Modeling the Golf Swing

Dave Tutelman  --  January 16, 2012

Starting 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 swing.

One reassuring thing is that the main findings of the earlier, coarser models are not generally contradicted by the later, finer 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".

 Researcher Time Frame The Model Reference Alastair Cochran & John Stobbs 1968 First proposal of the double-pendulum model, along with a really good argument for doing any kind of mathematical modeling of the swing. Book, "The Search for the Perfect Swing," Chapter 2 and subsequent chapters. Theodore Jorgensen 1994 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", Chapters 1-5. Sasho MacKenzie & Eric Sprigings 2009 Extended 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 freedom. Article, "A three-dimensional forward dynamics model of the golf swing," Sports Engineering, 2 July 2009. Steven Nesbit & Monika Serrano 2005 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. Article, "Work 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, No.4. 2012 and beyond Finally, I will try to put things in perspective and forecast where modeling of the golf swing may go in the next few years.

Let 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).
• Mandrin 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.
• Models 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.