Fletcher wrote:MrV .. I am trying to step back and visualize some type of simple thought experiment that could prove your N3 break and a surplus of mv and KE by default.
Precisely what i used in the calculations - an ordinary elastic collision, unadulterated by gravity - ie. horizontal to it.
Then perhaps that thought experiment could translate into a sim and then real world build as POP ?
Picture a slack length of rope between the two masses. Their internal displacement's 9.81 meters, so make it 9.82 m long.. as soon as the sim ends the rope is pulled taut by the lower mass's higher velocity relative to the upper one's, jerking the upper one into motion and so divvying up the momentum difference, per N3.
I realize that you say it takes either 5 Reactionless Accelerations (RA's) to go from accumulated UU to OU by your reckoning. Why not go for 6 and have internal losses well and truly covered ?
Or 8 for that matter? 200% OU sounds pretty sweet huh?
Anyways, the point being that if you assume the expanding Jack is massless it makes it easier to isolate and simplify. BTW this can pretty much be done in a sim by using massless rods as connections between small nodes (of very little mass). Also you don't need large numbers of cross segments in the Jack. Any linear advantage (accelerations etc) can be duplicated by just two segments, a small one for the Effort and a large dimensions one for the Load. The results are the same as having multiple segments and duplicate what a normal multi-segment Jack does.
Cool tip, will keep it in mind ta.
Alternatively, pulleys fixed to a lightweight linear slider would also suffice - it matters not that the actuator has mass; it's preferable to optimise the ratio of course, but the effect has high margins of tolerance.. like i say,
any degree of cancellation of counter-momentum generates a corresponding momentum gain, of speed-invariant cost.
To continue .. usually if you can visualize a rotary system for a design then there is usually also a reciprocating / oscillating variant you could come up with.
I've tried to stress that angular-angular inertias seem the most build-friendly route. I've only been using exaggerated linear-linear displacements for clarity in demonstrating the gain principle. A build
could be based on the linear-linear examples, but it'd likely be inelegant, with a poor power density. An angular-angular or angular-linear interaction would also eliminate the need for the lower mass and thus any net change in GPE during the inertial interaction.
For the thought experiment what about 2 systems on a see-saw. One Jack extends and extra mv and KE is generated. At the same time the opposite mech on the otherside of the see-saw is contracted and reset. Then they reverse actions. IOW's they pump and reset alternatively.
LOL you basically just ninja'd my "MT 143" observation above..
Yep linear-linear systems are defo viable, but frankly the options are almost limitless - we haven't scratched the surface yet.
Any inertia in
any potential plane or axis (3 of each, 6DoF total), under
any static force field (magnetic, centrifugal (ie. mounted to the wall of a horizontally-rotating drum), inertial (ie. atop a rocket), bleedin' knicker 'laccy for all i know; force is force, F=mA, and whichever one's used, no net work's required from it.. it just has to hang there and set the atmosphere for the ensuing inertial interaction, which could be between anything from permanent magnets to solenoids to freakin' fluids), the process is scale-invariant so could be tiny or huge, the potential power densities almost arbitrary.
So look beyond the linear-linear examples - get the concept, and apply it imaginatively to try come up with an elegant implementation. As i've said, much smaller bites of momentum / angle seem the way to go - what would seem ideal would be an angular-angular interaction, possibly featuring a radial component to coordinate the interaction and feed back a CF workload. This could then be segmented around a wheel in the usual manner, also facilitating any interactions between mechanisms that might be useful.
An oscillating system OTOH (per "MT 143" would have to generate and harness all its momentum gain on each alternate stroke, presumably, so won't be able to build up much, unless its amplitude is governed by a spring or something. It could benefit from the single-strike-to-OU interaction i demonstrated above tho.. Prime it to its unity threshold with any kind of PE and generate excess momentum on the first stroke.. and every subsequent one..
If there is a surplus of mv and KE then one side of the see-saw should be able to reset the other, should it not ? Does this get us any closer to a POP concept ?
Strictly, there's no 'excess' of momentum until the moment it's rectified / consolidated.
Prior to that closing elastic collision, there's exactly the right amount of momentum, corresponding to the energy invested.
It's just unidirectional - of one sign only.
And because of that, a subsequent regular N3-consistent collision between them equalises their disparity to a positive sum...
That is the gain; when the unidirectional-but-fully-accounted-for momentum difference is equalised between both masses, their new shared ambient velocity has a higher KE value than when all of that momentum difference was manifested on one mass only, due to the v^2 multiplier (increasing energy magnitude when ascending, and decreasing in descent, so 1 kg-m/s
more is a greater change in net energy than 1 kg-m/s
less).
So it settles the system (of two interacting inertias) into a new equilibrium velocity - instead of say a 10 kg purchase of momentum being served as +5 and -5 kg-m/s portions, ie. a net zero change, we've bought it all of one sign only; +10 on one mass, and fat zero on the other. Thus a subsequent elastic collision equalises that unidirectional momentum between them, and both end up with +5 kg-m/s, accelerating the net system by this amount, and so causing a corresponding KE rise, for the same
internal work we'd've done - the same amount of momentum we'd've raised - whether the resulting division of momentum had been symmetrical or not. Either way, we paid for 10 kg-m/s, but obviously, if it's of one sign only then it's
worth a bit more (per #RA's / 4 or whatever Marcello's formula was) - and because the RKE is subject to the RPM^2 multiplier, but the inertias generating the momentum gain are speed-invariant, the unidirectional momentum gain is worth more as RPMs accumulate.
The take home point is simply that the energy
cost of generating this momentum gain is purely that of the inertial interaction, which costs the same energy regardless of whether it's gravitationally-skewed or not, and regardless of the ambient velocity at which the masses are moving (since inertia's not speed-dependent), whereas the energy
value of the accumulating momentum gains are a function of their rising velocity.
By the innate mathematics of nature, 4 consecutive reactionless accelerations applied this way result in precisely the same energy and momentum conditions as would be arrived at via any conventional, N3-compliant form of accelerating mass.
Less than 4 RA's result in an under-unity result, and more than 4, over-unity. 8 make 200% OU.
It
is a lot to digest i guess (esp. the way i ramble on) but stick with it, it's actually pretty straightforward, and can be reduced to just two inertias, one of which is subject to gravity during part of a full cycle. The rest is just the energy feedback system.
As i've shown, we have two basic design options - i'm kinda thinking of them as "2-stroke" vs "4-stroke" - not so much to do with the actual stroke counts, as one's quick'n'dirty and needs a good bump-start, with the other being more controlled and methodical, but also OU from a standing start.
We've all the time in the world to let it sink in and start trying to apply it, just familiarise yourself with the I/O energy asymmetry for now, check my maths, replicate the method and results, error remains possible (if not likely), understand the bounds and flexibilites, and with a little imagination our systems should largely design themselves.
I've tried to give as lucid an account of these findings as i'm able, so while it
is a bit a sprawl, if you've any glaring questions there's a good chance i've addressed them on recent pages already, tho anything i can help clarify, just ask..
