Wednesday, July 29, 2009

The Good Professor Vs The Jabberwock

That last paragraph in the previous post bears repetition and reexamination: The back reaction of a particle's own field on itself is necessary to explain the friction on charged particles when they emit radiation.

Now, in order to understand the gyro's behavior in purely electromagnetic terms, it is necessary to see understand how this back reaction works when a gyro is inductively suspended.

Refer to "Roll Isaac Roll" (Laithwaite, 1979): In Professor Eric Laithwaite's paper, "back reaction" are the very words used by him to describe the behavior of a gyroscope.

So if the spin momentum remains constant (-for a spinning gyro-and why shouldn't it, if we postulate a wheel in perfect bearings ), then a torque T is seen to give rise to an angular velocity omega** (and for an electrical engineer it can easily be seen as the reverse way around, for a current can be seen as the cause of a voltage in a series circuit). Since when has an angular velocity been capable of producing a back reaction? I thought only an angular acceleration could do that...
End quote

What the good professor is asking is this: Since the gravitational torque produced the angular velocity, any back reaction that the initiating gravitational torque suffers (in this case, it was gravity that initiated the torque and the back reaction cancelled the gravitational torque and kept the gyro horizontal) can only have been initiated by the product of the torque. Since the product of the torque was simply a precession, it means it was the back reaction of the precession that canceled out the torque – ie. A velocity produced an acceleration. That back reaction is identical to the back reaction of a particle's field on itself. That is, inductance is a field element. This is complementary to the fact that capacitances behave as point particles – both in Brillouin's development of electrical/mechanical filters and in Newtonian/Laplacian analysis of lattices. The two together form a harmonic circuit or arrangement – one which we can harness.

We can still continue to follow the analogy- this time with Quantum Electro Dynamics (QED). I propose that what we are seeing here is more commonly called the ‘jangle fallacy’ (Thorndike, 1904) -it refers to cases where two different terms are used for the same entity. That is, self-induction (in EM)/ back-reaction (of gyros) are analogous words describing one and the same phenomenon. ("the jangle") (As opposed to the “jingle” fallacy where we give two different phenomena the same name – thereby introducing another of confusion.)
There are already tantalizing hints that this is correct, from previous attempts to bring Gravity into the framework of the Unified Field Theory. For example, for one such alternate theory to work, it would need a decidedly non-Newtonian head-start:
The Abraham-Lorentz theory had a non-causal "pre-acceleration". Sometimes an electron would start moving before the force is applied. This is a sign that the point limit is inconsistent. An extended body will start moving when a force is applied within one radius of the center of mass.
end quote

This is strikingly similar to the fact that pure Newtonian predictions would be unable to account for the 'lead' of the precession resulting from a torque over the torque itself. Newtonian predictions will likewise be unable to explain how a Relativistic Machine can function, as in pure Newtonian terms, such a machine would appear as in the Abraham-Lorentz theory, to be “moving when a force is applied within one radius of the center of mass”.

This it would seem that Abraham-Lorentz theory is lacking exactly the same thing that Newtonian Physics is lacking (when it attempts to explain how precession could lead the torque that produces it), when it comes to explaining how gravity fits with the strong and electroweak forces. Solve one and you just might solve the other.

It would also seem that the Abraham-Lorentz theory is implying that in order for a framework to be possible that integrates gravity successfully with the other three forces, it would also have to be possible to have “non-causal preacceleration” i.e. Precession leading torque as well as “an extended body to start moving under certain circumstances when a force is applied within one radius of the center of mass” i.e for a Rel Machine to be able to fly.
What is also being proposed in this discussion is that a gyro ( or a spin angular momentum which has been suspended inductively)'s behavioral response is the ultimate measure of "extension" or space. Therefore it dominates any discussion of field related formulations of gravity in ways that are analogous to inductive elements in Electro-Magnetism. In addition, all non-gyroscopic objects/arrangements can be dealt with as pure point masses without spatial extent.

Now, just how did QED solve the renormalization infinities problem when it first appeared?

The main idea that QED brought to renormalization is to correct the original Lagrangian of a quantum field theory by an infinite series of counterterms, each one of which is labelled by the Feynman Graphs that encode the perturbative expansion.

In this methodology, the divergences appear in calculations involving Feynman diagrams as closed loops of virtual particles in them.

A Feynman graph consist of loops and edges and satisfies certain conditions. Each loop or edge represents a segment of the worldline of a particle.
End quote

Vacuum bubbles for instances are represented by a simple loop. Since in graph theory, a loop is an edge that connects a vertex to itself. Note that Feynman graphs consist of edges representing segments of the worldline of the particles involved.

Now, the spinning wheel going a-b-c- and-so-on-and-on and then forms vertices like at b over and over.
Thus spinning (a loop) is represented by a vertex with cyclical sets of two edges. (Figure 1) But in the case of a spinning wheel, we know that if a RelMachine is possible then it means that with respect to a stationary inertial frame (whose worldline is represented by a-c), the zig-zagging wheel and its support arrangement will move in space away from the inertial observer, ie there will be a divergence between them(Figure 2). Thus over time, the two sets of lines appear to diverge. Now, suppose both objects have spin – one simply has more spin than the other. Then, in such a case, we can modify the relativistic diagram further to Figure 3.

This is a cumbersome way of representing whats going on in the situation. A simple way would be as in Figure 4, where a single cycle of the spin is shown to both give the frequency and to represent that there is an inductive process at work. Then, we would connect the middle points d and b to indicate that there is an energy exchange going on. We could further encode information into the diagram by using color to indicate whether there is a large amount of energy interchange, color of the edge represents the type of energy of the particle, etc etc. This is infact what a feynman diagram does. (Figure 5)

Thus, the divergences that appear in calculations are implied to be inductive energy exhanges and are symbolically represented by loops. There are further useful deductions to be drawn from the analogy of spin/rotation with the Feynman Rules for loops in QED

Feynman Rule: Incoming and outgoing lines carry an energy, momentum, and spin.
Interpretation: The are inductive + capacitive arrangements (and therefore determined by their harmonic behavior.

Feynman Rule: each vertex where lines meet gives a factor derived from an interaction term in the Lagrangian
Interpretation: Each inductive interaction has its own entry in the Lagrangian.

Feynman Rule: A point where lines connect to other lines is an interaction vertex, and this is where the particles meet and interact--- by emitting or absorbing new particles, deflecting one another, or changing type

Interpretation: The vertexes are situations where an event is occuring with a decay and a collision (analogues of emission and absorption) involved.

It is proposed here that these closed loops that Feynman Diagrams refer to are nothing but representatives of the energy involved in harmonic arrangements involving inductively suspended spinning objects coupled to the capactive elements (point objects), in calculations.

Thus, in a Feynman diagram, the capacitive objects are being shown are particle (localizable) inputs and outputs, while the inductive objects/arrangements are shown only abstractly as a loop similar to a spinning wheel's spacetime trajectory. This would mean that the Feynman diagrams/graphs would be the best choice of schema to sketch the behavior of both inertio-gravitational oscillators and electro-magetic oscillators. Having a single schema to analyze both behaviors, in a way harmonizes the analysis of both phenomena.

A (inductively suspended object) gyro's behavioral response is the ultimate measure of "extension" or space. All non-gyroscopic objects can be dealt with as pure point masses without spatial extent.

The inductive property is disruptive to theories which are based on capacitive rules of interaction. Instead of the object going this way like a billiard ball would, it might go some other way, photons create virtual particles, etc etc. That is why they appear as "violations" of interaction rules which have been capactively founded. Much of this distortion has to do with the fact that most of our 'intuitive' interactions in daily life proceed capacitively (and all purely capacitive interactions can be analyzed using Newton's Laws).

We would expect that accordingly, the need for these 'loops' would coincide with situations where a large amount of inductive capability is trapped inside the particles/arrangments involved in the interactions. Indeed this is the case. Those situations for which the interactions are divergent are significant, which have large momentum/energy values.

While virtual particles obey conservation of energy and momentum, they can have any energy and momentum, even one that is not allowed by the relativistic energy-momentum relation for the observed mass of that particle. (That is, E2 - p2 is not necessarily the mass of the particle in that process (e.g. for a photon it could be nonzero).) Such a particle is called off-shell. When there is a loop, the momentum of the particles involved in the loop is not uniquely determined by the energies and momenta of incoming and outgoing particles.
End quote

i.e., the inductance of the arrangement and the backreaction of that inductance's field upon itself, in response to local forces can play a large enough role in moulding the results of the interaction, even though they are largely invisible to the capacitively oriented Newtonian (Euclidean) laws.

A variation in the energy of one particle in the loop can be balanced by an equal and opposite variation in the energy of another particle in the loop. So to find the amplitude for the loop process one must integrate over all possible combinations of energy and momentum that could travel around the loop.
These integrals are often divergent, that is, they give infinite answers. The divergences which are significant are the "ultraviolet" (UV) ones. An ultraviolet divergence can be described as one which comes from- the region in the integral where all particles in the loop have large energies and momenta.
End quote

(i.e. only those inductive processes are significant which have significant/large rotation energy and momentum) For such large energy processes (dubbed Ultraviolet divergences), the net result is calculated by adjusting the capactive laws to include an inductance (and the resulting harmonic oscillations term and its consequences) and its effects. The Feynman diagram is serving to document that adjusted calculation.

- very short wavelengths and high frequencies fluctuations of the fields, in the path integral for the field.
- Very short proper-time between particle emission and absorption, if the loop is thought of as a sum over particle paths
end quote

Interpretation: Thats exactly what we showed for the rotating wheel- cyclical emission and absorption.
The faster the rotation, the shorter the proper-time between particle emission and absorption, if the loop is thought of as a sum over particle parths.

That is, the loop implies rotational motion of the object or constituents of the object. Infinite answers imply that for large inductances, the resulting precessive output will be large and is going to cause divergences. The formulas are merely reflecting the unsuitability of capactive techniques to analyze inductive phenomena.

Some Final Remarks
The Higgs field has a non-trivial self-interaction, like the Mexican hat potential, which leads to spontaneous symmetry breaking:
End quote

That is, the Higgs field involved self-induction, which appears in the case of inductively suspended objects harnessed by a local variable torque.
So expect that in analogy, the weak gravitational force, inertia (capacitance) and induction arise through a Unified Field Theory with spin/rotation as a major driver.

In particle language, the constant Higgs field is a superfluid of charged particles, and a charged superfluid is a superconductor. Inside a superconductor, the gauge electric and magnetic fields both become short-ranged, or massive.
End quote

Thus, this formulation of the Higgs field as the resonant excitation of inductively suspended spinning objects can form explanations of SuperConductivity as well.

Please note that n this and the previous post, I have quoted from

Monday, July 27, 2009

The Problem With Gravitons

Although Albert Einstein spent the last two decades of his life seeking to unify Gravity with Electro-Magnetism, he did not succeed in building a Unified Field Theory. Even now, the current state of unified field theories is that there is as yet, no accepted unified field theory. Gravity has yet to be successfully included into the framework of such theories.

One of the best weapons in science is analogy. We apply the template of existing known processes in discovering/understanding new processes. Nature seems to somehow agree with us. Wave motion is one example of such a concept. In mathematics and physics, the Laplace operator is a differential operator used in modeling of many different kinds of wave propagation. It is used in formulating equations for acoustics, fluid dynamics heat flow, forming the Helmholtz equation, all the major equations in electrostatics, Electro-Magnetism, and in representing the kinetic energy term of the Schrödinger equation in Quantum Theory.

Harmonic motion is one other such concept - it has been applied successfully, over and over again in various (intially, to an untrained eye atleast) unrelated fields like mass-spring arrangements, a molecule inside a solid, an electron stuck in an atom, a car stuck in a ditch being rocked out, a pendulum and the earth in its orbit. (source:

Capacitors and Inductors are also such a concept. While capacitors alone interact as point objects, inductive objects (consisting of a spinning object in suspension about an orthogonal axis) behave as objects with a finite extension. Thus, viewing all interactions as being either capacitive or inductive can become a generalized technique that sorts the spatially extended objects (ie field elements) from point objects in both Electro-Magnetism (EM) and Iner-Gravitation (lets say, IG). This analogy raises spin/rotation to a unique postion of being the progenitor of both effects via capacitive and inductive suspension. By giving us the ability to distinguish between capacitive and inductive interactions, this analogy can also help harmonize IG with EM by solving the problem of renormalization when combining gravitons with strong and electroweak interactions. The following section explores one way to resolve this current problem in Physics.

All this means we can guarantee a unified structure to natural laws that would still look very familiar, but with a twist (almost literally) of the third derivative. In merging Mechanics and Gravitation with EM, we open the door to the Unified Field Theory. Lets not forget that Gravitation has long been the wedge that kept the whole structure from beautifully fitting together. In addition, it will give us the key to building ships that can cross space and make green transportation a reality.

In order to incorporate gravity into the Unified Field Theory framework, we have to work to replace curved spacetime as in general relativity with a situation where the gravitational interaction is mediated by gravitons. However, attempts to replace general relativity (GR) with gravitons have run into serious theoretical difficulties at high energies (processes with energies close to or above the Planck scale) because of infinities arising due to quantum effects (in technical terms, gravitation is nonrenormalizable).

Even just trying to combine the graviton with the strong and electroweak interactions runs into fundamental difficulties which boil down to the non-renormalizability of the results. The incompatibility of GR and quantum mechanics (QM) is another current problem in physics. Both these problems involve mechanics/gravitation's relationship with the remaining three forces.

In physics, although in principle we can predict the behavior of matter by keeping track of each atom, it is often more practical to treat matter as a continuum and then taking the continuum limit. Newton for example considered that air could be modeled as a lattice of mass points. He assumed the simplest possible lattice – equal masses spaced equally along the direction of propagation.

Laplace used this conception of air to successfully calculate the speed of sound. In fact, the entire set of Newton's Laws of Motion as well as wave theory itself can be deduced by taking the continuum limit of this simple lattice. If you wish to see the derivation, it is available here:

In Quantum Field Theory (QED), renormalization refers to a collection of techniques used to take a continuum limit of space and time.


When describing space and time as a continuum, certain statistical and quantum mechanical constructions are ill defined. In order to define them, the continuum limit has to be taken carefully.

Renormalization determines the relationship between parameters in the theory, when the parameters describing large distance scales differ from the parameters describing small distances.
end quote

However renormalization when gravitons and strong or electroweak interactions are combined is hindered by the problem of infinities. The problem of infinities is an old one dating back to the 19th and early 20th century. Back then, it arose in the application of classical electrodynamics to point particles. This first version of the problem was solved by QED (as will be discussed immediately below) and the second version of this problem that has arisen with respect to Gravitons is currently unsolved and is an open problem.

To put it simply,
the mass of a charged particle should include the mass-energy involved in its electrostatic field. Assume that the particle is a charged spherical shell of radius re. The energy in the field is
end quote

mem = q2/8*P*re


mem = electron mass

Now normally, this classical electrodynamics formula performs well for all electromagnetic interactions for which quantum mechanics is not relevant. However, notice what happens when the radius falls to zero. The energy becomes infinite when re is zero. This directly implies that the point particle would be infinitely massive and could never be moved - an absurd conclusion.

Max Born, Werner Heisenberg, Pascual Jordan, and Paul Dirac discovered that in perturbative calculations many integrals were divergent.

Further,(source: when calculating the electromagnetic interactions of charged particles, it is tempting to ignore the back-reaction of a particle's own field on itself. But this back reaction is necessary to explain the friction on charged particles when they emit radiation. If the electron is assumed to be a point, the value of the back-reaction diverges, for the same reason that the mass diverges, because the field is inverse-square.
end quote

Sunday, July 19, 2009

Estimating Power Supply Required For The Flywheel Motor

Selecting a motor with the right amount of torque to spin up your flywheel to the desired speed is a good first step. However, you still need to figure out how much power (voltage and current values) you will need to get up to the desired operating speed.  There are several factors to consider in such a situation. The motor's speed/torque characteristics and its nominal values of voltage and speed play an important role in all this. Note that we are assuming it is a DC motor that we are using. The required voltage can be estimated as follows:

ML = Operating Torque [mN-m]
NL = Operating Speed [rpm]
V0 = Nominal Voltage of the Motor Selected [V]
N0 = No Load Speed of Motor @ Nominal Voltage [rpm]
Dn/DM = Speed/torque gradient of the Motor [rpm/mN-m]
VL = Operating Voltage [V]

VL = (V0/N0)*(NL + (DN/DM)*ML) [V]

For instance: For the example flywheel we used in sizing the motor,

ML = 2400 mN-m
NL = 16000 rpm

Suppose a motor we are considering has the following additional characteristics

V0 = 24 V
N0 = 8000 rpm
Dn/DM = 8.69 rpm/mN-m

Then we can calculate

VL = 110.6 V

You will need to make sure that your motor is designed to accept this much voltage. If not, you will likely damage it and/or injuring yourself if you feed excessive voltage into it.

If you are going to be using a brushed DC motor along with a controller, it is highly likely it will use PWM technique to do so. In such a case, you will need a somewhat modified formula . Additionally, we also need the following extra data

PC = Pulse Width Modulation Cycle [in %]
VD = Max Voltage Drop Across Controller

VL = (V0/N0)*(NL + (DN/DM)*ML/PC) + VD [V]

So if I had a controller with say an 85% PWM cycle and a 3 V voltage drop across the controller,

VL = 124.6 V

You will need to make sure that your controller is designed to accept this much voltage. If not, you will likely damage it if you feed this voltage into it.

This calculated voltage and the nominal value of current of a given motor together will better guide your motor search/selection.

[Important: The calculations are meant for approximate 'ball parking' of the required power supply only. If you want the exact values, you will need to consult your motor's manufacturer's specs or technical support. Please also note that the actual numerical values used here are for illustration purposes only and do not represent the prototype.]

Sunday, July 12, 2009

On Assigning Cause and Effect in Inductively Suspended Objects

First it might be useful to take a quick refresher of forced oscillations in harmonic systems. Here's a good one I found after a quick search. Although written for a spring pendulum, this applies also to spinning wheels in inductive suspension.

Forced oscillations. (@
Begin Quote
On the whole you can see three different types of behaviour for forced oscillations:

case A: If the exciter's frequency is very small (this means that the top of the spring pendulum is moved very slowly), the pendulum will oscillate nearly synchronously with the
exciter and nearly with the same amplitude.

case B: If the exciter's frequency agrees with the characteristic frequency of the spring pendulum, the oscillations of the pendulum will build up more and more (resonance); in
this case the oscillations are delayed about one fourth of the oscillation period compared with the exciter.

case C: If the exciter's frequency is very high, the resonator will oscillate only with a very small amplitude and nearly the opposite phase.
End Quote

The gyro and the Rel(ativistic)Machine (that I contend will be able to fly) contain spinning objects. Any spinning object that is inductively suspended can be harmonically excited. Now, case B is the case of such a harmonic system at resonance. i.e. a Rel(ativistic)Machine.

Case C is the gyro in precession under the influence of gravity. The precession exists only as long as a rate of change of angular acceleration exists and cannot exist in the absence of it. Gravity is a high frequency exciter when compared to the characteristic resonance frequency of a spinning gyroscope suspended in inductive conditions. Thus, the gyro responds with a small steady precession & in the opposite phase (i.e. cancelling out gravity).

How do we assign cause and effect in analyzing interactions? Lets analyze the situation. Under routine Euclidean conditions, in analyzing cause and effect, if Y changed when I meddled with X, it can perhaps follow that X is the cause of Y. You would be correct in the sense of action - reaction (i.e. for a gyro on the ‘Eiffel Tower’, even if the physics says precession came first – that’s what the cross-product formula is saying- we 'know' that we initiated the torque first).

Now normally that would work just fine. But in looking at the Gyrscopic interaction that way, one will have missed the temporal aspect. What really matters in interaction analysis in this situation is that we label what comes first as the cause and what comes second as the effect.(And NOT to assign what you 'know' yourself to be doing as the cause and what happens then as the effect.) Otherwise one is liable to become deceived.

Lets look at how, if it were up to a Newtonian, s/he would label the energy flowing into the gyro (in the form of the torque) - S/He would call the torque the cause (lets label it event 1) and the precession as the effect (lets label it event 2). S/He would insist that event 1 came first and event came next.

But according to relativistic analysis, the energy contained in the wheel has a shorter spacetime distance between events. Therefore the energy of the wheel in the present is not in the same phase as the energy of the frame. (Recall this diagram below from this post.

That phase difference bestows upon it (the spinning wheel) a variable ability to oppose or reinforce the arriving energy that can be harnessed most efficiently, if we make the appropriate arrangements (i.e. it’s normally a limited ability but can be harnessed best when we induce resonance), in analogy with a properly tuned LC circuit. For a rate gyro its case C, the lower limit that is applicable.

For spinning wheels (for instance, for the prototype), under inductive resonant conditions, we activate case B, the ability of the momentum in the wheel to push more and more strongly (like a rider on a swing kicking the ground at exactly the right intervals over and over to acheive high amplitude.) against the frame, even as the frame's momentum attempts to act upon the wheel and drag it around, under the influence of that vertically mounted motor.

That means that under resonance conditions of reinforcement, momentum is delivered cyclically from the 'future' of the spinning wheel to the 'present' of the frame. (alternatively we can say energy from the 'present' of the spinning wheel is adding momentum to the 'past' of the frame. - the naming is irrelevant - also note that this doesn't imply time travel of any extended degree, in practical terms its merely saying the phase can be adjusted to be positive). If the cause is in the future and the effect is in the present, then it means the effect will be recorded first and only later the cause, when viewed from an inertial frame.

Since there is a process in play in which energy from the 'future' is influencing energy in the 'present', the actual temporal order of occurrence would be 2-1. That is the energy transfer would temporally execute in the order 2-1, but action-reaction accounting of the entire interaction would run 1-2. THAT mismatch in the ordering (between the temporal perspective and the interactive perspective) of events leads conventional Newtonian analysis of Gyroscopic action into a dilemma.

An analysis that conforms to Newtonian Laws can be achieved only by hewing to the temporal order of manifestation (so as to record the forces, play by play as and when they are manifest). However, in such an analysis, since we label what comes first as the cause and what comes second as the effect, that would also mean accepting cause as effect and vice versa in the situation of the gyroscope. (et voila,.. there you have the formula t = w X L - you gave up what should be the logical choice of what is cause and what is effect in order to preserve Newton's Laws)

Alternatively, we can break the temporal order i.e. we adopt the method of assigning the torque as the cause and precession as the effect. We say torque is the cause and precession is the effect - in which case, we would be forced to write w = t X L which does not conform to actual behaviour, thereby bringing all Newtonian analysis to a halt.

One cannot keep both logical and Newtonian order.
The formula t = w X L allows us to keep the Newtonian mathematics by reversing the logical order i.e. the order that we 'know' we are initiating and replacing it with the opposite situation i.e. the formula assigns cause and effect by temporal order - but at least in return for it, a simple Newtonian analysis of the motion is possible. (But only a limited one any how, because while Newtonian analysis has allowed the exploitation of the gyroscopic principle in constructing useful rate gyros etc, it has also deceived us into thinking we understand everything about the phenomenon. In fact, much more can be done if we harness the effect in a resonant mode.)

Suppose we assigned the torque as event 1 and the precession is then event 2 AND we also incorporated the insight (provided by relativistic analogy between Electro-Magnetism and Accelero-Gravitation) that the effect is in the present and the cause in the future, then we can do a modified Newtonian analysis as follows:

We introduce a fourth force (temporarily) right at the beginning of the interaction, equal in magnitude and direction to the effect, to hold the place of the effect until it actually kicks in. This force would account for the fact that the effect will itself be recorded (and will itself be the cause of activity) BEFORE the cause is manifest. In such a situation, we could preserve Newtonian analysis. Such a situation has striking similarity to the No-Nutation condition:

Lets say you have a spinning toy gyro. You've just spun it up. You are positioning one end of the axis on top of the 'Eiffel Tower'. How do you set off the gyroscope without causing nutation? The way to set a gyro precessing without causing any nutation is to give it a gentle nudge even as we let it go and the nudge but be equal in magnitude and direction to the precessive velocity that will be introduced. Then there will be no nutation.

That is we are compelled to introduce that exact 4th force we discussed in the above paragraph. i.e that fourth force is not a mathematical artifice but an actual, real necessity if you want behavior that strictly and smoothly adheres to Newtonian prediction. If not you MUST deal with perturbation (nutation) - perturbations open the door to Quantum Mechanical analysis of energy exchange.

By giving the gyro a velocity in the exact direction and magnitude as the precession would, we are ensuring that temporally, there is in existence exactly as much momentum and angular velocity -both amplitude and direction - identical to what would exist if there was precession - i.e. precession already exists before there is torque effect - or at least a
simultaneous start)

Finally let us analyze the situation where we don’t give it that fourth force. What happens if you don’t give it that extra push? Then you get nutation i.e. precession + torque.

In the best case (we give the push), the precession is already in existence before the torque came into existence. And in the worst case scenario, they are simultaneous. What we deduce is that the physical situations all work smoothly in accordance with Newtonian predictions only when built in the following fashion:

The precessional velocity comes FIRST and THEN the torque appears OR they are simultaneous. Its also how the precession formula works too - with the counterintuitive cross product.

Thus the real test that helps assign cause and effect in a way that conforms to Newton’s Laws is what comes first and what comes second. And in this we find that the precession comes first and then the torque OR they come together. But never torque first and precession second.

Tuesday, July 7, 2009

The Paradox In Gyroscopic Behavior

The vector cross product is an operation on two vectors which yields a third vector in a direction which is perpendicular to the plane containing the two input vectors.

Strictly mathematically speaking, the cross product is a binary operation there by involving two operands, an operation and a single output.
The cross product, A x B, gives a third vector, say C in a direction perpendicular to the plane containing the two input vectors. A famous example of such a vector cross product is the Biot-Savart Law.

The Biot-Savart Law relates the strength of a magnetic field, dB produced due to the current flowing in a length of wire dl, at any distance r from the wire.
                            ®  ®
dB = [μ0/(4*p*r3)]*(I X r)

So, there are two inputs: a current vector and a distance vector that have to be specified. Then, the cross product operation will produce the output, i.e. a magnetic vector that results from that combination.

Note that since the magnetic force is the product of the two inputs (namely current and displacement unit vector) it will figure on
the left side of the equation and the inputs on the right side of the Biot-Savart equation.
Now let us look at the basic operation of a gyroscope. Take a look at the diagram below showing the two inputs (the black arrowheads) and the output (the blue arrow head).

Given that we need two simultaneous inputs - a spinning wheel and a torque upon its axis- to produce the output i.e. precession AND given that a cross product is involved, we might be forgiven for guessing that analogous to Biot-Savart's Law above the formula would read

w = t X L

However we would be wrong. As the correct formula reads:

t = w X L

well, I thought we APPLIED the torque to RECEIVE the precessive velocity?

The actual formula however seems to be implying that the precessive velocity is an input just as much as the spinning wheel's angular momentum is. The two act upon each other, producing torque.
This is a reversal of cause and efect.

How to understand this? There is no conventional explanation for this reversal. It is merely illustrated without explanation in books. Some attempt vectorial gymnastics in an attempt to show that all is in aacordance with "physics", but none have acknowledged this weirdness and offered an explanation. The simple explanation for this reversal lies in the underlying relativistic analogy between electricity and spinning wheels.

The most useful mnemonic in ElectroMagnetism is probably 'ELI the ICE man'. (The following is excerpted from Resnick & Halliday, Fundamentals of Physics)
ELI contains the letter L (for inductor), and in it the letter I (for current) comes after the letter E (for emf or voltage). Thus, for an inductor, the current lags the voltage. Similarly ICE (which contains a C for capacitor) means that the current leads the voltage. You might also use the modified mnemonic "ELI positively is the ICE man" to remember that the phase constant f is positive for an inductor. End Excerpt

The way to translate "leads" in electricity to the mechanical version is to substitute "appears to be the cause of" in its place.

Note that according to the mnemonic, we can see that Current Leads Voltage in capacitive conditions - that is in our mechanical analogue, we would say Force Leads Angular Velocity or Force appears to be the cause of Angular velocity (via acceleration) and since almost all interactions we see in daily life are capacitive, we have viscerally absorbed the idea that force causes a change in velocity.

Voltage Leads Current in an inductor - so in the Mechanical analogy for a spinning wheel, Precessive Velocity Leads the Applied Torque i.e. Precessive Velocity appears to be the cause of the Applied Torque. Thus inductive behavior is characterized by this kind of a reversal of cause and effect.

While in electricity (due to the wonderful conformation of Maxwell's Laws with the Lorentz Transformations and therefore Relativity), this behavior results in another area of study (ElectroMagnetic oscillations and RLC circuits) and another mnemonic for the student to remember, in mechanics it has been ignored up until now. We are discovering here that in the inductive condition, the precessive velocity appears to be the cause of the force i.e. a gyroscope is an inductive element, unlike non-spinning objects, which are capacitive elements.

Our daily experience with the physical world involves only capactive encounters. There for we have a very hard time 'instinctively' grasping the differences in the behavior. They can appear are riddles at time. Gyroscopic behavior is a perfect example. The analogies between ElectroMagnetism and mechanics & gravitation are deep and the path lies through Kron's Generalized Machine Theory.

The gyroscopic formula is merely reflecting that fact.

Sunday, July 5, 2009

Sizing A Motor For A Gyroscopic Wheel And Other Objects

OK, so you have the gyroscopic wheels and you want to know how powerful your motor has to be in order to power them up to a certain speed. Or perhaps you have some motors and you want to know which one best suits your situation. I hope you have the specs sheets for the motors you are using. If so, the specific formula to calculate the operating torque demanded of a motor spinning a rotating disk is:

T = (1/2)*M*r2*(n/60)*2*p/t [Nm]

n = speed (rpm)
T = Torque (Nm)
I = moment of inertia
a = angular acceleration (rad/s2)
M = Mass
r = disk radius (m)
t = time to top speed (s)

For instance: If your wheel had an MI of 0.2875kg-m2 and you wanted to spin it up to 16000 rpm in 20 seconds, you would need a motor capable of delivering approximately 24N-m of torque.

Note that although the formula implies that you can use a really small motor if you’re willing to wait a long time to get up to maximum speed, this is not necessarily the case if you the mismatch between the motor and load is too large. That’s because motors will stall if hooked up to a load too heavy to start –static friction in the bearing is one contributor to such a stalling torque which prevents the motor from being able to start up the rotation.

You can also use the above formula to calculate the amount of time it will take for your motor to spin up a load of a specific inertia to a specific speed.

The formula above is derived from the simple equation

T = I*a

We substitute
a = w/ t
I = (1/2)*M*r2 for a cylindrical disk
w = (n/60)*2p

Please be careful to substitute the right formula for the moment of inertia to match the object being spun up, for instance if you were spinning up a sphere, the moment of inertia would have to be changed to (2/5)* M*r2. For a ring, with most of the mass on the outer edge, the moment of inertia would be M*r2.