# How stringy is the shaft at impact?

Dave Tutelman  -  January 23, 2022

Since the 1990s, I've been saying, "The shaft is a string at impact!" to anybody who will listen. I still believe it is a good rule of thumb. Is it absolutely true? Of course not! Here we explore the limits of approximating the shaft as a string during the moment of impact.

## Executive summary

 We look at the question from three different points of view: Where does the concept come from anyway? It's pretty silly if you take it literally. What it means is during impact your hands on the grip have no more control over clubhead/ball impact than if the shaft were a string. We can see from pretty simple physics that almost all the force the clubhead applies to the ball comes from the inertia of the clubhead. During impact considerably less than 1% of that force can be attributed to the hands applying a combination of force and torque to the grip at that moment. Both your effort and the shaft definitely affect ball flight. But not during the instant of impact. You need to do everything with your hands, body, and the club before impact occurs. From the start to the end of impact, there is no way your hands can transmit anything to the clubhead. What part of the mass of the shaft contributes to ball speed at impact? While the classic momentum transfer assumes all the mass involved is the clubhead, we know that there is at least a small bit of shaft tip bonded securely into the hosel. Does that contribute, and how far away from the hosel does shaft mass continue to matter during impact? It turns out to be less than 2.6 grams, no more than the portion of the shaft tip actually bonded into the hosel. Usually it is less than a hosel-filling dose. So far, we have just talked about creating ball speed with good center impact. What about things like gear effect and smash factor when the impact is not on-center? Does shaft stiffness have a role to play there? In this case, the shaft might have more influence than a string. If the strike is off-center, the impact point's distance and direction from the center will cause rotation in some direction with some speed. Some of that rotation will require the shaft to twist about its own axis (torsion), and some rotation will require the shaft to bend along its axis (flex). It turns out that the shaft is still a string in torsion, but has some effect of limiting rotation in flex. Because the shaft is at a lie angle of 56°-60°, both sidespin and backspin will have contributions from both torsion (little to none) and flex (could be noticeable if the shaft is stiff enough).

## The basic physics

Yes, I know it is pretty silly to compare the shaft to a string. The shaft is a rigid rod, right? Well, no. And the harder you look the less sense that characterization makes. The very notion that fitting involves shaft "flex" or "stiffness" gives the lie to perfectly rigid.

A much more accurate way to look at a shaft than a rigid rod is as a spring. It takes a certain amount of force to deflect a shaft a certain amount. The stiffer the shaft, the more force it takes. In fact, the ratio of the force to the deflection it produces is what engineers and physicists call the shaft's "spring constant".The overall spring constant of the shaft is not the only flex characterization, but it is the first and most important one.

We measure the spring constant of a shaft with a device called a "deflection board". Clubfitters and clubmakers have used deflection boards for well over a century. Here is an almost modern deflection board sold by by GolfWorks. You clamp the butt of the shaft horizontally, hang a known weight from the tip, and measure how far the tip moves. A more modern board may use electronics to measure the force, rather than hanging a calibrated weight. But they measure the same thing: a force and a deflection whose ratio is the spring constant of the shaft.

I just went down to my basement and measured a few full-length driver shafts in my NeuFinder 4.1. (That is a deflection board I co-designed over 15 years ago for shaft flex trimming and profiling.) The stiffest among them had a spring constant of less than 1 pound per inch. (I measured it at 0.90.) Knowing the ranges of stiffness for golf shafts, the stiffest driver shaft of normal design at this length is probably less than 1.2 pounds per inch, and it is hard to imagine a driver shaft as stiff as 2 pounds per inch.

Now we come to the way I think of "string at impact". During impact, the clubhead is applying force to the ball; this is how ball speed happens. The question becomes: How much of this force is due to simple momentum transfer from the mass of the clubhead ("inertial force") and how much is transmitted from the hands through the shaft to the clubhead? If the hands represent a negligible portion of the force, then the shaft is doing no more at impact than if it were a string.

### The total force

This is simple momentum transfer. The clubhead applies force to the ball to accelerate it according to F=ma. We know the mass of the ball. We can compute the acceleration, which is the change in velocity divided by the time it took to achieve the change. So:

F = m Vball / t

For this first set of calculations, we will assume a drive struck with a driver head traveling at 100mph. Everything else is very vanilla for a golf ball and driver. See the spreadsheet for actual numbers if you are interested.

When we do this calculation, the force is 1528 pounds. This is an average force, averaged over the whole duration of impact. It could be twice this for a peak force.

### Tweaking

So far, the shaft seems very stringy indeed! Is there anything we are justified in changing about this model, that might make force through the shaft a little more significant? Let's try tweaking it a little.

First off, let's try tweaking things in the spreadsheet. We'll only tweak things that make sense, not something like the mass of the ball (every golf ball is within a gram or two of 46g).
• Clubhead speed: This does nothing for the "stringyness ratio". It stays at the same value, 12532, for every plausible clubhead speed.
• Coefficient of Restitution: Same thing. You can crank it all the way down to 0.7, roughly what it was 40 years ago with wooden driver heads and balata balls, and the stringyness ratio stays at 12532.
• Duration of impact: 450 microseconds is the median of the values in the studies I have seen. But maybe cranking it up to the longest from any of the studies, 500 microseconds, will make a difference. That does move the needle, all the way to 10151. But the shaft is still providing less than 1/10,000 of the force -- still awfully stringy.
• Shaft stiffness: Let's postulate an XXS shaft (double-extra-stiff). That would be something like 16% higher spring constant than our S-flex slaft, or a spring constant of 1.04 pounds per inch of deflection. The ratio in that case would be 10804. Still less than 1/10,000 of the force, and still very stringy.
OK then, how about just making some unreasonable assumptions and seeing what they do. Let's assume you can apply a 2-inch backward bend to the shaft during impact. No, I don't believe for a second that you can, and neither should you. But let's suppose, for argument's sake. Then, let's double the spring constant of the shaft, to 2 pounds per inch. Again, that is a completely unrealistic number, but we'll do it for the sake of argument. A simple calculation gives us a force of 4 pounds applied to the clubhead through the shaft. But even with that, the force provided through the shaft is only about 1/400 of the total force. Still a string!

### What's next?

Now we understand why it is reasonable to say, "The shaft is a string at impact." The forces transmitted to the clubhead via the shaft during impact are a negligible part of the forces actually involved in impact. The shaft might just as well be a string, for all the good it does. Of course, the shaft plays a very important part in getting the clubhead to impact in the first place. That's why shaft fitting is important. But during impact itself, it might as well be a string.

But there are still a few interesting related questions.
1. We calculate impact based on the mass of the clubhead itself. But shouldn't some shaft mass also figure in that calculation? At the very least, the tip of the shaft is epoxied rigidly in the hosel; in essence, it is part of the clubhead. Why doesn't the mass of the tip count, and perhaps more of the shaft as well?
2. We have looked at generating ball speed during a sweet spot impact. But what about off-center hits? Does shaft stiffness play any part in gear effect, either vertical or horizontal? Or is that 100% stringy as well?
In November 2021, Jay Messner brought to my attention a couple of papers that contain possible answers to these two questions. The authors of both studies were Erik Henrikson of Ping and several researchers from the University of Waterloo. Here is what that research tells me.

## Mass of the shaft

The usual assumption for the club-to-ball momentum transfer is reflected by the well-known equation:
 Vb  =  Vh 1 + e 1+ mb/Mh
Note that the mass of the shaft is not even considered; only clubhead mass is considered part of the transferred club momentum. It is hard to believe this assertion, since there is at least a small portion of shaft tip inextricably bonded inside the hosel. How could this not act exactly as the clubhead mass does? And if it is part of the momentum mass, then how much more of the shaft is involved before there is too much distance and too much flex to be part of the momentum transfer?

A 2020 paper can give us considerable insight into this question. (Erik Henrikson, Behzad Danaei, William McNally, John McPhee, "Adjusting a Momentum-Based Golf Clubhead-Ball Impact Model to Improve Accuracy") The basis of the paper was to test the accuracy of the classical momentum transfer model between the clubhead mass and the golf ball. From the introduction:
Two [simplifying] assumptions that affect the accuracy of the momentum-based impact models are neglecting the deformation of the golf ball during the impact and the influence of the golf shaft on the impact dynamics. The purpose of this study was to account for the aforementioned factors by making two simple and physically meaningful adjustments to the conventional momentum-based impact model. The first involves making a slight adjustment to the center of mass of the golf ball to account for golf ball deformation, while the second involves adding mass to the clubhead in proportion to the shaft mass.
The way they did the study was:
1. Measure the club and ball extensively, and prepare a simple momentum-transfer model based on those measurements.
2. Do experimental runs using multiple golfers and measuring launch characteristics with a GCQuad launch monitor. The results for both spin and ball speed were fairly close to the simple model, but not exactly the same.
3. Tweak a few parameters -- OK, we'll use their word, "adjust" -- to make the results come out right on the money. It turns out that, for ball speed, the parameter to be adjusted was the clubhead mass, and the adjustment was an added mass. Moreover, the size of that mass adjustment was a function of the shaft mass.
• The other parameter they adjusted was the position of the ball's center of mass. This turned out to affect spin considerably, but ball speed not at all. Why adjust the ball's CoM? Because the ball compresses on the clubface during impact, so the ball changes shape and the CoM is not the same distance from the clubface.
The experimental runs included three drivers with identical clubheads but shafts of different weights. The mass adjustment (Δm) was computed separately for each shaft. The adjustment was not quite proportional to the shaft's weight, but it was monotonically related. That is, for each higher mass shaft, the adjustment Δm was higher.

Here is a graph of Δm, the dashed blue curve, plotted against the total mass of the shaft. Even for the almost-75g shaft, pretty heavy for a driver, the mass adjustment to the clubhead was less than 3 grams. That number just begs for the answer to, "How much shaft mass is firmly inside the hosel?" Let's try to answer the question.

The red curve shows the length of shaft tip containing Δm grams of mass. It assumes that there is a uniform mass distribution over the length of the shaft. That is probably not a terrible assumption, but we'll revisit it in a moment. Not that the length of shaft that gives Δm grams is less than an inch for the two lighter shafts and about an inch and a half for the heaviest shaft.

How does that compare with the hosel insertion depth for a driver? What we are really asking is, how quickly does the contribution of shaft mass to impact fall off as we get farther from the hosel. I measured all the unoccupied driver heads in my basement, and found that the insertion depth is close to 1.5 inches for most of them. I did find heads from 1.25" to 1.75", but most were much closer to 1.5" than that. Therefore, Δm struggles to even fill the hosel. There seems to be no shaft mass outside the hosel that is participating in the momentum transfer.

I promised we would get back to the assumption that the mass distribution is uniform along the length of the shaft. Most driver shafts have a balance point a fraction of an inch above the geometric center. That might mean the mass density is a little lower near the tip. But more than offsetting that is the fact that driver shafts typically have reinforced tips. The wall thickness inside the hosel is several times more than the average wall thickness of the shaft. So it is even less likely that any shaft mass outside the hosel might be part of the mass adjustment.

This paper shows that any shaft mass that is contributing to the momentum transfer is unlikely to be outside the hosel of the clubhead. The diagram above might more accurately be drawn as shown here. The full effect of shaft mass stays inside the hosel, and there is no momentum provided to the ball by any shaft mass outside the hosel.

Bear in mind that the tiny contribution of the shaft mass is due to the flexibility of the shaft -- its tendency to mimic a string at impact. In a 2009 paper, Rod Cross and Alan Nathan estimate the contribution as a quarter of the shaft mass, a much larger number. But their calculation is based on the assumption that the shaft is perfectly rigid, so it contribution would be due only to mass distribution.

So far, we have been talking about an on-center impact. The clubhead remains in a stable orientation through impact, and just applies a simple deflection to the shaft. But what happens if the impact is off-center, either heel, toe, high, or low? Here is a figure that shows an off-center impact, in this case toward the toe.

The force applied to the ball by the clubface is off-center; more precisely, it does not go through the center of mass (CoM) of the clubhead. There is a moment of this force, equal to the magnitude of the force times x (the distance of the force from the CoM), trying to rotate the clubhead clockwise around the CoM. The second picture shows the result of this rotation. The rotation itself is in blue -- the radius from the CoM to the place where clubface meets ball, and a circle representing the rotation at that radius. The clubface moves tangent to the rotation circle, in the direction of the red arrow. We have resolved that motion into two orthogonal motions:
• Parallel to the clubface (yellow) - This movement is tangent to the ball's surface, and causes the ball to spin. The spin is due to what is known as "gear effect".
• Perpendicular to the clubface (green) - This movement is away from the ball. This movement subtracts from the overall clubhead movement toward the target, and thus reduces ball speed. Another way of putting it is we have lost some "smash factor".
This combination, giving spin and taking away speed, increases with distance from the center of mass and decreases with clubhead moment of inertia, which resists the rotation of the clubhead.

If the impact had been through the CoM, the the clubhead would not rotate and it would only exert deflection on the shaft. We have seen that the shaft is close to a string at impact for pure deflection. Can we treat the shaft as a string for clubhead rotation as well? I tried to address the problem in my 2009 article on gear effect. The first several pages treat the shaft as a string, then ask the question about whether shaft stiffness can impede gear effect by limiting clubhead rotation. I dismissed axial twist of the shaft (usually represented by the shaft's "torque rating") by a similar study showing that it could not provide a lot of resistance to clubhead rotation in the moment of impact.

But it is not nearly as easy for shaft flex. The reason is that impact creates two opposite deflections fighting against one another.
• On one hand, the force itself wants to push the clubhead back and deflect the shaft backwards.
• But there is also the rotation created by the moment of the force. In the picture, the impact is above the center, so the moment of force wants to rotate the clubhead face-up. That rotation tends to bend the shaft forward.
With these two bends possible working against one another -- large bending efforts lasting less that half a millisecond -- there isn't an easy way to know what the shaft's influence on head motion might be. (I tried in 2009, and never got an answer I was confident in.) A proper analysis would be a computer-intensive Finite Element Analysis (FEA); I do not have access to software like that.

Fortunately, someone with that sort of resources became interested in the problem. A 2018 study (Henrikson, McNally, McPhee, "The Golf Shaft’s Influence on Clubhead-Ball Impact Dynamics") was done by pretty much the same team that did the shaft mass adjustment work discussed in the previous section. What they did was:
1. A model of the clubhead and shaft, detailed enough for FEA, was derived from direct measurement of the components. The model was calibrated by comparing its kinematics with actual motion capture of multiple swings from ten elite golfers. The match was good, except for shaft stiffness. The shaft, nominally a stiff flex, was softer in calibration than the manufacturer's spec indicated. So the rest of the study was done with the shaft at 1.5 times the manufacturer's nominal stiffness. (I'll get back to that in discussion of the results.)
2. The calibration had been done using motion capture during the downswing, especially the last 10 msec before impact. But motion capture at 723fps is not nearly fast enough to get any detail on impact itself; the entire duration of impact is only one third of a sampling interval. So impact was modeled by simulating it with with FEA software. Two models were used:
1. A free body model, just the clubhead with no shaft. Consider that a 100% "stringy" shaft.
2. A full club model, including FEA analysis of the shaft firmly attached to the clubhead.
3. The key findings (a set of graphs in figure 5 of the paper) is the difference between the free body simulation and the full club simulation. That difference is a direct measure of how close to a string the shaft is during impact.

### Results

Let's start our review of results with the ball speed graph from the free body impact model.

Here is Fig. 4b from the paper superimposed on an actual driver face, with a few things I added: a pair of blue centerlines in the center of the face, and a red circle inscribed in the locus of 255m/s ball speed. Things to note:
• The maximum ball speed of 260km/h is slightly toward the toe from the center of the face. This is pretty well known; it arises from the fact that point is traveling a bit faster than the center of the face.
• The curves of constant ball speed fall off in speed as you get further from the max-speed curve near the center. This is also well-known. As impact moves away from the center, the clubhead rotates more during impact -- meaning the face falls away from the ball more, so not as much force is applied to the ball.
• The curves of constant ball speed are not perfectly circular. If you compare the 255 curve with the red circle, it is a horizontally-aligned ellipse. It isn't too far from a circle, but it is definitely not a circle. Why? I discuss this in detail in comparing horizontal and vertical gear effect. Very simply, the moment of inertia against rotation in the horizontal plane is greater than that in the vertical plane. So the clubhead resists horizontal rotation better, and impact can be more off-center for the same ball speed.

Now let's do the same thing for the full club impact model. Now the ellipses of constant ball speed are longer, and they are slanted. Why should that be? I added a red dashed line where the major axis of the ellipse is. Note that it is almost parallel to the shaft. That tells us that the shaft is resisting rotation of the clubface where it flexes, but not much where it twists.

The red line is not perfectly parallel to the shaft. It may be a precision problem with my graphing, or it may be real. If it is real, it is probably due to the fact that the free body model didn't give us perfect circles either. There was an ellipse due to different moments of inertia around the two axes, which is probably biasing the full club ellipses. We should be able to get a better feel for the effect of shaft itself from the difference graphs -- the difference between behavior with and without the shaft. Let's look at that.

On this graph, the curves are the differences between the full club model and the free body model. When you think clearly about it, this shows the effect of the shaft and golfer. If the shaft were truly a string at impact, there would be no difference at all, no curves on the graph, just a blank sheet of graph paper. What can we conclude from looking at this graph?
1. The curves are pretty close to a set of parallel lines. The space they define is a valley whose floor is a vaue of roughly 2 meters/second; that is the difference in speed at the center of the clubface between having a shaft and not having one. The valley is long and narrow, and slanted on the face rather than straight across the face.
2. I drew a red line tangent to the curve nearest the center of the clubface. A perpendicular to that line is 57° from vertical. Note that it is extremely close to parallel to the axis of the hosel and shaft. That tells us that the shaft is offering almost no resistance to clubhead rotation when twisted about its axis -- which shaft makers specify as "torque". But, since the difference rises as you move parallel to the shaft, we must conclude that the shaft is limiting clubhead rotation with its cross-axis deformation -- which we usually refer to as "flex".
3. Sanity check time! Here are a couple of tests of gross ballpark validity of these conclusions.
• The curves of the graph are not all perfectly parallel. So I tried a near-center line (in blue) on the other side of the center. In order to stay close to the center of the clubface, I visually bisected the "2m/s" curve at the upper left of the graph. It was 60° instead of 57°. Still pretty close.
• The paper does not mention the lie angle of the driver that was used. So I looked in online catalogs for a half-dozen club manufacturers. Most of their drivers had lie angles of 58°-59°, and all fit in the range of 56°-60°. That matches the angle of the curves on the graph. We can be pretty sure that we are looking at a difference between shaft twist (pretty much a string) and shaft flex (not so much a string).
• Finally, just in case you wondered, the picture of the clubhead (which was not part of Figures 4 and 5 from the paper; I don't even know what brand and model of driver was used) is oriented perfectly horizontally. I performed a measurement that showed the lines on the face are within a half degree of horizontal.
The paper's Figure 5 has difference graphs not only for ball speed -- which we have been examining -- but also for: launch angle, azimuth (initial angular direction of the ball), side spin, and back spin.

Here are those graphs, together with a centrally measured direction of the curves. In each case, the direction is pretty close to the lie angle of a driver. That means that we can continue with our provisional judgment that the effect of the shaft is very different in flex than in torsion. Specifically:
• As we move along a direction perpendicular to the shaft (along the red line), we are moving along a single value of what the graph is about -- for instance, the same amount of side spin. We start on a curve and we follow it, or perhaps leak off it, but not far from the same value. Since this is a difference graph between with the shaft and without one, that tells me that the shaft does not do much during impact to determine the ball flight.
• As we move along a directon parallel to the shaft (perpendicular to the red line), the value changes more quickly. So the shaft does have an effect in this direction. In particular, the flexural stiffness of the shaft limits the rotation of the clubhead, even noticeable during impact. So the gear effect in this direction is reduced.
Let's compare this to my attempts a dozen years ago to analyze the shaft's ability to limit gear effect.
• I made the mistake of equating horizontal gear effect (sidespin) to shaft torque, and vertical gear effect (backspin) to shaft flex. Perhaps that is the "normal" way to think about it, but it only works for a vertical shaft -- that is, a 90° lie angle. Drivers have a lie angle of 56°-60°, so the underlying assumption is not valid. True, the majority of shaft influence is flex for backspin and torque for sidespin, but it is a small majority at best -- perhaps a bit more than 3-to-2.
• I concluded that a rather stiff shaft might have a 14% reduction of gear effect. This study says it is a lot more, and I am very prepared to believe that. But... Let's go back to the 50% adjustment of the shaft's spring constant. The increase in stiffness from the manufacturer's specs were to make the results agree with the measurement in the "calibration" phase of the study. That may well have been a proper adjustment, but 50% is a huge increase in stiffness. Consider: if you increase the S-flex to XXX-flex -- triple-X -- that would only be a 30% increase in spring constant. So there is an off chance this analysis is using an unrealistically high shaft flex. If that were the case, the influence of the shaft might not be all that large.
• In support of the idea that shaft stiffness may account for more than a 14% reduction of gear effect, noted clubfitter Dana Upshaw cited some anecdotal experience in 2005. Going to a much more flexible tip shaft helped two of his clients get lower backspin and hence more distance from their drivers. (Not a study, no controls, so completely anecdotal.) If true, this might be used in clubfitting. But Upshaw acknowledges that it only works with a very consistent swing with an upwards angle of attack and impact above the center of the clubface; other impacts will be hurt, not helped.

## Conclusions

A shaft is a string at impact! That means that, during the less than a half millisecond of impact, there is nothing your hands can do at the grip that would produce a different ball flight than if the shaft had just been a string. Any strength you want to exert on the clubhead needs to be done during the downswing; there is nothing you can do to help during impact. And shaft fitting is about letting your swing provide the best position and velocity to the clubhead as it enters impact -- and nothing about what happens during impact.

This article explored the limitations of that statement, and there are a couple of limitations:
• The statement is mostly true for a center strike. But even there, the shaft can have a small effect. It is still a string, but a string with mass. Part of the mass of the shaft tip participates in the momentum transfer of impact, as part of the clubhead. But it is only about the amount of shaft that is firmly bonded into the hosel. No big surprise there, if you have thought about it at all.
• On an off-center hit, the clubhead wants to rotate as well as simply slow down. Such rotation adds gear effect to the spin, and reduces the ball speed compared to a center impact. If the shaft were truly a string at impact, the only limit to clubhead rotation would be the moment of inertia of the clubhead itself. The shaft is indeed string-like for clubhead rotation that requires the shaft to twist around its own axis. But that part of the rotation that requires the shaft to flex along its axis is somewhat limited by the tip stiffness of the shaft. To that extent, there may be some implications for shaft fitting -- but if so, any use of it requires a golfer who already has good and very consistent impact.

## Acknowledgments

I would like to acknowledge very helpful comments from several reviewers of the first draft: Erik Henrikson, Jay Messner, Russ Ryden, and Don Johnson . Thanks for your help.