All about Gear Effect  2
Dave
Tutelman 
February 19, 2009
Vertical
gear effectOK, we have a
handle on horizontal gear effect. Now let's look at vertical gear
effect, which increases or decreases the backspin a driver puts on the
ball. This involves rotation around the red axes in Figure 12, caused
by
impact above or below the center of gravity. The only thing that is
different is the moment of inertia; it is I_{v}
instead of I_{h}. Well,
we
should probably change the horizontal miss distance x
to a vertical miss y. So
equation
2, adapted for vertical gear effect, becomes:
s 
= 
58,830 V_{b} 
C y_{
}
I_{v} 


(Equation 3) 
That presents a problem. I_{h}
is widely discussed and is sometimes published, so we have a pretty
good idea of its
magnitude. I_{v} is
not. How do we go about estimating a good value for I_{v}?
Unfortunately, I'm not set up with a Finite Element Analysis program.
Every shop that does actual clubhead design uses such programs the way
a carpenter uses a Skilsaw. It computes as you design, keeping track of
total mass, moment of inertia, stresses, flexes, etc. But I don't have
that tool, and I'm not going
to compute by hand the MOI of a complex shape like a driver head.
Figure 21
(From the FEKO
web site)
So we'll have to find some way to approximate it. For a first try,
let's assume the clubhead is similar to an oblate spheroid, shown in
Figure 21. An oblate spheroid
is
a sphere that has been flattened; one "radius" (c
in the figure) is smaller than the other two (a
in the figure). If a portion of one edge were chopped off to form a
clubface, this would be pretty similar to a driver head. In
fact, measuring a driver head shows more similarity than you would
think just by looking. Let's try an oblate spheroid 4.6" by 2.5". The
4.6" approximates the width and depth of a driver head, the 2.5"
approximates it's maximum height, and the total volume is 454cc  very
close to the 460cc legal limit that everybody builds to today.
It is possible to compute the I_{h}
and I_{v}
for an oblate spheroid. Assuming the spheroid is not solid, but rather
a shell of uniform thickness, the equations are:
I_{h}
= 2/3 m a^{2
}I_{v}
= 1/3 m (a^{2} + c^{2})^{
}
Using a typical driver mass of 200g, the computed moments of inertia
are:
I_{h} =
4550 gramcm^{2}
I_{v} =
2950 gramcm^{2}

This is pretty good. The computed I_{h} is
very close to the average of the moments of inertia we got from the inpakuto.com
web site (4575).
But,
before I injure myself patting myself on the back, I better note that
it is well below the norm for the values in the Golfsmith catalog.
Moreover, it does not show much promise of getting near the legal limit
set by the Rules. Here is my explanation why the formula should give
realistic
but lowend values of MOI.
Until quite recently, 460cc heads had significantly higher moments of
inertia than previous, smaller heads. So designers were happy to sell
them as improvements. A few years ago, there was competitive parity at
that point  4200 to 4500 gcm^{2}  so designers
looked for new ways to increase MOI. They did it by abandoning the
uniform thin shell (the assumption in our oblate spheroid model), and
lightening some parts of the shell so they could add weight elsewhere
and increase the MOI. (In the business, this is called "discretionary
weight": weight that is not necessary for structural integrity, so it
can be moved to affect things like MOI or CG position.)
Figure 22 Using discretionary
weight produces
a higher I_{h} than
our model accounts for. But what does this do to I_{v}?
Figure 22 shows:
 Where you want to add weight to increase I_{h}
(the airbrushed yellow band around the periphery),
 And where weight
will increase I_{v} (the
red band around the top and bottom).
Moving weight to increase I_{h} does
not necessarily increase (nor decrease) I_{v}.
What designers do
to increase I_{h} 
What that does
to I_{v} 
Move weight to the rear of the clubhead. 
Same increase in moment of inertia. 
Increase
the depth of the clubhead. 
Same
increase in moment of inertia. 
Move weight to heel and toe. 
No effect on I_{v}. 
Remove weight from crown to obtain discretionary
weight for elsewhere. 
Reduces I_{v}. 
Face
and sole are hard to take weight from; weight is needed there for
structural reasons. And adding weight there does little for I_{h},
so that is not a candidate. 
No effect on I_{v}. 
Make the head wider (requiring it to
be flatter to stay at 460cc). 
Reduces I_{v}. 
There are two distinctly different possibilities of the best way to
estimate I_{v}.
Without actually measuring heads  which I am not equipped to do,
and such measurements are not widely published  it is hard to tell
which of the two views of I_{v} is
closer to fact. FWIW, here they are:
 The changes in the above table that increase I_{h}
tend to be somewhere between neutral and a reduction in I_{v}.
So, while the oblate spheroid model shows I_{v} to
be about 2/3 of I_{h},
it is probably less than that for driver heads with I_{h} of
5000 gcm^{2} or more. I am just guessing, but I
would use
a value of about half I_{h} for
a head that pushes the USGA limit of 5900 gcm^{2}.
Note that half of 5900 is 2950  exactly the same I_{v} that
we computed for the oblate spheroid. Coincidence or not? I don't know.
In any event, it looks like somewhere around 3000 gcm^{2} is
a reasonable value of I_{v} to
use for our sample calculations.
 Let us remember that the ratio of C
to I_{h} is
remarkably constant over a lot of drivers with a variety of moments of
inertia. That suggests that the prevalent tool used by designers to
boost MOI involves moving weight rearward. That is one of the few items
in the table above that increases I_{v}
as it increases I_{h}.
So perhaps the ratio of C to I_{v} is
similarly constant. If so, the ratio runs at about 3/2 that of C
to I_{h},
making vertical spin computable as 1.5*16.4V_{b}y.
I
am going to use #2 to compute vertical gear effect, not because I think
it is the more correct way (I really don't know), but because it is
easier. Besides, if I start with 3000gcm^{2} as
the moment of inertia, I will have to pick a CG depth, which we know
varies a lot from driver to driver. So we will go with:
s = 25 V_{b
}y
(Equation 3a)
Once again, let's run some real numbers as a sanity test, and compare
them with any data that might be around. We will start with our old
standby of 150mph ball speed and 11º of loft. We will assume a face
roll of 12" radius, same as the bulge. A centerface hit (y=0)
will give
a distance of 241 yards, as we expected.
Let's
move up on the face as high as we dare without losing COR. I'd estimate
that to be about 5/8", based on measuring the same drivers I did
before. Equation 3a becomes
s = 25 V_{b }y
= 25 * 150 * .625 = 2344rpm
That's a lot of spin! And it comes straight off the
backspin applied by loft. (Assuming, of course, that the clubhead can
rotate freely  a question we address at the end of this article. But
for
now let's assume that clubhead rotation is unrestricted.)
How
does this square with other investigators of vertical gear effect?
Almost everyone I know of has reported much smaller numbers, seldom
above 500rpm. Almost everyone.
But Dana
Upshaw has published data
suggesting numbers in this general range. His launch monitor numbers 
3300rpm difference in spin for a 1.5" difference in height at a ball
speed of about 140mph  are about 63% of the equation's estimate. Not
the same, but in the ballpark.
Spin optimization
Since the contribution of vertical gear effect spin is obviously
substantial, we need to ask what it does to the trajectory and the
distance. We know that many (perhaps all) golfers will get more total
distance with a higher launch angle and
less spin than they'd get from
a normal center hit on a driver of the "best" loft for their clubhead
speed. But that's "and", not
"or"; if you increase launch angle at the
expense of higher spin, or vice versa, you will lose distance, not gain
it. And there are only two ways of independently adjusting launch angle
and spin to accomlish this:
 Increase the angle of attack! This increases launch
angle without changing the spin.
 Vertical gear effect! This decreses the spin with
minimal change of launch angle. (And that minimal change is in the
right direction.)
So let's see
what happens when we subtract the 2344rpm due to gear effect. At 5/8"
above the center, face roll increases the loft to 14º and the launch
angle to 12°. So the backspin due to launch angle is 3987rpm, and we
have lost 5 yards of distance, back to 235 yards. But that is
without factoring in the gear effect!
What happens if we keep the launch angle, but cut the backspin by the
amount of gear effect? 3987rpm backspin minus 2344rpm gear effect is
1643rpm net backspin? Whoops! We
just lost even more distance. Now we're only 230 yards, down 11 yards
from the original 241. We obviously needed some of that spin. But not
all of it. A quick optimization with TrajectoWare Drive says a gear
effect spin of about 1000rpm would be optimal for that 12° launch. It
would cut the spin to just under 3000rpm, add a few yards to the
carry, and keep the angle of descent at a reasonable 37°.
There must be some reason we are told to hit high on the face. A few
possibilities:
 We went too far when we went to 5/8".
 The
shaft restricts the rotation of the head, so not as much spin is
produced. Less than half of that gear effect spin would be just right.
 That
is where the engineers designed the face for maximum COR. Not likely!
It is harder to do than at the center of the clubface, and the
engineers' real goal is to make the COR the maximum allowed over the
whole clubface. Let's dismiss this out of hand.
We will
investigate (b) later in the article. For now, let's look at (a). Here
is a table of what happens as we move above the center of the face.
Height
of
impact 
Loft
at
point of
impact 
Backspin
due to
loft 
Topspin
due to
gear effect 
Net
backspin 
Distance
(yards) 
Angle
of
descent 
Center 
11° 
3135
rpm 
0 
3135
rpm 
240.5 
34° 
0.2" above 
12° 
3416
rpm 
750
rpm 
2666
rpm 
239.3 
32° 
0.4" above 
12.9° 
3668
rpm 
1500
rpm 
2168
rpm 
235.3 
30° 
0.6" above 
13.9° 
3947
rpm 
2250
rpm 
1697
rpm 
230.0 
28° 
Conclusions:
 A
lot of topspin is generated (actually, a lot of backspin is
eliminated). It is too much gear effect to increase
the carry for the increased
launch angle. The good news is the reduced angle of descent,
which will increase the
roll after landing.
 We need to see what is behind "door b"  the
restriction of clubhead rotation by the stiffness of the shaft tip. We'll look at that later.
Or maybe there is yet another explanation!
It has
been said that god is in the details. There is one detail we
have been ignoring.
Figure 23
So far, we have treated the force as being in the direction of the
clubhead's movement, as shown in the left image of Figure 23. Let's
remember that the force is Newton's "equal and opposite reaction" to
the departure of the golf ball. A more accurate picture would have the
force exactly opposite the departure direction of the ball  the
launch angle  as shown in the right image of Figure 23. This will
make a very small difference in C
and a much larger difference in y,
both in such a direction as to reduce the gear effect. Specifically:
C = D cos
a
y = H  D sin a
Where
 a = launch angle
 D = depth of CG
from the center of the face (what we had been calling C
until now.)
 H = height of
impact above the center of the face (what we had been calling y
until now.)
That is not going to make much difference in C
(only about 2%, for launch angles below 12°), but it will make a big
difference in y. Let's redo the
table above, with this more accurate picture. (We will use D=1.3",
consistent with the driver data from inpakuto.com.)
Of course, we are going to have to include the launch angle and the
actual value of y. We won't
bother recomputing C, as the
difference will be minimal, way less than any precision in our estimate
of D.
Height
of
impact 
Loft
at
point of
impact 
Launch
angle 
Actual
y 
Backspin
due to
loft 
Topspin
due to
gear effect 
Net
backspin 
Distance
(yards) 
Angle
of
descent 
Center 
11° 
9.7° 
0.22" 
3135
rpm 
825
rpm 
3960
rpm 
237.2 
41° 
0.2" above 
12° 
10.5° 
0.04" 
3416
rpm 
150
rpm 
3566
rpm 
241.6 
39° 
0.4" above 
12.9° 
11.2° 
0.15" 
3668
rpm 
563
rpm 
3105
rpm 
244.5 
37° 
0.6" above 
13.9° 
12° 
0.33" 
3947
rpm 
1238
rpm 
2709
rpm 
245.7 
35° 
0.8" above 
14.8° 
12.7° 
0.51" 
4198
rpm 
1913
rpm 
2285
rpm 
244.9 
34° 
This is more like it! This
is what we should
have expected. We can wring quite a few extra yards of carry out of a
strike 1/2" to 3/4" above the center of the clubface  and get a bonus
of more roll after landing (a consequence of the lower angle of
descent).
A few points to carry away from this table:
 A
center strike will create vertical gear effect to increase the
backspin. That is because the force passes below the CG and rotates the
face downward.
 In fact, you have to strike almost 1/4" above
center face just to be geareffect neutral  no spin due to gear
effect. You need that just to get the "nominal" distance out of the
driver.
 The topspin due to vertical gear effect looks much
more
reasonable here. It is still higher than most estimates, but not by
nearly as much as before. But...
 If we looked at the consequence of a low hit, we see
oodles of gear effect backspin.
If we go between 3/4" high and 3/4" low (a total of 1.5") we still see
the same spin difference of 5600rpm. But it is biased more
towards backspin, since the spinneutral point is almost 1/4" above the
center of the face.
The conclusion from this is that vertical gear
effect is a very good reason to try to hit your driver high on the
clubface.
Before
we leave the subject, I'd like to point out that the real gains will
probably be smaller than those in the table. The table was based on a
ball speed of 150mph
for all the rows. But there are small losses of ball speed as impact
moves up the face, due to:
 As we can see in Figure 13,
the sideways velocity of the face (which creates the gear effect) is
accompanied by a backwards velocity. This is essentially a loss of
clubhead speed. For a strike 0.6" above center, this is a loss
of 1.5%. (But that is not a loss compared with a center
strike,
which also has a gear effect loss of about 1%. The lossless strike
occurs where y=0, about 0.22"
above center.)
 There
may be some falloff of COR away from the center of the face. How much?
That depends on the clubhead designer, and how well it was designed to
keep the maximum COR over as much of the face as possible.

Reality check
As we did with the
horizontal gear effect, we ought to check the
mathematical model against realworld numbers. I was unable to come up
with any real data to check against the model when I wrote the article.
During subsequent discussion on the TWGT forum,
Darryl Green
pointed to some recentlypublished data. There was an article in the
Feb 2009 issue of Golf Magazine, disclosing a very
useful set of data taken by Hotstix.
They
set up a robot to hit a driver from various tee heights, resulting in
different impact heights on the clubface. They used a driver that looks
like it might be a TaylorMade model, but did not identify it. They
described it as a 460cc driver with nominally a 9.5° loft. All
the test data was taken
at a clubhead speed of 100mph. They measured hits oncenter, a quarter
inch above and below center, and a half inch above and below center.
The data they published for each of those positions were:
 Launch angle in tenths of a degree.
 Backspin in rpm.
 Distance in yards. From the numbers, we can
safely assume that was total distance, not just carry distance.
This is sufficient information to give the model a workout. In fact, we
don't even need the distance data. The important thing is to determine
if a 100mph clubhead speed and the given face heights and launch
angles, when fed to the model, gives the same backspin that Hotstix
measured.
Here's a table of how the model fared. 
Input
from Hotstix article 
From
TrajectoWare Drive 
From
model 
H 
Launch
angle 
Measured
total
backspin 
Loft
at
impact 
Backspin
due to
loft 
Ball
speed 
y 
Gear
effect
backspin 
Computed
total
backspin 
Error
(rpm) 
0.50" 
4.3° 
3165 
4.6 
1283 
148 
0.60" 
2210 
3494 
10.4% 
0.25" 
5.3° 
2971 
5.8 
1617 
148 
0.37" 
1369 
2986 
0.5% 
0 
6.8° 
2564 
7.5 
2088 
148 
0.15" 
570 
2657 
3.6% 
+0.25" 
8.4° 
2098 
9.4 
2613 
147 
+0.06" 
221 
2392 
14.0% 
+0.5" 
9.4° 
1862 
10.6 
2943 
146 
+0.29" 
1050 
1893 
1.7% 
And here's where each column in the table comes from:
 Green
columns are raw data, numbers copied from Hotstix article.
 For
the blue columns, I cranked 100mph ball speed into TrajectoWare Drive.
Then I changed the loft until I got the same launch angle as the data.
The blue columns record that loft, as well as ball speed and spin.
 The yellow columns are the predictions by the
mathematical model
 The value of y
was computed from
y = H  D sin a. I assumed a
1.3" CG depth for the driver, and used the measured H
and launch angle.
 The gear effect spin was
computed from equation 3a: s =
25 V_{b
}y. (For this table, I used
the convention that backspin is positive and topspin is negative.)
 The computed total backspin
is the sum of the backspin due to loft and the gear effect backspin.
This is the number to be compared with the
measured total backspin in
the Hotstix data.
Figure 25 We can look at the comparison numerically in the table, and also look at it graphically in Figure 25.
Given
the coarseness of the assumptions, there is a remarkably good match
between data and model. Three of the five data points are within 4%,
which is
scarygood considering how we approximated the MOI and CG depth. The
other two points are off by only 10% and 14%. And they are not
clustered together at one end of the data, which might have suggested a
trend that the model was missing.
Bottom line: the data strongly suggests the model is close enough to reality to be very useful.

Improving the match
Given how good the match is, I should probably just give it a rest and
declare success. But let's see if sharpening our assumptions could make
the match even better. What parts of our model should we
allow to change, and what parts are just Newtonian physics and
obviously correct? Let's look at equation 3, the basis of the model:
 Gear effect spin is proportional to ball speed.
That's basic, so leave it alone.
 Gear effect spin is proportional to y,
the distance between the force of impact and the CG of the clubhead.
Again, that is basic; it is the moment arm for the torque on the head.
 Gear effect is proportional to the center of gravity
depth C. Fairly basic, but the
value we used for C was based on
some rough assumptions.
 Gear effect is inverstely proportional to the moment
of inertia I_{v}.
Totally basic, but again the value for I_{v}
was based on rough assumptions.
The most suspect thing about the model (as expressed in equation 3a) is
the assumptionheavy way we approximated the vertical moment of inertia
I_{v} and
the depth of the center of gravity C.
If we play around with these values and watch the error column, we can
try to find values that minimize the error. I set up a spreadsheet and
did that. Specifically, I played with the coefficient 25
in equation 3a and values for C
other than our 1.3".
With a CG depth of 1.1", all the errors
are below 10%. No amount of fooling with I_{v} did
much to reduce the error. That is a plausible set of values. The CG
depth is at the low end of the data we used earlier (from the Alba
magazine study), but still within range. Now, remember that we
don't know much about the specific driver Hotstix used for the test. If
it had a relatively short CG depth, then the model should be using
something like 1.1" for C.
Could it be even better? Not with the raw data
we
have. I think we are bumping up against the limits of measurement error
at this point. There are "wiggles" when we plot the data  spin vs
face height and launch angle vs face height  that suggest
inaccuracies in the measurements. To test this, I smoothed the data using a bestfit
secondorder
polynomial for spin and launch angle. (I didn't want to force a bestfit
straight line, in case the real underlying effect is nonlinear.
Including a secondorder term allows curvature but not
enough wiggling to be able to fit any five data points
arbitrarily.) The results are shown
in Figure 26.
Figure 26
The figure shows the actual data points, the bestfit "line", and the
equation for that line. (The bestfit lines actually have some
secondorder curvature, as shown by an xsquared term in the equation.
The fact that curvature is hard to see indicates that the data is
nearly linear.)
We see that the differences between the data and
the line are less than 100rpm for the spin data and less than 0.2
degrees for the launch angle data. If you are familiar with how spin
and launch angle are measured, you know that these could easily be
measurement error. It is really quite creditable how good the
measurements are.
What I did next was replot the model's output
using the "smoothed" data  the bestfit curves  instead of the raw
data. Now the model fit even better.
With a CG depth of 1.1" and a coefficient of 25, only one point was
over 4% error, and that was 6.6%.
And, if I played more with the model, at 1.0" and 23.5 the errors were
all under 3%.
Conclusion: With the match between the model
and the data, the model is accurate enough to be very useful.
For
the remainder of this article, we will continue with the model using a
coefficient of 25, because that is plenty close enough to the data. We
will also stay with C=1.3" for
the "generic" driver, though we will use the actual C
if we know it. 
Last modified  Mar 24, 2009
