Placement of Center of Gravity
for Best Spin and Launch Angle
Dave Tutelman  August 16, 2013
We have seen the reasons for the CG placement to affect spin and
launch angle. Now for some numbers. Let's look at a few cases, and see
what moving around the center of gravity does to spin and launch angle.
I built a spreadsheet to do the calculations. You can
download a
copy for yourself if you want to play with cases and get a
better feel for what is going on. Let's cover that first, before we
review what playing with the numbers teaches us.
Spreadsheet
I have made an Excel spreadsheet that will plot the spin and launch
angle for two different placements of the center of gravity. You can
set up the two placements; the reason for two is to be able to compare
and note the effects. Here is an
overview screenshot of the calculation sheet, in case you decide to download
and use it:
The screenshot shows the Calculations page. There is also an
Introduction page, which reviews the meaning of each parameter, and
gives
the formulas used in the calculations.
At the upper left is the input area. You can only enter data in the
light blue cells of the input area. The rest of the spreadsheet is
protected from inadvertent entries, so you don't accidentally corrupt
the formulas.
You can input the data for two different placements of the center of
gravity. The calculations, results, and graphs are colorcoded, green
for placement #1 and red for placement #2.

The spreadsheet is
based on coordinates shown in the accompanying diagram. We require some
coordinate system, because first and foremost we need to describe the
position of the CG; that is the primary focus of our study.
The origin (the [0,0]
point) of our coordinates is the center of the clubface. Specifically,
it is the face at exactly the point where the loft is the nominal loft
of the clubhead. If we are studying a 10.5º driver, then [0,0] occurs
where the face is slanted 10.5º from vertical. In the diagram the X and
Z axes are green dotted lines.
The center of gravity is therefore specified with X being the
horizontal distance behind the midpoint of the clubface, and Z being the
vertical distance above the midpoint of the clubface. The example in
the diagram shows the CG about 1.3" behind the clubface and 0.2" below
the axis. So the coordinates of the CG would be X=1.3 and Z=0.2, or [1.3, 0.2].

All input
is done in the blue cells at the upper left of the worksheet, shown in
the screenshot below. The input
parameters are:
X, Z

The coordinates of the CG,
in inches. There are two columns here, one for each of the CG locations.

I_{v}

The vertical moment of
inertia of the clubhead. Modern drivers have this value close to 2950
gramcmsquared, so that is the default in the spreadsheet. Why two
columns? Because some methods that clubhead designers use to move the
CG may also change the MOI, so the two CG locations may need different
values for MOI.

L_{o}

The nominal loft of the
clubhead, in degrees.

R

The face roll, specified
as the radius of curvature in inches. The default of 12" is around the
middle of the range for modern driver heads.

C_{sb}

Coefficient of shaft bend,
in degrees per inch. This is a simple number that Jeff Summitt tells me
is a pretty good approximation. It will change with clubhead speed and
shaft flex profile, but probably not by a factor as much as two.

V_{c}

Clubhead speed at impact,
in MPH.

SF

Smash factor. The default
is the theoretical maximum for a 200g driver head.

F

The horizontal distance
from centerface to the shaft centerline. In coordinate terms, it is
the Xcoordinate
of the shaft centerline. The default of 0.7" is a typical value for a
modern driver. It might be lower for, say, an offset driver or a
fairway wood.

F_{age}

The fraction of the angle
of loft change during impact, due to gear effect,
that gives us the effective loft for
computing launch angle. The analysis says it should be 1/4, which is
the default value. (That is, 1/4 of the angle change during impact is
the effective angle for computing the change of launch angle.) But you
can tweak it to a higher number if you want to test the
assumptions I made in the analysis.


Results
Let's
look at some graphs from the spreadsheet, and see what they tell us
about how we affect spin, launch angle, and ultimately distance by
moving the
clubhead's center of gravity.
Spin
Here are spin graphs for two runs of the spreadsheet. One moves the
center of gravity rearward in the clubhead by 0.1", the other downward
by 0.1". Except for the specific CG locations in the diagram, the
spreadsheet was primed with the default values shown in the previous
section.
Before we talk about the results, a few words about the graphs
themselves. Each graph has two sets of green curves (CG position
#1) and two sets of red curves (CG position #2). The solid curve is the
total spin, and definitely the most interesting. But there is also
something to be learned from the dashed curve, that I call "loft spin"
for short. The dashed curve is the spin created by the loft of the club
as it comes to impact, including both the loft built into the clubhead
and the added loft due to shaft bend. The solid curve is the dashed
curve minus the gear effect topspin.
Now for the results themselves...
Moving
the CG rearward
has a mixed value, but most of it has the effect of increasing the
backspin. This is counter to the older conventional wisdom, which holds
that a readward CG reduces backspin. The higher on the clubface you hit
the ball the less the spin penalty, but the curves don't cross until
more than a half inch above midface. At midface, the rearward CG has
about 160rpm more
backspin. That is significant, though far from huge.
Of course, we know that a midface strike is not where you want to hit
the ball to maximize distance. Suppose we try 0.4" above the center.
Looking at the curves, we can see that the spin difference between the
green and red curves has almost disappeared. It is only 65rpm.

X
= 1.3

X
= 1.4


Ball
Speed

Spin

LA

Carry
Dist

Ball
Speed 
Spin

LA

Carry
Dist

h
= 0

145

3642

9.9

230

145

3801

10.0

229

h
= .4

144

2797

11.6

232

143

2862

11.8

230

Let's
see how that translates into distance. For the table at the right, we
plug the computed launch parameters into TrajectoWare Drive
and find the carry
distance. In each case the rearward CG results in less distance, but
only by a yard or two.
The bottom line is that moving the CG rearward makes small, almost
unmeasurable differences in distance  and the difference might go
either way.


Z
= 0.1

Z
= 0.2


Ball
Speed

Spin

LA

Carry
Dist

Ball
Speed 
Spin

LA

Carry
Dist

h
= 0

145

3642

9.9

230

145

3274

10.0

231

h
= .4

144

2797

11.6

231

144

2415

11.7

229

Moving
the CG downward,
on the other hand, has a pretty large payoff in terms of spin. We knew
this just from
the visual description on the previous page, but let's quantify it
now. The total spin lines for positions 1 and 2 are almost parallel and
are 370400rpm apart. Of course, the lower CG enhances favorable gear
effect and gives lower spin. Let's see how this measures up in terms of
distance, for a centerface strike and a highface (h=0.4") strike.
The results shown in the table are very surprising. Despite the lower
spin for the low CG position, it seems to lose distance. Not a lot of
distance  only 1or 2 yards  but distance that we thought
we would gain instead of lose. We will discuss the reason for this
surprising result on the next page. It involves the shape of the "launch space",
the plot of distance vs launch angle and spin. The conventional wisdom
says you always
want lower spin and higher launch angle, but we have demonstrated here
that is not always the case.

Launch angle
There is a lot less to be said about the variation of launch angle with
CG motion  and that in itself is a surprise.
 Fore
and aft movement of the CG
changes the shaft bend by .15º for every .1" of CG movement. That will
change the launch angle, but by less than .15º. It appears to be
roughly .1º of launch angle change for every .1" of CG movement. Barely
measurable, it's just one place in the tenths column.
 Up
and down movement of the CG
doesn't change shaft bend at all. The only way it can change launch
angle is through head rotation during impact. That is not a very
efficient way to affect launch angle; a .1" CG motion gives less than a
.1º change of launch angle.
Both these are much less than I had expected. Perhaps clubhead rotation
works by a different mechanism than simply changing the loft during
impact. In fact, I postulated a completely different mechanism and
tried some backoftheenvelope calculations. I got almost the same
result, so perhaps it is true. Unfortunately, I don't have any good
experimental data relating CG placement to launch angle; therefore, I
don't have any yardstick to measure whether my launch angle analysis
corresponds to reality.

On the next page, we summarize, discuss, and try to explain these
results. This leads to a discussion of how to design and fit a driver
for a given golfer.
Last updated
 Mar 11, 2014
