# Vibration to Torque: Magic Motor

In the this video the person cuts 6 grooves on a wooden pencil and mounts a paper propeller using a pin on the soft eraser side.

As he starts rubbing the groves vigorously, the paper propeller starts spinning clockwise.

The mysterious thing is this: torque is generated on the paper from just translational motion (the rubbing of the grooves)! Where does this torque come from?! My current hypothesis is that the pencil traces a small circle if you hold it with your hand while rubbing the grooves, but I've disproved this theory when I mounted a replicated device on a clamp and performed the rubbing - the rotations still happens! I've ran computer simulations on this and have yet to determine the origins of the torque, which means the torque generation lies in non-ideality. An ideal system will dissipate the vibrational energy perfectly, generating no torque.

How can sound waves in a wooden media generate any torque as it leaves through a pivoted rigid body (the paper propeller)?

Can anyone shed some insights to a theory of how this motor works, and provide quantative/numerical solutions to this problem?

A numerical solution that quantizes all relevant parameters would be extremely helpful to explain this mysterious phenomena!

P.S. IMPT: Please provide numerical solutions to back your theory up! Please do not just provide qualitative solutions!

My current sentiments: It has something to do with acoustics which may tell us the efficiency of energy transfer to propeller and an non-ideality which generates torque, however the latter mechanism of origin of the torque remains shrouded in mystery.

• Your request for a numerical solution sounds like a few weeks research project. I suggest you change it to a request for a sketch of how someone might go about setting up an analysis: what kind of effects one would look at Aug 11 '16 at 1:29
• I do think this is an excellent question but make the following comment: at a very top down level, the torque is not mysterious. Firstly, the system is cantilevered so that there is no problem with conservation of AM. Secondly, you're looking at the interplay of very large forces (reaction force from cantilever, stroking of the pencil by an animal that weighs probably $10^5$ times the little propellor) acting on a very light body. The slightest asymmetry in their application will result in torques that can move such tiny things. ..... Aug 11 '16 at 1:34
• .... My guess is that the vibration sets up cyclic circular motion of the "fulcrum", "pivot"? (whatever one calls the tiny shaft that the propellor spins on). This in turn will set up cyclic reaction forces between propellor and shaft and attendant friction that is directed predominantly in the direction preferred by the circular motion. Aug 11 '16 at 1:37

I expect that the issue is quite similar to the one that I described in my answer about a vibrating wire - namely, that there is some anisotropy in the pencil, which means that it has a different resonant frequency in the direction parallel to the grain and the direction perpendicular to the grain.

I could not find published numbers on the bending strength of wood parallel and perpendicular to the grain, but since the compressibility changes, and wood that is bent will compress, it stands to reason there is a difference.

This means that your pencil, in response to the impact of the vibrations, will start to vibrate. The vibration will be at a different frequency in the parallel and perpendicular direction - and this will give rise to rotation of the tip. If the motion is sufficiently damped (which it should be, being held in a hand), then the vibrations will never get so far out of step that the tip would start rotating in the other direction.

For the detailed math, see my other answer.

It occurs to me that the presence of the grooves will affect the response frequency of the pencil as well - but in a direction that is aligned with the stimulus. I can imagine that the unevenness of the grooves may be sufficient to explain things - but I like the "anisotropy of wood" idea better.

If this is correct, then it would lead to a couple of experimental predictions:

1) if you use an isotropic rod (like a thin plastic rod) you may find it harder to get rotation
2) if you put the notches in a face of the pencil that puts the vibration along a principal axis of the grain (either across, or along), it may be more difficult to excite rotation

If you have the chance to do the experiment, you may be able to confirm or refute the hypothesis...

• So the non-ideality lies in the material itself. However, in my experiments, the motor only spins clockwise. (Clockwise is mysterious! It doesn't ever go anti-clockwise, no matter where you start rubbing or how many grooves you cut.) And if on one axis the vibrational motion is greater due to different resonant frequency, won't there still be 0 net torque as the forces are equal in magnitude in the opposite directions? Besides, the vibration is at the pivot, so any torque generated won't be too far from the pivot, which is also why in my simulations the propeller does not spin at all. Aug 10 '16 at 1:38
• P.S. Did you make the animation using Mathematica in your answer to the vibrating wire question? Aug 10 '16 at 1:38
• @Lagrangian - Right now I'm not at the computer I used to make that animation - but pretty sure I would have used Python (with an outside chance it was Matlab). Aug 10 '16 at 2:30
• @Lagrangian I used Python for the animation. Aug 10 '16 at 12:16
• Regarding anisotropy (as seen in the vibrating wire) I think you might be getting carried away with a pet theory, as I was with the Wilberforce Pendulum. There is plenty of anisotropy here in the method of excitation and the asymmetric pinning/hanging of the propeller without any need to invoke anisotropy in the wood. Aug 11 '16 at 11:28

The "fixing" where he is holding the pencil in his other hand is actually very flexible. I would guess the result is that the end of the pencil is actually moving in a circle or an ellipse, without any significant bending of the pencil itself. Since the paper is not exactly symmetrical about the pin, the elliptical motion of the pin will make the paper rotate.

Wood is indeed anisotropic, but from the video the stiffness of the wood itself is very large compared with the (lack of) stiffness in the hand and arm holding the pencil. In any case, the pencil with notches cut in it is no longer a beam with a symmetrical cross section, and the neutral axis of the notched pencil is not at the geometrical center of the un-notched pencil.

I don't think this experiment would work at all if the pencil was rigidly clamped, instead of being held in the experimenter's hand.

• I clamped it firmly onto a table and put a spinning DC motor at one of the grooves to quantify the generated vibrational waves. the propeller still spun clockwise. Aug 10 '16 at 15:57

This mechanical toy is described in wikipedia where it is called Gee-Haw Whammy Diddle, also on the Harvard Maths website where it is called The Hui (or Hooey) Machine. Elsewhere it is called The Magic Windmill.

A better demonstration is given in this video which also explains construction and operation.

A list of references is given on The Flying Circus of Physics website, and another with links (with limited access) in Which research papers are referred to for the toy mentioned in the Arvind Gupta's TED Talk video?.

In operation this toy is very similar to twirling a Hula Hoop with a stick, or exciting a pendulum by oscillating the point of suspension. I think this has nothing to do with sound waves. As alephzero points out, the pencil itself is relatively rigid, as are most online versions. The pin in the rubber is not rigid, and will amplify the vibrations of the pencil even when the latter is clamped.

Rubbing a horizontal stick over the notches causes the pin to oscillate vertically. Rubbing the side of the notches causes a horizontal oscillation. The two oscillations combined cause the pin to follow an ellipse. Contact forces (including friction) between pin and propeller generate torque which rotates the propeller.

The crucial feature is that the propellor must not be pinned at its CM - which can be avoided by making the hole in the propellor at least twice the diameter of the pin. The larger the distance of the CM from the pin, the larger the amplitude of vibrations of the pin needed to rotate the propeller.

As with the hoola hoop, starting rotation requires a relatively large amplitude circular motion, but a small amplitude linear oscillation is enough to keep it going. Demonstrators explain that rubbing the side of the notched stick with a finger helps to start the propeller turning, and that changing which side of the stick is rubbed causes the propeller to change direction.

The Harvard Maths webpage makes the observations that :
* the geometry of the device, the frequency and vigor of the excitation are all unimportant in the operation of the machine; and
* the position of the damping finger and its distance from the point of excitation are important in starting or changing the direction of motion - perhaps because this controls the phase difference between vertical and horizontal oscillations of the pin.

The 2 parts of the toy can be analyzed/modelled separately.

The wikipedia article (more so the Schlichting-Backhaus article on the Harvard Maths webpage) indicates how the spacing of the notches and the placing of the finger lead to an elliptical driving force. This part of the model is trivial, I think, except that the connection still needs to be made between the orientation of the ellipse and the preferred direction of rotation of the propeller, and what part the rubbing finger plays in this.

How to Twirl a Hula Hoop can be adapted to model the excitation of the propeller. The waist corresponds to the pin or nail, and the hoop corresponds to the hole in the propeller. (See diagram after eqn 3.)

Alternatively, the propeller can be modelled as a rigid pendulum attached to an oscillating support.