Mechanical Build (Contd.)

 

Mechanical Build is Complete

Limit Cycle Build Step 1: Inner Gimbal

Limit Cycle Design

Conceptual Design Of Limit Cycle Inertial Transporter Device

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?v=X1eDLSEMMyI

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).

Start Up Behavior:

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.

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ఓం శాంతి శాంతి శాంతిః !

The Offering is made to the all-manifest Supreme Being
The Altar of the Offering is the manifestation of the Supreme Being
The Offering too is the manifestation of the Supreme Being
The Fruits of the Offering thus Made,
Are also verily the Supreme Being's manifestation.

May the Supreme Being bring us all to peace among all the peoples, peace among all the creatures of the Supreme Being's creation and peace too in the Heaven administered by the Supreme Being!

Finally, Some Proof!

Dear Friend,

As Gabriel Kron wrote in his work on Generalized Machine Theory, there are similarities between the ElectroMagnetic(EM)/Mech. and other sciences and these similarities are not a coincidence nor do they mean that the concepts are fundamental to these sciences. In fact there are concepts that appear in all the sciences and are fundemental to all aspects of matter and fields of all sorts. One such concept is the idea of organizing any functioning system into inductive, capacitive and resistive elements. (For more on this, I refer you to my blog post: http://relmachine.blogspot.com/2011/02/would-kron-recognize-in-concepts-behind.html)

While the idea of an inductor may have been most clearly defined in EM systems, I propose that it is a fundamental property of Spacetime (as Einstein defined it). I propose that the gyroscope is an inductor-like device, with an impedance too. It can be shown that according to Einstein's Special Theory of Relativity, an object travelling at near-light-speeds traverses a shorter path in Spacetime than one at rest. I propose that in fact Mother Nature is the Ultimate Karma accountant and so even for speeds not approaching light speed this theory is true. I propose that a spinning wheel is the most elegant proof of this. The rim of the wheel has a larger linear speed than the center (which, technically is a singularity with zero linear speed) and that difference operates by Relativistic principles to produce a phase-difference in the energy distribution across the spinning wheel, and that is the source of the gyroscope's peculiar behavior. I further propose that Magnetism and all inductive phenomena have their roots in this specific theory. (For proof of the relativistic path difference between objects travelling at different speeds and more information on my thoughts on the subject you can refer to my blog post:
http://relmachine.blogspot.com/2009/05/spin-angular-momentum-and-worldlines.html
and also
http://relmachine.blogspot.com/2011/09/on-gravitomagnetism-and-compound.html)

Professor P. and I propose that the exact behavior of the gyroscope can be modeled as a nonlinear oscillator of the Van Der Pol type, while the fundamental modes can be analysed using linear state space techniques. We propose that the gyroscope really has two modes of oscillation only one of which is being exploited commercially today.
(For proof on the analysis of the modes of oscillation of a gyroscope, please refer to my blog posts: http://relmachine.blogspot.com/2015/04/analysis-of-analogous-networks-page-1.html
and also
http://relmachine.blogspot.com/2015/07/complete-characterization-of-gyroscopic.html )


I propose that the second mode is harder to invoke, but when activated, it is equivalent to the second mode of the electrical inductor, which is invoked when the inductor is coupled to a capacitor and resonated with a forcing function of the appropriate frequency. (Please note that although traditionally, Electrical engineers do not see the inductor as having two modes, I propose that its behavior w.r.t. DC current is a manifestation of its first mode and its behavior w.r.t. to an AC current is its second mode.

I propose that when commercially developed, the gyroscopic oscillator (G.O.) is capable of producing motion of a hitherto unseen kind - Large velocities without acceleration (i.e. inertial motion, unlike today's conventional wisdom which holds that an engine that produces velocity will concommitantly produce acceleration). This new kind of G.O. engine also has other peculiar properties dervied by analogy to EM inductors, such as ability to minimize friction, ability to behave in ways that violate Newton's Laws but obey the Lorentz Transform (i.e that it is Relativistic). Such engines which have much higher energy efficiency and also the ability to avoid collisions through precessive reaction to all collision-causing encounters.

Here is the summary of my proof:

1. Picture & Initial Video: You might want to first take a clear look at the picture of the basic prototype I am testing now. It is imaginatively titled "prototype august 2015".

First example: showing you that for a 'DC' torque, i.e. a torque that is not varying in its direction or amplitude, a smooth precession is produced.

2. "Sinusoidal Torque Right Frequency" video: Here I apply several sets of 6 cycles of torques each. The torque waves are analogous to AC current in that they vary sinusoidally, increasing from 0 to a maximum to zero and then reversing their direction for the rest of the cycle. This happens 6 times per cycle.

I can tell you that the friction on the lazy susan at the base of the rims has variable friction (what with me being a very delicate operator and all, and having bent it a bit over the months), so there is variable damping which muddles the picture....

But watch the video: It is 20 seconds of beauty - The beast is caught at the smoothest portion of the lazy susan, with even damping, and you can see that the gyro executes continuous circles! I can also tell you that continuous circles are only possible for this specific time period of the torques and will happen for any maximum amplitude torque at this specific frequency (within reasonable limites).

3. "Sinusoidal Torque - Wrong Frequency" video:  Any other frequency of the torque will fail to consistently produce full circles and will instead produce behavior similar to that shown in this video. I can prove to you that the power input to the gyro is the exact same as that of the previous video. The gyro just doesn't like the frequency. That is exactly how an inductor behaves - Right frequency - Open Sesame! Wrong frequency - Sorry, Wrong Answer.

I challenge anyone to prove me wrong by building a model that can produce smooth rotations of the gyro under arbitrary-frequency "AC torque" conditions! It is not possible! For DC torque, yes, you can produce smooth rotations of any frequency by adjusting the magnitude of the torque. But NOT AC torque.

I have found other peculiar things about gyroscopic behavior all of which are documented on my blog (http://relmachine.blogspot.com), but this has probably already taken too much of your time.

Thank You
Ravi

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