Optimizing double pendulum action using mgr/I

[babolat_rocks]

The past few days I have been reading Travlerajm's posts about viewing the hitting structure of the arm & racket as a double pendulum. I find it fascinating and quite possibly a brilliant insight. However, I am also confused. As I understand the argument, a player can tune the weighting of the racket, so that for a given swingweight, the speed of the hand & tip of the racket have the best match.

The main implication is that weight is added not at the butt end, but towards the top of the handle, around 7". This will not change the moment of inertia much for the racquet part of the double pendulum, but apparently will increase the moment of inertia for the arm pendulum, slowing down the hand relative to the racket. Is this the right understanding?

If so, then I am wondering:
(1) How much would 10g, 20g or 30g really change the moment of inertia for the arm pendulum? I would think the arm itself & hand would dominate the relationship so that the small amount of weight would not really matter.
(2) Empirical experiments show that the speed a racket, golf club or bat can be swung depends varies with 1/(I0)^n, where I0 is the moment of inertia around the butt end, and n is an exponent that varies in a narrow range between 0.25 and 0.27 in all of the experiments conducted. To maximize speed, we would want to counterbalance in the butt end, not at 7" up the handle. Here is the reference:
http://webusers.npl.illinois.edu/~a-nathan/pob/jwuk_jst_88_web.pdf

Does anybody have further insight here? Any references that backup the idea of adding weight higher on the handle to counterbalance the 2 pendulums?

 

[babolat_rocks]

Some interesting calculations

Out of curiosity, I ran some calculations to test the theory a bit. I calculated the moment of inertia of the arm+hand+handle pendulum as follows:

(1) Take the elbow as the axis of rotation.
(2) Treat the forearm as a uniform bar to calculate its moment of Inertia (Ia).
This overestimates Ia, as more of the mass is near the elbow. As you will see, this won't matter much. Ia = 1/12 * Ma * La^2, where Ma=mass of arm and La=length of the arm.
(3) Calculate the moment of inertia of arm + hand (Iah) by:
Iah = Ia + Mh(La^2), where Mh is the mass of the hand.

• The mean forearm length of humans is La = 27.84cm.
• The mean forearm mass of humans is 1.113KG with standard deviation Ma = .271KG.
• The mean hand mass of humans is Mh =.400Kg with standard deviation = .091KG.

With these numbers, we can calculate the moment of inertia of the forearm rotating around the elbow as: Ia = 71.87 Kg-cm^2 and the moment of inertia of the forearm + hand as: Iah = 381.9. notice that the hand weight completely dominates the effect of the forearm.

If we add 30g of weight in the wrist (at the end of the forearm where La=27.84cm), then the new moment of inertia of the combined forearm+hand+30g weight Iahw = 407.17 Kg-cm^2. So, the weight does affect the moment of inertia quite alot!

Now, suppose we move the 30g weight to 7" (18cm) up the handle. Using the property of a right triangle, we can calculate the Length from the elbow to the weight as the hypotenuse h = square root( La^2 + 18cm^2). The new Iahw = 411. Moving further to 25cm makes Iahw = 418.3. By increasing the weight to 50g at 18cm, we get Iahw = 430! Compare this to the original Iah=381.9.

A few observations:

• Adding weight in the butt increases the moment of inertia of the arm lever quite alot!
• Moving the weight up to 7" or even further also increases the moment of inertia further, but only by a few percent. Not sure how much difference this makes.
• The real length of the arm lever is somewhere between the elbow & shoulder. I took a rough measurement in my own case from my shoulder socket to the handle, and its about La=40cm. Using that figure, the moment of inertia Iah = 788. Realistically, I guess La is somewhere in between.
• If we take the length of the arm lever in between 40cm and 27.8cm, then the moment of inertia of the arm and the racket moment of inertia around the butt can be similar magnitudes (for my racket, the moment of inertia around the butt cap, I0 = 503.
• Its not practical to calculate the value that is optimal, because we don't have precise measurements of the swing path, hand weight, and axis of rotation. However, we can see that weight in the handle definitely affects the moment of inertia of the arm axis relative to the racket moment of inertia. Furthermore, putting weight at 7" could further enhance this effect albeit by a small amount. By playing with the amount and position of weight, you should definitely be able to feel significant differences and find an optimal value, though I would argue the amount of weight is more important than exactly where.
• Perhaps the exact placement (between say 4cm and 18cm) has more impact on the vibrational dynamics than the moment of inertia? That is just pure speculation on my part.
• Note that some players with very long swing paths might prefer less weight in the butt, because La is bigger, yielding already very large moments of inertia. Think of Federer or Nadal's extended arms with almost straight elbows vs. players who have bent elbows.
• An interesting question would be the variation of swing path lengh La itself from stroke to stroke. Since it has a quadratic impact on the moment of inertia of the arm pendulum, Having a repeatable swing path length La is probably much more important for balanced hand/arm/racket coordination than weighting. But, for top players with consistent swing path La, it can make a big difference!

Thats how I see it, but I am an electrical engineer by training and not a physicist or mechanical engineer. Hopefully it makes some sense, and would much appreciate any feedback to guide our understanding.

 

[pyrokid]

Wow, this is really interesting. I've always been curious about the mgr/I equation.

Watching this with interest.

 

[babolat_rocks]

A few more notes:

• I assumed the wrist was bent at 90 degrees (angle of the wrist is taken as the angle between the forearm and hand, bigger means straighter). 90 degrees is typical of a pro during the critical rotational phases of the swing.
• Most amateurs would have a wrist angle closer to 135 degrees or even more. This would place the mass of the added weight further away from the elbow, increasing its effect. So, de-polarizing would help more in this case!
• Some pros, like Federer, have wrist angles of less than 90 degrees!, In that case, polarizing the racket might be better, because the weight at 7" would actually be closer to the axis of rotation! You can see this angle in Federer's forehand when he is at the bottom of his stroke and begins to "PULL" the racket butt towards the ball. As he pulls forward, the wrist snaps back the other way, decreasing the angle, and effectively decreasing the moment of inertia of both pendulums.

 

[travlerajm]

A few comments:

1. I use the MgR/I formula to get my racquets close to where I want, but doing some fine-tuning while hitting against a wall is crucial. With a litttle experience, it becomes easy to tell when MgR/I is too low or too high. When MgR/I is too low, I tend to push the ball wide right of my target on the wall (on my forehand); if it's too low, I tend to pull it left.

2. Keeping track of MgR/I becomes especially handy for players with 2hb. A good tuning procedure is to first tune MgR/I for the forehand, then adjust the balance to separately tune MgR/I for the 2hb. For the 2hb, substitute R - 10cm for R, and use the axis 10cm up the handle for calculating I. I find that my optimum MgR/I for my 2hb is about 22.6. Once you find both MgR/I values for a given swingweight, you can go back to your spreadsheet and calculate the static weight and balance that meets both MgR/I constraints. Without knowing both MgR/I values, it is virtually impossible to find a weight distribution that is tuned simultaneously for both forehand and 2hb.

3. The calculations in the posts above demonstrate that the arm lever swing frequency is affected by the amount of mass in the handle. Becasue of this, the optimum MgR/I is slightly different for different swingweights. For me, my forehand optimum is about 21.0 for a SW of 360, but 21.4 for a SW of 300. In any case, tuning against the wall is necessary. The effect of handle mass on the arm lever swing frequency compounds the sensitivity of the swing to small changes in balance.

 

[Yesudeep]

For me, my forehand optimum is about 21.0 for a SW of 360, but 21.4 for a SW of 300. In any case, tuning against the wall is necessary. The effect of handle mass on the arm lever swing frequency compounds the sensitivity of the swing to small changes in balance.

Have you considered calculating Mr^2/I? I notice a swingweight-relative optimum for Mgr/I as well, but I get a constant optimum for Mr^2/I (torque/shock/elbow crunch/wrist crunch/shoulder crunch factor expressed as a ratio of the moments of inertia about the center of mass and the swing point): all the racquets that "feel great and accurate" to me,
have a constant Mr^2/I, but different balance points, MgR/I, and swingweights. I believe the magic constant you are trying to look for is your personal optimum for Mr^2/I. This link describes it as an accurate index of racquet quality. I'd go a step further and say, find your optimal Mr^2/I. Could you correlate player rankings with Mr^2/I and share your findings please?