Leecommotion, the Right-Side Swing

Non-Circular Model: Upper Bound

This note in green tells you how to calculate the torque profile for a given non-circular swing. It depends only on high school math and college freshman physics. But if you're not comfortable with algebra and never expect to run the computational model yourself, you won't miss anything if you skip the note entirely.

Let's start with notation:

• T_o  = The actual shoulder torque that the golfer is capable of exerting for a golf swing.
• T(t) = The torque profile used as input to the model, as it varies over time.
• R_o  = The circular radius in the latter (release) portion of the swing. If the swing is not circular at this point, the model is inaccurate.
• r(t) = The distance from the path to the center of rotation, as it varies over time. Obviously, it becomes R_o late in the downswing.
• I_o  = The moment of inertia of the arms, hands, and club when fully extended and still with the initial wrist cock. Basically, the moment of inertia as it would have been for the early phases of a normal, circular-path downswing.
• I(t) = The moment of inertia of the arms, hands, and club, when folded to the lever arm of r(t), also as a function of time.
• α(t) = Angular acceleration of the arms, hands, and club, as a function of time.
• M  = The mass of the arms, hands, and club.
• K  = An arbitrary constant used in computing the moment of inertia. Its value depends on the shape of the body. For most shapes, the formula takes the form I=KMr² as a quick visit to any table of moments of inertia will attest. Luckily for us, we don't need to compute its value, because it will cancel out of our calculations.

The goal is to compute T(t), the torque profile to apply to a standard double-pendulum model.

We depend heavily on the elementary piece of physics, T=Iα, the rotational analogy of F=Ma. But we will see it in the form:

α  =

T I

(1)

First of all, the angular acceleration over time will be given by the equation (1), where the torque is what the body produces T_o and the moment of inertia is the actual moment of inertia due to the changed lever arm I(t). In other words:

α(t)  =

To I(t)

(2a)

Our goal is to compute a new torque function T(t) that, when applied to the simple circular double pendulum model, gives the same angular acceleration over time. So:

α(t)  =

T(t) Io

(2b)

Since we are constraining both cases to the same acceleration profile α(t), we can equate (2a) and (2b) to get:

α(t)  =

To I(t)

=

T(t) Io

T(t)   =   To

Io I(t)

(3)

Now we need to see how the moment of inertia varies during the early part of the downswing. The only thing that is varying, fortunately, is the radius from the center of the torque to the hands. If the club were flying out from the center, releasing the wrist cock angle, we would have major complications here. But, since it is not, we can compute the moment of inertia to a reasonable approximation by the simple, well-known:


Plugging this into the moments of inertia in equation (3):

T(t)   =   To

K M Ro2 K M r2(t)

T(t)   =   To

Ro2 r2(t)

(4)

Equation (4) is our answer, and it is really simple. The details of the mass and shape of the arms, hands, and club have canceled out; we don't have to worry about them. The formula may not be trivial to use, because we have to find r(t) for the actual swing. That may require a frame by frame analysis of video of the swing. But at least we know what we need from the video, and how to use it when we have it.

As a "sanity test" of equation (4), we would expect T(t) to revert to T_o in the latter phases of the downswing. And it will, as long as r(t) reaches R_o before the wrist cock angle releases much. And, if you remember, that was one of the constraints we knew up front we would need to impose on the model.