Frequency Profile Shaping Without A Meter
Dave Tutelman
Copyright June 21, 1998  All rights reserved
I'd like to thank Paul Nickles of Clubmakers' Heaven for providing
me
with a lot of actual measured data that supports my calculations. I
know
he won 't agree with all my conclusions, and I apologize to him in
advance
for that.
Here's a situation that may become common soon for amateur
clubmakers,
or even custom shops that are too small to have a frequency meter.
You are willing to put up with the PRECISION of your favorite shaft
manufacturer,
but you don't like the FREQUENCY PROFILE that you'll get if you follow
their
tiptrim instructions. You want to choose your favorite frequency
profile
for a set of irons, but you don't have a frequency meter and don't have
access to one? Well you can do it based on formula, with much the same
accuracy
as if you had followed the tiptrim instructions.
The biggest problem is that you are at the mercy of whatever
precision
the manufacturer builds into the shaft. Which shafts should you use,
and
which should you stay away from, when you care about frequency and you
don't
have a meter?
 The data in [WISH] indicates that the majormanufacturer steel
shafts are probably best for procedures like this.
 Don't use a sheetwrapped graphite shaft for this, unless it is a
model delivered with controlled or selected frequency. There's entirely
too much variation to be at all predictable. [WISH] (At the first
writing of these notes in 1996, that was generally true for all
sheetwrapped shafts. Today, there are companies that will
guarantee or select shafts with narrow frequency ranges, that can be
depended upon.)
 The data in [WISH] shows that filamentwound graphite from some
manufacturers can approach steel shafts in predictabilty. However, that
predictability can fall off for a tiptrim of more than 2 inches
[NICK].
That out of the way, let's look at how we would determine the
tiptrim
to get a particular frequency profile. To do this, we need two
properties
of the shaft:
 The tiptrim sensitivity of the shaft, in cpm per inch. This
tells us how many inches to tiptrim the shaft for each club, to effect
a given frequency change.
 The frequency profile of a set of irons, given that they're all
tiptrimmed the same (for instance, not at all), instead of tipped
according to manufacturer's instructions.
If we have both #1 and #2 for a shaft, we can adjust the untrimmed
profile
#2 by tiptrimming according to #1, to get any profile we want.
(Caveat:
that's any profile until we run out of shaft. We can run out of tip to
trim,
or we can trim so much tip that there isn't enough butt left for the
length
we need. But this isn't a property of the method proposed here; it can
also
happen when trimming frequency with a meter, or even when following the
manufacturer's instructions.)
As it turns out, both #1 and #2 above seem to be be relatively
independent
of the shaft used.
The next sections will examine:
 The tiptrim sensitivity of commercially available shafts.
 The frequency profile of an untrimmed (or identically trimmed)
set of irons.
 Using #1 and #2 to give clubs with "interesting" frequency
profiles.
 Errors in the process.
 Derivation of #2 for the analytically inclined.
1. TipTrim Sensitivity
The tiptrim sensitivity of a shaft is the number of cpm stiffer the
shaft becomes when tiptrimmed an inch. This assumes, of course, that
the
shaft remains the same overall length, and is measured with the same
clubhead.
The best way to find the tiptrim sensitivity of a shaft is to
measure
it with a frequency meter. But this whole article is predicated on the
fact
that you don't have one. So can we make use of published data to deduce
the tiptrim sensitivity?
If a shaft is sold in a "combination flex" (e.g. "R&S"),
then we can learn enough to deduce its sensitivity. We can find out:
 The actual 5iron frequencies of the shaft for the "R" flex and
the "S" flex, which are published in the [SUMM].
 The tiptrim DIFFERENCE between the "R" and "S" flex, in inches,
which is part of the manufacturer's trimming instructions for the
shaft.
In all the numerical examples that follow, the frequency data is
from
[SUMM] and the tiptrim data is from the manufacturers' instructions
(as
reprinted in the Dynacraft and other catalogs).
 For instance, consider the FM Precision Microtaper R&S shaft.
 Its "S" flex is 298 cpm for a 5iron, tiptrimmed 4 inches.
 Its "R" flex is 287 cpm for a 5iron, tiptrimmed 2 inches.
 The frequency difference is 11 cpm, and the trim difference is 2
inches. Therefore...
 The sensitivity is 11/2 = 5.5 cpm per inch.
Steel shafts
Below is a table I compiled for many of the currentlyavailable
steel
combinationflex shafts.

Higher Freq

Lower Freq

Freq Diff

Trim Diff

Sensitivity (cpm/inch)

Apollo 





AP44 R&S 
314

301

13

2"

6.5

Shadow R&S 
303

298

5

2"

2.5 *

Spectre A&L 
295 
290

5

1"

5.0

Spectre R&S 
306

299

7

2"

3.5 *

FM Precision
& Brunswick 





LFA A&L 
283

277

6

1"

6.0

MFA R&S 
318

308

10

2"

5.0

Microtaper R&S 
298

287

11

2"

5.5

True Temper 





Dynalite A&L 
290

280

10

1"

10.0 *

Dynalite R&S 
317

304

13

2"

6.5

Dynamic A&L 
278

273

5

1"

5.0

Dynamic R&S 
323

310

13

2"

6.5

TT Lite A&L 
297

293

4

1"

4.0 *

TT Lite R&S 
323

312

11

2"

5.5

Note that more than 2/3 of the shafts show sensitivities in the
range
of 5.0 to 6.5. I've flagged the "outliers"; but note that the
method Dynacraft used to determine their frequency was such that it
could
be explained by relatively small sampletosample variation (3 cpm in
all
but one case), rather than a large error in the tiptrim sensitivity.
Indeed,
if we look at the manufacturers' "intent":
 Most manufacturers want a 1013 cpm difference between "R" and
"S".
 Most manufacturers (and ALL steel shafts that I've seen) specify
tiptrimming a 2" difference between R and S in combi shafts.
 This implies a tiptrim sensitivity between 5.0 and 6.5.
In the absence of reliable data to the contrary, I'd use a
sensitivity
of 5.5 cpm/inch for any of these shafts. Moreover, many of these
combination
shafts have singleflex (i.e. noncombination) cousins; for instance,
the
Dynamic combi R&S is the same pattern as the Dynamic Gold R and the
Dynamic Gold S. I'd use the same number for them, on the grounds that
the
shaft pattern is so similar that the tiptrim sensitivity is probably
the
same.
I have a limited verification of this theory for the Dynamic Lite; I
tiptrimmed a sample shaft and measured its frequency for a constant
length
and head weight, and tiptrims of 0, 1", 2" and 3". Each
inch of tiptrim raised the frequency between 5 and 6 cpm on my
frequency
meter.
Graphite shafts
My first cut at this dealt only with steel shafts, because graphite
shafts,
especially sheetwrapped graphite, is not sufficiently consistent in
the
raw shaft to consider. After all, why aim at a specific frequency
profile when you have no idea of any shaft's base frequency?
But more recently, a few graphite shaft manufacturers are doing a
lot
better in delivering shafts in a tight frequency range. Some are
doing
it with tighter processing, and others are measuring the manufactured
shafts
and sorting them by frequency. Either way, the customer gets
shafts
of a consistent frequency across the set.
So I decided to try to compile a similar table for graphites, and
see
whether there is a useful estimate of tiptrim sensitivity. I
went
to a more recent (1997) version of [SUMM], in order to look at more
modern
graphite shafts. Even so, it was hard to find good data points,
because
not many graphite shafts come in combinationtrim models. Of
course,
none of the models with guaranteed frequency come in a
combinationtrim.
Here are the models that could lend some information to the tiptrim
sensitivity
of graphite shafts.
[F] = Filamentwound

Higher Freq

Lower Freq

Freq Diff

Trim Diff

Sensitivity (cpm/inch)

Aldila 





CLW A&L 
271

250

23

2"

10.5

CLW R&S 
280

261

19

2"

9.5

LoTorque A&L 
276 
259

17

2"

8.5

LoTorque R&S 
295

279

16

2"

8.0

VL (A&L) 
274

260

14

2"

7.0

VX (R&S) 
292

279

13

2"

6.5

Harrison 





Boron Gold R&S 
302

299

3

1/2"

6.0

Tour Classic II R&S 
287

281

6

1/2"

12.0

Paragon 





Lite Touch A&L 
283

270

13

1"

13.0

Lite Touch R&S 
292

286

6

1"

6.0

Rapport 





Synsor R&S 
328

308

20

1"

20.0 *

True Temper 





Assailant [F] A&L 
256

248

8

1"

8.0

Assailant [F] R&S 
283

275

8

1"

8.0

The graphite shafts are a little harder to draw a conclusion
about.
There are fewer combination models than steel, and they vary more in
their
tip trim instructions. As a result, the spread of data is wider
than
the steel, even if you ignore the one obvious outlier. For the
rest
of the data:
 The median value is 8.0, and the average is 8.6.
 The two values for filamentwound shafts are both 8.0.
This suggests using 8 cpm/inch as a guesstimate of tiptrim
sensitivity
for graphite shafts. But a better strategy would be to ask the
manufacturer
if they can recommend a number for your shaft. And a still better
strategy would be to use a frequency meter, unless you have good reason
to believe in the low tolerances of the shaft you have chosen.
For the examples in this article, we will deal with steel shafts
only.
It should not be hard to duplicate the calculations for a number other
than
6.5 (for instance, 8.0), provided you trust the shafts to be as
consistent
as steel.
2. Frequency Profile of an Untrimmed Set
According to [COCH], the frequency of a club is proportional to:
 The square root of the stiffness of the material of the shaft.
 The square root of the stiffness of the geometry (the
crosssection) of the shaft. This may have to be averaged in some way
over the length of the shaft.
 INVERSELY to the 3/2 power of the club length.
 INVERSELY to the square root of the head weight.
 It is also affected by the shaft weight, but the effect is
smaller than the above factors.
The first two (especially the second) are a little difficult to deal
with if you're trying to predict the absolute frequency of a club based
on measurement. But if what you're trying to do is predict the
VARIATION
of frequency of the same kind of shaft going from one iron to the next,
you can lump the first two parameters together in some easy way;
they're
the same for all the clubs in the "series". It turns out that
some nominal known frequency for some club in the set will be quite
sufficient
to pin down the shaft parameters.
So we're left with:
 Club length, and its variation between clubs.
 We'll use a nominal length of 37 1/2" for a 5iron.
 We'll use 1/2" as the clubtoclub variation; this is quite
standard.
 Head weight, and its variation between clubs.
 We'll use a nominal weight of 256 grams for a 5iron.
 We'll use 7 grams as the clubtoclub variation; this is very
standard.
 Shaft weight, and its variation between clubs.
 Since this effect is much smaller than the others for
reasonable shaft weights, we'll ignore this for now and examine what we
lose in the section on errors.
The frequency variation from club to club is a function of these
variables,
and also of the frequency of the "base" club. We'll use a 5iron
for our base club (I hope nobody's surprised), and a nominal frequency
of
300 cpm (a middling "R" shaft). In fact, the variation is directly
proportional to this base frequency.
If we reflect the proportionalities above and use partial
derivatives
to obtain the total differential, we're left with the equation:
Plug in the values above: M=256 grams, dM=7 grams, L=37.5", and
dL=0.5", and we get:
For a set with a 5iron frequency of 300 cpm, this gives a
clubtoclub
variation of 1.89 cpm.
If we look at the variation of the number as we change the
assumptions,
we get numbers like:
Conditions

ClubtoClub Frequency Variation

Nominal

1.89 cpm

Light head (253 grams)

1.86 cpm

Heavy head (263 grams)

2.01 cpm

Underlength club (36 inches)

2.13 cpm

Overlength club (39 inches)

1.63 cpm

Flexible shaft (280 cpm)

1.76 cpm

Stiff shaft (320 cpm)

2.02 cpm

There isn't all that much variation in simply assuming a nominal 1.9
cpm or so, unless we're talking about errors all building in the same
direction.
But errors in the same direction aren't likely. For instance, the
biggest
pair of errors would be a shaft that's very short and very stiff, not a
likely combination. Someone with a strong enough swing to need a very
stiff
shaft would almost certainly go for the extra distance afforded by a
longer
shaft.
The empirical data I've seen suggests that this number is something
like
1.5 to 1.8, so this isn't far off the mark.
So let's use a nominal variation of 1.8 cpm per club. If we're off
by
0.2 or 0.3 cpm, there are only four clubs each side of a 5iron anyway,
so the maximum error will only be 1 cpm in the 9iron and 1iron.
What we've found is the frequency profile of the set if all
the shafts are tiptrimmed the same. Not to the
manufacturer's
spec, but the same for all clubs. Here's the frequency profile it gives
for a set with a 300 cpm 5iron.
Club

Freq (cpm)

Club

Freq (cpm)

2I

295

6I

302

3I

296

7I

304

4I

298

8I

305

5I

300

9I

307

Now that we know:
 What the profile looks like with CONSTANT tiptrim, and
 How much of a frequency change we get from a specific tiptrim...
We are ready to specify a tiptrim to give us any frequency profile
we
want.
3. Creating Your Own Frequency Profile
We know that:
 With no tip trimming, or the same tip trimming for all the
shafts, the clubtoclub difference in frequency is 1.8 cpm, increasing
as the clubs get shorter. (If we have a sufficiently unusual club, we
know how to adjust this number up or down.)
 For the same shaft, clubhead, and club length, tiptrimming an
extra inch increases the frequency by 5.5 cpm. (This is a good estimate
for most shafts, though it's probably the least reliable shaft
parameter.)
Using this, we can see how to tiptrim to get any frequency profile
we
want. For example:
Sloped at 1.8 cpm per club:
This is just the clubtoclub frequency variation of an untrimmed
shaft.
For the shaft you're going to use, find the tiptrim that gives a
5iron
the frequency you want; say that trim is 1.8" for the 5iron. Then
trim 1.8" from EVERY shaft tip, and butttrim the clubs to length.
According to Paul Nickles [NICK], this isn't a bad slope for people
who
aren't convinced either way between the sloped and the
constantfrequency
schools, and therefore want to play it safe with a small slope.
Sloped at the "Brunswick slope" of 4.2 cpm per club:
This requires additional tip trimming, because the slope is more
than
the constanttrim slope of 1.8 cpm per club. We need an additional
But we know the tiptrim sensitivity is 5.5 cpm per inch, so each
successively
shorter club needs to be trimmed an additional
For all practical purposes, this is 1/2", which is the usual
tiptrimming
instructions for the major steel shaft manufacturers. (I consider it
something
of a vindication of this technique that it gives the same tiptrimming
instructions
that Brunswick recommended for their nonfrequencymatched shafts.)
Suppose we want the same 5iron frequency as above, so we know to
tiptrim
the 5iron by 1.8". Then our tiptrim chart would be:
Note that there isn't room on the shaft to get the full flexibility
we
want in the 1iron. We are 0.2" short; that is, at zero tiptrim the
shaft is still trimmed 0.2" more than we specified. By now we should
know that this means the frequency of the 1iron is higher than we
wanted
by:
Since no golfer can feel a 1 cpm difference, we won't worry about
it.
Constant frequency, zero slope
For this, we need to tiptrim to ELIMINATE the 1.8 cpm per club.
That
means we do more tiptrimming on the longer clubs; this
is
backwards from the way we usually think of tip trimming. Each longer
club needs to be trimmed an additional
For all practical purposes, this is 1/3".
It is worth noting that Paul Nickles [NICK] has told me that 1/4"
is a good rule of thumb for a first cut at a constant frequency. The
difference
between 1/3" and 1/4" per club is at its maximum in a 9iron,
where the difference is 2/3", or 3.6 cpm. This is right at the limit
of what a golfer can feel, so the difference probably isn't important.
If
you tune the frequencies with a frequency meter, you can tell the
difference
 but of course you wouldn't need a rule of thumb.
Suppose we want the same 5iron frequency as above, so we know to
tiptrim
the 5iron by 1.8". Then our tiptrim chart would be:
This time we have enough tip to make the long irons flexible enough,
but we probably don't have room on the butt to make the long irons long
enough. For instance, consider that a 1iron is usually 39.5" long.
Imagine we bought a shaft whose raw length is the common 41". Once
we tiptrim 3.1", we are left with 37.9" Now, suppose the clubhead's
bore ends an inch above the ground. That means we need a cut shaft
length
of
So we're 0.6" too short. We may have to leave a little more tip,
and go for a slightly softer 1iron.
Finding the base frequency
One point is still a mystery, even to the careful reader: how did we
know that our base 5iron tiptrim was 1.8" to get the frequency we
wanted?
In order to figure this out, we need to know our target frequency
and
the frequency of the shaft for some known 5iron and tiptrim. For
instance,
suppose we are using an Apollo AP44 shaft to make clubs built 1/2"
over length, with standard weight clubheads. We want a 5iron frequency
of 295 cpm.
Here are the calculations:
 From [SUMM], we know that at standard length and 2" tip trim
(standard for "R" flex), the club should vibrate at 301 cpm.
 Frequency varies by 10 cpm for each inch of added length,
assuming the same clubhead weight and same tip trim. (This follows from
the equation in the last section, but isn't obvious from anything
that's been said so far. Ten cpm/inch is pretty constant over the
irons, and independent of the shaft.) So 1/2" overlength takes us to
296 cpm.
 We need another 1 cpm slower. Let's leave the tip 0.2" longer, at
5.5 cpm per inch of tip trim.
That gets us down to 295 cpm, with a tip trim of
Summary table
Let's end this section with a table of the frequencies of each of
the
tiptrim strategies above. We'll go with the model we've been using:
 Apollo AP44 shaft
 5iron trimmed 1.8" to give 295 cpm

Constant
Frequency

Constant
TipTrim

Brunswick
Slope

TipTrim Amount

1/3"
per club

1.8"
all clubs

+1/2"
per club

Freq diff per club

0

1.8 cpm

4.2 cpm

1 iron

295

288

278

2 iron

295

290

282

3 iron

295

291

287





4 iron

295

293

291

5 iron

295

295

295

6 iron

295

297

299





7 iron

295

299

303

8 iron

295

300

308

9iron

295

302

312

Range
3i to 9i

0

11 cpm

25 cpm

A few points about this table:
 If you use a different shaft or a different base (5iron)
tiptrim, you can simply assume the table "scales" by just finding
adding the same number to all frequencies. (That number would be the
difference between your 5iron frequency and 295.) Your errors won't be
large compared to the other errors in the process.
 The constantfrequency column assumes the raw shaft is long
enough to take all the trimming we need to do. This isn't the case; the
AP44, like most R&S combi iron shafts, is 41" long. As we saw,
this will give a 1iron that is either too short or too flexible.
 The sloped systems give frequency differences that are very
substantial over the set. Remember that a difference of 5 cpm should
make a difference to most golfers, and a difference of 1013 cpm is a
full flex grade for most manufacturers. Even if you leave out the
1iron and 2iron, the range of even the modest constanttiptrim set
is a full flex grade. Most sets trimmed to manufacturers' instructions
cover a range of two and a half flex grades.
With this much variation across a set, I wonder why there isn't
complete
agreement within the industry on whether frequency should be constant
or
sloped. Oh, well......
4. Errors in the Process
Since the whole point of this exercise is to achieve a particular
frequency
profile WITHOUT the feedback of measurement, we really need to have
some
idea of how errors in the process are likely to propagate into
frequency
errors. We also need some idea of what is an acceptable frequency error.
First, what should our target error be? It's generally accepted that
frequency differences under 3 cpm are not felt by the great majority of
golfers. On the other hand, differences over 5 cpm will make a
difference
in performance and feel for most golfers. A full "letter flex grade"
(say, the difference between an "R" and an "S") is 1015
cpm for most shaft designs.
The process errors we'll have to take into account are:
 Shafttoshaft variations in stiffness and tip stiffness.
 Inconsistencies in the length of the club.
 Inconsistencies in the weight of the clubhead.
 Inaccuracies in the formula, including what we lost by ignoring
shaft weight.
Let's examine each in turn:
Shafttoshaft errors
This is the biggie. Data in [WISH] suggest that there is a 3 cpm
error
at most in Apollo and True Temper steel shafts. This number is a
composite
of raw shaft accuracy and tiptrim accuracy, which is what we're
depending
on in this process.
However, clubmakers who have worked extensively with frequency
meters
tell me the [WISH] data is hopelessly optimistic. They observe big
variations,
especially in the tiptrim sensitivity (which is not generally a
specified
or controlled parameter of the shaft). [NICK] and others have told me
that
every shaft model "has its quirks", completely aside from
shafttoshaft
repeatability; the tiptrim sensitivity isn't smooth across the trim
range
in the same shaft. (I might add that my own experience, while
more
limited than the clubmakers I'm citing, has borne out theWishon data.)
The only solace I can offer is that the results are no worse than
you
could expect from the same shafts following the manufacturer's tiptrim
instructions. But this way, you get to decide the overall shape of the
frequency.
Whatever the actual number, it may be a little bigger for
filamentwound
graphite, and is much bigger for sheetwrapped graphite. There
are
some graphite shaft manufacturer that recognize this, and either
control
their process better or measure their shafts and sell them at the
measured
frequency. If you can get the frequency data you need from your
shaft
vendor, then this might not be a problem for you.
Club length errors
You are the clubmaker. You control these. This is something you CAN
measure.
Don't let them happen!
If they do happen, the error is about 10 cpm per inch.
Clubhead weight errors
With reasonable quality control, a clubhead's weight should match
its
spec within 3 grams. (This is the stated tolerance in the Golfsmith,
GolfWorks,
and Dynacraft catalogs.)
A 3gram error in clubhead weight gives a 1.8 cpm error in
frequency,
if the iron vibrates at 300 cpm. The error is proportional to the
frequency,
in case the 300 cpm assumption isn't a good approximation.
It's worth noting that a 2gram difference in clubhead weight
results
in:
 About one point of swingweight, and
 A bit over one cpm of frequency.
The message to me is that the first order of business is to make
sure
that the clubhead weights are right for the swingweight you want. This
is
one of the most important things you can do to get the frequency right.
Paul Nickles has told me this [NICK], but I have even more dramatic
evidence.
After I built my frequency meter, I went back and checked out the
frequencies
of several sets that I had built and still had in the house. In all the
sets built before I had a swingweight scale, and none of the sets built
after I was using a swingweight scale, the biggest frequency error
corresponded
to the club with the biggest swingweight error. Lesson: the
maximum
frequency error would have been smaller if the clubs had been properly
matched
in swingweight or moment of inertia.
Formula errors
There are three main sources of error in the "formula":
 We are ignoring the shaft weight.
 We use a single constant, 1.8 cpm, for the frequency difference.
 We "linearized" the frequency by using differentials.
Let's examine each of these in turn.
The basic formula for frequency in [COCH] accounts for shaft
weight.
The mass term, in which we included only head mass, is really:
This assumes the shaft is of uniform weight (probably reasonable, if
not absolutely precise) and uniform flexibility (not even close; we
know
the tip is much more flexible than the butt). The latter fact suggests
that
a lot less than 0.24 of the shaft weight is participating in affecting
the
frequency. Even so, I recomputed the formula using the shaft effect
suggested
by [COCH], and came up with a clubtoclub variation of 2.6 cpm, rather
than 1.8. Since 1.8 is much closer to the actual empirical data, I have
to assume that an improper accounting for shaft weight is worse than
none
at all. So I went back to the formula with no factor for shaft weight.
Another source of error in the "formula" is that we are using
a single constant, 1.8 cpm, for the frequency difference
between
clubs. This ignores the fact that the frequency difference between
clubs
varies with a number of parameters, as we saw in an earlier section.
The
most serious of these are the overall length of the set and the overall
frequency of the set. If the design of the set has the 5iron very
different
from 300 cpm and 37.5", then you'll want to adjust the clubtoclub
frequency difference from the nominal value of 1.8. (There is a table
in
a previous section that is a useful guide.)
There is another error implicit in the formula, due to the fact that
the clubtoclub frequency difference is actually proportional to the
frequency
itself. We have "linearized" the frequency difference
between clubs, when it is really proportional to the frequency.
Assuming
that the clubtoclub difference is a constant 1.8 cpm gives an error
that
grows as you get further from the 5iron. In the table below, we design
a set to a 5iron frequency of 300 cpm, and see how the approximation
of
a constant 1.8 cpm per club varies from the actual frequency (as
computed
by the exact formula for frequency [COCH], minus the shaft weight
component).

Actual

Constant

Error

2iron

295.2

294.6

0.6

5iron

300

300

0

9iron

309.2

307.2

2.0

Thus the approximation of a constant frequency difference over the
set
gives an error up to 2 cpm at the extreme, and less through the rest of
the set.
5. Experimental Results
I am indebted to Paul Nickles for providing most of these
measurements.
After most of the work for this article was complete, I finally got my
own
frequency meter working and was able to add some measurements of my own.
The table below shows the frequency accuracy of a number of sets of
irons.
The entries in the table were obtained by taking frequency measurements
for sets of irons, and plotting them on linear graph paper. A straight
line
was drawn to give a "best fit" to the data by eye. Then I tabulated:
 The slope of the bestfit straight line.
 The maximum error of any club from that straight line.
 The average error (magnitude) over all clubs from the straight
line.
This was done for expensive offtherack OEM clubs, both steel and
graphiteshafted,
and for several steelshafted sets that I built before I had a
frequency
meter. My clubs were trimmed to manufacturers' recommendations. It is
also
surmised that the OEM clubs were similarly trimmed (1/2" per club),
judging from the positions of the steps of the shafts.
I haven't identified the OEM clubs by name, because the purpose of
this
data isn't a "consumer reports" brand comparison but rather a
sample of what sort of frequencies you get when you trim to
manufacturers'
specs.
Measured Iron Sets (all data in cpm)
Irons

Slope

Max
Error

Average
Error

OEM
Steel




A

4.7

4

1.0

B

3.7

3

0.4

C

4.7

2

0.9

D

4.0

4

1.1

E

4.7

4

0.9

F

4.0

5

1.7

G

4.8

4

2.6

OEM
Graphite




H

3.3

8

5.6

I

4.7

19

5.3

J

3.3

5

2.0

K

4.3

18

3.9

My
Steel




L

3.6

4

1.2

M

4.3

2

1.0

N

3.3

5

1.9

O

7.0

2

0.8

P

4.2

2

1.0

What conclusions can we draw from this data?
 The general approach described in this paper is pretty good for
steel shafts, because it predicts a slope of 4.3 cpm per club and that
is pretty close to what the clubs are. (Ignore set "O"; those are
TrueTemper FlexFlow shafts, which have very different characteristics
and trimming instructions from most steel shafts.)
 Steel shafts lend themselves well to this approach. A few sets
had a single club that was as much as 5 cpm off; 5 cpm is the minimum
that most golfers can detect.
 Runofthemill graphite shafts do not lend themselves to this
approach. The best set of the graphites was about as precise as the
worst set of the steels. Half the graphites had at least one club that
was nearly two full flex grades in error. That's like ordering an "S"
and getting an "A" flex.
I'll venture an editorial opinion based on this data. I will NEVER
get
a set of graphiteshafted clubs unless I know for a fact that it was
frequencymatched
on a machine. The graphite OEM irons in the table that gave such poor
results
were topoftheline sets from very respected manufacturers, selling
for
big bucks. So people who say, "You get what you pay for," or "With
the big OEMs, you're paying for quality," have a lot of explaining
yet to do. This data is the best argument I've seen yet for custom
clubmaking.
6. Acknowledgments
First and foremost, I'd like to thank Paul Nickles of Clubmaker's
Heaven
(clubhvn@execpc.com), whose experimental data inspired much of this
work,
and whose encouragement and detailed criticism greatly increased its
quality.
I'd also like to express my gratitude to Dave Miko (now a PCS ClassA
Clubmaker;
congratulations!) and to Normand Buckle for reviewing drafts of the
article
and providing me with useful suggestions for its improvement. Finally,
thanks to Neil Daniels for finding a math error in the appendix. It was
sitting there for eight years before anybody noticed. Fortunately, it
does not affect any of the conclusions.
APPENDIX
Derivation of the ConstantTrim
Frequencies
From the appendix of [COCH], we know that:
where:
E = Material stiffness
I = Geometry stiffness (crosssection MOI)
L = Length of club
H = Head mass
S = Shaft mass
The conditions to make this true include an assumption of constant
crosssection,
which is clearly an incorrect assumption for real shafts. This figures
into
the equation in the factors of 0.24 and I. But we can finesse this
issue
because:
 The .24 factor doesn't require the same shape of cross section,
just the same crosssectional area. Until quite recently, this was a
good approximation to real shafts. It still is, unless you get a shaft
that's deliberately tipheavy, or deliberately "bubbled", or has some
other deliberate manipulation of the weight distribution.
 The I factor can be replaced by some sort of "average I". On the
surface, this would appear to mean we need to analyze every
diameterreducing step. But we could just assume we know it, and see
what happens to it in our analysis. (It turns out to go away altogether
in the variational analysis.)
Let's reexpress (1) as:
Where:
This will make it easier to differentiate, and the next step is to
take
partial derivatives with respect to the club parameters L and M. (In
the
math below, the representation is handicapped by the lack of upper and
lowercase "delta" on the standard computer keyboard. I use "d"
for itself and both deltas. Math majors will miss it, but most others
will
probably applaud the loss of a sometimesconfusing precision. :)
df 1.73 1/2
3/2 3/2 1 f
 =   (EI) L
M =    (3a)
dM 4pi
2 M
df 1.73 1/2
5/2 1/2 3 f
 =   (EI) L
M =    (3b)
dL 4pi
2 L
We didn't take partial derivatives of E or I, because we're looking
at
the variation with constant tiptrim. We don't need to know exactly
what
E and I are, just that they don't change. Since tiptrim is the thing
that
changes the flexibility of the shaft, no tiptrim means no (EI) change.
The actual values of E and I make themselves felt implicitly; note that
the actual frequency (which includes an EI factor) is part of the
derivatives.
We can find the total variation of frequency when we vary the other
club
parameters, by combining (3a) and (3b) as a total differential.
df df
df =  dM +  dL
dM dL
1 dM 3 dL
df = f (      
) (4)
2 M 2 L
Equation (4) is our fundamental description of clubtoclub
frequency
variation. When we plug in nominal values for a 5iron:
L =
37.5"
dL = 0.5" (Clubs get 1/2" shorter each club)
H = 256 grams dH = +7 grams
(Clubs
get 7 grams heavier each club)
f = 300 cpm (A nice round number, in the middle of the 5iron range)
We get df = 1.89 cpm per club.
The table of other values in the text come from plugging other
values
of club parameters into equation (4).
REFERENCES
COCH 
Alastair Cochran & John Stobbs, "The
Search for the Perfect
Swing", Lippincott, 1968. 
NICK 
Paul Nickles, private communication,
Dec 1995. 
SUMM 
Jeff Summitt, "1995 Shaft
Fitting Addendum", Dynacraft
Golf Products, 1995. 
WISH 
Tom Wishon & Jeff Summitt,
"The Modern Guide to Shaft
Fitting", Dynacraft Golf Products, 1992. 
