The Double-Pendulum Model and the Right Arm
-- November 30, 2010
instructors and some golfers criticize the double-pendulum model of the
golf swing as inadequate. The most frequent complaint is that
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
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 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
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
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:
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.
- 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.
Here's another critical place where you might not understand the model.
you sure you know what we mean by "wrist torque"?
What we're torquing about
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:
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.
- 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.
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
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:
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
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.
- 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.
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
The Right Arm
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
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
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:
All of these forces can theoretically be exerted by the right hand on
the grip of the club.
- Force in the swing plane -- the red arrow in the
- Torque in the swing plane -- the circular blue arrow
in the picture.
- Force perpendicular to the swing plane. (If we could
show it, it would be towards or away from the viewer of the picture.)
- Torque perpendicular to the swing plane.
- Axial force -- force along the axis of the shaft.
- Axial torque -- rotation of the club around the axis
of the shaft.
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.
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
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,
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).
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.
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.
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
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
- 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 right arm and
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
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.
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.
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
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
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.
can we use this to analyze non-circular swings? From
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
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.
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
- 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
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
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
line: The double pendulum is an effective model for what
- As noted above, any out-of-plane motion of the club.
- Left-right face angle at any point in the swing, including
- Clubhead path, or swing plane.
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.
could conceivably calculate in-plane shaft bend, though most existing
simulation programs don't produce the information needed to do this
But it does not deal with issues like out-of-plane motion, clubface
direction, or how the body and legs contribute to the swing.
- In the swing plane
- Between the neck/shoulders and the clubhead
the right arm.
Last modified -- October 24, 2011