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All about Gear Effect - 5
Dave
Tutelman --
February 22, 2009
The role of roll
After the previous pages were posted to my web site, Marcel
Bal
asked me about face roll. Specifically, he wanted to know if vertical
face curvature was a good thing or a bad thing. He mentioned Tom
Wishon's GRT (graduated roll technology) drivers, which are
specifically designed with minimal face roll
except fairly high on the clubface -- and even then then a relatively
small roll.
What a wonderful question! And why didn't I think to
ask it before I posted the article? It would seem a logical culmination
of all that work, because it requires almost everything to give a
complete answer. In fact, it probably calls for correcting for clubhead
rotation; even though the rotation is small during impact, it can have
a non-neglible effect on the answer to this question.
Let's see if we can come up with an answer.
General approach
We clubmakers tend to think of a single loft number as the
ideal loft
for a specific swing, where a swing is a combination
of clubhead
speed and angle of attack, and perhaps other effects like shaft bend
and wrist position at impact. I won't debate exactly what goes into it,
but rather address the idea that the ideal loft is a single number at
all heights on the clubface.
When we look for the ideal loft, we
vary the loft while looking at launch conditions -- ball speed, launch
angle, and backspin -- and how the combination affects the distance.
Our simple model of ideal loft therefore assumes that, for a
given swing, the launch conditions (especially the backspin) is the
same wherever on the clubface you strike the ball. Now we
know that assumption is wrong!
Figure 5-1
Figure 5-1 shows the clubface angle (the loft) at various heights on
the face, specifically at the center and a quarter-inch and half-inch
above and below the center. The black line on
the left is a constant-loft face (in other words, no curvature, no face
roll). Are the launch conditions going to be the same at each height?
In a word, no!
- The backspin will vary quite a lot. The data we used to validate
vertical gear effect showed a variation of 1300rpm over this same range
of heights.
- The ball speed and launch angle may vary a small amount,
because of face rotation caused by a high or low impact.
So we can't just assume that the same loft is the ideal, or
optimum, loft at each height on the face.
The red and blue segments are a more realistic picture. We can look at
the launch conditions at each face height and optimize the loft for
that height, taking into account face rotation (and thus gear effect).
We already know that the backspin will be less as we move up the face.
A little work with any trajectory program tells us emphatically that
decreased backspin calls for increased launch angle, and vice versa. So
we can expect that high on the clubface (low backspin) the best loft
should be higher than the center of the face. Conversely, the
backspin is higher at the bottom of the face, so we can expect the
optimum loft there to be lower.
If we take these ideal loft
segments and string them together, as we do at the right on Figure 5-1,
we see that the progression of lofts requires a curvature of the
clubface. So, if we have a collection of ideal
lofts, that implies what the ideal face roll should be.
How
can we go from a set of optimum lofts at various face heights to roll
radius? Actually, that's the simple part. (The hard work was finding
the optimum loft at each face height.)
Figure 5-2
Figure 5-2 shows a driver head, with the loft specified at two points.
The two lofts are L1
and L2,
and they are separated by a height difference of ΔH.
What we will do is find a series of lofts L1,
L2, etc,
at height increments of ΔH.
Those lofts constitute a curvature statement. But we are used to seeing
the curvature in terms of radius -- as roll is usually
specified. We will have to turn that series of lofts into a series of
radii of
curvature. If they are all pretty similar, we can use the average as a
fair description of the roll radius for the whole clubface. If there is
a lot of variation, then the ideal clubface has a graduated roll. This
may
be the same as Wishon's GRT graduated roll -- or different. We're going
to find out which.
The figure hints at the computation, but let's be explicit. We have
drawn two radii, perpendicular to the face at the two points of
interest. Where they meet is the radius of curvature, at least for the
section of clubface between the two points. The angle at which they
meet is the difference between the lofts
ΔL =
L2
- L1
The angle ΔL as a
proportion of 360° is the same as ΔH as
a
proportion of the circumference 2πR,
or:
Solving the proportion for R, we
get:
| R
= |
360
2π |
ΔH
ΔL |
=
57.3 |
ΔH
ΔL |
So the steps are:
- Choose a set of heights on the clubface. We will choose
-0.8" to +0.8" by 0.2" increments.
- Optimize the loft at each point.
- Look at the optimized lofts, and draw some conclusions
about face roll.
Let's do it!
Equations we will need
If you're interested in the
results but not the math, skip this
sub-chapter.
We already have, or can easily find from what we've already done, the
equations we need. They are summarized in the table below.
Now
we have enough to create a spreadsheet to give us launch conditions for
any loft and clubhead speed. We will build the spreadsheet using the
above formulae, and then
iteratively solve for the optimum loft. That is:
- We will start with a constant clubhead speed (102.7mph, to
give 150mph ball speed) and constant loft across the face (11°).
Further assumptions about the driver:
- We
will assume a constant COR across the face at 0.83, the rules-allowed
maximum. Designers have learned how to shape the face thickness to
preserve COR across a lot of the face, so this analysis assumes such a
driver.
- We continue (as in all the previous examples) with a zero
angle of attack.
- For each height on the clubface (H= -0.8" through H=
+0.8"), we will do the following:
- Use the spreadsheet to compute a set of launch conditions
(ball speed, launch angle, and backspin).
- Plug the launch conditions into TrajectoWare Drive (TWD)
and
find the carry distance and angle of descent.
- If
this is the maximum distance we are likely to get, note it and the loft
-- then move on to the next H. Otherwise, choose a new loft
(higher or lower) and go back to step 1.
Optimized roll results
After we go through those calculations, we will have found the optimum
loft for each height on the
face. Then we will use that to picture the face roll curvature. And
here is the result of those computations.
| Inputs |
Computed intermediate results |
Backspin |
Outputs |
| H |
Loft |
Ball speed |
Launch angle |
y |
Correct'd
ball
speed |
Correct'd
launch
angle |
Due
to
loft |
Due to
gear effect |
Net |
Carry
Distance (TWD) |
Angle
of
Descent
(TWD) |
Roll
radius
R |
| -0.8 |
4.7 |
152.2 |
4.4 |
-0.90 |
148.3 |
2.9 |
1346 |
3331 |
4677 |
213.9 |
31.9 |
--- |
| -0.6 |
5.8 |
152.0 |
5.3 |
-0.72 |
149.4 |
4.2 |
1660 |
2692 |
4353 |
221.7 |
31.8 |
10.4 |
| -0.4 |
7.2 |
151.6 |
6.5 |
-0.55 |
150.1 |
5.7 |
2059 |
2057 |
4117 |
228.5 |
34.0 |
8.2 |
| -0.2 |
8.4 |
151.2 |
7.6 |
-0.37 |
150.5 |
7.0 |
2400 |
1396 |
3797 |
233.9 |
34.2 |
9.6 |
| 0 |
9.6 |
150.7 |
8.6 |
-0.19 |
150.5 |
8.3 |
2740 |
728 |
3469 |
237.9 |
34.1 |
9.6 |
| 0.2 |
11.4 |
149.8 |
10.0 |
-0.03 |
149.8 |
10.0 |
3247 |
99 |
3347 |
241.0 |
35.4 |
6.4 |
| 0.4 |
12.8 |
149.0 |
11.1 |
0.15 |
148.9 |
11.4 |
3640 |
-553 |
3087 |
242.6 |
36.8 |
8.8 |
| 0.6 |
14.3 |
148.1 |
12.3 |
0.32 |
147.6 |
12.8 |
4058 |
-1192 |
2867 |
242.8 |
37.4 |
7.6 |
| 0.8 |
16 |
146.9 |
13.6 |
0.50 |
145.7 |
14.3 |
4529 |
-1803 |
2726 |
242 |
38.6 |
6.7 |
Figure 5-3
- Take a look at the formula for roll radius. It is
proportional to ΔH
/ ΔL,
which is the
slope of the height vs loft curve. But we are restricting our search to
the best loft to a tenth of a degree. That restricts ΔL to
the nearest tenth of a degree, which limits the computed roll to a few
discrete values, not a continuum. That is the major reason for "noise".
- By way of further explanation, the distance vs loft
curve is fairly flat near its optimum -- and we are working near the
optimum. That means that we could be off by a tenth of a
degree
of loft. As we saw, numerical differentiation amplifies error. A tenth
of a
degree error in two successive readings could throw off the face roll
by 2 inches.
- In fact, even the trend might be wrong; the correct
answer might be a constant roll or, for that matter, a bigger
difference between the top and bottom of the clubface. Why? The lower
quarter of the clubface (H≤ -0.4)
shows very high net spin -- over 4000rpm -- because the loft spin is
adding to a strong gear effect backspin. But TrajectoWare
Drive starts to lose accuracy at spins over 4000rpm. So we cannot take
very seriously any
results from the bottom of the clubface.
- Finally,
it is worth remembering that we assume a 0.83 COR across the entire
height of the clubface. Any "sweet spot" effect is due entirely to roll
radius and gear effect. If the face is hotter in some spots
than
others, the COR profile has to be superimposed on our work, and may
affect the optimum loft at some of the points.
Even with all these caveats, there is a reasonable conclusion to be
drawn from the data. The face that the best possible loft progresses
with height on the face tells us that the best possible face must be
curved. Thus... Face roll is useful in keeping
the distance up for high and low mishits. How
useful? Let's run our spreadsheet a few more times and see.
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How sensitive is distance to the roll radius?
This
time, instead of finding the loft (L)
at each face height (H), we will
assume a roll radius (R) and
compute the distance at each face height. The loft at each height is
easy enough to compute, knowing the roll radius. Remember the equation
we used earlier:
R = 57.3 ΔH
/ ΔL
If we choose some base Ho
and Lo
for the clubhead, we can find the loft at any face height by solving
that equation.
We will continue to use a clubhead speed of 102.7mph, giving a
center-hit ball speed of 150mph. From our previous work, we know that
the maximum carry distance occurs at about H=½"
or perhaps a fraction higher,
and the best loft at H=½"
is 13.6°. Let is use these as Ho and
Lo.
That way, we have the same maximum distance no matter what the roll
radius.
Figure 5-4: Ball speed = 150mph
Figure 5-4 is a graph showing how carry distance
varies with H for different
values of the roll radius R. The
graph shows several things pretty clearly:
- A
radius of 8" (the yellow curve) seems to give the best preservation of
distance as the height of impact varies. It makes for the most
forgiving clubhead -- at least forgiving of height errors.
- For R=8", the
distance is within a yard of the maximum anywhere from ¼" to ¾" above
the center of the face.
- Things are still quite good with a roll radius of 10"
(the green curve). We still lose only a yard from
¼" to ¾" above the center. A lower strike, say on-center, is a yard
shorter than the 8" roll -- not bad, but probably measurable.
- We see a little more loss of distance at 6" and 12".
Now we are losing three yards from a center hit.
- A nearly flat face without roll (R=40")
drops off quite severely when struck either high or low of the optimum.
This tells us that roll matters!
It is a good thing. It "makes the sweet spot bigger".
A
flat face is way too sensitive to the height at which the ball is
struck. Yes, it may preserve a nice-looking trajectory; a low strike
will not produce a hot worm-burner. But, given the extra backspin due
to gear effect with a low strike, the worm-burner needs that spin and
will give more
distance -- and the "nice trajectory" from the flat face
will balloon and fall short.
But is an 8" roll radius really best?
It may well be. Then again, we
don't see that much curvature on the clubheads that are being
sold. Perhaps the gear effect model may be a little off. The
physics is
definitely sound, but our estimate of things like vertical moment of
inertia may be a little off. Certainly not by an order of magnitude.
Most likely
not even by a factor of two. But it might be off enough so that the
ideal radius is more like 10" or even 12". I would be surprised if it
is off by more than that. And anything approaching a flat face has got
to be wrong.
You might ask about another possible source of
error in our conclusion. TrajectoWare Drive, the software that computed
the distance, loses accuracy at higher spin rates. Is that an issue
here? No, it is not. Spin rates high enough to give
inaccurate
distances did show up, but only for distances under 230 yards, lower
than the bottom of the graph. None of the data shown on the graph is
affected by TrajectoWare Drive error. |
One roll fits
all?
| Is
8" -- or, for that matter, is any
roll radius -- the
best for all golfers? We haven't addressed that issue at all so far.
Let's test the one-size-fits-all assertion by trying higher and lower
clubhead speeds and seeing how the ideal roll varies. So far, we have
assumed in every case that the golfer has a 102.7mph
clubhead speed, generating a ball speed of 150mph. Now we'll try ball
speeds of 180mph and 120mph. |
Figure 5-5: Ball speed = 180mph
First we'll try a higher clubhead speed. Figure 5-5 is based on a
clubhead speed of 122.5mph, which gives a maximum ball speed of 180mph.
This would be a pretty big hitter on the PGA Tour, but not the top ten
and definitely not a long drive competitor. For this clubhead speed,
the best distance occurs at H=0.6" and a loft at that height of 12.5°.
The
8" roll radius is still the best. Looking at the second best,
this time
it is the 6" roll. (At 150mph it was the 10" roll.) So perhaps the
ideal
roll is a little more curved for higher ball speeds -- a bit
above 8" at 150mph and a bit below at 180mph. But it is not a very big
difference, considering that 20mph is a huge difference in clubhead
speed,
and 30mph a huge difference in ball speed.
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Figure 5-6: Ball speed = 120mph
How about lower ball speeds? Let's try a clubhead speed of 83mph,
giving a ball speed of 120mph. All the drivers modeled had a 17.2° loft
at a point 0.6" above the center of the face. We see the result in
Figure 5-6.
This time, the 8" roll shares top billing with the 10" roll, suggesting
that perhaps a 9" roll might even be a little better. Still, that is
not all that different from the higher clubhead speeds. |
Conclusion
Face roll makes a significant difference in how forgiving a driver is
to high or low misses. A flat face has a very small "sweet spot" in
terms of height; the carry distance falls off sharply if impact is
above or below this height.
The optimum face roll varies a bit with clubhead speed and ball speed.
But not much. I would estimate the best face roll for a 180mph ball
speed is less than 2" different from that for 120mph ball speed. This
might be worth working the problem for a long drive competitor, but
probably not for someone playing competitive golf. If the roll
is optimized for 150mph, the lost distance is less than a yard for any
ball speed between 120mph and 180mph (compared with a roll optimized
for that ball speed), for any strike at the middle of the face or
higher.
The calculations show an optimum face roll in the vicinity of 8". This
may or may not be correct, depending on the accuracy of the model and
the assumptions involved.
This invites a discussion of sensitivities of the
conclusion.
The main sensitivity is the actual amount of ball spin due to vertical
gear effect. For the range of interest, more gear effect spin requires
more face curvature. In particular:
- If the clubhead moment of inertia is higher than the model
estimates,
then the optimum roll radius may be more (a flatter face).
- If the CG is farther back in the head, then the optimum
roll radius may be less (a more curved face).
- If the effect on launch angle and ball speed due to
face rotation is higher than we used, then the optimum roll radius may
be more (a flatter face).
- If the shaft tip is actually stiff enough to significantly
limit
vertical gear effect, then the optimum roll radius may be more (a
flatter
face).
We noted earlier that the model disagrees with two reports of the
amount of gear effect spin, Wishon and Upshaw, which certainly do not
agree with each other in any event. The model predicts spins much
higher than Wishon reports, and closer to Upshaw's. But I was unable to
duplicate analytically Upshaw's reported dependency on shaft tip
stiffness. Looking at the sensitivites above:
- If Wishon's
much smaller estimate of gear effect spin is correct, then the optimum
roll radius would be much greater (a much flatter face). This explains
his GRT driver heads, with a 15" roll for the top third and a 20" roll
for the bottom two thirds.
- If Upshaw's shaft-sensitivity of
gear effect is correct, then the optimum roll radius would become
heavily dependent on the shaft. It might be anywhere from 8" to nearly
flat, depending on the shaft used.
Barring either of these extremes, the bullet-points above are unlikely
to cause even a factor of two
difference in the optimum roll. In real terms, the optimum roll is
almost certainly between 8" and
12". For drivers! Hybrids or even fairway woods might be quite
different, because their MOI and depth of CG are quite different.
Last modified - Mar 24, 2009
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