The Double-Pendulum Model and the Right Arm

Dave Tutelman  --  November 30, 2010

Many instructors and some golfers criticize the double-pendulum model of the golf swing as inadequate. The most frequent complaint is that it fails to reflect the role of the right arm (in the right-handed swing). Interestingly, physicists and engineers seldom offer this criticism, because they know how to incorporate the effect of the right arm. Here's what they know that the instructors don't. This article covers what the model is and what it isn't, what it can tell you and what it can't.

The Model

The double-pendulum model is the basis for every physical analysis and simulation of the golf swing that I have seen. It holds that the moving parts of the swing are an inner member (green) representing the arms and shoulders, and an outer member (black) representing the club.

The green member rotates around a hinge at the top of the spine, and carries a hinge itself (the wrists), to which is fastened the black member. Both hinges may be driven by torques; most analyses call these the shoulder torque and the wrist torque. These torques can be positive (counterclockwise, causing the swing to evolve in the usual direction), negative (clockwise, retarding the evolution of the swing), or even zero. In fact, the swing many physicists refer to as the "standard swing" uses a zero wrist torque for most of the downswing. (For a refresher course on torque, click here.)

It is probably important to mention that the model -- the double pendulum -- is the picture on the left. It is two rigid "sticks" connected by hinges, which may be powered at the hinges. The picture on the right is redrawn so a golfer will recognize it more easily. But don't attribute any anatomical detail to the way it is drawn. Ultimately:
  • The tan body (legs, hips, torso, and head) are really just a stand that can exert a torque on a hinge pin. Other than that, the entire body is rigid.
  • The green triangle (shoulders and arms) are really just a rod hinged at each end. The body can exert torque on the rod at one end, and the rod can exert torque (or not) on the club at the other end.
So it is probably safer to think about the model than the nice drawing that reminds you what's what. You don't want to get too attached to the idea that you can see arms or legs in the drawing; they are not part of the model.

What we're torquing about

Here's another critical place where you might not understand the model. Are you sure you know what we mean by "wrist torque"?

We better make sure. Too many golfers, even instructors, hear the word "torque" and immediately assume we are talking about twisting the shaft around its axis. That is not necessarily what an engineer or physicist means by torque. To someone who's familiar with torque (which is covered in my physics tutorial), it is a turning force. The important points are:
  • The force causes the object it acts on to turn, rather than just move.
  • The magnitude of the torque is the force times the distance between the action of the force and the axis of the turning.
The picture shows what we really mean by wrist torque, when we use the double pendulum model. It is not a force that tries to twist the shaft about its axis, but rather a force from the hands that tries to turn the club down to the ball -- a force that releases the wrist cock.
There are certainly many variations using this model. For instance, the "standard swing" applies a positive wrist torque early in the downswing, for the purpose of keeping the wrist cock angle from falling into the middle of the swing. This torque is usually represented by a "stop" that holds the wrist cock angle; but it is really a torque that peaks very early and drops to zero about a tenth of a second into the downswing. By that time, centrifugal force on the club is more than enough to keep the wrist cock angle from shrinking; in fact, the club wants to fly out -- to release. So no further wrist torque is needed for the stop.

This model is simple enough to set up differential equations that can be simulated pretty easily on a computer. In fact, it doesn't have to be a workstation of the class used to design clubheads using Finite Element methods. Any PC can run Max Dupilka's SwingPerfect program at visually instantaneous speed.

But is the model useful? Does it model enough detail of the swing so the results of the model are not just an academic exercise? I think so. Most physicists and engineers feel the double pendulum represents reality well enough to:
  • Evaluate changes in club construction -- at least those coarse enough to be characterized as mass positioned on a rigid body. These include club length, the masses of the shaft, clubhead, and grip, and the shaft balance point.
  • Evaluate some advice on how to swing, or even to train for golf; i.e.- how much clubhead speed comes from increased shoulder torque or wrist torque.
In the final analysis, we have to remember that it is a model. We don't expect it to be reality, but it has to reflect reality at some level. The level may not be useful for instruction -- often not even close. But the model is useful to evaluate some kinds of swing changes, to find their effect on clubhead speed and loft at impact. What kind of swing changes? Those that can ultimately be described in terms of shoulder torque and wrist torque.

Many instructors feel the model is inadequate to analyze their favorite detail of the swing. In some cases they are correct. But the most commonly cited shortcoming is the folding and unfolding of the right arm -- and the model has a very effective way to deal with that. Let's see how we can deal with complex right arm action in this very simple model.

The Right Arm

Most instructors feel that the folding/unfolding of the right arm (or pointing/unpointing, as one put it recently) is an important part of the swing that the double pendulum fails to model. Yes, it is an important part of the swing. But the double pendulum is quite capable of modeling the important thing it does -- add forces to the grip with the right hand.

One instructor actually insisted that I change my analysis to the model in this picture, changing the two hinges of the double pendulum model to six powered hinges. Honestly!

Fortunately, the real effect of the right arm produces something the model was designed to handle: wrist torque. Let's look at this in more detail.

The right arm and wrist torque

The point of the analysis is to compute the progress of the swing, in terms of club position and speed. So any additional complexity due to the right arm boils down to what the right hand does to the club. In three dimensions, it can do six things to the club:
  1. Force in the swing plane -- the red arrow in the picture.
  2. Torque in the swing plane -- the circular blue arrow in the picture.
  3. Force perpendicular to the swing plane. (If we could show it, it would be towards or away from the viewer of the picture.)
  4. Torque perpendicular to the swing plane.
  5. Axial force -- force along the axis of the shaft.
  6. Axial torque -- rotation of the club around the axis of the shaft.
All of these forces can theoretically be exerted by the right hand on the grip of the club.

But let's remember what a double pendulum model is used for. It computes the progress of the swing, in the swing plane. It is a two-dimensional analysis.
  • Motion of the club (or golfer) perpendicular to the swing plane is not interesting to the analysis, nor is axial rotation of the club. Both are very interesting to the direction the ball will travel, but you'll never see a double-pendulum analysis that deals with clubface direction, except for wrist cupping or bowing that adds or subtracts loft at impact. Most issues of clubface direction and clubhead path require a totally different approach to analysis.
  • As for axial force, it turns out to be the way shoulder torque is conveyed to the club. It has already been accounted for in the shoulder torque.
So, from the entire list above, only #1 and #2 (in-plane force and torque) are relevant to a double pendulum model's job. It is easy to see how the in-plane torque is accounted for; it is simply part of the wrist torque. But we still don't see where a double pendulum is able to account for the force.

This picture shows that the model can also handle the in-plane force. Any force exerted by the right arm in-plane is a push or a pull on the grip. (The picture shows a push.) That force works against the pull or push of the left arm. In other words, the left hand acts as a fulcrum or pivot, and the right hand's force tries to turn the club around this pivot.

But what is such a turning force? It is a torque. And how big is it? It is the size of the force, times the distance between the force and the fulcrum. So any action of the right arm can be factored into a double pendulum model as wrist torque, because either it is a torque (the blue arrow above) or it can be computed as one (the red arrow above).

Here's a bonus observation:
  • If your favorite swing uses this sort of positive wrist torque to cause clubhead release, for either control or power, then a ten-finger grip (or baseball grip) is appropriate. That's because you increase the distance between force and fulcrum, thus increasing the wrist torque.
  • If, on the other hand, you believe the "standard swing" model is right for you (with no wrist torque), then an overlap grip is preferable. The less distance between the hands, the less opportunity for unwanted hand action to produce wrist torque. In fact, some good players (Jim Furyk comes to mind) use a double-overlap grip, which further reduces the torque-multiplying separation between force and fulcrum.
But the picture above is not the only way the arms can work to produce wrist torque. The opposite sense is also feasible. Suppose the right arm is pulling and the left is pushing. That creates a "negative torque", a torque that tends to prevent the club from releasing. It will encourage holding and perhaps even increasing the clubhead lag.

This is not mere speculation. Kelvin Miyahira has been looking at lag from a biomechanics viewpoint rather than the physics approach I am comfortable with. By studying videos of golfers (including a lot of Tour players), he has identified a number of "micro-moves" that encourage the retention of clubhead lag. Two of the important micro-moves are left arm extension and a right elbow tucked down in front of the body through almost the entire downswing. That is exactly the diagram we show here:
  • The extended left arm is a pushing force at the butt of the grip.
  • Tucking the right elbow prevents it from extending, "shortening" the right arm. Do this assertively and not just passively, and the right hand is pulling the grip.
The net result is a negative torque, one that prevents the release of the lag until inertial forces are irresistible in releasing the clubhead. Kelvin's observation is consistent with physics theory in this regard.

The right arm and shoulder torque

For a long time, I didn't think the right arm could affect shoulder torque. But some recent analysis suggests that it can. No, it doesn't actually affect the shoulder torque itself. But you can model certain right arm actions as a change in shoulder torque. And, since the double pendulum is not reality but rather a model useful for analysis, that is as relevant as anything could possibly be.

The diagram at the right is a re-drawing of the double-pendulum model. The important thing here is that the length of the inner member of the pendulum (representing the arms) defines a circular arc of radius R, and the hands move along that arc as if they were on a track. They are driven around that track by the shoulder torque. Expressed more accurately...
They are driven around the track by a force due to the shoulder torque, shown in the diagram to the left. Since the path is a perfect circle, the magnitude of the force is
Force = ShoulderTorque / R
and the force acts exactly tangent to the arc.

Anybody familiar with physical modeling can see immediately that both pictures of the model -- the double pendulum and a circular track for the hands -- gives an identical result for any analysis. So, even though it looks different and the equations you get might look different, the answers will come out the same. Anyplace you could use a double pendulum analysis, you could use a curved-track analysis.

Why is this even interesting? Because one of the criticism's leveled against the double pendulum model that the path of the hands is not circular. Due to the folding of the right arm, the radius may be shorter at the top of the backswing. (May be. But not "must be". It depends upon which muscles transfer the shoulder torque to the hands, which is an issue of swing keys and technique.)

Does this invalidate the model? Probably not. For years, it has given realistic, accurate results when modeling the swings of real golfers. Cochran & Stobbs noticed this as early as 1968. Jorgensen quantified it in the mid-1990s. Other researchers have been similarly successful using it to model the real golf swing.

But sometimes it is necessary to take into account the folding of the right arm. When that happens, the circular track becomes a useful modeling tool, more useful than the original pendulum representation. If the path does not vary too radically from a circular curve, you can use the diagram below to represent the hands as a carriage moving along a track.

In the diagram:
  • The dotted line shows the direction of the center of rotation (the shoulder pivot). The center is a radius r from the hands. (The radius is lower-case this time, to indicate that it varies over the course of the swing.)
  • The black dashed line is the actual path of the hands. The important thing to notice here is that it is not a circle around the shoulder pivot; if it were, then it would be perpendicular to the dotted-line radius. So there is some angle between the actual path and an ideal circular path.
  • Shoulder torque produces a force along the ideal circular path; that is what torque does. The force is shown in blue, and is of a magnitude equal to the torque divided by r.
  • The force can be resolved into components, shown in aqua. One component (the bold one) is parallel to the path of the hands, and so accelerates the hands along their path. The other (the very pale one) does nothing to aid or hinder the progress of the hands, and is rather small as well; we can ignore it for most analysis purposes.

How is this diagram useful for analysis purposes? It tells us that we can get a rather good approximation of reality by using the double pendulum model and tweaking the shoulder torque profile (that is, shoulder torque vs time) to provide the actual accelerating force that the hands see.

How can we use this to analyze non-circular swings? From photographs or slow-motion measurements of the swing, plot the path of the hands as it varies during the downswing. The key piece of information needed is r(t) , the distance from the path to the center of rotation, as it varies over time. Using r(t), calculate the shoulder torque as a function of time that would give the same accelerating force Fa if the model were a conventional double pendulum of fixed radius R. Then just run the double-pendulum model using the newly calculated shoulder torque, and you will get a behavior that mirrors the non-circular swing, especially in the vicinity of impact.

Actually, that is an oversimplification. It would work if all the mass of the arms, hands, and club were accelerating linearly, so we could apply simple F=ma. Since it is rotating, we have to calculate the varying moment of inertia of the arms, hands, and club as r(t) varies. That sounds complicated, but it isn't bad at all in practice. The diagram changes to the one at the left, stressing moment of inertia instead of forces. If you are interested in more detail of the altered model, including an example of its use, you can get it in my article on the right-side swing.
But there is a constraint on where you can use this modification of the model. The "strobe" diagram on the right is adapted from the SwingPerfect computer program. The circular path of the hands is clearly apparent as the collection of green and red dots, representing the hands at each "snapshot". I have modified SwingPerfect's diagram to color-code the dots: green while the initial wrist-cock angle is still intact, and red once the club swings out and releases the wrist cock. As long as the wrist cock angle is not changing (green dots), our modeling is quite good. But, once centrifugal force starts to release the clubhead (red dots), accuracy depends on the hands being on the circular path. There are a few reasons:
  • The double-pendulum model reflects that most of the clubhead speed is due to the release of the wrist cock transferring energy from the hands and arms to the clubhead. How much energy is transferred depends on the curvature of the path of the hands during the release. If we change the path of the hands, we will get a different clubhead speed. So the path of the hands once the wrist angle starts increaing must be the circular path being calculated by the model.
  • During release (once the wrist angle starts increasing), tension in club's shaft is exerting a force on the hands that slows them down. Notice that means the force is one that opposes the shoulder torque, which is accelerating the hands. But opposing shoulder torque is what moment of inertia does. So, since we are varying the model by playing with moment of inertia, our formula for MOI would need to get a lot more complicated during release in order to reflect this force. Better we get to the circular path before release, so we can use the proper double-pendulum model and a relatively simple formula for moment of inertia.

So we have a condition for the altered model to work: By the time the wrist cock reduces significantly from its original angle, the hands must have reached the circular path assumed by the model. That is true for many interesting variants of the swing. In fact, the so-called "standard swing", where there is no wrist torque applied except to keep the club from falling in to the center of the swing, has essentially no change in wrist cock angle for about 60% of the downswing time.

So What Is Double Pendulum NOT Good For?

It would be a bit much to assume that as simple a model as the double pendulum is the way to answer all possible questions about the swing. And indeed it isn't. While it is a very good predictor of clubhead speed and loft at impact, there are some things it can't begin to handle. Here are a few examples of things it cannot tell you:
  • As noted above, any out-of-plane motion of the club.
  • Left-right face angle at any point in the swing, including impact.
  • Clubhead path, or swing plane.
  • How any action below the shoulders affects the shoulder torque. For example, how about the "X-Factor", the angle between hips and shoulders during the downswing. That has to be analyzed separately. The double pendulum model uses shoulder torque as an input to the analysis, not an output from it.
  • Similarly for questions like whether stack-and-tilt is a good idea, or whether a constant frontal spine angle is a good or bad thing. Both of those are inputs to the double pendulum model, not outputs. If you can calculate their effect on shoulder and wrist torque -- which the model doesn't tell you how to do -- then the double-pendulum model can tell you how those torques translate into clubhead speed and effective loft.
  • It could conceivably calculate in-plane shaft bend, though most existing simulation programs don't produce the information needed to do this calculation.
Bottom line: The double pendulum is an effective model for what goes on:
  • In the swing plane
  • Between the neck/shoulders and the clubhead
  • Including the right arm.
But it does not deal with issues like out-of-plane motion, clubface direction, or how the body and legs contribute to the swing.

Last modified -- October 24, 2011