Placement of Center of Gravity
for Best Spin and Launch Angle
Dave Tutelman  August 16, 2013
Here is the analysis behind the spreadsheet. Feel free to skip this if
you're not interested in the physics and math.
Calculations for spin
First
thing, we need a coordinate system so we can describe the center of
gravity placement, the impact of ball on clubface, etc. Let's use a
twodimensional system with axes x (horizontal
along the target line) and z
(vertical). We will set the origin at the point on the clubface where
the nominal loft is measured. If we talk about a 10.5º clubhead, then
[0,0] is the point on the clubface where the loft is actually 10.5º.
(Because of face roll, there is only one height on the clubface where
the loft is exactly 10.5º.)
Within this coordinate system, we can begin to describe the parameters
of our analysis.
 We designate nominal loft as L_{o}. The [0,0] point is
where the black line at an angle of 90ºL_{o} is tangent to
the blue clubface.
 The placement of the CG is at [X, Z]. X is a depth
from the face, and Z
is height above the nominalloft height. A low CG (as in the diagram) corresponds to a
negative Z.
 Impact occurs h above the
nominal loft, measured along the clubface. If it occurs lower on the clubface than nominal loft, h is negative.
 The effective loft where impact occurs is L_{i}.
The diagram shows L_{i}
as incorporating the effect of face roll, but it will also include
clubhead rotation due to shaft bend. The face roll is specified by a radius
of R. In most driver heads, R is between 10 and 14 inches.

Having
established a coordinate system, we can use it to compute C and y, the
quantities that will determine the spin. Our strategy will be to find
the [x,z]
coordinates of two points in space, [i,j] and [s,t]. Then:
 C
is the distance between [i,j] and [s,t].
 y
is the distance between [s,t] and [X,Z].
Let's go through the steps.
First compute L_{i}.
L_{i}
= L_{o} + arcsin (h/R)
+ 1.5 (XF)
The arcsin(h/R)
accounts for face roll for impact h inches above
nominal. The "1.5" factor accounts for loft change due to shaft bend,
explained later on this page.
Once we have the effective loft, we use the "standard" formula to get the
launch angle.
A = L_{i} (0.96  .0071 L_{i})
We can also get good values for [i,j].
i = h sin (L_{o} + 1.5(XF)) + h^{2}/2R
j
= h cos (L_{o} + 1.5(XF))
In each case, the sinecosine factor accounts for position along the
black line. The h^{2} factor for i
accounts for the small space between the black line and the blue curve
at impact. (We aren't going to worry about the tiny difference in j due to the
launch angle.)
Now let's get values for [s,t]. This is a
little harder, but we can use the fact that C is measured
along the launch line and y is
perpendicular to the launch line. Let's start by defining the slope of
the launch angle as m, then use the
pointslope method (remember, back in high school algebra?) to define
the lines of C
and y.
This is why we want a coordinate system; we can treat the whole diagram
like a graph.
m = tan (A)
Now we can write the equations for the two green lines. The equations are easily found using the pointslope form. The line for
C
(along the launch line) goes through the point [i,j] at slope m. The line for
y
goes
through the point [X,Z]
at slope 1/m
(the perpendicular to m). So the lines in pointslope form are:
z  j = m (x  i)
z  Z = 1/m (x  X)
Next we need to find the point [s,t]. We know that the two green lines intersect at [s,t]. So let's solve the two equations for x and z, and set those values to s and t. The solution is:
s = (m / (m^{2}+1)) ( j + mi  Z + X/m )
t = m (is) + j
So now we know the coordinates of the points [i,j], [s,t], and [X,Z]. What we really want is C and y, but those are just distances between the points. We find those distances with the Pythagorean Theorem.
C^{2} = (si)^{2} + (tj)^{2}
y^{2} = (sX)^{2} + (tZ)^{2}
However,
y can be negative as well as positive. The Pythagorean Theorem gives
the magnitude but not the sign. So let's use trigonometry instead.
y = (tZ) / cos(A)
Now we have everything we need to find the spins, using formulas from basic impact and gear effect.
V_{ball} = V_{club} * SmashFactor * cos (L_{i})
SPIN_{GE} = 58830 * V_{ball} * C * y / I_{v}
SPIN_{loft} = 160 * V_{club} * sin (L_{i})
SPIN = SPIN_{loft}  SPIN_{GE}
This is very close to the calculations in the spreadsheet. (We took a few shortcuts and consolidations in the spreadsheet.)

Calculations for launch angle
Head rotation from gear effect
(This
is the weakest rationale for any of the calculations in this article. I
assert here that the mechanism at work is the change in loft due to the
rotation of the clubhead during impact. If there is some other
mechanism at work, the conclusions here may be wrong.)
One factor
that changes launch angle is the head rotation due to gear effect. The
head is the big gear that
drives the little gear  the
ball. So the head will spin at a rate proportional to the ball's spin
rate (just the gear effect
spin) and also proportional to the "gear ratio", the ratio of diameters. Expressed
as an equation:
BallSpin * BallRadius =
HeadSpin * HeadRadius
What are these equal to?
 BallSpin:
we computed that already as S_{GE}. We'll use the
number we computed.
 BallRadius:
0.84 inches, controlled by the Rules of Golf. (Actually, the Rules
controls the diameter at 1.68 inches.)
 HeadSpin:
what we want to find.
 HeadRadius:
by definition, the distance from the center of rotation (the CG) to the
face. We recognize it as the computed value of C.
So the equation becomes:
HeadSpin S_{h}
= 0.84

S_{GE}
C

The unit is RPM. We're looking for loft change, and we measure loft in
degrees. Actually, degrees per second would be a much more convenient
unit for HeadSpin. So we want to multiply by 360 (to get to degrees)
and divide by 60 (to get to seconds). The equation becomes:
S_{h}
=

0.84
* 360
60

S_{GE}
C 
= 5.04

S_{GE}
C 

S_{h} is
a spin rate, an angular velocity, measured in degrees per second. But
what we really want is a total rotation in degrees. That is the amount
that the loft can be considered to change, thus changing the launch
angle. Just as distance is the accumulation of velocity ("integration"
in calculus terms), the rotation is the accumulation of angular
velocity. So we need to know the angular velocity function during
impact, and integrate it to get the rotation of the clubhead.
The angular velocity does not jump to S_{h}
from the beginning of impact, just as ball spin doesn't jump instantly
to S_{GE}. Let's assume
it climbs on a straight line. This is a plausible approximation, though
it is more likely a slight Scurve; that is, the angular acceleration
on the gear pair  the clubhead and ball  probably builds as the
ball gradually compresses and falls off as the ball releases. But let's
keep things simple and assume a straightline increase.
The graph
on the right shows the angular velocity in blue, as it increases from
zero at the beginning of impact to S_{h}
when the ball releases .0004 seconds later. The equation for the
straightline
angular velocity function is:
In order to find the angle of rotation at any time T after the
beginning of impact, we have to integrate the angular velocity.
θ(T)

=

T
∫
t=0

S_{h}
.0004

t dt







=

S_{h}
.0008

T^{2}


The function for angle of rotation θ(T) is the red
curve. We can see from the equation and the graph that it is a
squarelaw parabola. At the moment of separation (T=.0004 sec), the clubhead has rotated .0002S_{h} degrees.
But is that what we really want? The loft as the ball leaves is not
likely to be the angle that most affected the launch angle; the effective loft is probably
somewhere in the middle of impact, e.g. the
point of maximum compression of the ball on the clubface. So what is
the clubhead rotation in the middle of impact? That is T=.0002 seconds.
We evaluate it as:
θ(T=.0002)
=

S_{h}
.0008 
(.0002)^{2}
= .00005 S_{h}

This is only a quarter of the rotation we found for separation. Not a
half, but a quarter. If you're familiar with quadratic functions  or
even take a good look at the red curve in the graph  this is not
surprising.
We will use this value as the loft change due to gear effect, as we
estimate the change in launch angle. If you want to see what a higher
fraction of the head rotation would do, the spreadsheet allows the
fraction as a parameter; the default is 0.25, but you can adjust it.

Head rotation from centrifugal pull
Centrifugal
force pulling on the clubhead's CG creates a turning moment, trying to
rotate the face upwards. The moment is equal to the magnitude of the
centrifugal force times the "moment arm", the distance between the
centerline of the shaft and the clubhead's center of gravity. This
moment is resisted by the stiffness of the shaft. So how do we factor in
all these parameters?
Jeff Summitt has given me a rule of thumb that he says works over a
variety of clubheads and shafts. The clubhead rotates to increase the
loft by 0.15º for each 0.1 inch increase in the moment arm. That is a
rate of 1.5º per inch, though Jeff is quick to remind me that moving
the CG by an inch is huge.
Yes, that rate varies a bit with the velocity, clubhead mass, and shaft
stiffness,
but the difference is small enough that you can ignore the variation.
LoftChange
= 1.5º * MomentArm
 LoftChange
in degrees
 MomentArm
in inches
 v
in miles per hour
If we want more resolution than this, the formula has to account for:
 Velocity, which is a squarelaw factor in centrifugal force.
 Radius of the clubhead's path at the moment of impact. That
is typically somewhat more than the club length  maybe 25% more for a
full swing with good release.
 Bending properties of the shaft. Not just the overall
stiffness, but the flex profile over the length of the shaft. This can
be characterized by an EI profile, a frequency profile, or a deflection
profile  and each has its own computational difficulties. The EI
profile is probably the most elementary flex profile for use in a
computation, but is rather rare to have around for a given shaft. (Russ Ryden has been characterizing shafts by EI profile.)

So there you have it, a set of calculations embodied in a spreadsheet,
to compute the effect of CG placement on spin and launch angle.
Last updated
 Mar 10, 2014
