App 4 
Modify the model to accommodate GRT
Here are the details of the factorbyfactor reevaluation of the
model, in an attempt to make GRT look like a good design choice.
Change the parameters of the model
The basis of the model is equation 3a:
spin = 25
y V_{b}
That, and the value of C
(which we set at 1.3, but could realistically vary between 1.1 and
1.5), are the major contributors to the way the model fits the
published data. (C figures in
because it is instrumental in determining y
in equation 3a.) The simplest, and undoubtedly the most effective, way
to vary the model is to play with the value 25
 the coefficient in equation 3a, so we'll refer to it as coef
 and with the value of C.
I played with C and especially coef
until the model gave an optimum roll that was close to GRT.

Original
Model
coef=25
C=1.2 
Modified
Model
coef=12
C=1.1 
Actual
GRT 
Figure A41 
Optimum
highface roll 
8" 
16.4" 
15" 
Optimum
lowface roll 
10" 
19.1" 
20" 
Figure A42
This is a good match to the
actual face roll of a GRT clubhead. The
next question becomes: How does this modified model fare against the
test data for vertical gear effect. Let's compare it with the Hotstix
data.
Let's review what we are doing here. We have modified the parameters of
the mathematical model so that the optimal face roll is essentially the
same as GRT. This gives us a new model  an asyet unvalidated model. We have data that we can use to
validate any model for vertical gear effect. We just have to
plug the launch angle and clubhead speed into the mathematical model,
and see what backspin we wind up with. If the backspin matches the
measured data, then the model is valid; if not, the model is not a good
representation of reality.
So let's try it with the new, modified model (coef=12,
to match GRT). Figure A42 is the comparison. The modified model
(yellow
line) does not match Hotstix data at all, while the original model (red
line) is a good match. The yellow line does not even show backspin
decreasing with the height of impact; it is nearly constant  and
increasing if you look closely. So it cannot possibly be a valid model.
And, if it is not a good model for vertical gear effect, it cannot be
used to justify GRT. We will have to look elsewhere. 
Shaft tip flex
Suppose shaft tip flex
limited
the gear effect. If it limited it enough, then GRT might be a
superior face roll design. Let's explore this a little. Points to
consider:
 The Hotstix test data agrees with the original
model,
unfettered by shaft stiffness. So, for this assumption to be tweaked
enough to make GRT superior, we would need to assume that Hotstix used
a flexibletip shaft on the driver, but GRT users have to use a much
stiffer tip.
 Therefore, it would require one of two
assumptions:
 That Hotstix took 100mph
clubheadspeed data with a shaft illfit for most 100mph clubheadspeed
golfers. That is not very likely. Hotstix is a competent
organization, whose raison d'être is clubfitting.
 That golfers using a GRT clubhead would use a
shaft
that was much more tipstiff than normal. I can think of no sound
clubfitting reason for this to be the case.
 How much stiffer, and how much good would it
do? Even
if it were the stiffest you ever tried to use, it would still only cut
gear effect by 15%. (We looked at this a few pages ago.)
In order to get to the range where GRT works, we need to cut gear
effect by 50%. So even a hugely tipstiff shaft would get us less than
a third of what we need.
If we assume that a GRT
user will probably want a shaft not wildly stiffer than Hotstix used,
even the 15% is way more than we would see in reality. Think about it;
do you want to use a shaft that is otherwise too tipstiff for you,
just
so that GRT will work for you? If course not! You want a shaft that
fits you, not a shaft that allows you to fit a particular face roll
profile. So this would work only for a golfer who naturally needs a
shaft whose
tip is a lot stiffer than normal for his clubhead speed.
Bearing that in mind, the limit is probably well under 5%, nowhere near
the 15% theoretical maximum.
What would be the influence on roll of a 5% reduction of gear effect
due to shaft tip stiffness? (We're being very generous here; there are
few golfers that would benefit from a shaft change of this magnitude.)
When we go through reoptimization, we find that:
 At 0.8", the ideal loft goes from 4.7° to
4.9°, a
difference of +0.2°.
 At +0.8", the ideal loft goes from 16.0° to
15.6, a
difference 0.4°.
 At +0.23", there is no gear effect spin; the
ideal
loft remains at 11.7°.
Coincidentally, the zerogeareffect height divides the face quite well
into top third and bottom two thirds. So we will be able to see the
effect on roll in the zones that GRT divides the face. 
Our formula for
backspin
due to loft
I am using a formula to go from effective loft to backspin. The formula
is not from basic physics by rather a simple equation (second degree
polynomial) that gives a good fit to
data from several sources. (Cochran
& Stobbs' book, the TWGT trajectory
profiler, and Max
Dupilka's trajectory model program.) It is possible that the
sources are incorrect. Alternatively, it is possible that they are
correct, but based on measured backspin  with the gear effect spin
already factored in.
The backspin formula is not part of the model of gear effect. But it is
(a)
part of the calculations by which we validate the model against
realworld data, and (b) part of the calculations to optimize face
roll. So an error here would cast doubt on both the validity of the
model itself and the optimality of the face roll computed from it. So
let's see how much it might be off, and whether the error would lead to
a model that makes GRT a more likely candidate for optimal roll.
Spin for 11° loft
at 100mph clubhead speed 
Wishon

2860 
Our
formula 
3053 
Dupilka 
3102 
The formula's
origin
is described in my article on launch conditions. It is a curve that
lies between the Dupilka and Wishon numbers for backspin. The table at
the right shows typical numbers for the range where we are looking at
face roll. The formula gives a bit more spin than the Wishon numbers,
but less than 200rpm out of 3000, or 7%. This is not much. Still, let's
see what it would mean if we were overestimating the loft backspin by
7%.
I went back and recomputed the validation against the Hotstix data,
using the loft backspin reduced by 7%. This required a reduction in the
gear effect spin for a best match to the data. The best match occurred
with coef=21.5 (instead of the
original model where coef=25).
With these changes, the fit to the data was even better than the
original model. Not a lot better, but it was better. So it is entirely
possible that that the formula for backspin from loft should be 7% less.
As we did with shaft tip flex, let's see what this does to the optimum
loft high and low on the face.
 At +0.23", there is no gear effect spin, but
the
backspin is now lower; the ideal
loft becomes 12.0°. We will use this number, rather than 11.7°, for
computing the differences. In other words, the differences will be
0.3°
from what they appear at first glance.
 At 0.8", the ideal loft goes from
4.7° to
5.8°, a difference of +0.8°.
 At +0.8", the ideal loft goes from 16.0° to
16.1°, a
difference 0.2°.
Note that this is the most plausible of the changes that
produce a significant difference, so it has been cranked into the
calculations of GRT performance on page 6. The lowerspin parameters (7%
less backspin, and coef=21.5)
were used to produce Figure 62 and the related tables.

Corrections
for clubhead rotation
Figure A43
Figure A43 shows the driver head rotating as the result of an
abovecenter impact. There is a black and a red combination of an arrow
at impact and a tangent to the face at impact. The black represents the
clubhead at the beginning of impact, and the red as the ball leaves the
clubface. When we look at the differences between the black and red, we
see three things:
 There is some upward movement from black to
red. Of
course! That is what gives us gear effect in the first place.
 There is some backward movement from black to
red.
The clubface is "falling away" from the ball. The effect is a loss of
some of the clubhead speed. That is the wx
vector in Figure
13.
 There is some increased tilt (loft) from black
to red.
Numbers 2 and 3 represent corrections
that should be applied
when we compute the distance as we optimize the lofts. #2 is a
correction to the clubhead speed, and #3 is a correction to the loft.
The problem is, how much of a correction should be made? Without any
correction, the black image is used for the computation. But we don't
want to just use the red image; that represents things at a time that
the clubhead no longer has influence on the ball. The best,
most accurate
correction should be somewhere in between.
It is easy to make a case for using the average of the before
(black) and after (red) conditions. That is,
 Take half the rotation and used it as
the
correction for loft.
 Take half the wx
vector as the ball leaves the clubhead and used it as the correction
for clubhead speed.
The
rationale is that the maximum force between ball and clubhead occurs
roughly midway through impact, so that is the point of maximum effect
for the corrections. The corrections fall off on either side of the
maximum; while not exactly symmetrical, they are close enough, given
that the corrections are a small percentage of loft or speed.
The size of the corrections will have an impact on the computed optimum
lofts, which determine the optimum roll. In particular:
 Loft
correction, for both high and low hits, look like increased effective
roll. So, the bigger the correction, the less actual roll you need in
the clubface to accomplish the change in loft over the face. So a
bigger loft correction will yield a flatter curvature for the optimum
roll.
 Speed correction, for both high and
low
hits,
requires a higher loft for optimum distance. That is because the lower
the clubhead speed, the more loft you need. So a bigger speed
correction will result in a flatter face low and a more curved face
high.
To enhance the likelihood that GRT is valid, let's try an
unrealistically high correction  the full correction as
the ball
leaves the clubface  and see how that would affect the optimum roll.
I modified the corrections (loft and speed
separately) to the red conditions, and optimized again.
For the loft correction:
 At 0.8", the ideal loft goes from
4.7° to
6.3°, a difference of +1.6°.
 At +0.8", the ideal loft goes from 16.0° to
15.2°, a
difference 0.8°.
 At +0.23", there is no gear effect spin; the
ideal
loft remains at 11.7°.
For the speed correction:
 At 0.8", the ideal loft goes from
4.7° to
4.9°, a difference of +0.2°.
 At +0.8", the ideal loft goes from 16.0° to
15.9°, a
difference 0.1°.
 At +0.23", there is no gear effect spin; the
ideal
loft remains at 11.7°.
So
increasing the corrections to the [unrealistic] maximum possible, we
see the loft correction causing a flatter face. The speed correction
also
causes a flatter face, but not nearly as much so.

COR change over the clubface
Figure A44 So far everything in this article assumes that
the clubhead engineer
has done a good job of designing the face flex to maintain the
coefficient of restitution (COR) at 0.83 (the rulesallowed maximum)
across the entire range of interest. Designers have gotten pretty good
at grading the face thickness to this purpose, but not yet perfect. Figure A44 was plotted from data in Wishon's eTECHreport from May 2007.
It shows a falloff in the first half inch of less than 0.02 of COR for every head tested.
What would COR falloff do to the ideal loft at each height, and thus the optimum roll radius? The result would look
something like the speed correction we saw above. Again, the speed will
fall off at the top and bottom, compared to the middle. So we should
expect the
result would be a flatter face.
Let's
compute the effect on roll, using a model chosen for easy computation
rather than strict realism. We will assume the middle third of the
clubface (actually from +0.23" to 0.23") is the full .83 COR, and it
falls off on either side. For the worst falloff in the data in Figure a44, the COR would still be above 0.79 at
+0.8" and 0.8". So let's use 0.79 at ±0.8", to see how flat we can optimally make the face with this change.
 At 0.8", the ideal loft goes from
4.7° to
4.9°, a difference of +0.2°.
 At +0.8", the ideal loft goes from 16.0° to
15.9°, a
difference 0.1°.
 At +0.23", there is no gear effect spin; the
ideal
loft remains at 11.7°.
 At 0.23", we will use the ideal loft from
optimization on the original model, 8.2°.
(When
we compute the roll later, we will have to remember that the
"lowerface roll", unlike the other modifications in this section,
begins at
0.23" and 8.2°.)
The result is a small step in the direction of a flatter face.
The
irony is that the faces with the least COR falloff in Figure A44 are
TWGT clubs with GRT faces. So the nonWishon drivers are a better match
to a GRT roll. (But only very incrementally  less than a quarter inch
of roll radius difference.) 
Adding it all up...
Figure A45Here is a graph of all the factors and what each
contributes to
roll radius.
For each factor, we have calculated the ideal loft at the top and
bottom of the range (+0.8" and 0.8"). We can find an overall roll for
the face, using our equation:
R
=
57.3 ΔH
/ ΔL
For ΔH,
we use 1.6", the distance from top to bottom of the range. ΔL is
the difference in ideal loft between top and bottom.
The only factor that contributes more than an inch to the roll radius
is the headrotation loft correction. It contributes just over two
inches. And we can't possibly get all of that, because it would require
all the correction effect to be taking place as the ball is leaving the
clubhead. It is unlikely we could get more than half this correction in
reality.
This does not hold much promise that incorporating these factors will
get us very close to GRT. 
Let's take a closer look. This time we'll use a table with numbers
instead of a graph. We'll also look separately at the top 1/3 of the
face and the bottom 2/3.
Besides the ideal loft at ±0.8", we also know that the
zerogeareffect height is +0.23", which is 0.57" from the top and
1.03"
from the bottom. At that height, the ideal loft is 11.7°. Now it is
easy to calculate the radius of roll
curvature. The equation becomes:
R_{topThird}
= 57.3 * 0.57 / (Loft_{+0.8} 
11.7)
R_{bottomTwoThirds} = 57.3 *
1.03 / (11.7  Loft_{0.8})
This is not a pointbypoint optimization, but rather the average roll
over the top third and, separately, the bottom two thirds of the
clubface. From the
graphs we have already plotted, it is likely that this average is
representative of the roll we would see from a pointbypoint
optimization, but without the "noisy" radius variation.
The table below shows the ideal roll radius if each of the assumptions
is relaxed (in a direction to favor GRT), individually and in some of
the more likely combinations:
Assumption 
Ideal loft
at top
(+0.8") 
Change
in loft
from
original 
Ideal loft
at bottom
(0.8") 
Change
in loft
from
original 
Implied roll:
top 1/3 
Implied roll:
bottom 2/3 
Original model 
16.0° 
 
4.7° 
 
7.6" 
8.4" 
Gear
effect limited (5%) by shaft tip stiffness. 
15.6° 
0.4° 
4.9° 
+0.2° 
8.4" 
8.7" 
Loft
spin reduced (7%), and model readjusted for best fit to data.
(Turned out to be coef=21.5) 
16.1° 
0.2°
^{[1]} 
5.8° 
+0.8°
^{[1]} 
8"
^{[1]} 
9.5"
^{[1]} 
Loft
correction for rotation (raised from half to full) 
15.2° 
0.8° 
6.3° 
+1.6° 
9.3" 
10.9" 
Clubhead
speed correction for rotation (raised from half to full) 
15.9° 
0.1° 
4.9° 
+0.2° 
7.8" 
8.7" 
COR
rolls off away from center of face 
15.9° 
0.1° 
4.9° 
+0.2° 
7.8" 
9.9"
^{[2]} 
Collection of all factors ^{[3]} 
14.4° 
1.6° 
7.7° 
+3.0° 
12.1" 
14.8" 
Collection of reasonable factors: ^{[3]
}
 Loft spin reduced
 Loft correction for rotation (but only
half)
 Speed correction for rotation (but only
half)
 COR falloff

15.25° 
0.75° 
6.6° 
+1.9° 
9.2"" 
11.6" 
Notes:
 Calculation based on a loft of 12.0° at +2.3",
not
11.7°.
 This is just the bottom 1/3, not the bottom 2/3.
 The collections of factors (the blue rows) are
computed by adding up the changes in loft, then
applying those changes to the original model loft to get a new loft.
Then we compute the implied roll from the new loft.
The first blue row in the table is the combination of every
factor we have considered here. All added together, they push the
radius of curvature up over 12", but:
 The 12"/15" (high/low) curvature is
still closer to conventional roll than to the 15"/20" of GRT.
 It includes some rather implausible and unlikely factors.
If
we limit ourselves to the more plausible factors, we are left with
thesecond blue row. When we adopt this plausible set of
corrections, we get a 9" radius
high and less than a 12" radius low. Very conventional. Very unGRT.

