The following discussion is prompted by JC's closing conclusions on his latest blog; i wanted to address the issue here however as it's broadly relevant to everyone involved in the effort.
I suspect i'm also partly responsible for perhaps sowing seeds of misconception in my earlier analyses - conclusions from which i myself have moved on, but from which others may still be somewhat waylaid, and thus i feel obliged to clearly summarise the issue and its conclusions using rigorous logic and plain English.
We know Bessler's one-way wheels were under static torque when stationary, and were tied off to a ground fixture in order to hold them still.
In a statorless wheel - with no means of torquing the wheel against the earth via an external stator - it would seem the only possible source of static torque would be over-balancing torque; ie. the wheel's rotation allowed a weight to be lowered, which presumably had to be riding the wheel down, as if set against a rim-stop.
In support of this interpretation, Wolff was of the impression that the collisions he heard were indeed over-balancing weights landing on the descending side of the wheel (albeit perhaps imbued with more force or momentum than had been gained from their passive falling alone).
However there is a serious limitation in the way the energy yield from over-balancing evolves with RPM, which i'll now try to lay out straight:
• input GPE is determined by gravity * mass * height
• neither of these three components are time-dependent, and because of this, GPE is not speed-dependent; it makes no difference how quickly or slowly we raise our weights, we're going to be performing the same amount of work, spending the same amount of energy regardless
• thus at 1 RPM or 10 RPM, the input energy cost of GPE is constant and unchanging
• if we look at how output kinetic energy evolves with regards to RPM though, it's a completely different story:
• at say 10 RPM, the weight spends half as much time over-balancing as it does when at 5 RPM; gravity however is a constant acceleration, so if you spend half as long under its influence, you gain half as much momentum..
• so the per-cycle momentum yield (from gravity and time) is a function of time and thus velocity, irrespective of the fact that the gravity, mass and height parameters aren't changing; at twice the RPM, an over-balancing wheel gains half as much momentum for the same drop..
• we can further refine this conclusion by factoring in the consideration that our net mass and angular inertia is also constant over a cycle; since momentum is a function of inertia and velocity, if the former is necessarily constant, then the specific component of momentum gain that's falling off with RPM is simply speed; as RPM's increase, we're gaining less speed from each successive drop, because we spend less time per cycle under gravity's constant acceleration. Capiche? Simples.
• hence each lift costs the same input energy / work done, but the increase in RPM's we get from dropping them each cycle is necessarily decreasing as net RPM's accumulate
• if we look now at the kinetic energy equation - i'll just use the linear form for simplicity - it's 'half the mass times the velocity squared', ie. for a given amount of mass / inertia, each doubling of velocity corresponds to four times the energy..
• NOW CONCENTRATE! - if input energy (our GPE) is constant, then the amount of velocity rise each successive drop is going to yield must necessarily diminish inversely to the half-square of whatever the current velocity; in other words, PE:KE symmetry is being enforced by giving you less velocity per cycle, the faster you go..
• to put it another way, after an overbalancing 'power-stroke', your new system KE will be whatever it was prior to the drop, plus that constant per-cycle GPE.
Thus in conclusion, it is mathematically impossible to break unity via over-balancing torque alone.
If Bessler's wheels did only apply passive over-balancing toque, then there must be an effective GPE asymmetry on offer somewhere, ie. reducing the effective input energy cost of GPE with rising RPM. That is, if you can't gain energy on the output side of the interaction, then you're left with just the input side..
Although i currently still believe the prospect of discount GPE is slim to nil, it's only right to at least summarise any factors that might lead to an exploit in this direction:
• for one thing, at local or 'mechanical' scales at least, gravity can be considered effectively constant regardless of any constant vertical speed
For example consider a simple paternoster - counter-balanced lift cars, one descending as another rises - so long as the speed is constant, both cars experience a constant 1 G, invariant of their rising / descending speed. Inertia and Newton's 3rd law ('N3') still apply of course, but perhaps there's some way of applying an effective N3 violation, such that say a 1 Joule GPE input relative to the lift car performs more than 1 J of lift, specifically with regards to the absolute 'height' component in the ground reference frame.
To be sure though, it is N3 that is enforcing relative vs absolute GPE symmetry in this example. In other words, such an exploit - buying cut-price GPE - would ultimately depend upon an effective N3 violation, albeit manifesting as an asymmetry between relative vs absolute changes in input GPE / work done.
I simply have not had time to completely eliminate such a possibility as yet, hence why i mention it. However, again, this is the only circumstance in which passive over-balancing torque could ever be mathematically consistent with over-unity efficiency.
That proviso dealt with, i want to move on to other considerations, now assuming that no such GPE discount is possible; instead, looking at what would necessarily be involved in gaining excess kinetic energy from the drop:
• as noted, velocity-gain per cycle is a direct function of effective G-time per cycle, itself directly a function of RPM
• our objective is thus implicitly deigning to decouple the velocity-gain yield from the drop time / available G-time at a given RPM
• we can raise G-time only by slowing the RPM's - as by inertial torque from the ice-skater effect perhaps, but this necessarily increasing our moment of inertia ('MoI'); any extra momentum we thus gain each cycle is in the 'inertia' component of momentum, rather than the all-important 'velocity' component with which KE squares..
• ..furthermore, converting that momentum gain's inertial component back into velocity requires pulling mass back in against centrifugal force, the gain in rotational kinetic energy precisely equal to the work so done against CF force.
Again, this results in the usual impasse in which your net system energy, after using the ice-skater effect to speed up, is inescapably whatever you had before, plus whatever work you've just performed against CF force. Decoupling the cost / benefit relationship there seems intrinsically impossible, because CF force also squares with angular velocity, just as KE does.
In summary of the above points thus far, then, the final action in an OU interaction cycle cannot be a passive over-balancing torque, and also cannot be work done against CF force per the ice-skater effect. Both interactions are inevitably and necessarily bound to solve to unity.
However, there's another obvious alternative to manipulating G-times:
• what if the weight wasn't riding the wheel down, but was instead torquing the wheel (or else accelerating some other part) against an internal 'pseudo-stator'?
• this pseudostator would obviously be in the pseudostatic reference frame of the gravity vector, sinking counter-angular-momentum to gravity and time while applying torque to the wheel
• in other words, another way to try to decouple velocity / KE gains from a given drop-time is to torque the wheel against gravitating weights, such that the wheel is going faster than they are, ie. no longer dependent on their passive drop time / RPM for the per-cycle gain in velocity and thus KE
Just by way of crummy example, in the mechanism below scissorjacks are employed as linear levers cranking the wheel against a co-rotating part:
..as you can see it's utterly futile as-is, but for perhaps demonstrating that conventional OB torque might not be the only game in town; non-OB torque is possible, and can decouple wheel speed (or that of some other internal part) from whatever effective G-time of the weights as a function of their independent RPM. Evidently something more artful than the above cycle is required however, presumably involving collisions and better coordination..
In conclusion then, the take-home is this:
• if you're designing for passive OB torque, you need discount input GPE to break unity, necessarily implying an effective N3 violation while lifting. IOW if your supposed 'trick' isn't an effective discount in one or more of GPE's 'GMH' components, you're going nowhere.
• if however you feel you've eliminated the possibility of cut-price GPE (ie. getting more lift than actual work done) for that fixed amount of input work, you must squeeze an excess of output work, which can only be in the 'velocity' component of momentum and KE.
Presumably this excess velocity has to be accumulated piecemeal via internal inelastic collisions..
However there is one last point that i believe really precludes passive OB as the main form of torue:
• the speed-dependent characterstic of the exploit
B's wheels had a preferential RPM, below which they gained momentum (specifically its 'velocity' component), and above which, they shed it.
The source / sink can only have been gravity * time.
Over-speeding a passively over-balancing wheel would not incur resistance - it'd quite happily overspeed as fast as you're willing to torque it, you'd think..
But moreover - and here's the real clincher - B intimated that while time-consuming to build, it was possible to embody the exploit in a wheel that turned very slowly, but with great torque, all while emitting a steady chatter.. (paraphrasing)
Suffice to say these characteristics are starkly at odds with those of passive OB torque.
Rather, they can only be consistent with an effective N3 exploit sinking counter-angular-momentum to gravity and time; each cycle involves production of both torque and counter-torque; it is the balance of the asymmetry between them, as regards the momenta sourced or sunk from or to G*t, that is inverting either side of whatever their preferential RPM.
IOW, the hypothetical slow torquey wheel B alludes to is the key to all of this; whatever exploit you're considering, it has be able to explain that high-torque, low-displacement wheel he tells us is time-consuming, but possible. Thus if such a wheel does involve over-balancing weights, it also necessarily involves production of internal counter-torques that prevent the wheel from simply keeling freely to lower the weights. This implies an ongoing asymmetric interaction between co-rotating angular inertias, the balance of which is somehow a direct function of system RPM.
Again, for clarity, i believe we can be certain that the constancy of speed under positive or negative load as when unloaded per Wolff's report can likewise be extrapolated as inherent behaviour to the low-speed / high-torque embodiment B describes; it'll resist overspeeding with just as much negative torque as positive when under-speed.
TL;DR - passive over-balancing torque cannot be the main source of torque in an over-unity wheel, unless you have a relative vs absolute input GPE asymmetry resulting from an effective N3 violation while lifting, however even this could not explain the preferential RPM of B's wheels, which instead is only reconcilable with the RPM determining the sign of an effective N3 break sourcing and sinking momentum (velocity!) from and to gravity and time. In particular, OB torque seems entirely inadequate to explain the hypothetical high-torque, low-speed embodiment B alludes to, or the preferential RPM under positive or negative load as reported by Wolff.
ETA: - i'd also mention MT 41 in relation to this point - the original print using an inflected 'A' on one side to denote that the interaction - although still employing a conventional (and thus inappropriate) stator - can be considered in either direction - radial lifts with angular drops (IE. over-balancing torque), or else angular lifts with radial drops...
