it up. ;-)> Oh, well..... Dynacraft does include a throwaway remark about that, but no data and no analysis. OTOH, my articles (mentioned above) are long on analysis of exactly that issue. I conclude that, for "swingers", there's very little to be gained by extra club length unless you go to lighter components to compensate for the increased swingweight/MOI. And only the purest "hitter" will gain distance proportionately to length with the same components. An article in a recent Clubmaker presents data without explanation that's strongly supportive of my analysis. Again, let me assert that I've separated "heft" factors of feel from "flex" factors, and only concentrated on the "heft" factors. I believe the "flex" factors can be treated separately, and I know the "control knobs" at construction time can be adjusted separately. It may be that they interact in determining feel, but I suspect they shouldn't be treated that way. You should determine your ideal "flex" factors separately, then clone clubs from one with that "flex" that also feels right -- by cloning BOTH the "heft" and "flex" of that club. (BTW, I probably should use a word other than "clone" here. I mean copy the flex and MOI; this has nothing to do with a "look-alike clone".) Hope this answers your questions. Dave

For completeness, here is what I wrote in February of 1995 about building MOI-matched clubs. It is the basis I used for building the clubs in the experiment, along with a brief version of the analysis on which it is based.

BUILDING AND MEASURING MOMENT-OF-INERTIA-MATCHED CLUBS

Dave Tutelman - February 1995

Since I began writing about moment of inertia (MOI) as an alternative to swingweight (SW) for golf club design, I've been in discussions with people about the possibility of MOI-matching a set of clubs instead of SW-matching them. The first to approach me with this was TJ Fields of Iowa State University. He sent me the output from a number of runs of a computer program that showed that: - If successive irons vary by 1/2" (the standard design) they will have very similar swingweights. - If successive irons vary by 3/8" they will have very similar moments of inertia measured about a fulcrum 4" from the butt. This was a very interesting first result, even though I wasn't satisfied with it. TJ assumed that the pivot would be the middle of the hand grip on the club. But my analysis was that the pivot was the wrists, just about at the butt. Besides, I wanted to go to a program that would match MOI exactly, club by club. I have since written that program, built a set of clubs from the numbers it gave me, and played around with a lot of designs. With all this experience behind me, let me just say that TJ had the right idea, even if the wrong numbers (IMHO). I've come to the conclusion that the most practical way to build a set of MOI-matched clubs is the same as building a SW-match set; only the numbers are different. The rules below apply to building a matched set of irons, using fairly standard clubheads that differ in weight club-to-club by seven grams. (This is true for every set in every catalog I've checked this evening.) I haven't worked the problem yet for woods, but it should yield to the same analytical techniques at the end of this article if anyone wants to get ahead of me on this. SW Matching MOI Matching =========== ============ Trimming to length .5" between .4" between clubs clubs Measured swingweight All clubs 1/2 SW Point the same between clubs (longer lighter) That's all there is to it. If you usually build clubs by trimming to designed length and trusting it, just change the trim. The 0.1" difference isn't much for a single interval, but it adds up to a 3/4" difference over a set -- and nearly a 5 SW point difference from the shortest club to the longest. If you measure your clubs and correct the swingweight as you build them, you can use the same swingweight scale you're used to. Just don't make the swingweights all the same; instead, aim for each longer club to be 1/2 point lighter than the last. This does raise an interesting point, though. What happens when the golfer approaches the new set KNOWING the swingweight he/she wants? The way I'd handle this (the way I DID handle this when I built my own MOI-matched set) is to ask, "What's your favorite club in the bag?" Copy that club's SW, then work up and down the set with a .4" trim and the half-point rule. BTW, that .4" does sound a lot like TJ's original recommendation (which works out to .375"). Well the actual number I get is .438", which is very close to 7/16". So what? The .4" I recommend is a good hedge of bets between TJ's analysis and mine. The rest of this article is an analytical treatment of the problem. If you just want the pragmatics of MOI-matching a set of irons, you've already seen it and don't need to read the analysis. Variation of Length for Constant Swingweight What should the length interval be between irons, to give a constant swingweight? We can find this by differentiating the swingweight equation, and keeping the differential swingweight zero. SW = L (H + S/2) - 14 (H + S) Differentiating: dSW dSW --- = H + S/2 --- = L - 14 dL dH Or, expressed as differentials: dSW = (H + S/2) dL dSW = (L - 14) dH To keep SW constant, make sure dSW due to dH is equal and opposite to dSW due to DL: - (H + S/2) dL = (L - 14) dH (L - 14) dH -dL = ----------- H + S/2 Variation of Length for Constant Moment of Inertia We can do the same thing to the equation for MOI, giving us the conditions for constant MOI from club to club. I = L^2 (H + S/3) Differentiating: dI dI -- = 2L (H + S/3) -- = L^2 dL dH Or, expressed as differentials dI = 2L (H + S/3) dL dI = L^2 dH To keep I constant, make sure dI due to dH is equal and opposite to dI due to DL: - 2L (H + S/3) dL = L^2 dH L dH -dL = ----------- 2 (H + S/3) Table of Club-To-Club Increments The following table gives the club-to-club increment of length, computed about each iron in the set using the equations we derived above. We're using standard club lengths, and a fairly standard set of head weights (with a VERY standard 7 grams between clubs; i.e., dH = 7). We're assuming a 120 gram shaft, center-balanced. dL To Produce: Club H L Const SW Const MOI ==== === ==== ======== ========= 2I 235 39 .59 .50 3I 242 38.5 .57 .48 4I 249 38 .54 .46 5I 256 37.5 .52 .44 6I 263 37 .50 .43 7I 270 36.5 .48 .41 8I 277 36 .46 .40 9I 284 35.5 .44 .38 --- --- Average dL .513 .438 In the rules of thumb above, I rounded this to .5" for constant SW and .4" for constant MOI. Any clubmaker who can trim closer than the .1" difference is working with tools and skills well beyond mine. But the difference in average dL of .085" is significant; it is almost exactly 1/12 inch (.0833"). Note that a SW point corresponds to 1/6 inch in the irons. That means that the constant-MOI trim, if it were exact, would give an exact SW difference of 1/2 point from one club to the next. So, by measuring and adjusting the SW accordingly, any errors in trim can be minimized in the club's final MOI-match.