# Matching the double rod pendulum

## Main Question or Discussion Point

I'd like to understand how the double rod pendulum works and how it changes when a point mass is added to one of the rods. Especially I'd like to understand whether it's possible to match the two rods with the point mass so that the movement is stable under sinusoidal applied angular acceleration.

The problem set is this: Assume two rods of masses $M_1$/$M_2$ and lengths $L_1$/$L_2$ attached together by a frictionless hinge. The mass is distributed evenly on the two rods. The motion is restricted to 2D by putting the rods on a frictionless table. The other end of the first rod is attached to a perfectly secured lossless attachment point. The question is: If one then applies a sinusoidal angular acceleration $\alpha(t) = \alpha_0 \sin(\omega t)$ at the attachment point to the first rod, what is the condition under which the second rod tends to follow the first rod? By following I mean that the rods either keep aligned on the same line or the rods both move periodically with the second rod only moving over a larger range of angles, but without flipping? Can the point mass be used on either or both of the rods to match the rods to the applied acceleration in this regard? If yes, where should the point mass(es) be added? And what is the logic how the double pendulum changes when the point mass is added to: 1) tip 2) tail 3) center of the first or the second rod?

How does the situation change when the applied angular acceleration is not sinusoidal, but a short burst of acceleration? Can the rods be forced to follow each other in this case?

My own intuition says that the moments of inertia and the centers of masses of the two rods are key parameters here, but I cannot understand what is the logic behind the movement, i.e what part of the movement slows down when mass is added to any of the three points on the two rods.

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One cannot apply (angular) acceleration. One can apply force (torque). Did you want to say that the first rod is forced to oscillate with the specified acceleration?

One cannot apply (angular) acceleration. One can apply force (torque). Did you want to say that the first rod is forced to oscillate with the specified acceleration?
Yeah, that's what I meant. I thought it would be easier to analyze the problem in this way. With an applied periodic torque, one should know the effective rotational inertia at all time instants in order to find the acceleration from torque.

This problem set is related to real life. In many sports, e.g. golf, tennis, baseball, there's a requirement to swing a double pendulum of arm/racquet. I'd like to understand how this double pendulum changes when a point mass is added. And the condition for the rods following each other is related to a swing that does not require any torque applied from the wrist.

I do not think that the double pendulum, as stated, is a good model for the sports. The coupling between the "pendula" in the sport setting, which is the fingers, is not a frictionless pivot. It is a rather complex and "powered" joint. Of all the parts in the human body, the hand (fingers + wrist) has got the largest motor cortex in the brain: http://en.wikipedia.org/wiki/File:Human_motor_map.jpg

The mechanical problem as stated is probably best attacked using the Lagrangian approach.

Yeah, human body is more complicated than that. But the objective here would be to mimimize the stress on those tiny wrist muscles with the rod matching. When the racquet follows the arm without active force from the wrist, most of the work can be done with larger muscles. I'd like to experiment this rod matching by adding mass to a tennis racquet, but I don't fully understand what'd happen if I add mass to 1) handle, 2) center or 3) tip of the racquet.

Just thinking, but maybe if I add mass to the tip of the racquet, it'll make the racquet lag behind the arm due a lot of added moment of inertia to the racquet? And maybe if I add mass to the handle, it'll slow down the arm relatively due to wrist holding the extra mass? But what happens when I'd add mass in the center of the racquet? Any theoretical ideas to analyze the problem? Ideally I'd like to write a Matlab/Octave code to analyze this, but what are the equations?

The heavier the racket is (or the greater its moment of inertia is), the more stress the hand will experience when accelerating the racket. That is unavoidable. By adding mass to the racket anywhere, everything else being equal, you increase its mass and moment of inertia.

Things become (a lot) more complex when you add to the system hand + racket an object that is hit by the racket.

Sure, accelerating a heavier object requires more work as a WHOLE. But placing weight in the handle only increases the moment of inertia of the double pendulum relative to shoulder. As the weight is in the handle it does not increase the moment of inertia of the racquet relative to wrist. Shouldn't this lessen the lag of the racquet relative to arm, due to shoulder joint feeling more rotational inertia, but the racquet having the same rotational inertia?

Adding mass to the handle does not increase the moment of inertia with regard to the wrist, but why do you want to add mass anywhere to begin with? If you mean redistributing, then you can reduce the moment of inertia w.r.t. the wrist, but, again, if you can remove the mass from the non-handle parts of the racket, why not just lose that mass altogether instead of moving it to the handle?

I just came across this, while doing a search for topics on my other passion - tennis. EMguy, you may be interested in this thread from a tennis forum I post on. Link

The OP has posted many experimental customized rackets and string setups, inspired by studies on tennis from Howard Brody, Rod Cross and Crawford Lindsey. The OP's Mgr/I theory backs up the EMguy, that adding mass to the tip of the racquet will make the racquet lag behind the arm and adding mass 7" up the handle will do the opposite.

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Namaste,

I'm intrigued by the question as well. For anybody taking a jab
tennis:

According to http://www.racquetresearch.com/ [Broken]

Mr2 / I

appears to be a good indicator of racquet quality. It is a multiplier
in the http://www.racquetresearch.com/formulas.htm [Broken] for shock, elbow crunch, wrist crunch, and shoulder crunch.
Minimizing this while keeping a higher mass in the handle than the hoop of
a racquet (what tennis players refer to as a head-light balance) appears
to produce racquets that help reduce the shock on the joints in the arm.
Reducing the mass of a racquet would take the "punch" (momentum) out of your strokes.
The goal behind having a high moment of inertia is punchier shots (momentum transfer)
while reducing shock; greater "plow-through" in tennis terms.

2. A close observation of the Federer forehand, for example,
shows that the racquet is pulled linearly until the last few milliseconds
before contact, so angular acceleration/torque may only affect the wrist
during contact and the wrist relaxation phase. (Most tennis coaches
emphasize "locked" wrists, by the way, i.e. until contact the "double pendulum"
of the arm and the racquet is "locked" at a constant angle and the lower
pendulum is only "released" at or after contact.) May be something to think

A question arises in my mind. Does Mgr/I (degree of polarization
of a tennis racquet), therefore, explain how comfortable the wrist feels
during the forward swing of a tennis stroke? Players like Federer and Nadal
who use racquets with higher polarization tend to have wristier strokes in
their arsenal too, which could explain their ability to hit more swerving shots
than players using the more traditional depolarized racquets.

Cheers,
Yesudeep.

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