Modeling the Golf Swing
-- January 16, 2012
with the double-pendulum model discussed by Cochran & Stobbs before
1970, mathematical models and engineering models have given us insight
into how the golf swing works. Here is a survey of important models of
the swing, and how each further refinement of the model refined what we
know about how to swing a golf club.
This is a multi-page article, each page summarizing an analytical model of
the golf swing. Each successive model
adds some detail, making it closer to what the body is doing in the
One reassuring thing is that the main findings of the
earlier, coarser models are not generally contradicted by the later,
models. The earlier lessons are refined, but not invalidated. That reinforces
what I have learned over a lifetime of engineering and research: A
well-thought-out but crude and easily analyzed model will give you an
80-90% correct answer. If you want the additional 10-20%, you will have
to refine the model considerably, and put in a lot of analysis effort.
(Think of this as yet another manifestation of the 80-20 rule: the first 20% of the effort
gives you 80% of the answer. That was Joseph
Juran's extension of the Pareto Principle.)
Below is a table of the models I will cover, together with links to the
page giving the details of what each model has taught us about the golf
swing. Before going on the the details of the models, we have a one-page executive summary of the material in the article -- along with what we mean by a "model".
& John Stobbs
|First proposal of the double-pendulum model, along
with a really good argument for doing any
kind of mathematical modeling of
|Book, "The Search for the
Perfect Swing," Chapter 2 and subsequent chapters.
|Described the double-pendulum
model in equations. Then instrumented a good golfer, and found the
coefficients so that the equations duplicated the good golfer's swing.
Finally, did sensitivity studies to determine how changing the
coefficients would change the results.
|Book, "The Physics of Golf",
& Eric Sprigings
the double pendulum to a triple pendulum, and extended it from a two-dimensional model to 3D. The left shoulder pivots
around the spine, and the left arm moves separately
from the rotation of the shoulders, giving two additional degrees of
three-dimensional forward dynamics model of the golf swing," Sports
Engineering, 2 July 2009.
Nesbit & Monika Serrano
|A completely different type of
model. A full-body model using 14 joints was computerized, and matched to the swings of four very different golfers. Then the
computer calculated, for each of the swings, the work done by the
torques at each joint, plotted against time.
and Power Analysis of the Golf Swing." Also "A Three-Dimensional Kinematic and Kinetic Study of the
Golf Swing," both in Journal of Sports Science and Medicine, Vol.4,
I will try to put things in perspective and forecast where modeling of
the golf swing may go in the next few years.
me take this opportunity to head off any flak you might be tempted to
shoot my way because of my omissions. Here are things that are
deliberately left out of this survey:
- Homer Kelley and "The Golfing Machine". It is
certainly a detailed discussion of the bones, muscles, joints, and club
during the golf swing. But it falls short of a model
in that it is all qualitative. There is nothing there that tells you
how to analyze it mathematically, nor has that analysis ever been attempted (to my knowledge, anyway).
did a limited double-pendulum analysis as an investigation of Kelley's TGM book. Aaron Zick
extended that analysis in 2005. Neither did anything beyond what
Jorgensen had already published in his book nor Max Dupilka programmed
in SwingPerfect, both in
the 1990s. So I won't spend time on it here.
of shaft bend during the swing. Both MacKenzie's model and Nesbit's model include shaft flex. Paul Braunwart at Virginia
Tech and Betzler et al at Edinburgh Napier University have also addressed shaft flex. I consider it a different
subject, and it will be dealt with separately at another time.
modified -- February 28, 2012