I was just finishing up a more polished version with telemetry but then the sim crashed, so this'll do - and it's obvs a no-go.
What's immediately obvious is that the initial counter-rotation of the net system from the spin-up phase subtracts from the angular velocity of the second axis on the diametric lever, balancing its effective AM at all times against that of the counter-AM on the floating base axis.
If the anomaly had been real, spinning up +1 kg-m²-rad/s on the radial axis would've applied -1 kg-m²-rad/s of counter-AM to the base, and then switching over to the diametric axis and braking would've unloaded +2 kg-m²-rad/s, ending with a final system AM of +1 kg-m²-rad/s from a standing start. Wouldn't that have been magic? (precisely eh)
• a conserved quantity of angular momentum is given to the longer (diametric) axis..
• ..which then coasts around, holding that AM, until it aligns with the shorter, radial axis..
• ..at which point the radial axis unlocks, and then immediately after the diametric axis locks
Hence the wheel is free on its axis prior to locking the lever, thus in principle able to receive and express / embody all of the conserved AM transferred to it the instant the diametric axis locks; ie. preventing that AM simply grounding via the wheel axis.
If all of the AM was so conserved, the final 2 kg-m² of inertia would've had a speed of 1 rad/s, and hence 2 J of KE - 4x more than the startup energy.
Equally interesting however was the null result, insofar as net system AM halves even though the axis is free-floating and thus inertially isolated (ie. not exchanging torques with its environment). Similarly, even though the inverse sequence involves grounding 1 kg-m²-rad/s upon locking the wheel axis, the net system AM remains at 2 kg-m²-rad/s, ie. showing that the remaining system AM on the lever has doubled.
From the stationary frame as modelled, and since each axis was only moving one at a time independently (no compound rotations), the angular velocities remain valid with respect to the absolute or ground FoR, so ie. 1 rad/s indicated is the actual objective speed, and likewise 1 or 2 meters of radius is also absolute not relative, hence the angular momentum distributions being metered remain valid - it really is spontaneously doubling or halving, in an ostensibly-closed system.
The prospective utility of such an effect would be in providing a potential source of momentum that doesn't require torquing against the world outside - ie. some kind of stator - since the energy cost of momentum sourced that way necessarily squares with rising angular velocity, thus enforcing PE:KE symmetry. Thus in a true PMM everything must, of necessity, go around together - and something like this would offer an ideal form of reactionless acceleration, its unit-energy cost of accumulation summing over successive cycles and rising net system RPM, whilst KE squared. In effect the reference frame of the onboard workload sourcing the momentum would've been rotating with the system, like dragging your reaction mass along with you and cyclically re-using it (ie. 'boot-strapping').
In reality of course the only identifiable source that might fit this description is gravity and time - spend more time under its constant acceleration and less under its constant deceleration, and you gain momentum equal to the I/O ±dp/dt. Per-cycle yields are thus necessarily RPM-dependent, and Bessler's solution somehow embraces this, having positive G-time below a given wheel's design speed, negative above it and obvs no gain over rolling losses when at that speed, which he stresses may be arbitrarily-slow (if more demanding in construction). I know that's vague and general, but it's also the closest we can get to a peak inside. To this end, the above quirky inertial interaction is just a passing distraction (there's no net momentum anomaly when the startup counter-AM is included in the closed system, which is what the final sim demonstrates).
The actual solution is necessarily going to fulfil certain characteristics:
• it fixes the unit energy cost of momentum
So a given amount of input work achieves the same amount of system AM gain, invariant of rising RPM
• the system's G-time symmetry starts out positive, approaches unity minus losses at whatever the design-speed, and inverts to negative if over-sped
This to satisfy details of speed-regulation under varying loads as noted by witnesses, as well as direct claims from Bessler (ie. the system 'gained yet more advantage from applied loads' as they protract the G-time asymmetry that's causing the net torque. Same reason the wheel would lift a big fellow off his feet when forcibly-stopping it).
I think the mechanical solution involves oscillation of diametric levers, each swing phase representing a pair of inertial interactions with the wheel axis and thus potential opportunities for manipulating components of ± momentum distributions via overlaid gravitational interactions. Basic logic alone however tells us that 'KE gain' is a misconception (a system can only have 'the right amount' of energy for its given inertia and velocity distributions), hence the only effective CoE break that can be coherently formulated is some form of PE discount, ie. breaking PE:KE symmetry, which means gaming the FoR's of the input workload's 'velocity' and/or 'displacement' metrics. The Toys page is depicting a cycle involving two inelastic collisions between equal inertias, each thus dissipating 50% of per-cycle input energy in succession, for a 25% CoP of mechanical work conversion / AM gain. This accumulating system RPM is transposing the KE value of the internal PE expended - ie. the FoR of the input workload has diverged from the ground / absolute frame of the KE - thus increasing its efficiency by 25% with each elapsed cycle. This sequence reaches PE:KE unity at the fourth cycle, and 125% at the fifth. I think the lower hammer toy depicting children represents the diametric levers, and that the upper toy featuring the thresher and scholar represent alternate strokes of an MoI variation, per the kiiking effect.
The wall i've hit is simply this:
• we can gain AM from G*t by doing work against gravity, and / or against CF force
If we go for GPE workloads, then we face the problem of G-time yields being inverse to RPM (less time under gravity's constant dp/dt each cycle) - they drop off by the inverse square of velocity, thus enforcing PE:KE symmetry regardless, by throttling the effective CoP as a function of RPM (but note the prior observation that B's solution somehow embraces RPM-dependence)..
..whereas if we choose CF workloads, then all else being equal, CF force also squares with angular velocity and thus rising system RPM, again enforcing PE:KE symmetry.
So, there's evidently some means of negotiating these constraints and consolidating dp/dt at fixed unit-energy cost, at least up to some predetermined RPM, above which the system will likewise destroy momentum and energy at the same linear, speed-invariant CoP, in trying to settle back to its 'punctum quietus' of minimal or tick-over ± G*t asymmetry at that design-RPM.
What regulates the design RPM such that it can be arbitrarily-slow, yet with many more interactions per cycle? This, it seems to me, is the most profitable avenue of enquiry. Some counter-torqued component that thus counter-rotated when below a certain system speed - at least on the return stroke of an oscillation - and achieved relative stasis when at that speed, then turning positive when over-sped? Or else one might turn to G-time symmetry as a function of an over-balancing system that runs at finite speed, hence as RPM's approach that speed the angle of over-balance starts to drop down towards BDC, then climbing back up the other side if over-sped further.. Yet things can't be this simple, since a statorless system will keel freely under over-balance.. hence how could the design-RPM be arbitrarily-slow?
These are the real issues to be grappling with, the conceptual waypoints towards mechanising the maths of a coherent solution. The 'inertial quirk' explored above is just a distraction from that self-illuminating path..
Last edited by MrVibrating on Sat Nov 19, 2022 3:38 pm, edited 7 times in total.
