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Driver Head Weight and Club Length

Dave Tutelman -- October 28, 2012

Constant Club Length

Cochran & Stobbs' study

In their wonderful 1968 book, Cochran and Stobbs attacked the problem head-on. The accompanying graph from the book shows their estimate of how clubhead speed varies with head weight. The curves are based on a formula derived from "plausible but not verified" assumptions (their terms), but they do fit the measurements available at the time. They show how, for a given golfer, the clubhead speed goes down as the clubhead mass increases.

The table below combines these curves with the well-known formula for ball speed:

Vball = Vhead

1 + COR

1 + m/M

which reflects the fact that higher clubhead mass (M) means more momentum transfer. The table shows how ball speed varies with clubhead mass, for a low-handicap golfer swinging each clubhead as fast as he can. (The table is adapted from a table in the book. Only the units have been changed, to more familiar units for US clubfitters.^[1])

Clubhead mass		Clubhead speed (MPH)	Ball speed (MPH)	Carry (Yards)
Ounces	Grams	Clubhead speed (MPH)	Ball speed (MPH)	Carry (Yards)
1	28	132	85	<100
2	57	126	118	156
4	113	117	140	206
6	170	110	146	218
8	227	104	147	219
10	283	100	145	216
12	340	96	142	210
16	454	90	138	200
24	680	81	128	178

What this table tells us is that there is a rather broad maximum at a little over 200 grams. We should not be surprised that the maximum is about 200g. Remember, golf has been played for hundreds of years. Whether or not they know the physics, clubmakers have a huge collection of experience. If most driver heads today are about 200g, that must be for a reason.

The exact position of the optimum head weight doesn't matter, because the distance is within a yard of the maximum from 170 grams to perhaps 240 grams. This says that the exact head weight should not make a difference you'd notice in distance, so choose the weight that the golfer seems to swing the best. That is probably where the clustering around 200 grams comes from, at least as much as maximum distance.

A computer study

Such an interesting question probably calls for another pass at an answer. Not many had access to a computer in 1968; just about everybody has one today. I used two computer programs:

SwingPerfect for going from a club and swing to impact conditions (clubhead speed and wrist angle at impact).
TrajectoWare Drive for going from impact conditions to ball speed and carry distance.

The first thing I did was set up SwingPerfect for a neutral 100mph clubhead speed with a flat wrist at impact. I tweaked input conditions, and eventually got a clubhead speed of precisely 99.6mph and a wrist cock at impact of precisely -0.1º. (Negative wrist angle means the wrist is cupped. But 0.1º cupped is flat for all practical purposes.) Here are the settings I used to get those impact conditions:

Clubhead mass: 200g
Club length: 45"
Shaft mass: 65g
Arm angle at transition: 180º
Wrist cock at transition: 90º
Shoulder torque: 46 foot-pounds constant during downswing
Wrist torque: zero until 200msec into the downswing, then 3.9 foot-pounds
Golfer: 176 pounds, with a 24" arm length

At 99.6mph clubhead speed, carry distance is maximized with a dynamic loft of 12.5º. This assumes a 0º angle of attack. We know we have a flat wrist, so the wrist is not contributing to the dynamic loft. If we assume a shaft bend that increases the loft by 2º, that means we need a loft built into the clubhead of 10.5º -- a very standard driver head. For each row, we optimize the dynamic loft to get the maximum carry. It ranges from just under 11º to just over 13º -- a small enough change that we won't bother to show it with a loft column on the table below.

Clubhead mass (Grams)	Clubhead speed (MPH)	Ball speed (MPH)	Carry (Yards)
100	121.7	150.3	240
110	119.0	151.0	241
120	116.2	151.1	242
130	113.8	151.1	243
140	111.5	150.8	242
150	109.2	150.1	241
160	107.2	149.3	240
170	105.1	148.3	238
180	103.2	147.1	236
190	101.4	146.1	234
200	99.6	144.8	232
210	98.1	144.0	230
220	96.3	142.5	227
230	94.8	140.8	224
240	93.4	139.8	222

Comparison

A look at the tables -- or, to see more easily, the graph on the right -- shows that the models do not give the same result. Not even close.

The most apparent discrepancy, which can be seen in the tables, is that the total driving distance is not the same at all. The drives in Cochran and Stobbs are much shorter. However, that is easily explained and compensated for. Cochran and Stobbs did their research in the 1960s, well before spring faces extended the Coefficient of Restitution to 0.83. It is generally agreed that a driver with no spring effect will give a COR of about 0.78. If you apply that COR to the computer model, you get about 223 yards, plenty close enough to the C&S data.^_[2]

In the graph, we have scaled the C&S data as if it were gathered with a more modern driver and ball. We chose the scaling factor so the two would be the same distance at the conventional driver head weight of 200 grams.

Even then, the models give drastically different results! The maximum ball speed and distance occur at a much lower head weight: around 130g rather than C&S' 200g. On top of that, the C&S maximum is much broader and the computer simulation much sharper. The shape of the two curves is very different, not just the scale.

Something is wrong here. Can we get any more information that might shed some light on the difference?

Daish's study

Rod White (author of the article on Golf Swing Physics) brought my attention to a similar study by C. B. Daish in his classic book "The Physics of Ball Games". Daish's approach was to measure the variation of clubhead speed vs clubhead weight, using four human test subjects who spanned a variety of phsyical strength and golfing talent.

Here are the curves for the four golfers individually. The vertical axis is clubhead speed and the horizontal axis clubhead mass, but both axes are a logarithmic scale. In playing with the data, Daish had seen that the curves would be straight lines on a log-log plot like this, which gives some insight into how we can express things mathematically. In particular, a negative-sloped straight line on a log-log plot can be expressed as:

M_head V_headⁿ = constant

The slopes are similar -- almost the same for three of the four golfers, and not very different for the fourth -- at an average value for n of 5.3. So the equation for clubhead speed vs clubhead mass is:

M_head V_head^5.3 = constant

What is the value of constant? Well, it stays constant for any golfer as the head weight varies, but its value depends on the strength and skill of the golfer. A bigger hitter has a higher value for constant.

By doing some algebra on this, we can solve for ball speed as a function of clubhead mass.

V_ball = K

M^(1-1/5.3)

M + m

where K is a constant that depends on the golfer's strength and the clubhead's coefficient of restitution, and m is the mass of the ball. In the graph at the right, we use this formula to add the Daish model to the two we had before, Cochran & Stobbs and the computer simulation. I selected K to give a carry distance of 232 yards at 200 grams of head weight, the same as the other two studies at this reference value.

The Daish study looks a lot more like the C&S study than the computer simulation. Like the C&S study, Daish has a very broad "peak" at about 200g. Since we know that Daish measured clubhead speeds for real golfers, and they all looked pretty consistent, it is probably the most trustworthy answer. But why is it so different from the computer model? Here are some thoughts on the difference:

There is a considerable difference between the clubhead speeds of the Daish study and the computer simulation study. The variation of clubhead speed with head weight for the computer study is about twice that for the Daish study.
The same model was used in all three studies to go from clubhead speed to ball speed. While different models were used to go from ball speed to distance, they produce fairly similar results over the ball speeds encountered in the study. So the difference is clearly in going from head weight to clubhead speed.
Therefore, the difference must be based on a difference in the torques applied during the swing.
The Daish study requires that the torque applied by the golfer is greater with heavier heads and less with lighter heads. That is, the swing adapts to the load. I do not question that real humans might well adapt in this manner.
The reason may have to do with the speed more than the load. Recent swing modeling efforts have used the biomechanical notion of a torque-velocity curve. This says that the torque necessarily goes down as the joint it is driving turns faster. For our constant-length variable-headweight study, any increased clubhead speed due to lighter headweight is accompanied by faster turning of the joints -- which would result in a lower driving torque.

Let's follow up on this thought, and see if the torque-velocity relationship can account for the difference.

Torque-velocity computer study

We will use the torque-velocity curves from Sasho MacKenzie's model of the golf swing. That model is:

T = T_o

ω_max - ω

ω_max + Γω

The omegas are rotational velocities, the Ts are torques, and Γ is a "form factor". We use a double-pendulum model, and the torque is completely shoulder torque (which is torso and shoulder rotation in the MacKenzie model). For torso and shoulder rotation, ω_max=30radians/sec and Γ=3.

So the torque reduction becomes:

T = T_o

30 - ω

30 + 3ω

The graph to the right shows what the torque reduction looks like. It is a pretty serious reduction. The torque is reduced by half at 6 radians/sec; that's the same as 344 degrees/sec. Torso/shoulder rotation during the swing can easily be twice that, so the torque-velocity reduction is very significant.

I do not have the tools at hand to do the simulation exactly, but I can certainly do a back-of-the-envelope study to see if the effect is the right order of magnitude to explain the difference between the studies.

The graph at the left is produced by SwingPerfect, and is the angular velocity of the arms' rotation during the downswing. For all the swings of interest, this curve is roughly the same shape; only the scale changes. That means we can get a first-order approximation by picking a representative velocity and using that in our torque-velocity curve. Since we get the biggest effect on torque at high velocities, and the torque is most important later in the downswing (where the velocity is the highest), we will pick the dip just before impact. That will be our reference velocity.

For this study, we will start by finding a constant torque that, when reduced by the torque-velocity curve and applied to SwingPerfect, gives a 100mph clubhead speed. (Due to the shape of the curve and the "granularity" of setting torques in SwingPerfect, we will slope the torque reduction for the first two segments of the swing.) This takes some iteration, because changing the torque gives a different reference velocity, which gives a different torque reduction. Then we will take that constant torque, apply it to SwingPerfect with different head weights, and again iterate on torque-velocity to get a consistent clubhead speed.

Here is the graph we get when we go through the computation of carry distance. The green curve is the Daish study, and the red curve is the original computer simulation. The blue curve is the modified simulation, where we reduce torque based on the torque-velocity curve. Accounting for the biomechanical reduction of torque does not give us the Daish (nor Cochran & Stobbs) results, but it does cut the headweight advantage in half.

I tried a variation of the torque-velocity study, where I used the entire reference velocity to reduce the torque during the entire downswing. This is not completely valid, because it takes 100-150 milliseconds for the angular velocity to get up near the reference velocity. For that first 100-150 milliseconds, the torque-velocity curve does not provide nearly as much reduction. The simulation above accounted for this, but let's see what happens if we apply the full reduction for the full downswing.

What happens is shown in the graph. The carry distance falls off as the clubhead weight is reduced. So the torque-velocity curve is capable of reducing the clubhead speed down to the Daish level, or even lower. But we have to apply more of it than is physically accurate.

But wait! This suggests something else that might be going on. In fact, it is very likely going on. What this study does is tone down the torque applied to body rotation early in the downswing. That may well be happening, even if the torque-velocity curve is not the reason. It is probably unreasonable to expect full body torque to be applied the very first millisecond after the transition from backswing to downswing. Mackenzie's biomechanical model has built-in a time constant that says it takes muscles 60msec to build up close to full strength. If considerations of balance in the transition slows this down more, the torso/shoulder torque might take 100-150msec to build up -- and the result might well look like Daish's study.

Conclusions

Here are the conclusions I draw:

The Daish study is the most likely to be factual. The reasons are:

It is based on actual measurement of human golfers swinging with varying head weights.
It comes up with an optimum at 200g. Over centuries of golf experience, driver heads have always been around 200g, give or take less than ten grams, adding some completely non-analytical support to any study with an optimum around 200g. (We'll get to the exception -- substantially longer clubs -- next.)

The difference between the Daish study and the constant-input-torque computer simulation is due to the fact that human beings don't apply the same torque in the swing as the head weight varies. The reasons are::

Going from head weight to clubhead speed involves human biomechanics, not just physics. That is the most problematic step.
Going from clubhead speed to ball speed is physics, and done with the same simple formula in all the studies. No difference there.
Going from ball speed to distance is also just physics. Admittedly, it is done a little differently in each study, but the resulting differences are very small -- not enough to significantly affect the results.

The human biomechanical differences in producing clubhead speed are:

Partly a torque-velocity relationship. The faster clubhead speed would be due to a faster body turn. But a faster body turn implies less torque available to produce the turn. So the speed increase is not as much as you might have expected. This part is probably roughly half of the discrepancy.
Some of the remaining difference is probably the fact that full rotational torque is probably not applied at the very instant of the swing's transition, but somewhat ramped up.
And some is possibly due to the body applying more strength if it has a higher load to move.

Last updated Jan 16, 2013