Precession & Relativity

Experiment Update: The design of the new parts has been finalized down to the screws. P.H. has taken the lead to get this done. Vielen danke P.! Das ist wunderbar! Here's one small part of the new designs. I expect assembly in 2 weeks.



Motor Harness

 Prof. James Hartle makes clear in his book "Gravity: An Introduction to Einstein's Relativity" that gyroscopes experience precession in curved spacetime. The curvature maybe caused either due to the sheer presence of the mass (in which case the gyroscope is described as being in de Sitter or geodetic precession) or due to the rotation of mass. Since it would take a whole lot of mass to curve spacetime even slightly, we can safely discard de Sitter precession as having much to do with the behavior of the rel.machine prototype in experiment 4.60. The situation is less clear when we look at Lense-Thirring precession.

Just as in Lense-Thirring precession, in experiment 4.60 also, the spinning wheels' axes are shifting their Azimuth.

In just the way a lense-thirring precession effected spinning gyro orbits the spinning earth, each flywheel is orbiting the central axis of the machine in such a way that each flywheel interprets the other wheel as rotating about it. Could it be that this thereby causes each flywheel to experience Lense-Thirring style precession due to its orbiting of the other spinning flywheel.

The gravity probe B experiments have now shown that the cyrogenic precision gyroscopes in orbit around the Earth in a satellite experience about 0.037 degrees per year of precession due to this effect.

Is it possible that the spacetime curvature produced by a large slowly rotating body is comparable in some way to the spacetime curvature produced by a small fast rotating body? Could the earth serve as a good model for the former case and a spinning flywheel serve as a good model for the latter?

[Afteral, the rotation velocity of a point near the earth's surface (due to the rotation of the earth about its own axis) is 32792.16 cm/second. The velocity of a point on the flywheel's surface at around the experimental speed is around 4149 cm/second. That is to say the flywheels when they're spinning at the speeds at which experiment 4.60 was conducted, have 13% of the velocity and experience a centrifugal force about 9%, of that experienced by them as consequence of being situated on the earth's surface as opposed to free, empty space].

If so then, is it possible that the latter could cancel the former? Could it also be that the Lense-Thirring style precession also has a flip side to it, i.e, that as the gyro of the gravity probe B precesses (due to the rotation of the earth), the axes of the cryogenic gyros in the satellite also cause the combined earth-satellite frame to translate in space?

Only, since the earth is so vey massive this effect upon the earth from the gyros as they hurtle through space, is so tiny as to be not measurable by any instruments available. In the case of the rel.machine, given that the frame of the machine has about the same range of mass as the spinning wheels, is it possible then, that the frame experiences this translational momentum more strongly than the earth does due to the gravity probe B gyroscopes? Could it be that we could safely ignore such effects in the gravity probe B, but that we cannot ignore it in the case of the rel.machine? If so, could it be that we can exploit this precessive inertial motion of the flywheels to do useful work?

From this point of view, experiment 4.60 is but the tip of what can be achieved by the arrangement. The two flywheels are in Lense-Thirring style precession around one another.This is accompanied by a net angular momentum that would, if it were free to, take the flywheel system along a path radially away from the local gravity source.  Each flywheel is executing a spiral path upward. And that is why we see the flywheels coming in as they do alternately in experiment 4.60. It is the circular movement in the horizontal plane of a spiral trajectory. The vertical ascent part of the movement is experienced by the main frame frame and cage holding the flywheel because any effort on the part of the flywheels to rise will be transmitted to the main frame and cage due to their mechanical attachment along the z-axis. However, since the amount of lift induced in the arrangement is still relatively weak, we see the entire frame rising but unable to do more than move the center of mass to a slightly higher level but not taking off the ground. It seems we still need to amplify the lift considerably if we wish to see the frame lift clear off the ground.Right now, we have a very flat spiral.

de Sitter or Geodetic Precession

The following is quoted from James Hartle "Gravity: An Introduction to Einstein's General Relativity"

Begin Exceprt
First consider the behavior of a gyroscope in orbit around a nonrotating spherical body of mass M. For simplicity let's consider a circular orbit in the equitorial plane. An observer riding with the gyro will see its spin precess in the equitorial plane. In the observer's frame, where the gyro is at rest, the spin has only spatial components, its magnitude is constant, and the symmetry under reflections in the equitorial plane shows that it remains in the equitorial plane if it started in it. Thus limited, precession in the plane is all the gyro can do.
Suppose at the start of an orbit the observer orients the gyro in a direction in the equitorial plane (say in the direction of a distant star). General relativity predicts that on completion of an orbit, the gyro will generally point in a different direction making an angle delta phi (geodesic) with the starting one. That change in direction is called geodetic precession.

A gyroscope in orbit about a spherically symmetric, nonrotating body with an orbital velocity small compared to the speed of light. In this spacetime diagram, time points upward and space in horizontal. The scale of time has been made about a factor of five smaller than the scale of space to get the diagram to fit on the page. The tube is the world sheet of the surface of the body about which the world line of the gyro twists. The spin s is perpendicular to the four-velocity of the gyro u, although that relationship is not so evident with the reduced scale of time. The spin remains fixed in a local inertial frame falling with the gyro but precesses with respect to infinity because of the curvature of spacetime produced by the body. This is called geodetic precession.

End Excerpt

Now, since geodetic precession is a product of mass and we do not have significant enough mass to cause relativistic spacetime curvature, we may safely discard geodetic precession as having a major part to play in the theoretical construct we seek to explain experiment 4.60 and to also design a better prototype that might exploit the behavior we have seen to perhaps do useful work.

Lense-Thirring Precession refers to the precession experienced by the spin axis of a gyroscope in orbit (i.e. physically bound to the object whose spin is causing the test gyroscope to spin, but still having a certain degree of freedom atleast to alter its azimuth say, wrt a distant star) around a ROTATING massive object (the Earth for example). I have capitalized the word 'rotating' to emphasize that it is not the mass of the earth that directly produces Lense-Thirring precession of the spin axis of the orbiting gyro (that effect is discussed above under the topic of geodetic precession), but rather it is the rotaton of the massive object about which the test gyroscope happens to be in orbit.

General Relativity predicts that this rotation of the mass of the earth also causes a curvature of spacetime around it. That curvature is responsible for the precession of the spin axis of the test gyro in addition to the geodetic precession that the earth's mass will produce in the spin axis of the test gyroscope. (The total curvature being a sum of the two.) The following is quoted from James Hartle "Gravity: An Introduction to Einstein's General Relativity"

Begin Excerpt

Gyroscopes in the Spacetime of a Slowly Rotating Body To illustrate how the effects of rotation on the geometry of spacetime can be studied with gyroscopes, we consider the thought experiment shown schematically in figure 14.3. A laboratory carrying a gyroscope falls freely down the rotation axis of the slowly rotating Earth. Initially the spin axis of the gyro is oriented perpendicular to the rotation axis pointing in an azimuthal direction phi.

Were the Earth not rotating, the guro's spin axis would remain fixed as it falls- always point along the same azimuthal angle phi. This can be verified by solving the gyroscope equation (14.6) but it follows mor eimmediately from the symmetry of the Shwarzschild metric under phi -> -phi. The gyso could not precess without breaking this symmetry. The geodesit precession is. therefore, zero for this orbit. But the rotation of the Earth breaks this symmetry and the gyrscope precesses with time, as we now calculate. The precession of the gyro on its downward plunge is determined by the gyroscope equation (14.6) in the metric (14.22) because it is following a geodesic. We expect the rate of precessionto be small for the Earth.

End Excerpt

The following are important relavant illustrations regarding Lense-Thirring Precession.

Note: The Gravity Probe B experiment was launched shortly after this edition of the book was published. http://einstein.stanford.edu/ lists the results of the experiments. Final results of the GP-B experiment were announced at NASA HQ in Washington DC on 4 May 2011. The experimental results are in agreement with Einstein's theoretical predictions of the geodetic effect (0.28% margin of error) and the frame-dragging effect (19% margin of error).

Thus while frame dragging has been confirmed, its lower value (~37.2) as against its prediction (~.39.2) leaves open the question of what happened to the remaining energy? Could it just be that that energy was infact transformed into translational energy of the earth-gyro frame about their solar orbit?



Experiment Update:

One or more of the parts I procured for the upgrade need to be augmented or replaced. This pushes back the time table by at least 3 weeks probably longer...

On GravitoMagnetism And Compound Objects

Much of the known physics I quote in this entry is humbly borrowed from James B. Hartle's epic book Gravity: An Introduction to Einstein's Relativity. The speculations at the end on the other hand are solely mine. I clarify not so much as to declare zealous ownership but rather to spare the eminent professor of any misattribution of such speculations of mine, which prove not to be to his taste. I intend neither to mislead nor obfuscate. Yet progress for me is impossible unless I advance theory to suit observation and therefore I speculate. Sometimes I need a theory to foresee my future actions and therefore I speculate. At the root is the desire to explore a corner of rotational dynamics that still remains benighted in my brain.

I unreservedly recommend Hartle's book to those all amateur scientist-activists out there, even for those without any mathematical introduction to basic differential calculus provided that they have the intrepidity to grapple with the calculus aspects of some of the Jabberwockies they are sure to encounter in this doozie of a book. This is because often the real details, the real beauty of the concepts Hartle lays out are hidden in the differentials of first- and second- order differential equations involving angular velocity, angular momentum and angular acceleration.


Begin Excerpt from Gravity: An Introduction to Einstein's General Relativity by James B. Hartle

The geometry outside a non-rotating black hole or a star is given accurately by the spherically symmetric Schwarzschild geometry. However, no object in nature is exactly non rotating. The Sun for exmple, is rotating at the equator with a period of approximately 27 days. As a consequence of the resulting centripetal acceleration, the Sun is not exactly sperically symmetric but is slightly squashed along the rotation axis. But it is not very much out of round; an equitorial diameter is less than a part in a 100,000 longer than a diamter along the rotation axis.  The small value of that difference is why the schwarzschild geometry is an excellent approximation to the curved spacetime geometry outside the Sun.

The curved spacetimes produced by rotating bodies have a richer and more complex structure than the Schwarzschild geometry, as discussion of rotating black holes (in the next chapter, of the book Gravity by James Hartle from which this passage is being quoted) illustrates. But there is one limiting case that is accessible. This is the case of slow rotation, when the body is rotating sufficiently slowly that only deviations from the spherically symmetric Schwarzschild metric that are first order in the angular velocity or angular momentum are of significance. Since centripetal accelerations are second-order in the angular momentum, the shape of the body is not rotationally distorted to first order. It remains spherical. Why then is there a change in the exterior geometry of spacetime? The answer is that general relativity predicts that curvature is produced, not only by the distribution of mass-energy, but also by its motion. When the curvature of spacetime is small and the velocities V of the sources are also small, these effects are typically of order V/c smaller than the GM/Rcsquare effects of the mas distribution itself. This is not unlike electromagnetism, where fields are produced not only by charge distributions but also by currents. Pursuing this analogy, these (V/c)(GM/[Rcsquare]) effects are sometimes called gravitomagnetic. (...) We explore one simple example of a gravitomagnetic effect- the dragging of inertial frames by a slowly rotating body. In this chapter the dragging is small;  in the next chapter on rotating black hold it will be large.

Rotational Dragging of Intertial Frames

Consider my post of... where we noted that inertial frames of special relativity are not rotating with respect to the frame in which the distant matter in the universe is at rest. Were all the distant matter somehow to start rotating, the local inertial frames- those in which the plane of the Foucault pendulum would not precess- might be expected to rotate along with it. If only a small part of the matter in the universe is set into rotation, then the inertial frames might be dragged along slightly. General relativity predicts such rotational dragging of inertial frames.

Even the rotation of the Earth drags the inertial frames in its vicinity slightly. Dimensionally, at the surface of the Earth, the induced angular velocity of the inertial frames with respect to infinity, w, might be guessed to be related to the Earth's angular velocity omega earth by
w ~ (GMearth/[Rearth*csquare])*Omegaearth,
where Mearth and Rearth are the mass and radius of the Earth, respectively. Later in this chapter (not included in this brief excerpt- please refer to the original book by Hartle), we will confirm this estimate, which gives
w ~ .3"/yr
The inertial frames therefore rotate each year by an angle that is rougly that subtended by a football field on the Moon. Even so, at the time of writing, satellite experiments are underway to detect this small effect predicted by general relativity.

The gyroscope is a natural test body with which to observe the dragging of inertial frames because the spin of a gyro points in a fixed direction in an inertial frame. A discussion of gyroscopes in curved spacetime is therefore an appropriate place to begin the discussion of the dragging of inertial frames.

End Excerpt

Speculation: As Hartle points out, rotation produces its own curvature of spacetime. This curvature causes frame dragging. Now, theoreticians have been comfortably ignoring third order deviations, stopping with second order deviations in angular acceleration as the last distinct phenomena to study. Certainly second order deviations in angular accleration have been proven capable of producing not first order shape distortions, but rather second order shape distortions (i.e. the Sun remains more or less round, but with some deviation, or 'squashing' at the poles and some bulging at the equator.

Can it, perhaps be so that third order deviations in the prototype are capable of producing neither first order shape distortions, nor second order shape distortion, but simply third order position distortion, i.e. inertial motion? For I propose that what the prototype is doing in experiment#3 is attempting to maintain an inertial frame in the face of the disturbance imposed on the spinning wheels by the constant torque.

At present I am confronted with the dilemma of how to explain the results of the experiments I posted in my last entry. How do we explain that reversing the spin of one of the wheels cause such a radical change in the behavior of the prototype? Let me recap the results of the experiments where we impose a constant 2A torque for 30 seconds - observe the video closely, several times and you will notice the following facts about the video of the experiment:

a) When the wheels weren't spinning the wheels' z-axes remained radial w.r.t the center of the main rotation we are imposing upon the cage and the cage speeded up, theretically indefinitely but practically, the experment was suspended after about 20 seconds as the the speed was reaching unsafe levels
b) When the wheels are reinforcing each other's spins in the initial condition, there is no more indefinite acceleration. The steady state result is a tangential positioning of the wheels with the spin axis of the wheels aligned tangentially (as in along the vectorial direction of the instantaneous velocity  of any object undergoing circular motion).
c) When the wheels are not reinforcing each other's spins i.e., they are cancelling each other out, there is also no indefinite accleration, but this time the wheels perform a dance, with each wheel taking its turn to perform a self-rotation i.e. each wheel's motor platform can be seen coming inward into the center of the rotation and swinging out. The two wheels take turns and never come together into the center. They are scrupulous in this behavior which I have documented and verified several times.

Now why would they do that? Why would they execute these self-rotations, when according to Newtonian dynamics the easiest thing for them to do is simply behave as if there is no net momentum in the system, which there isn't, when you analyze the initial condition? Lets go one step further and also ask, why the second case i.e. the experiment which had the wheels' spins reinforcing each other, didn't do this? Afteral, with rotational friction force almost zero, there is no difference in the physical restrictions imposed on the configuration for these experiments.

It seems to be that if we understand the prototype as not a simple object, i.e. an object with a single center of mass. We need to introduce the brand new concept of a 'compound object' we can explain this dilemma easily. I have proposed earlier in a different post that the prototype consists not of one center of mass, but rather multiple centers of mass.

A compound object has primary and secondary centers of mass. The prototype contains one primary center of mass and two secondary centers of mass. During the third experiment, we have two secondary centers of mass (the centers of each of the rotating wheels) with net spin, and one primary center of mass, the center of mass of the overal prototype without a net spin - remember that in the initial condition, the two wheel spins are oriented opposite to one another, thereby giving the overal prototype a net zero spin.

Under such conditions, we can say that third order deviations of angular surge manifest neither as a shape change in the primary center of mass, but nor as as a shape change the secondary centers of mass, but rather as a position change ie. uniform motion of the spinning wheels i.e self-rotation. This self-rotation is identical for the two wheels, (as they are identically impressed upon by the constant torque and both wheels have identical moments of inertia as well as spin velocity) and therefore appear to take turns in their self-rotation, with each motor seeming to come into the center and swing out alternately.

Viewing the phenomena in this way allows us to see that while theoreticians have analyzed rotation and its effects on spacetime geometry accurately to first and second order, they have hitherto neglected the third order effects. It now seems that experiments are proving to us that even third order effects are significant, especially in the case of gyroscopes with reasonably fast rotors and small masses (compared to a star or a planet).
Why then does the second experiment NOT show this effect? Because, in the case of the second experiment, the net spin of the primary center of mass is not zero - witness that the entire object has a net spin (given by the sum total of the angular momenta of the two wheels in the initial condition.) This being the case, it is the primary center of mass that is to be analyzed as the one with a net angular momentum and the
two secondary centers of mass that are seen as being without spin, therefore the effects are seen as a position change of the primary center of mass, i.e. the constant gyroscopic reaction of the main frame of the machine, causing it to lift up about a constantly changing point situation on the circumference of the base of the prototype. The spinning wheels therefore simply 'fall' into a tangential position and maintain it as itis the lowest energy position.


Experimental Update: Upon carefully considering the results of my previous experiments, I have decided that there is a need to make one more modification to the the prototype. All major parts necessary have been procured. I expect to begin testing the modified prototype within one week. </div>

Experiment4.60: Response of Prototype At Constant Torque

Empirical Demonstration of the Angular Momentum Dependence of the Prototype's Behavior

Here is a short film showing clips from the crucial experiments cited in the graphs in the previous post. The short introductory text in the film explains exactly which of the experiments is shown. Corrigendum: The set of experiments labeled to be @ 8Amperes in the previous chart were actually conducted at 7 Amperes. This became clear after some investigation. I investigated it because I became suspicious of how close to the 6 Ampere graphs, the '8 Ampere' graphs looked and realized that they were infact at 7 Amperes. Nonetheless, the important thing is that they continue to confirm the angular momentum dependency that is being demonstrated by the prototype!

Trend Continues At 8 Amperes of Torque

As you can see from the graphs, the greatest amount of impedance seems to be right around the 3000 RPM mark.

Trend Confirmed At 50% Higher Torque (6 Amperes)

Intent upon confirming the results of the experiments @ 4 Amperes, I put in a mammoth day with the experiments and reran the entire set (well, almost) but this time at 50% higher power in put, i.e. @ 6 Amperes.

Here is a plot combining the last set of graphs with the new set of graphs so that we can see side-by-side, the behavior of the imbalance in the rotations in the CCW (counterclockwise) and the CW (clockwise) directions, on the left for a maximum torque of 4 Amperes and on the right for a maximum torque of 6 Amperes. As can be seen from the plot, the trend for the 6 Amperes is similar to the 4 Amperes scenario, with some quirks of its own. The most negative zone seems to be between 3000 & 3500 RPM for the 4 Ampere case. The most negative zone seems to be between 2500 & 3000 RPM for the 6 Ampere case.

Existence of Inflection Point Has Been Determined

The following graph has been plotted from the data from the previous post. (The 2000 RPM and 2500 RPM data are new and are from experiments conducted since my last post.)

The data points are generated by a simple formula:

(Number of Counterclockwise Rotations in the cycle - Number of Clockwise Rotations in the cycle)

Please note that)

a) The first cycle is a sinusoid @ Maximum of 1 Ampere.

b) The second cycle is a sinusoid @Maximum of 2 Amperes.

c) The third to tenth cycles are sinusoids @ Maximum of 4 Amperes.

I've already noted most of the trends seen here, except for one important one: As we increase the speed of the wheels, and perform the same experiment over and over again, we notice an inflection point around 3000 RPM when the system experiences the most clockwise rotations. 

On either side of this speed (3000 RPM), we see that that number of CCW rotations exceeds the number of CW rotations significantly, but while in the inflection zone the number of CW rotations exceeds the number of CCW rotations significantly! 

This inflection in a zone of speed where the CW rotations exceed CCW rotations is a sign of that the saturation point indeed exists.

In Search of a Saturation Speed

Well, a lot has changed since my last post - I found a mistake, I lost my conviction, I had a conversation, I performed more experiments and I found my conviction.

I found a mistake:
The last post I made, (July 17) has a major mistake in it. It happened because I myself did not realize it until after I made the post that I had compiled a table that compared the results of an experiment involving 10 second cycles of torques (for the 0 RPM case) with results of an experiment involving 8 second cycles of torques. When I fixed the mistake and coded the proper videos to get the right data, the chart looked more like this.

The results are much less impressive and I wasn't sure some of this wasn't just a one-off difference and that ultimately there might be nothing different at all. One good result of this mistake is that, I've decided to record more basic data on the online record, not just the aggregated data. I've also decided many more multiples of the same experiment in order to provide crosschecks for data.

One consequence of these results shown by the new prototype was what...

I lost my coviction:
I was pursuing with conviction, the idea that there will be an assymmetry between the clock-wise and the counter-clockwise rotations due the same quantity of torque. Those last three posts showed that such was not the case. At 4900 RPM, the data is showing that there is no assymmetry. After the transient crest, there is no assymetry.

When I had the conversation, I had lost my conviction that this problem was soluble in any non-trivial manner. The symmetrical movement of the wheels laid bare the truth. There was only a fast diminishing amount of lift in the system. When free, the system seemed to be within Newtonian parameters. I thought that perhaps I had seen things in the wrong way. That perhaps a gyroscopic system cannot dissipate any energy afteral.

I had a conversation:


I remembered then that Sandy Kidd had warned me that there was nothing to be had at such high speeds. He was the one who had originally said that spinning wheels can move laterally if rotated about a center located about an axis perpendicular to the spin axis of the wheel. I hadn't really paid attention, but when I built the earlier prototype, there it was. And it remained a problem until I solved it the only way possible - drilling a through-hole and putting a bolt to permanently secure the spinning wheel to the axle on which it was mounted.

So I had an email conversation - with Sandy Kidd. He's a wise old man. He's had his own tussle with this problem and he's still at it. He's been public with his own information already a long time and has personally tried to steer me away from experimenting at high wheel speeds. Something Sandy said in email stuck with me - He said "Consider a typical gyroscope system of the twin opposed gyroscope configuration being rotated at a fixed rotation speed with NO gyroscope rotation. At this point, the system is delivering the maximum angular momentum it can.

By strategically fitting strain gauges to the system and coupling to an oscilloscope or modern equivalent it will be found that angular momentum (or centrifugal force for anyone who is happier with that) diminishes as the gyroscope rotation speed is increased. This loss of angular momentum or centrifugal force begins as soon as the gyroscope starts to rotate and at a point farther up the gyroscope rotation speed range, diminishes to zero at a point I called the "Saturation Point".

No more system rotation speed or gyroscopic rotation speed will affect the system other than increase the gyroscope's upward and inward acceleration, hence saturation (and broken machines)"


I found my conviction:
I think I have a way to test this proposition. If this is infact the way that gyroscopic systems behave, then would it not be true that if I take my current prototype, at run the same experiment (say, 8 seconds per cycle, 4 ampere maximum, 10 cycles of sinusoidal torque) at different fixed speeds of the wheels.I should see that the number of rotations of the system might show interesting variations if there is in fact a phenomenon to study. I'm half way through this, at the moment at it seems a good time to summarize events and release the fresh data.

Consider the following spreadsheet image containing the raw data from 19 experiments over the last few days. I have done multiples of 3 experiments for every unique speed of the wheels to provide verifiability of the results:

Now, it seems from this data that:
1. When the arms are in (i.e., when the wheels are positioned with the motors pointing inwards at the beginning of the experiments and the system therefore having the least moment of inertia in this configuration), the number of rotations received is higher, while when the arms are out, the number of revolutions received for the same torque is lower. This is happening for the same reason that a dancer who is spinning speeds up when she pulls her arms in and slows down when she moves her arms out.

What is not clear however, is why it is that often and especially in the CCW (counterclockwise) direction, we are receiving even fewer rotations than the maximum moment of inertia condition would allow - Somehow the gyroscopes seem to be soaking up the torque, that would be the only way that would be possible. However it seems to be happening only for that particular direction of rotation too...!!

2. When the wheels are NOT spinning, the cage holding the wheels has a significantly larger number of rotations, (almost twice as many) for the first cycle of 4 Ampere worth of torque! You can see from the experimental data that I performed the 0 RPM experiment 5 times. Each time, in the end we see that the number of CW rotations is equal to the number of CCW rotations, thereby indicating to us that the transients have been ironed out of the system. However, with the wheels spinning that is not always the case. In fact the data shows that the lower the wheel speed, the more likely it is that there will be assymmetry in the CCW and CW rotations. So far.

By dropping wheel speed from 4900 RPM to 4500 RPM to 4000 RPM to 3500 RPM to 3000 RPM, we are seeing increasing trend toward assymmetry in the clockwise versus counterclockwise rotations of the cage!

3. Most intriguingly, there seems to be an increase in the 'flightiness' of the machine at lower speeds, especially for on direction of the rotation of the cage and this whatever you want to call it, jumpiness, flightiness - a tendency of the machine to seem to perform a little flightlike manoevor that can look like a mini jump- this is what is responsible for lower rotations for that direction of rotation of the cage! Could it be that it will keep increasing as we keep lowering speed?

4. In addition, even though we dropped the wheel speed drastically from ~5000 RPM to 3000 RPM, we do not see any big change in the number of rotations say, in the number of rotations of the cage during the first cycle of its 4 Ampere phase (stays at around 3.5-4 rotations of the cage during those 8 seconds)! In comparison, the zero RPM condition shows us that during that 1st cycle @ 4 Amperesthe number of rotations go up to 7-8 rotations. Not only that, we have performed dozens of experiments with wheels speeds gradually decreasing from 4900 to 3000 RPM and we do not see a trend of increasing rotations ...yet. 5000 to 3000 RPM is a dramatic drop, so its not clear at what speed of the wheels, the system will start reaching the 7-8 rotations.

In summary, it might yet be that there exists some saturation speed for the system and it might yet be that that speed lies somewhere between 0 RPM and 3000 RPM. It would be that point where those cycles would creep up from 4 to reach 7 or 8.

Thats where we're going right now......! To the Saturation Speed!

Experiment 4.8:

These results confirm the trend of the previous experiments, as do other experiments I have conducted that are not posted on the blog. The bottom line is this: Once the transient factors vanish, the system displays symmetry in its counter clockwise (CCW) and clockwise (CW) rotations.

The interesting thing about the data, however is the consistently lower number of rotations we obtain for the case where the wheels are spinning(~3)  than for the case where the wheels are NOT spinning(4.5).

Experiment 4.4

These two experiments (4.3/4.4) show us that there is a small amount of transient torque in the system that dissipates over 40-50 seconds. The last cycles of torque produce roughly the same amount of rotation in both directions.

Something else that is also interesting and not yet fully explainable is that the same amount of torque produces less rotation of the system if the wheels are spinning, than if they are not. As can be seen from experiment 4.4's data, when the wheels were not spinning, an application of the sinusoidal torque caused 4.3-4.7 revolutions and with the wheels spinning, an application of the same sinusoidal torque caused only 2.6-3.2 revolutions.This parallels what the Inductive Effect experiments I conducted in Jan 2010 revealed.

Experiment 4.3:

Tensor Model of Impedances

So what's so special about a flywheel suspended inductively? The key is that if infact gyroscopes show inductive behavior that has certain implications in tensor mathematics. This mathematics might be used to understand how the machine may achieve flight analogous to how certain electrical circuits resonate EM waves at certain frequencies or transform voltages and currents up and down etc. In other words tensor mathematics can be used to analyze the machine even as we are still building it and help us in better designing it.

We must assess the nature of impedance (inductance and capacitance are the two kinds of impedance found in nature) in a mathematical way in relation to other physical entities. Tensors offer a way of understanding impedances that sets them naturally apart from other electrical parameters. As we move forward, I have quoted so extensively from Kron in his book Tensors For Circuits that I have found it expedient to italicize his words to keep my train of thought comprehensible to myself.

Wherever in this entry there is reference to voltage, the reader should be aware that it represents also velocity in the analysis of purely mechanical or electro-mechanical systems and current to angular acceleration. See my entry on the analogy between mechanical and electrical behavioral parameters.

Further, as my entry on Grabriel Kron makes it clear, he himself saw all his analysis as being fully applicable to mechanical machines, indeed all and any machine. Thus, although we are flying through Kron's tensor model of electrical rotating machinery, we are also able to trace in the same model, information regarding our own machine. The flywheels suspended the way they are in the new prototype are the live inductors with a pulsating torque going through them at a certain frequency and the rest of the machine that is spinning under the influence of the Y-axis torque is the capacitor being charged up and their combined impedance is related to the velocity and the acceleration of the overal system as velocity = Z*acceleration (analogously to voltage, e = Z*i in the electrical scenario), where the impedance is a term with time units in the mechanical scenario. A single closed network in the electrical model can be envisioned as the mechanically invariant closed system of components capable of expending energy through torque or other methods in a different kind of machine.

Impedances are mathematically reprensentable as tensors of valence 2. The tensor of valence 2 is a collection of 2-way matrices describing a physical entity. A tensor of valence 1 like voltage e or of current i is called a vector.

Impedances thus form their own class of devices of valence 2 whose tensor laws of transformation require two transformation matrices C1 and C2.

Energy is a tensor of valence 0, i.e. a scalar. A vector of valence 1 is a vector such as a voltage or current and they require only one transformation matrix C, power or energy requires no C's. Because of this "chemical" property of a tensor of attracting a different number of C's the expression "tensor of valence n" originated.

Impedances, being of valence 2, would have transformation tensors with 2 C's, C= C1*C2.

Now, if the variables describing a specific impedance have been changed from i to i' by C1, then from i' to i" by C2 , then again from i" to i"' by C3., the successive transformations may be performed in one step with the aid of one transformation tensor C= C1. C2.C3... This important property of C is called a "group property". (Kron, Tensors for circuits, "WHy tensors")

With the help of a group transformation tensor C, one may model an inductor's behavior as follows.

The fact that power, P is a valence 0 tensor means that it does not undergo transformation and doesn't need a C to transform it from the reference frame of one machine to another. Kron uses this property of tensors to first extract information about the behavior of voltage in such a system.


The law of transformation of the voltage vector may be found from the physical fact that in going from one reference frame to another the instantaneous power input e*i (a linear form) remains unchanged or "invariant". That is P= P' or e*i = e'*i'. This relation is the physical link that connects all networks together (for any given machine)
Now let the current change from i to i' by i = C*i'
Substituting, e*C*i' = e'*i'
Cancelling i' e*C = e'
Hence e' = Ct*e
and e = Ct(inverse)*e'
It should be noted that though both e and i are vectors, they are transformed to a new reference frame in a different manner. But both being tensors for valence 1, they require C once only.

Kron then follows this up with a brilliant analysis of the impedance tensor itself.

Tensor analysis requires that if the equation of a system in one reference frame is e= Z*i, it should have the same form in every other frame. This property will give the law of transformation of Z. In the old reference frame let e = Z*i
Express i and e along the new reference frame. That is, replace i by C*i' and e by Ct(inverse)*e'.
Ct(inverse)*e' = Z*C*i'
Multiplying both sides by Ct
e' = Ct*Z*C*i'
If the following definition is introduced as the law of transformation of Z
Ct*Z*C= Z'
then the equation in the new reference frame becomes e' = Z'*i'
The equation of the new system is of the same form s that of the old system...


In summary, Kron proved that, if there is at all an impedance in the system, then an analysis of the system at any and all levels must involve an analysis of the impedance, with the invariant form of the law of impedance, e = Z*i (where e is voltage or velocity depending on the machine and similarly, i is current or angular acceleration depending on the machine).

We can see darkly from this statement, for example the answer to the questions: Why could this prototype fly? or Why does a gyroscope precess?

The answer, Kron's discoveries seem to say, is that tensor theory mandates that e = z*i is a permanent invariant law in any system involving an inductor or a capacitor (sources of impedance). So then we may further propose that
a) the purely capacitive impedance condition is addressed in traditional newtonian rotational dynamics,
b) the purely inductive impedance condition is met by gyroscopes. (This was something Eric Laithwaite partly recognized when he stated that the gyroscopes are like inductors. He also said that gyroscopes do not do any work. They rotate without being able to accomplish any energy transfer on a meaning level. It is clearer now that, that is because the system is purely inductive without any significant capacitance. The gyro rotates in a set of gimbals that isolate the flywheel's spin so that there are no levers connecting it to the external frame.)

c) the LC resonance condition is what is being targeted in the new prototype's tests.

Kron also gives certain guidelines for machine analysis. For example when coils, beams, wheels etc are connected into an engineering structure, the constrained reference axes are ignored.

Using his methods and his modeling techniques, I will attempt an analysis of the prototype in the next few entries. One of Kron's models is a good starting point for that analysis and is given below.

Experiment Update: Repairs complete. Testing begins tomorrow.

Deduction From Experiment 4.1

Proposition: Inductively suspended wheels, when suspended in symmetric pairs will (when given the freedom to do so) always prefer to orient themselves with their spin vectors anti-parallel to each other, when under the influence of a common centrifugal force about an axis perpendicular to the plane containing their angular momenta vectors.

I am going to represent this notion of antiparallel orientation of pairs of wheels with the following notation
-->
<-- This behavior is analogous to the atomic electrons which also settle in pairs, anti-parallel to each other and I am also proposing that the two phenomena have a common origin in the nature of spin itself. This proposition is based on the results of the experiment I performed on June 3, 2011 (and posted on June 5, 2011 on my blog http://relmachine.blogspot.com/2011/06/experiment-41-contrasting-prototypes.html). Here are the main points I wish to emphasize regarding the experiment of June 3, 2011:

a) The wheels do NOT show any intention whatsoever of orienting themselves in this way when they are not spinning.

b) The wheels do however voluntarily choose the anti-parallel orientation once significant angular momenta is introduced into them.

c) The wheels also seem to prefer the tangential orientation - their angular mometa vectors turn away from the center of rotation and become tangential to the circle that passes through the centers of the wheels and represents the path traced by the centers under the influence of the centrifugal force applied.


SUMMARY
So far my experiments have revealed 3 important facts:
1. Spinning wheels under the influence of a centrifugal force applied about an axis perpendicular to the plane containing the angular momenta vectors will move inwards toward the center of rotation. This I have named the Kidd Effect after Sandy Kidd, who was the first to discover it.

You can see that experiment here: http://relmachine.blogspot.com/2009/09/kidd-effect.html


2. For configurations similar to those needed to observe the Kidd Effect, the higher the centrifugal force, the lower the resulting displacement of the angular momenta vectors. This I have named the Inductive Effect (after the fact that this is analogous to the behavior of inductors).

You can see that experiment here: http://relmachine.blogspot.com/2010/01/experiment-28-inductive-effect.html


3. Inductively suspended wheels, when suspended in symmetric pairs will (when given the freedom to do so) always prefer to orient themselves with their spin vectors anti-parallel to each other, when under the influence of a common centrifugal force about an axis perpendicular to the plane containing their angular momenta vectors.

experiment 4.1: Contrasting the prototype's behavior @ different levels of angular momenta of the wheels

I have disassembled the right half. It will be 2-3 weeks before I will be able to complete the repairs and restart.

Prototype Is Ready

Testing in 1 week!

Progress

Progress

In my opinion a diligent researcher should entertain a healthy level of skepticism towards preconceptions that he/she brings to the research- I do, therefore entertain the idea that this might be only a transient minor wrinkle in my own understanding of science. Nonetheless, given my skill set and the cost-benefit analysis of the potential technological breakthrough it was my choice to meld art and science, theory and experimentation, hypothesis and application, logical roots and real world implications and run with this idea as a tech-startup. The end product being offered for sale is the resulting machine.

Right now it is to me, a project management challenge involving the engineering of the technology and the development of the theoretical map (implicit in this is the validation of the idea itself - is this idea worth anyone else's time/money?) behind the technology along with all the other things that I must do to organize a business behind this idea, from legal/marketing/finance/operations purposes. The rest (sales/H.R./I.T./infrastructure management) are moot ... for now.

New prototype assembly is in progress.

A Tiny Bit Of Heaviside

I spoke too soon. The last two of the parts for the prototype were in fact mailed by P. (Vielen Danke, P.! Das ist ein wunderbares geschenk und Ich war total ueberrascht mit deine freundliche hilfe!) but they hadn't arrived at my place. In the meantime, I've prepared by disassembling the old prototype to refit the new parts. They're finally here. Assembly clock starts now.

I'm currently reading a biography by Paul Nahin titled Heaviside: A Sage in Solitude. I recommend this book to all amateur physicists as a great introduction to the drama of 19th century science becoming the modern physics that we know now. Heaviside knew, corresponded, argued and propounded theories with a good number of those whose names now fill the textbooks of electromagnetism - Wheatstone, Maxwell, Poynting, Helmholtz, Faraday, Kirchhof, Boltzmann, J.J.Thomson, Searle, Tesla, Wiener etc etc.

Heaviside himself in his last days attempted to draw an analogy between gravity and electromagnetism (source: Nahin, Paul; Oliver Heaviside: A sage in solitude, pg 307, 1987 IEEE Press) "with the key link being the localization of energy in a field." Einstein had better luck than Heaviside in finding a formulation for gravity within 12 years of Heavisides own attempts, nonetheless, it is instructive to see what Heaviside made of the link between electromagnetism and gravitation.

Begin Quote "If one brings two charges together, energy is required to overcome the repulsion, and it is this energy that goes "into the field" (giving a positive field energy density in space). Two masses, on the other hand, attract each other and it takes energy to keep them apart, leading to the (strange) result of a negative field energy density for space in the case of gravitational models. This result, implying the presence of less than no energy in space, so bothered maxwell he gave gravity up as beyond 19th century physics. Heaviside too reached the same conclusion:"...it must be confessed that [negative energy density] is a very unintelligible and mysterious matter."

Heaviside based his analogy on an ether ("It is as incredible now as it was in Newton's time that gravitative influence can be extended without a medium..."), and reached the conclusion that gravity effects most likely propagate "immensely fast," probably much faster than the speed of light. This is all anti-relativistic, but of course Heaviside was writing this 12 years before Einstein published his Special Theory of Relativity. "

End Quote

One might wonder what less than no energy in a spatial region might be -perhaps, to have less than no energy in space might for example be energy that moves and carries mass away from the region of space, altering the local gravitational potential of the space itself.

Perhaps gravity's negative energy field is an acknowledgement that the energy is trapped in an 'inductive' field configuration and is therefore a reacting-entity that operates upon processes causing energy exchange in the local region of the inductive field.

We know that we can classify capacitances as positive impedances and inductors as negative impedances already. To associate positive and negative field energy correspondingly to the two kind of elements is only natural then. And we would see that a gravitational field is an energy field holding a spinning, inductively suspended earth as the central motor of the sun gyroscopically attempts to spin the earth out of the solar plane (in which our planet executes its orbits around the sun) and reaps the gyroscopic result that the input and output channels are orthogonalized - the spins the earth out the soalr plane and the earth responds by executing an orbit in the solar plane. Maybe we have difficulty comprehending gravity with our electricalized physics models because we have not used the concept of inductively suspended angular momentum as the equivalent of the electrical coil. Just a thought.

Another counterintuitive fact about electricity that Heaviside reasoned out regarding electromagnetism concerns a topic researched by Poynting first and slightly later, completely independently by Heaviside - the direction of flow of energy in a wire carrying an electric current. Like in the case of a gyroscope (my earlier prototype, whose results you have seen, most prominently in the Inductive Effect video) or in the case of an inductively suspended spinning wheel(as in my new prototype I will be testing soon), the direction of the energy's expressed movement and the direction of the wire (if it is assumed to be straight for the purpose of the analysis)are orthogonal to each other.

Heaviside and Poynting's research on this topic created a revolution in the field of eletromagnetism by changing the way physicists would model electromagnetism forever. Here is a passage regarding this important singular research with a suggestion of gyroscopic overtones of great importance to us as seekers of flying machines, from the book "Oliver Heaviside: A Sage in Solitude" by Paul J Nahin.

Pages 115-119

Begin Quote

Energy And Its Flux

By 1884 the principle of the conservation of energy was well established, but it hadn't been many years before when the idea of just energy, alone, was a new and strange one. The concept of force was the prominent one as late as the 1850s, for example, and it seemed to be intuitively the 'thing' that should be the hinge pin of dynamics, whether the system under consideration be mechanical or electromagnetic. The development of thermodynamics in the early and mid parts of the 19th century, however, began the process of elevating energy and changes in energy to the level of importance we attach to them today. Writing in 1887 Heaviside expressed this as, "There are only two things going, Matter and Energy. Nothing else is a thing at all; all the rest are Moonshine, considered as Things."

The ability to store what seemed to be astonishing amounts of energy in the newly perfected (1881) lead acid battery (by Camille Faure) led to a special flurry of interest in the matter, among even the general public. There was something about storing and transporting electrical energy (although a battery is really a box of chemicals) that was special, and particularly appealing to the Victorian mind. Coal was just a dirty rock out of the ground, while electricity was modern!

So, with all this interest in electrical energy, it is not surprising that people were also paying attention to its more abstract properties such as its conservation and even how it moves about. There is more to the conservation of energy, however than may be apparant at first glance. As Heaviside put it in 1891,

The principle of the continuity of energy is a special form of that of its conservation. In the ordinary understanding of the conservation principle it is the integral [total] amount of energy that is conserved, and nothing is said about its distribution or its motion. This involves continuity of existence in time, but not necessarily in space also. But if we can localize energy definitely in space [my emphasis- this is a most important idea, one we'll pursue with interest], then we are bound to ask how energy gets from place to place. If it possessed continuity in time only, it might go out of existence at one place and come into existence simultaneously at another. This is sufficient for its conservation. This view, however, does not recommend itself. The alternative is to assert continuity of existence in space also, and to enunciate the principle thus: When energy goes from place to place, it traverses the intermediate space.

And then a little later in the same passage, writing of the mathematical result that precisely specifies just how electromagnetic energy "traverses the intermediate space", he said,

This remarkable formula was first discovered and interpreted by Prof. Poynting, and independently by myself a little later. It was this discovery that brought the principle of continuity of energy into prominence.

Heaviside was referring, of course, to John Henry Poynting (1852-1914). Professor of physics at the University of Birmingham, Poynting combined his considerable ability in physics with that f a skilled mathematician and this double edge to his powers led to the writing of many papers which Oliver Lodge called "sledge-hammer communications". This certainly was the right way to describe the impact of Poynting's powerful paper "On the transfer of energy in the electromagnetic field," published by the Royal Society in its Philosophical Transactions in 1884. Starting with the Maxwellian idea of localized field energy Poynting was able to derive the elegantly simple vector expression E x H, now called the Poynting vector, for the flow of electromagnetic energy through space.

Poynting's paper, as well as some of the odd implications of the result, attracted a good deal of attention. Oliver Lodge, in particular, was tremendously impressed by it and wrote a curious paper (with a very long title!) in response. Lodge was particularly fascinated by the idea of being able to track a individual "bit of energy", writing ".. the route of the [bit of] energy maybe discussed with the same certainty that its existence [is] continuous as would be felt in discussing the route of some lost luggage which turned up at a distant station in however battered and transformed a condition." This semi-metaphysical paper seems not to have had much impact, but its opening words, describing Poynting's paper were prophetic, calling it "a paper which cannot but exert a distinct influence on all future writings treating of electric currents."

One of its most profound influences was the complete overthrowing of how people think of energy flowing in a wire carrying an electric current. In fact according to ExH the electromagnetic energy doesn't flow through the wire but into it, sideways from the fields surrounding the wire! This seemingly "crazy" conclusion was not greeted with Lodge's excitement by many of the "old-time" electricians. In 1891, for example, Silvanus Thompson and John Sprague became embroiled in a dispute over the nature of energy flow in electric circuits. Sprague held tot he old view of energy transfer through a wire, while Thompson argued for the revolutionary new viewpoint. The debate appeared over an extended period of time in the Correspondence sectin of The Electrician, and finally the journal felt it necessary to terminate the issue with an editorial: ..although we undoubtedly side with Prof. Thomson's views, there is no doubt much which appears, at first sight, highly artificial in the elaborate structure of lines of electric and magnetic force and induction, complicated still further, as it is, by the more recently discovered lines of energy-flow .. the idea that energy is located at all, and that, when it changes it position, it must move along a definite path, is quite a new one. The law of the conservation of energy implies that energy cannot disappear from one place without appearing in equal quantity somewhere else; but although this fact has long been accepted, it is only within the last few years that the idea of transference of energy has been developed, or that anyone has attempted to trace out an actual path along which energy flows when it moves from place to place. The idea of an energy current is of more recent date than the electro-magnetic theory, and is not to be found explicitly stated anywhere in Maxwell's work. I believe that the first time it was applied to electrical theory was in the pages of The Electrician, by Mr. Oliver Heaviside, to whom so much of the extension of Maxwell's theory is due. The idea as also independently developed and brought to the notice of the Royal Society in a Paper by Prof. Poynting.

In fact, The Electrician was perfectly correct in this proud claim for the priority of Heaviside. Poynting's paper certainly did not appear in print until sometime after June 19, 1884 and yet, in the June 21, 1884 issue of The Electrician Heaviside wrote (in a passage entitled "Transmission of Energy into a Conducting Core"):

The direction of maximum transference [of energy] is therefore perpendicular to the plane containing the magnetic force and the current directions, and its amount per second proportional to the product of their strengths and to the sine of the angle between their directions.

These words are not remembered today, and it wasn't until Jan 10, 1885 that Heaviside published the same result as is found in Poynting's paper (which is why historians today always write of Heaviside's discovery as dating from "the year after" Poynting's). Heaviside took a somewhat different view of history, however, and while he never disputed Poynting's credit, he also took care to remind his readers of the June 21 date, as when he wrote (in March 1885):

The transfer of energy in a conductor (isotropic) takes place not with the wire, but perpendicular thereto, as I showed in The Electrician for June 21, 1884, thus being delivered into a wire from the dielectric outside.

It is not clear when Heaviside first learned of Poynting's paper, but there is an interesting note on one of his copies of Nature (dated March 26, 1885) which was prompted by a report on a mechanical model (made of wheels and rubber bands) invented by FitzGerald, "illustrating some properties of ether." In particular, this model showed how "the energy of the medium was conveyed into" was a wire and "not along its length according with that Prof. Poynting has recently shown to be the case in all electric curents." Heaviside's note shows he was by them most familiar with Poynting's work, and thought his own more comprehensive:

But it is only true for conduction current, not for all currents. Not true in the dielectric [where the displacement current cannot be ignored]. The general formula for energy current ...[was] proved by me for conductors in the summer of 1884, and in January 1885 extended to all media non-homogenous as regards capacity, conductivity and permeability.

While Poynting may have beatenHeaviside into recognized print, and while Poynting's mathematics was impeccable, it is curious to note that his physics has a flaw which seems to have fone unnoticed, or at least uncommented upon, for the last one hundred years, except for Heaviside's own comments about it. Even with impeccable mathematics, however, many found Poynting's (and Heaviside's) ideas on energy transfer hard to believe, and not all of the skeptics were "old timers" like John Sprague. As Professor J.J. Thomson wrote two years after Poynting's paper,

This interpretation [the Poynting vector] of the expression for the variation in the energy seems open to question. In the first place it would seem impossible a priori to determine the way in which energy flows from one part of the field to another by merely differentiating a general expression for the energy in any region with respect to time, without having any knowledge of the mechanism which produces the phenomena which occur in the electromagnetic field...

These words show Professor Thomson, the bright young academic star of English physics at the time, was still committed to the Maxwellian goal of mechanical model building. But eventually even Thomson came around and in 1893 he called Poynting's result "a very important theorem" and "of great value." There was no mention of Heaviside's contributions to the energy flux theorem by Thomson, and I find this particularly ironic because this slighting by Thomson was to be his fate too, with another equally important result. And to make it doubly ironic, Heaviside was also involved in this (Heaviside's role is again forgotten,, along with Thomson's).

End Quote


Asa parting thought, let me remind you that as J. J. Thomson pointed out in 1893, the Poynting Vector product equation does indeed require a priori knowledge of the mechanism which produces the phenomena rather than being a general energy expression and such is also the fate of gyroscopic and inductively suspended angular momenta

This tells us that the true understanding of the phenomena rests on a higher level model that incorporates this a priori knowledge.

Would Kron Recognize The Idea Behind The Relativistic Machine?

When one realizes the enormity of his work, one might wonder why Gabriel Kron's name isn't more commonplace than it is. An Electrical Engineer by education and apprenticeship, he formulated an equation that applies to all types of machinery ever designed and ever will be designed in the future. He then used this tensor equation to analyze and systematically design induction motors. He is also responsible for a method called 'Diakoptics' which he used to design the automatic electrical load flow distribution system of the state of New York, the first such system anywhere in the world. However, one might wonder endlessly about a great many things that have come to pass, like for example, the strange fact that Kennelly's & Steinmetz's introduction of the concept of impedance didn't give rise to a wider scientific discussion of the nature of induction in electrical coils and a search for its mechanical analogue beyond the simple spring-and-block arrangements of the early mechanical physicists.

Throughout the later half of the 19th and the early half of the 20th century, so many important scientific discoveries were happening in so many different fields that one may be forgiven for pleading ignorance in certain esoteric areas. If the field is too vast however, one must seize aspects of the science that help us to focus on understanding the general plan of the field before plunging into the details as this leads to the common syndrome of being unable to see the forest for the trees. In this endeavor, Kron's Tensor Equation will be your everlasting friend, willing to venture with you in your searches into the most exotic designs for machines of your choice, no matter what your field or your time - always helping you design better machines by teaching you how to classify the components that form the machine. By neatly classifying the underlying components of the machine in question into purely inductive, resistive and capacitive components and using his trademark analytical method, Kron is able to set up equations and analyze everything from say, a steam turbine governing system (divided into the governor, the linkage, the pilot valve, and the turbine) to the electric speed drive/control (divided into the synchronous motor, the induction motor, and the stationary network).

Although his field of work was circumscribed by his assignments at GE (General Electric, where he worked), Kron sensed that he was playing with something much bigger than 'just' Electro-Magnetism. In reading Kron's work, I am convinced that he would approve of the relativistic machine's theory and design and see that it meshes with his own thinking. The following is an attempt to cite evidence for this supposition.

Inductive Angular Momentum And Kron's Tensor Equation

Gabriel Kron is the author of a method of analysis of rotating electrical machinery, in which one and the same tensor equation applies to ever conceivable type of machinery. - P. le corbellier in the preface to his book on the analysis of rotating electrical machinery.

IEEE Transactions On Circuit Theory, September 1968 noting the passing away of Gabriel Kron summarize that "Dr. Kron was the author of the classic paper entitled, “Non-Riemannian Dynamics of Rotating Electrical Machinery,” that became the basis of his theories covering all types of rotating machines and power systems. The pioneering work of Gabriel Kron demonstrated convincingly the superior organizational powers of the matrix-tensor notation in network theory."

All types of electrical machinery can be analyzed using the one tensor equation that Kron discovered in his intense theoretical and experimental forays. When one first come's across this promise in his work, one might wonder what kind of an equation can accomplish this mammoth task. There is however, more than just an equation involved in the analysis of a given machine. The equation is only relevant if you first break the machine down into components according to the rules that Kron gives, and in addition you set up the tensors for the various components. Kron found that he could reduce any conceivable machine (even a completely mechanical one)into a network of components which could be solved for the behavior of the specific machine using a single tensor equation. Kron himself went on to make clear that to him, "It is surprising how few ultimate types of elements there are that form the building blocks of the great variety of engineering structures. Most stationary networks consist of a collection of one-dimensional "coils" only; all rotating machines consist only of a collection of two dimensional "windings." The great variety of structures differ only by the manner of interconnections of these ultimate coils and windings, and the variety of theories differ only by the type of hypothetical reference frame assumed. It is only the study of the ultimate building blocks that requires analytical work. The interconnection of these units into a given system is a routine procedure."

The main tensor equation itself is analogous to the statement for Kirchhoff's Voltage Law in a circuit with tensors replacing vectors. As Banesh Hoffman notes in his paper (Kron's Non-Riemannian Dynamics, Hoffman, Banesh, Reviews of Modern Physics, July 1949) "The work of Gabriel Kron constitutes a significant enlargement of the domain of application of tensor analysis." Hoffman also wrote in the introduction to the book Tensors for circuits (authored by Kron) that he (Kron) uses tensors to unify great classes of physical systems. "With him, a tensor transformation changes, the equations of one electrical machine to those of another eletrical machine of a different type. He constructs (the equations of) a prototype machine -the primitive machines - from whose equations he obtains those of all other electrical machines by applying appropriate tensor transformations."

It turns out that Kirchhof's 2 circuit laws - the voltage law and the current law - can help us accomplish an unbelievable amount of modeling of the physical world. Nor is their usefulness restricted to electrical circuits. Kirchhoff's circuit laws laid the foundations also for the field of Topology. (Topology is highly relevant to General Relativity).

The Voltage Law turns out to also be at the heart of analyzing rotating electrical machinery (infact, all machinery ever made or to be made) and it requires a tensorial statement and methods of analysis. This is not a coincidence. Kron himself comments that in his research,he had found that it is interesting that Kirchhof laid the foundations of topology even while working simply with circuits. "It is not a coincidence but a consequence of some hitherto hidden relation between the properties of space and those of electricity that the science of electrical engineering and that of topology meet again on a common ground when both are viewed from an invariant point of view."

It was Kirchhof's laws that laid the foundations for Topology. Topology had hitherto used tensors and now Kron was using a tensorial version of Kirchhof's law to analyze rotating electrical machinery. Tensors used in topology were now also in the very heart of the analysis of rotating electrical networks;


It is worth our while to ponder what we are to make of the fact that nature has designed the phenomenon of electricity such that one equation can serve to model all manner of electrical machinery? Is it something innate to the phenomenon of electricity? Or is the breathtakingly sophisticated mathematical edifice describing all electromagnetic phenomena with the aid of just a few parameters such as impedance (inductive and capacitive), voltage and current and rate of change of current something that derives its seeming perfection from other, even more fundamental?

Kron himself is clear on this subject. He commented that "Although the method of reasoning will be employed [by him in his book Tensors for Circuits] only for stationary and rotating electrical networks, exactly the same reasoning applies also to mechanical and other physical systems. That is, all reasonings and all symbolic formulas to be studied are independent of electrical engineering. The electrical applications are only illustrations."

Kron emphasized that mechanical systems behave in ways that are identical to electrical systems and that they can therefore be studied analogously to electrical systems, using infact the same methods and tensor equation he discovered for (all )electrical machinery.

Impedance is one of that small set of basic building blocks making up the wide variety of electrical structures. It represents the opposition from the medium and is mathematically represented as a combination of three different kinds kinds of opposition possible - purely resistive, pure capacitive and purely inductive.

The one dimensional coils and two dimensional windings mentioned by Kron are in fact inductances. Now, while the purely resistive and purely capacitive opposition of the medium are known and extensively used in both electrical and mechanical machinery, the third kind of opposition possible, the inductance makes a prominent appearance hitherto only in electrical machinery - there it is everywhere. But there has not been the recognition that the inductive suspension of angular momentum is the mechanical equivalent of the one-dimensional coils in electrical machinery. The incorporation of this new type of opposition into mechanical machinery will bring the possibility of designing a new class of mechatronic machines that will revolutionize manufacturing systems in the way induction coils transformed electrical engineering into the modern electronics engineering. Putting aside the more arcane aspects of Gabriel Kron's work, looking directly at the main equation governing electrical machinery are presented by Kron in his book, we cannot but wonder what the shape such mechanical systems might take.


A passage from Kron's book Tensors for Circuits is reproduced below to illustrate not only Kron's mastery of the subject of the analysis of machines but also an inkling of how electromagnetic devices and mechanical devices are different manifestations of the same fundamental geometric and physical phenomenon. (Click on the images to open a larger version)

[Experimental Update: All parts are ready. Assembly starts now. Prototype unveiling in 3 wks. Testing in 4.]