Flexface irons can be more forgiving
than rigidface irons
Dave Tutelman 
March 4, 2009
On Nov 30, 2008, I had a debate on the Spinetalker's
forum with Tom Wishon and Richard Kempton. The subject was
highCOR irons. That is, irons whose faces flex enough to give
a
spring effect, which increases the Coefficient Of Restitution (COR) and
raises the ball speed. This is not an idle nor
hypothetical discussion; Tom Wishon Golf Technology (TWGT) offers such
an iron, the 770cfe, and is introducing another model this month.
Here were the arguments:
 My point: while drivers and 3woods are
about maximum distance, irons
need to be about repeatable, dependable
distance. It's OK if I can't hit my 7iron as far as you can hit yours,
but it's a disaster if I don't know how far my 7iron shot is going to
go.
 Tom's point: there are lots of golfers
who would love to hit their irons longer. Distance sells. Older golfers
want to get back the distance they lost. Etc, etc, etc.
 My point: There is going to be some
falloff of COR as the strike moves away from the center of the
clubface. With good design there may not be much, but there will be
some. So that adds to the loss already there for an offcenter
hit. Yes there is loss even for a rigid clubface, due to clubhead
rotation because the moment of inertia is not infinite. Adding to this
loss with COR falloff further increases the uncertainty of how far the
ball will travel.
 Richard's point: based on his
experience, his customers' experience, and Tom's assurances, there is
no more loss of distance with the 770cfe than with rigid face irons. In
fact, they were even more forgiving of an offcenter strike.
I
was skeptical of that last claim. This was my reasoning at the time...
The
COR of a rigid face club and a modern ball is about 0.77. COR increases
with the face flex. The COR boost due to spring effect is roughly
proportional to the amount the face deflects where the ball strikes it.
Let us look at the face deflection for a few possible clubface designs.
 The green
curve
is the rigid clubface. It is just a flat line at zero  obviously,
because "rigid" means the clubface does not deflect. So there is no COR
boost for this face; the COR is stuck on 0.77.
 The dark
blue curve
is a uniformly thin face. It flexes a lot in the center, but the COR
boost falls off fairly quickly as the hit misses the center. This would
not send the ball a reliable distance, because the COR would drop off
towards 0.77 quickly if the strike was not oncenter.
 The red
curve
depicts what I felt could reasonably be done by selectively thickening
the face and changing it flex characteristics.This retains most of the
high COR for some degree of offcenter hit, but falls off if you get a
bit farther from the center.

Remember
that the goal is to make the club as forgiving as possible, in the
sense that the distance does not change much in the event of a mishit.
Of the three curves shown, the rigid face is the best at meeting this
criterion. No, it does not give maximum distance, but it does give the
least loss of distance for an offcenter hit. The red curve is also
pretty good, as long as you don't miss the center by too much, but
still loses a lot of distance for a significantly offcenter hit. And
it was my belief that the red curve was about the best you could do in
this regard.
Let's remember that there is another reason we lose
distance when we miss the center of the clubface, even with a perfectly
rigid clubface  clubhead rotation. A high
clubhead moment of
inertia can minimize this loss, but even a highMOI clubhead will show
significant loss if the strike is significantly offcenter.
So whatever "rolloff" of ball speed we get from offcenter reduced
deflection is on top of the loss from clubhead rotation. 
I continued the discussion with Richard offlist. We came to the
conclusion that, for the 770cfe to be more forgiving
(not just longer), at least one of the following had to be true:
 The
MOI of the clubhead is greater than the normal iron. This was certainly
possible, because the thin portions of the face provide a bit of
"discretionary mass" to be moved to the periphery. But Tom insisted
that the bulk of the difference was in COR. So we are left with...
 The
face is designed such that the COR increases (not decreases) as the
impact point moves away
from the center. That is, the deflection of the face at the point of
impact must be more away from the center than at the center. This is
not just a constant COR (like a rigid clubface), but a COR that curves
upwards away from the center  totally unlike the red or dark blue
curves above.
I was very skeptical that this could be accomplished. I thought that,
to resolve the question, I would have to gear up to analyze the
deflection of a nonuniform beam with a noncentered load. While not
terribly difficult conceptually, it is a lot of work if you don't have
structural design software or finiteelement analysis (FEA) software. I
don't have software packages like that.
But
then I came up with a remarkably simple model that did exactly what
Richard and I needed; it increased deflection as the point of
impact moved away from the center. Consider the system shown in the
diagram. A rigid bar is supported by two identical springs, one at each
end. A force (representing the impact force of the ball) is applied
somewhere along the bar, x
distance from the center.
Let's start with an intuitive explanation of the behavior of this
system; then we'll proceed with an actual analysis.
 If the force is dead center (x=0),
then each spring gets to resist half the force.
 If the force is at either end (x= ±1),
then the entire force is resisted by just the spring at that end. So
the bar  where the force is applied  experiences twice the
deflection.
 Between the center and the ends, we would expect some sort
of smooth variation. This is not a forsure given, but there is no
reason to expect otherwise. And the analysis will bear it out.
If you're interested, the analysis is at the end of this article. The algebra is more work than the physics, and there isn't much
to the algebra either. When you're all done, the function for the
deflection is very simple:
deflection
= 
F
2K 
(x^{2}
+ 1) 
where F is the force and K
is the spring constant of each spring. You can see the deflection
function on the graph.
So, at least for the length of the rigid bar, deflection (and thus COR)
is an increasing function of
distance from the center. Not only that  it is a squarelaw function.
Why is that important? Because the loss of ball speed from clubhead
rotation on an offcenter hit is also squarelaw. So it should not only
be possible, but easy, for COR to compensate nearly exactly for MOI
losses. This was a big surprise to me.
But there is still a bit of a problem. This model does not look
anything like a clubhead. It certainly does not
look anything like a TWGT 770cfe iron head. And it looks like just the
sort of kludge the USGA would be ready to ban instantly. How do we turn
this physics existence proof into a golf club?
The
next few diagrams show the evolution of the concept from physics
textbook
to the Wishon catalog.
We
start with the original version of the
system. Then we remember that coil springs are not the only kind of
spring available. The second figure has replaced the coil springs with
leaf springs, like those that allow the wheels of a car to soak up
bumps in the road.
But what are leaf springs? They are just thinner, more flexible pieces
of steel. If they were the same steel and the same thickness as the
rigid bar in the diagram, then they would be more of a rigid bar, and
not
springs at all. So we could spring the bar by simply narrowing it to a
thin
strip of the same material at the ends.
Of
course, the bar isn't perfectly rigid; there will be some small amount
of flex in it. But compared to the thin section at the end the thick
section looks rigid  and the thin section looks like a spring. It is still the same as the
original model with the coil springs and the rigid bar, to the
tolerances that matter in a golf club. 
And that is essentially what
Tom Wishon has done. The third figure is a
picture adapted from the 2009 TWGT catalog. The face is a separately
fabricated insert, shown in crosssection and colorcoded in blue. It
has a thick (read that as rather rigid) portion in the middle, and
thinner
flanges that connect it to the conventional rigid steel frame. But the
thin
flanges are much wider than they need to be just to connect to the
frame. There is enough thin area there to be a spring edge for
the
rigid center.
So the thick center acts as the rigid bar, and the thin edge acts as
the leaf spring. As long as you hit somewhere on the thick part, you
will see the spring behavior where increasing COR makes up for the loss
from clubhead rotation. If the hit is so far offcenter that it is on
the thin portion, you will still get some "save" out of it  though
the COR must necessarily fall as the strike approaches the
conventional, rigid frame supporting the periphery of the face. 
So let me compliment Tom on a really innovative design. Richard says it
does what it intends, and now I have no reason to doubt it.
AnalysisFor those who want to see how we got the graphs, here is the physics and math behind the curves.
Deflection of
uniform beam
Let us find the deflection of a uniform flexible beam, loaded with a force F at a distance x from the center. We want the deflection at x, which is where the ball impact would be if it were a clubface. We use the wellknown formula
deflection =  Fab
6EIL  (L^{2}  a^{2}  b^{2})  =  Fab
6EIL  ([a+b]^{2}  a^{2}  b^{2})  =  Fab
6EIL  (a^{2} + 2ab + b^{2}  a^{2}  b^{2})  =  Fab
6EIL  (2ab)  =  Fa^{2}b^{2} 3EIL  
Let
us normalize everything so there is a unit distance between the center
and either end. So L=2. That means that a=1+x and b=1x. Substituting
back into the equation for deflection gives
deflection =  F (1+x)^{2} (1x)^{2}
6EI  =  F (1x^{2})^{2}
6EI 
Deflection of
rigid beam with springs
Let us find the deflection at x of a rigid bar suspended at the ends by two springs of spring constant K. In this case, the deflection is not the bending of the beam, but rather the motion of the beam restrained by the springs.
The forces at the ends must equal F for equilibrium. Let us denote them as aF and (1a)F. (We will worry later about the value of a. Let's just use it for now in the deflection curve.)
Since
the beam is a rigid bar, the deflection is a straight line (the green
dotted line in the diagram). It goes from the spring deflection at the
left to the spring deflection at the right. We will normalize the
length again, so the line goes from 1 to +1. If we call the distance
along the bar t (because x is already taken to mean the position of the force), then the deflection at t=1 is (1a)F/K and at t=+1 is aF/K.
We
can get the equation for the deflection curve by the slopeintercept
method I remember from tenth grade. The intercept is the average of the
two end deflections, or F/2K. The slope needs a little more work.
slope =  aF  (1a)F
2K  =  F (2a  1)
2K 
So the slopeintercept equation (using t for the xaxis index) is Next, we want to find a in terms of x, the offcenteredness of the hit. If we balance moments around the left end of the bar, the torque equation is
(1+x)F = 2aF
Plugging this back into the equation for the deflection curve, we get
deflection =  F
2K  {(2  1+x
2  1) t + 1} =  F
2K  (xt + 1) 
We are interested in the deflection at the point where the force is, so we evaluate at t=x and get
deflection =  F
2K  (x^{2} + 1) 
Last modified 
3/22/2009
