The angular inertia of a wheel being accelerated or decelerated instantaneously applies equal opposing counter-torque back to whatever the motor is affixed to.
However, the torque produced by changing the radius of an orbiting mass causes no instantaneous counter-torque - move the mass inwards towards the center of rotation and positive reactionless 'inertial torque' is produced, move it outwards and we get a similar negative torque.
The following experiment will resolve what happens when these two different sources of torque are perfectly counter-balanced against one another.
Some orbiting mass will be pulled inwards, generating a positive torque that would normally cause its angular acceleration. However at the same time, a motor will apply equal opposing torque between this orbiting mass and another, coaxial (ie. parallel) wheel.
Hence the inertia of this other wheel - its resistance to that acceleration - will cause the motor to apply equal counter-torque back to the orbiting mass, preventing its accelerations by the inertial torque.
The net result is that we can move the orbiting mass in and out, without causing any change in its angular velocity. Whatever its RPM, it is held constant despite its changing moment of inertia. Thus, only the wheel is able to respond to these orphaned inertial torques, and the orbiting mass effectively becomes a 'stator', albeit one in a rotating reference frame, against which we can apply unilateral torque to the 'rotor'.
This somewhat special condition is the particular subject of enquiry; what are the potential implications for CoE and CoM when counter-torque is perfectly counter-balanced by inertial torque? No immediate anomaly is anticipated - so no expectation of a CoE or CoM break - the outcomes could mostly be worked out from first principles, but it's quite a complex inter-reaction with non-linear feedbacks between the various torque factors.
For instance, we could choose a constant motor torque, and then vary the speed at which the orbiting mass moves inwards, to control the inertial torque and keep it matched to the motor torque..
..or we could do it the other way around, moving the mass inwards at constant speed, and varying the motor torque to counter-balance the changing inertial torque.
This latter option is more flexible, and allows for further elaboration such as varying the radial speed as well as the motor torque - so the radial motion could speed up or slow down, causing faster changes in inertial torque, which in turn could still be dynamically countered in real-time by motor torques, using the same feedback controls. This seems the best approach to proceed with.
The only aim is to see how things balance out, and obviously, check for any potential advantages or tricks that might be useful..
The Experiment
To begin with, we need two rotors; one with a fixed MoI, and another with a variable MoI.
Below, the blue wheel is a uniform disc with 2 m radius and 0.5 kg of mass, hence an MoI of 1.
Attached coaxially via a motor, the two red masses are 0.125 kg each, also at 2 m radius, thus the near-massless red beam they're attached to also has a net MoI of 1. The masses can be moved in and out along this beam as it rotates about the central axis, reducing or restoring its MoI.
Here, the motor supplies a constant 1 N-m of torque, causing equal opposing accelerations of these two equal angular inertias. The sim pauses at the 1 second mark, and again at 90°, to illustrate the relationships between the respective units and dimensions.
In the next sim the motor is now off, and both rotors have been set in equal motion. It is under this condition of mutual co-rotation that we'll be applying the inertial torque (by moving the red masses inwards), while simultaneously supplying just enough opposing torque from the motor to counter it.
This is going to require a simple kind of control loop, wherein we need to sense the angular accelerations of both rotors individually from an independent frame of reference, and feed that data back to the motor.
To this end, a third rotor is now added - this one is completely unconnected to the other two and doesn't interact with them in any way; it's purpose is to simply hold the 1 rad/sec spin it's been given and just sit there, rotating, unperturbed by anything else going on around it, so providing a neutral perch for our camera inside this rotating reference frame. We can thus use its constancy of rotation to detect any accelerations - any change in relative angle, however slight - of the other two rotors.
So our view is now rotating with the system, as if all three rotors were stationary and the outside world is rotating around them. In reality, the entire system's rotating uniformly anticlockwise, hence if either rotor accelerates they'll appear to rotate anticlockwise relative to the camera view, and if they decelerate they'll move clockwise.
Any such data can then be passed directly to the motor and scaled up via an appropriate multiplier to convert this angle-delta into torque, and thus counter-torque, from the motor. If the angle increases, so does the motor torque in direct proportion, regardless of the sign of the angle change or thus the responding motor torque sign. So we have a reactive self-limiting control mechanism that will cause the stator angle to track the position of this 'zero momentum frame' even if it began accelerating or changing direction...
..of course, it won't be doing either, but the point is that we can accurately hold the stator's rate of rotation constant, even while its MoI is changing and the rotor's also being torqued against it. Hence we have an effective freely-rotating pseudo-stator, against which we can torque up a rotor, and which appears from the motor's perspective to effectively possess infinite inertia, since its speed never wavers in response to these motor torques..
As a general point, If a motor that is already in a rotating reference frame could somehow apply torque to a rotor without applying equal counter-momentum back to the rotating reference frame itself, then the RKE of that rotor will be over-unity - it will have more KE than the energy spent / work done by the motor.
Obviously, if we apply gravity this way (as previously demonstrated), then we do indeed get a glimpse of this 'OU' performance from the motor... albeit, accompanied by a corresponding drop in GPE (and thus perfect unity).
A similar outcome is anticipated here - we're going to accelerate the rotor without counter-accelerating the 'stator', however this is not a reactionless acceleration in the strictest sense, since the effect is dependent upon reducing the stator's MoI whilst preventing its corresponding acceleration, hence reducing its momentum regardless, in equal proportion to the momentum added to the rotor. So, while we're not decelerating its angular velocity, we are nonetheless effectively 'decelerating' its angular momentum, just by other means..
In a truly effective N3 break, the net system momentum rises, because the momentum and counter-momentum from an applied force are not evenly distributed, hence summing to a net increase or decrease. Again, there's no reason to expect such a result here.
Nevertheless, the resulting RKE on the accelerated rotor should be substantially greater than the work done in terms of motor torque * angle.
However, we'll also be performing more work against CF as we pull the stator masses inwards..
Furthermore, just as we're reducing the stator's momentum, we're also slashing its RKE..
So energy-wise, the picture's a bit more nuanced - it's all swings'n'roundabouts no doubt, but while no energy gain is expected, i'm not so confident about energy loss, in which case, as ever, the question will be whether or not the loss is of the dissipative, or non-dissipative kind (ie. is the loss time-reversible).
Fundamentally, the ability to manifest mechanical energy gains depends upon being able to somehow successively raise the momentum of an otherwise 'closed system' of interacting masses (obviously, any OU system is implicitly open at heart, even if in really surprising ways..), but as the net momentum of this system appears constant (gravity's off, but would have no effect anyway), no opportunity for KE gains would seem to be on the cards..
Still, it's an interesting interaction - pitting these two raw elements against one another in perfect counter-balance (to six figures displayed, 16 under the hood) - it's also a tricky balancing act you'll probably never see anywhere else..
That is about its sum value though - so, should anyone now or hence ever have pause to wonder just what might happen in this particular arcane and unnatural deadlock of elementary forces (weirdo), may what follows offer some scant elucidation:
Success! At least the technique works, even if the outcome's useless..
Still, we could put a touch more polish on this turd with some of the embellishments mentioned above, such as varying the radial speed the masses are drawn in at..
Currently, the sim begins and ends with them moving inwards at a constant 1 m/s. The plots are still perfectly valid - we can accurately find the work done by the actuator and motor - but it's a bit rough'n'ready.. why not begin and end with zero radial velocity, smoothly accelerating them inwards at the beginning, and decelerating them to a full stop at the end? Then we can capture their cost of operation inside the energy plots:
Here the masses accelerate inwards up to 1.5 m/s for the first 33 ms, then hold that speed for a further 33 ms, before decelerating back down to halt at 1 m radius in the final 33 ms. Now the entire interaction is fully accounted for by the radial and angular work integrals.
Here's the same interaction again, having switched back out to the external reference frame:
There's one further variation that seems worth investigating, prompted by that first angled peak on the above work plots; it appears that significantly higher motor torque can be applied when the masses are accelerating inwards, rather than when moving in at constant speed..
..so what would happen if we just accelerate the masses all the way into the center? Let's try that:
Here, the stator masses begin with zero radial velocity, then accelerate smoothly up to 4 m/s at dead center. Whereas the previous run increased the net KE by 650 mJ, we now see a 3 J rise; we've added 1 J of RKE to the system (up from the previous 0.65 J), plus we still have 2 J of radial KE conserved on the stator masses.
So, that's the experiments done. All that remains is the energy accounting. On the one hand this seems kind of redundant, since we've already established that the net system momentum isn't going to budge. Still, having started, one might as well finish - i'll pull the data and Riemann sums so we can cross-reference consistently between all plots.
Again, there's no leeway in this system at present to admit a rise in momentum, precluding outright any prospect of energy gains from any variation of the current principles or apparatus. There remains a possibility of energy loss, in the form of negative work done in decelerating angular momentum, albeit by reducing MoI rather than angular velocity, but even then, such a phenomenon is useless unless it can somehow be reversed..
Still, if anyone thinks of any further variations or questions to test out..
Hopefully should wrap this by the w/e and get back to time-dependent momentum-from-gravity tricks, which seem much more promising..
Sims in the .zip, final data to follow..
