Modeling the Swing - Jorgensen

Dave Tutelman  --  January 16, 2012

Quantification of the Double Pendulum

Theodore Jorgensen was a professor of physics at the University of Nebraska. Analysis of the physics of golf was a lifetime passion for him. In 1994, he published the first edition of his book, "The Physics of Golf". I have a copy of the second edition (1998), which is basically the same material but an easier read, with much of the math reserved for the appendices. But all the math is there -- you just don't have to bother with it during an in-line read of the book.

The swing model Jorgensen used was the double pendulum, as introduced by Cochran & Stobbs. Here is Jorgensen's view of the model, labeled to show the quantities used in the equations. Don't be scared by the Greek letters or the "differentiate with respect to time" symbols (those little dots over some of the Greek letters). The important things to get from this diagram have to do with how much of the real world is included in the model:
  • The angle α is angular motion around Cochran & Stobbs' fixed pivot (O in the diagram). Think of it as the current value of the shoulder turn.
  • The angle β is the current value of the wrist hinge.
  • The length of the arms (R) is separate from the length of the club.
  • The club is not just a uniform rod. (Some have accused the double pendulum model of being limited to a uniform rod for the club.) The fact that the club's total mass is centered at a point on the club (the square labeled Mj) and the club's moment of inertia is explicitly identified (Ij) means that we can configure the club however realistically we like. Heavy head and light shaft. Light head and heavy shaft. Even a uniform rod, if that's what you'd like to ask about.
  • Likewise for the upper lever, representing the arms (Mi and Ii this time). In fact, by moving the center of mass (the square at Mi) and changing its value and moment of inertia (Ii), you could incorporate the entire rotation of the torso into the model.
  • One additional effect missing from this diagram (but present in Figure 2.3) is the horizontal acceleration of the "fixed" pivot (point "O" in the diagram). It moves to the right in the picture (the golfer would see it as a shift to his left), correponding to the shift of the left shoulder as the torso rotates.
I am spending time on this, not because I expect you to analyze the model in this detail, but because you should understand how rich the model can be, even as simple as it is. There have been criticisms that "the model doesn't take this or that into account". Sometimes the criticism is correct -- but sometimes it is easily incorporated in the model by playing with the parameters I mention above, or adjusting the shoulder torque or wrist torque.

For instance, in 2005 Aaron Zick responded to a double-pendulum analysis by Mandrin. Zick's refinements of Mandrin's model were:
  • A more realistic club than Mandrin's uniform rod. (We have already discussed how Jorgensen had that covered.)
  • Instead of a single rod for the upper lever, Zick had a triangle comprising full-size shoulders and separate extended arms. This is easily taken care of in Jorgensen's model by adjusting the center of mass and moment of inertia for the upper lever.
In other words, Zick's contribution was already incorporated in Jorgensen's model; it just remained to use those features of the model.
Jorgensen validated the model by matching it to a professional golfer's swing. He put reflective tape on critical points of the golfer and the club, then took a strobe sequence of the golfer's swing. The points of application of the tape included the clubhead (two there), several points on the shaft, the grip, the elbow, the shoulders, and the golfer's head.

The dots show the positions of these features at rather close intervals in the swing. Those time intervals were known, and were precisely identical over the whole swing. So, by measuring the distance between dots and knowing the interval between flashes, it is easy to calculate the velocity of any taped point at any time in the swing.

Jorgensen plotted all the relevant velocities during the downswing. Then he turned to the double-pendulum model. He tweaked the parameters of the model until it matched very closely the measured values. In particular, he got a very good match to the clubhead speed, for the entire speed curve during the downswing.

The agreement between model and real golfer should tell us that the model is valid, at least as far as we can tell. If more measurements, better measurements, or other swings do not fit the model, then that casts doubt on the model's validity. But remember that we want to validate the model for good swings, swings that result in effective shots. If the swings that do not fit the model are duffers with high handicaps, it is not useful to model their swings. Better to clue them in on the model they should be looking to emulate.

BTW, emulating the model is the approach of at least one instructor. Paul Wilson (whom we shall meet below) teaches his students by first showing them a mechanical model of a double-pendulum golfer. Then he picks out the important characterists of a good double-pendulum swing, and has the students emulate that.

Mechanizing the model

When Jorgensen was exploring the model in the 1990s, a computer on the desktop had become a pretty common thing. He was able to "mechanize" the model with a program that would do the computation. So he was in a much better position than Cochran & Stobbs had been 20 years earlier. He could plug in "what if?" values and see what the model told him would happen. We will review below the lessons he drew from the model.

But Jorgensen was not the only party that got busy mechanizing the double pendulum model. There were both software and hardware implementations of the model. Here's one of each:

Software - SwingPerfect Program
Serious swing model researchers wrote their own computer programs to exercise their model. It was inevitable that some of those programs would be sufficiently "well polished" to be offered as products. (What is surprising to me is that there have not been more of them.) The program I use is SwingPerfect, written by Max Dupilka. The image is a screenshot of the program. The program's features include:
  • The ability to adjust everything interesting about the golf club.
  • The ability to crank in shoulder torque and wrist torque, not just a constant for the whole swing, but a variable profile over the downswing. (A four-segment profile for the shoulders and a ten-segment profile for the wrists.)
  • Graphs of almost everything in the model, including all the accelerations and velocities.
  • Setting the time interval to as little as a half millisecond (for numerical studies) or larger amounts (for visualization; the image at right is set for 5 milliseconds).
  • An optional lateral movement of the fixed pivot. This gets a bit of the movement of the left shoulder into the model -- not with true accuracy, but with remarkably true effect.
If you are interested in how I use SwingPerfect for research, see my article on right-hand hit.
Hardware - Iron Byron
In 1963, the TrueTemper shaft company decided they needed a robot to test shafts. The objective was a machine with a perfectly repeatable swing, so differences between shaft prototypes could be measured using the same swing. They got George Manning and his team, of the Battelle Institute, to design a swing robot dubbed "Iron Byron" (after Byron Nelson, whose swing was notoriously repeatable). Many copies of Iron Byron were made, for R&D testing in the golf club and golf ball industry, and even for the USGA for conformance testing and research.

Iron Byron was designed directly from the double pendulum model of the golf swing. Over decades, it has proven its value, which is certainly a vote in favor of the value of the double pendulum model.

Paul Wilson is a golf instructor who uses Iron Byron as a teaching model, not just a testing device. In this video, he explains why the double-pendulum-based machine is a good enough model of the swing for a real golfer to copy (even though history has it the other way around; the robot's designers were trying to copy a human golf swing). The explanation is covered in the first three minutes of the video; it is an excellent description of why the superficial differences between robot and golfer are not important. The last portion of the video is an interview with George Manning, Iron Byron's inventor.

Lessons from the Double Pendulum Model

The best thing about having a mathematical model is that you can do "what if?" experiments with it.

Do you know what a "what if?" experiment is? Think of one of the most productive uses of spreadsheets. Once you have a spreadsheet set up to give you your answer -- whatever the subject matter -- you can just change a value or two and see what happens to the output. That spreadsheet is a mathematical model for something, and you can tweak variables and see what happens. Tweaking the input to the model and seeing what happens is the essence of a "what if?" experiment.

Jorgensen and others have done "what if"s, and have taught us something about the swing. (Well, about the model anyway. It is about the swing only to the extent that the model is valid, at least for that feature of the swing.) Below we'll list some of the conclusions that Jorgensen drew from the model. They are from Chapter 4, entitled "Variation of Parameters Brings New Understanding of the Golf Swing" -- the essence of a mathematical "what if?"
  1. An increase in the shoulder torque (the strength of the body rotation that provides the power to the swing) increases the clubhead speed. Not surprising so far. But the increase in clubhead speed  is not proportional to the increase in torque. You have to increase the torque by about 3% for every 1% increase in speed.
  2. All other things being equal, the greater the initial wrist cock angle, the higher the clubhead speed at impact.
  3. Reducing the amount of backswing (the body turn at the transition) leaves the clubhead speed almost the same as before. Moreover, it tends not to allow the wrists to over-release to a cupped position, but instead encourages a solid-hitting position with the hands leading the clubhead at impact. Another way of saying this: Overswinging leads to a bad impact position, with very little gain of clubhead speed.
  4. Wrist torque ("hand action") affects clubhead speed at impact in a very surprising way. So much so, in fact, that Jorgensen refers to it as "The Paradox". Here is the essence of what he found:
    1. The good golfer he measured used just enough wrist torque just long enough to maintain the initial wrist cock angle until inertial forces started throwing the club outward. That typically takes .1-.15 seconds. After that, the golfer used no wrist torque at all! Jorgensen recalls a gem from Bobby Jones' instructions that the club feels like it is "freewheeling through the ball."
    2. So the paradox: any wrist torque during the downswing that aids release will result in a lower clubhead speed at impact. Oh, it will indeed increase the clubhead speed through most of the downswing. But you don't care about that; you want the maximum clubhead speed you can get at impact. And using hand action to release the clubhead works against that aim.
    3. In fact you can increase the clubhead speed at impact by using a hindering hand action. This is paradoxical, counterintuitive -- but the model says it is true. And I know at least one instructor who gets very good results teaching a hand action that tries to hold the wrist cock right through impact -- a swing key that creates a hindering torque.
    I have written a whole article devoted to Jorgensen's paradox, in case you want to look deeper into it.
  5. Gravity provides about 8% of the clubhead speed.
  6. The forward shift provides almost 9% of the clubhead speed.


Let me close this section by emphasizing that the Double Pendulum Model, in spite of its simplicity, served the golf research community well for over 30 years. It is still often quoted as gospel by those who understand it, and it provides the underlying theory for all golf equipment robot testing.

It was after 2000 before the community felt the need to refine the model (i.e.- complicate it, with the aim of emulating a human golfer more closely). It is my distinct impression that the more complex models were developed not because researchers were unhappy with the double pendulum, but rather because:
  • The non-research community was uncomfortable with the counter-intuitive results -- The Paradox -- and the researchers felt the need to respond.
  • Instructors wanted to know how each part of the body should be contributing to the swing. The double-pendulum model tells us a lot about the arms and hands, but all it tells about the rest of the body is that the job is to produce rotation, shoulder torque. It doesn't tell us how to do that, muscle by muscle.
  • The computational tools were now common enough to run much more complex sets of differential equations.
  • A bumper crop of graduate students were available to do research, and needed topics for their dissertations. (The last may be surprising but seems realistic. I remember my own grad school days, and my own and my colleagues' search for thesis topics. Also, I have looked at where a lot of the new models are coming from.)
Let's move on and look at some of the newer models, and see what more we can learn from them.

Last modified -- January 28, 2012