|
| Home ⇒ golf ⇒ shafts ⇒ Current Article |
Frequency Profile Shaping Without A MeterDave 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 tip-trim 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 tip-trim 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?
That out of the way, let's look at how we would determine the tip-trim to get a particular frequency profile. To do this, we need two properties of the shaft:
If we have both #1 and #2 for a shaft, we can adjust the untrimmed profile #2 by tip-trimming 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:
1. Tip-Trim SensitivityThe tip-trim sensitivity of a shaft is the number of cpm stiffer the shaft becomes when tip-trimmed 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 tip-trim 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 tip-trim 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:
In all the numerical examples that follow, the frequency data is from [SUMM] and the tip-trim data is from the manufacturers' instructions (as reprinted in the Dynacraft and other catalogs).
Steel shaftsBelow is a table I compiled for many of the currently-available
steel
combination-flex shafts.
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 sample-to-sample variation (3 cpm in all but one case), rather than a large error in the tip-trim sensitivity. Indeed, if we look at the manufacturers' "intent":
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 single-flex (i.e.- non-combination) 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 tip-trim sensitivity is probably the same. I have a limited verification of this theory for the Dynamic Lite; I
tip-trimmed a sample shaft and measured its frequency for a constant
length
and head weight, and tip-trims of 0, 1", 2" and 3". Each
inch of tip-trim raised the frequency between 5 and 6 cpm on my
frequency
meter. Graphite shaftsMy first cut at this dealt only with steel shafts, because graphite shafts, especially sheet-wrapped 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 tip-trim 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 combination-trim models. Of
course,
none of the models with guaranteed frequency come in a
combination-trim.
Here are the models that could lend some information to the tip-trim
sensitivity
of graphite shafts.
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:
This suggests using 8 cpm/inch as a guesstimate of tip-trim 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 SetAccording to [COCH], the frequency of a club is proportional to:
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:
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 5-iron 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:
1 dM 3 dL Plug in the values above: M=256 grams, dM=7 grams, L=37.5", and
dL=-0.5", and we get: df = .0063 * f For a set with a 5-iron frequency of 300 cpm, this gives a club-to-club variation of 1.89 cpm. If we look at the variation of the number as we change the
assumptions,
we get numbers like:
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 5-iron anyway, so the maximum error will only be 1 cpm in the 9-iron and 1-iron. What we've found is the frequency profile of the set if all
the shafts are tip-trimmed 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 5-iron.
We are ready to specify a tip-trim to give us any frequency profile we want.
3. Creating Your Own Frequency ProfileWe know that:
Using this, we can see how to tip-trim to get any frequency profile
we
want. For example: Sloped at 1.8 cpm per club:This is just the club-to-club frequency variation of an untrimmed shaft. For the shaft you're going to use, find the tip-trim that gives a 5-iron the frequency you want; say that trim is 1.8" for the 5-iron. Then trim 1.8" from EVERY shaft tip, and butt-trim 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
constant-frequency
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 constant-trim slope of 1.8 cpm per club. We need an additional 4.2 - 1.8 = 2.4 cpm per club But we know the tip-trim sensitivity is 5.5 cpm per inch, so each successively shorter club needs to be trimmed an additional 2.4 cpm per club / 5.5 cpm per inch = 0.44 inches per club For all practical purposes, this is 1/2", which is the usual tip-trimming instructions for the major steel shaft manufacturers. (I consider it something of a vindication of this technique that it gives the same tip-trimming instructions that Brunswick recommended for their non-frequency-matched shafts.) Suppose we want the same 5-iron frequency as above, so we know to tip-trim the 5-iron by 1.8". Then our tip-trim chart would be: 1I 2I 3I 4I
5I 6I 7I 8I 9I Note that there isn't room on the shaft to get the full flexibility we want in the 1-iron. We are 0.2" short; that is, at zero tip-trim the shaft is still trimmed 0.2" more than we specified. By now we should know that this means the frequency of the 1-iron is higher than we wanted by: 0.2" * 5.5 cpm/inch = 1 cpm Since no golfer can feel a 1 cpm difference, we won't worry about
it.
Constant frequency, zero slopeFor this, we need to tip-trim to ELIMINATE the 1.8 cpm per club. That means we do more tip-trimming 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 1.8 cpm per club / 5.5 cpm per inch = 0.33 inches per club 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 9-iron, 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 5-iron frequency as above, so we know to tip-trim the 5-iron by 1.8". Then our tip-trim chart would be: 1I 2I 3I
4I 5I 6I 7I
8I 9I 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 1-iron is usually 39.5" long. Imagine we bought a shaft whose raw length is the common 41". Once we tip-trim 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 39.5"-1" = 38.5" So we're 0.6" too short. We may have to leave a little more tip,
and go for a slightly softer 1-iron. Finding the base frequencyOne point is still a mystery, even to the careful reader: how did we know that our base 5-iron tip-trim 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 5-iron and tip-trim. For instance, suppose we are using an Apollo AP-44 shaft to make clubs built 1/2" over length, with standard weight clubheads. We want a 5-iron frequency of 295 cpm. Here are the calculations:
That gets us down to 295 cpm, with a tip trim of 2" - 0.2" = 1.8" Summary tableLet's end this section with a table of the frequencies of each of the tip-trim strategies above. We'll go with the model we've been using:
A few points about this table:
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 ProcessSince 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 10-15 cpm for most shaft designs. The process errors we'll have to take into account are:
Let's examine each in turn: Shaft-to-shaft errorsThis 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 tip-trim 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 tip-trim 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 shaft-to-shaft repeatability; the tip-trim 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 tip-trim 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
filament-wound
graphite, and is much bigger for sheet-wrapped 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 errorsYou 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 errorsWith 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 3-gram 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 2-gram difference in clubhead weight results in:
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 errorsThere are three main sources of error in the "formula":
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: M = H + .24*S 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 club-to-club 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 5-iron very different from 300 cpm and 37.5", then you'll want to adjust the club-to-club 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 club-to-club 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 club-to-club difference is a constant 1.8 cpm gives an error
that
grows as you get further from the 5-iron. In the table below, we design
a set to a 5-iron 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).
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:
This was done for expensive off-the-rack OEM clubs, both steel- and graphite-shafted, and for several steel-shafted 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.
What conclusions can we draw from this data?
I'll venture an editorial opinion based on this data. I will NEVER get a set of graphite-shafted clubs unless I know for a fact that it was frequency-matched on a machine. The graphite OEM irons in the table that gave such poor results were top-of-the-line 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 Class-A
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 Constant-Trim Frequencies
From the appendix of [COCH], we know that: 1.73
EI where: E = Material stiffness The conditions to make this true include an assumption of constant cross-section, 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:
Let's re-express (1) as: 1.73 1/2
-3/2 -1/2 Where: M = H + .24S if we want
to include the shaft weight 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 lower-case "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 sometimes-confusing precision. :-) df 1.73 1/2
-3/2 -3/2 1 f df 1.73 1/2
-5/2 -1/2 3 f We didn't take partial derivatives of E or I, because we're looking at the variation with constant tip-trim. We don't need to know exactly what E and I are, just that they don't change. Since tip-trim is the thing that changes the flexibility of the shaft, no tip-trim 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
Equation (4) is our fundamental description of club-to-club frequency variation. When we plug in nominal values for a 5-iron: L =
37.5"
dL = -0.5" (Clubs get 1/2" shorter each club) 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
|
|
|