Modeling the Swing - Jorgensen
-- January 16, 2012
Quantification of the Double Pendulum
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
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
But all the math is there -- you just don't have to bother with it
during an in-line read of the book.
Jorgensen used was the double pendulum, as introduced
by Cochran & Stobbs. Here is Jorgensen's view of the model,
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:
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.
- The angle α
motion around Cochran & Stobbs' fixed pivot (O
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)
from the length of the club.
- The club is not just a uniform rod. (Some have
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
this time). In
fact, by moving the center of mass (the square at Mi)
its value and moment of inertia (Ii),
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.
For instance, in 2005 Aaron Zick responded to a double-pendulum
analysis by Mandrin. Zick's refinements of Mandrin's model were:
other words, Zick's contribution was already incorporated in
Jorgensen's model; it just remained to use those features of the model.
- A more realistic club than Mandrin's uniform rod. (We
already discussed how Jorgensen had that covered.)
- Instead of a single rod for the upper lever, Zick had
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.
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
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
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 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
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
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
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:
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:
If you are interested in how I use SwingPerfect for research, see my
article on right-hand
- The ability to adjust everything interesting about
- The ability to crank in shoulder torque and wrist
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
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.
In 1963, the TrueTemper shaft company decided they needed a robot
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.
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?"
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
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?"
- An increase in the shoulder torque (the strength of the
rotation that provides the power to the swing) increases the clubhead
speed. Not surprising so far. But the increase in clubhead
not proportional to the increase in torque. You have to increase the
torque by about 3% for every 1% increase in speed.
- All other things being equal, the greater the initial wrist
angle, the higher the clubhead speed at impact.
- 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
- Wrist torque ("hand action") affects clubhead speed at
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:
I have written a whole article
Jorgensen's paradox, in case
you want to look deeper into it.
- The good golfer he measured used just enough wrist
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
used no wrist torque at all! Jorgensen recalls a gem from
Jones' instructions that the club feels like it is "freewheeling
through the ball."
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.
- 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.
- Gravity provides about 8% of the clubhead speed.
- 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:
Let's move on and look at some of the newer models, and see what more
we can learn from them.
- The non-research community was uncomfortable with the
counter-intuitive results -- The Paradox -- and the researchers felt
need to respond.
- Instructors wanted to know how each part of the body should
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
complex sets of differential equations.
- A bumper crop of graduate students were available to do
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.)
modified -- January 28, 2012