I'd previously been considering the radially-translating masses as moving together, as if connected by a rod. Under such a regime, the minimum MoI state corresponds to both masses being equidistant from axle and rim, and CF wants to maximise the MoI by pushing one mass out to the rim while pulling the other into the center.
Any benefits of this method seem speculative at best. Really, i can't see a way to break symmetry if both the masses are contributing equal and opposite RKE variations - which is a direct consequence of their sharing a single axis.
But what if we add another, coaxially to the main center? The inbound and outbound masses can have their own individial pivots about the main axis.
So when we extend a mass outwards, increasing the MoI and decreasing the RKE, this can occur on a separate axis to when we retract the mass inwards, reducing its MoI and so causing an acceleration.
So consider instead a slightly different proposition: the masses still change radial positions at the same time. They could be attached to the ends of radially-oriented scissorjacks - exactly as shown in
MT 39, with its "very special" application of them. The key point is that while their states of extension or retraction are hard-coupled via the interconnecting pulley system, their relative
angle can vary independently!
You can probably now see where this is going..
Extending a mass radially induces a counter-torque. Retracting one induces positive torque.
All else being
equal, so is the work integrals of these two torques over their respective angles; the same positive torque applied to the same mass will cause an equal change in energy as a torque of opposite sign.
But what if, instead, the two torques are divided into different masses?
So as one mass moves inwards, its angular acceleration increases, and thus applies a poitive torque to a small mass, which thus undergoes a high acceleration and thus rise in energy. Meanwhile, at the same time, we extend the other mass outwards, causing a deceleration.. but this counter-torque is applied to the
whole system - the wheel itself, via suitable rim stops or whatever - and thus divided into a much greater mass, and thus a smaller velocity and thus energy drop.
An analogous mechanism would be the counter-rotating armatures from MT 143, the Roberval balance.
And the funny letter "A" littered throughout Machinen Tractate, as alluded to above.
So one arm undergoes a positive torque as its MoI goes down, the other undergoed a negative torque as its MoI goes up, and the former torque accelerates a small mass, while the latter decelerates the net system, but by less energy than has been invested into the smaller mass.. which then impacts the wheel, imparting its energy gain.
In short, equal and opposite torques can induce an unequal distribution of CW vs CCW RKE.
Gravity can assist in both retracting and extending the radially-moving masses. This contribution is balanced by having to relift them again, but while the net GPE remains neutral, the CW vs CCW distribution of RKE does not, and this results in the system having a natural directional bias - as the CW and CCW torques are distributed into unequal masses and thus energies.
As JC deduced, a bi-directional arrangement requires two sets of mechanisms biased in opposite directions.
In summary, gravity helps us decrease one MoI, while increasing another. But the work performed by the resulting positive and negative torques is distributed into unequal masses, and thus energies, and because these torques are self-induced (ie. statorless), the net system momentum and energy is unbalanced - there is a "preponderance" of positive, relative to negative RKE, and thus, in turn, momentum and torque.
Applying an additional load to the axle further divides the distribution of energy between the larger mass (now increased by whatever load's been attached), vs the smaller and unobstructed internal mass.. momentum is conserved, and thus the increased negative torque results in a commensurately-inreased positive torque applied to the smaller mass, thus perpetuating the asymmetric distribution of CW to CCW energies.
It is intrinsically load-matching. The slower the net system is braked, the faster the smaller internal mass is accelerated, reinforcing the asymmetry.
This would seem consistent enough with all the above clues, as well as the general course of this thread.. seemingly wrapping everything up neatly.
The solution, in a nutshell, is to convert the two opposing torques from an increasing vs decreasing MoI into unequal CW and CCW RKE's by dividing them into unequal masses. The two momenta will be equal, per N3, but the energies are not, and the greater energy of the smaller internal mass can easily be harvested by a simple collision.
It no doubt needs refining in the details, but that's the general picture. Will start work on it forthwith..
