## Monday, September 26, 2016

## Monday, September 19, 2016

### Limit Cycle Design

Behavior of a Toy Gyroscope as Executing a Limit Cycle

07/25/2016

**Dear Dr. P.,**

I need your help to model the behavior of a toy gyro as a Van Der Pol-like limit cycle. Even in this nascent form, I believe the model is applicable to the gyroscope. I believe we might be able to write a paper together, outlining a new understanding of gyro behavior. Please hear me out:

**Video Link**

You can see the video somebody put up, of a simple toy gyro being spun up, released and allowed to slowly spin down to rest, at this linkhttps://www.youtube.com/watch?

**Description of Behavior**

For the sake of simplicity, I will call the spin axis of the gyro, its z-axis. Further, lets call the most acute angle the Z-axis forms with the ground to be z1 and the angle the gyro axis forms with the ground when it is mostly upright to be z2 (which is almost 90 degrees during the part of its play when the spin velocity is large).

In the video, initially the gyro is spun up and placed at a very acute angle with respect to the ground [Time Index in Video: 0 minutes 5 seconds to 0 minutes 30 seconds]. The bottom point of the gyro axis executes increasingly smaller ellipses/circles on the ground, and as the curves tighten, the gyro becomes more upright.

Slow-Down Behavior:

Conversely, when a gyro starts to slow down [Time Index in Video: 1 minute 9 seconds to 1 minute 21 seconds], its spin axis which was previously stable, erect and perpendicular to the xy-plane, starts to teeter. Simultaneously, the top point of its z-axis starts to execute increasingly larger ellipses/circles on the x-y plane, as the gyro starts to slow down.

**Theoretical Proposition**

What if we consider a toy gyroscope's angle to the earth to be the dynamic variable being modeled by the Van de Pol equation?

This angle is important because it determines the amplitude of the torque being applied to it, with

i) the torque being maximum if the angle is close to zero, provided the frame of the gyro or the wheel doesn't hit the ground)

ii) the torque being minimum if the angle is close to 90, because the weight vector of the gyro is then passsing through its axis, causing the torque due to gravity to be nearly zero.

We can conclude that, if the Van der Pol eqn. were applicable, due to its non-conservative nature and nonlinear damping:

a) When the angle is small the damping is negative, i.e. energy is generated.

b) When the angle is large, the damping is positive, i.e. energy is dissipated.

That is exactly what I observe with the toy gyro. Please allow me to explain.

Start-Up Behavior:

1) When the gyro is released at a very acute angle to the ground, it 'self-rights', itself; That represents an increase in potential energy, as it means the center of mass of the gyro has risen from rSin(z1) to r.

Its potential energy has increased by mg[r- rsin(z1)]

, where r is the length from the bottom of the gyro axis, which is on the ground, to the center of the wheel, where the center of mass of the gyro lies.

Slow-Down Behavior:

2) Part (i): when the gyro is upright, it is constantly losing energy, which is usually explained away as due to bearing friction. But the truth I believe is that no matter if we had magnetic bearings, the energy loss would continue, because its determined not only by any friction losses but also dominantly by the positive damping associated with the large amplitude of the angle of its spin axis with respect to the ground.

Part (ii) Further, you can note that, when the gyro has really slowed down, it loses its ability to stay straight, and yet, as it starts to teeter, and the angle again falls, instead of falling straight to the ground, there is a small period of time, when it traces these increasing ellipses. i.e., there is an interplay betweeen energy dissipation and energy generation, and it tries to keep up a dynamic stability game that it eventually loses and only then can it come down.

Part (iii) Lastly, I would like to say that from my experience playing with toy gyros (which is a lot, actually haha! True that!) at those ending moments, when the gyro is spinning slowly and its axis is playing this plosing game between positive and negative damping, the spin of the gyro is so slow, that if I were to actually take a non-spinning gyro and invest it with that exact rate of spin, I would observe no precessive effect whatsoever. The thing would fall to the ground, like a stone.

Sincerey

R.

Experimental Progress

I am beginning the next design cycle. Here is the design as it evolves. The concept has already been demonstrated in the previous set of experiments. This round is conceptually superfluous, but I hope of interest to investors seeking to fully grasp at least a minimal application of the technology proven in the last round.

In view of the fact that this round is aimed at investors, I will keep the technical descriptions brief and instead speak to the business investment opportunity offered. Given that the applications are myriad, I would like to emphasize that potential investors should focus less on the IP rights and more upon the sectors they are interested in.

*R.G.*హరి ఓం తత్ సత్

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