Decoupling Per-Cycle Momemtum Yields From RPM

[MrVibrating]

Yes! You're both bang-on - esp. you Robin; how well do i really know what i think i know? Obviously all i have is a few waypoints, on an incomplete map..

So, 'back to basics'..


Here's a 1 m radius wheel, with a 1 kg weight attached.

Both have an MoI of 1 kg-m², for a net system MoI of 2 kg-m².

The GPE is thus 1 kg * G * 2 meters = 19.613 J.

The weight drops from TDC to BDC, ten times, incrementing the starting speed by 1 rad/s each run.

The angular momentum rise is then plotted, along with drop time and KE:

Code: Select all

Start Speed	Drop Time	AM Gain		AM Gain		KE Gain
						                     _________
						                     Drop Time	


 1 rad/s		1.288 s		7.080 L		5.497 L/s	19.613 J

 2 rad/s		0.961 s		5.719 L		5.951 L/s	19.613 J

 3 rad/s		0.769 s		4.698 L		6.109 L/s	19.613 J

 4 rad/s		0.637 s		3.935 L		6.178 L/s	19.613 J

 5 rad/s		0.541 s		3.359 L		6.209 L/s	19.613 J

 6 rad/s		0.469 s		2.915 L		6.215 L/s	19.613 J

 7 rad/s		0.412 s		2.567 L		6.231 L/s	19.613 J

 8 rad/s		0.367 s		2.288 L		6.234 L/s	19.613 J

 9 rad/s		0.331 s		2.061 L		6.227 L/s	19.613 J

10 rad/s	   0.301 s		1.874 L		6.226 L/s	19.613 J

As you can see, each drop converts the full GPE to KE.

The AM gain decreases as a function of rising speed and decreasing drop time, as expected.

However there is one new detail here i hadn't noticed before - if we divide the AM gain per drop, by its drop time, we see that the momentum gain per unit time is actually increasing with speed..(!?)

This must have a simple explanation, though it currently eludes me..


As ever, attaining OU efficiency requires fixing the unit energy cost of momentum invariant of RPM.. so this experiment is just trying to scope out the situation, and maybe get a better handle on the barriers between us and that goal.

Already tho we're seeing this increasing L/t dynamic that i'd never have guessed was there..


Not sure what to try next.. but thinking of maybe adding a vMoI, to increase or decrease the net system MoI by, say, 1 kg-m² over each drop. I could then plot the same data, plus CF-PE (+/- CF workload)..

..again tho, no idea what i'd be looking for, other than better understanding the dynamics and spotting anything else i've missed.


This increasing L/t with speed is a surprise tho, eh? I'd've thought it'd be a constant function of drop time relative to gravity's constant acceleration.. or if anything, decreasing with RPM, just as the momentum yield does.. For now, i've no idea why this is happening..

[MrVibrating]

Just double-checked: momentum per second peaks for the 8 rad/s starting speed, then slightly decreases again.. Weird huh? Is this a meaningless metric, perhaps? Or else, what does it mean?

[Fletcher]

I think only by revisting things over and over, do we have the opportunity to perhaps one day examine relationships and cause and effect in a different light and perspective. And maybe in that tedious analysis and familiarity find the chink and exploit it.

We need to look at every aspect with absolute objectivity and the biggest obstacle to overcome to achieve absolute objectivity is the illusion that we already have it.

I agree. It is a matter of perspective I think. How you look at what you are experimenting with and experiencing, and then how you in assimilate that information, into known Laws and Math. I'm not sure any of us can be completely objective, but we can try to be.

Personally I believe that Bessler's wheel was a simple gravity wheel. And this belief is at odds with MrV because he is equally sure it is not the answer. That's fine by me and I still read his threads because he is not short of a patience and the willingness to learn something from the dissection and analysis processes. And I might too.

If you remember B. said that he came across the right mechanical path to a working PM wheel (paraphrased). And he knew why ALL THE OTHERS had been wrong ! [ *** EVERYTHING else had been the wrong approach ! ]

This suggests to me that his epiphany moment was very clear. He foresaw the mechanical circumstances (arrangements) where a gain in system momentum could be had for free. Probably first a mechanical concept gleaned from his dream. Then a mechanical experiment to confirm a movement. Then a 'runner' as proof of the system gain reality with all its mechanical clarity and exactness.

You will note I don't suggest a hypothesis or theory approach, along the lines of the scientific process. Newton's Laws were published but torque and forces were not well advanced, nor discussed in those terms by B., or anyone else in those times.

I think he was a hands-on practical man who built things and tried them out.

We do that too but also with a tinge of insurance. In that we reference back to the Newtonian Laws and the Mechanical Laws (which prove the impossibility ?!).

B. was not encumbered or restricted by that circular thought process. And so he was free to explore something we would consider on-paper impossible, if not for him.

We have to free our minds and range to the same degree imo ! And to do that 'openly and objectively' seems at odds when not including the base Laws, lol.

[MrVibrating]

Notwithstanding that 10 rad/s is already over 95 RPM and well above our expected operating range, i was curious enough to raise the sim freq to 10 kHz and plot out the changing momentum gain / unit time to 20 rad/s, just to see where it goes:



Green highlights the increasing trend, and red for decreases.

The overall trend is obviously that it's converging to 6.252 L/s..

..yet doesn't the variation seem strange? At 10 kHz each run parks precisely at BDC, 0.5 turns on the dot (to the three sigfigs used), and the KE comes in right on the money every time.. despite this consistency however, the momentum rise per unit time seems to have these quite-empirical dips..

Maybe they're rounding artefacts? Either way it seems an insignificant variance.

What seems much more certain is the increasing trend towards this convergence at 6.252 L/s - up from 5.497 L/s at 1 rad/s.

On the one hand it seems somewhat reassuring that it's settling at a constant value as starting speed is raised.. but on the other, 1 rad/s is almost 10 RPM and a fair old clip by any measure.. yet the momentum gain / unit time is clearly starting out at significantly below that convergent value.

It's prolly a minor detail either way, yet doesn't seem to be in error, and i'm flummoxed on what might be causing it.

No matter; the general trend is that from 20 rad/s, the drop gains around 12% more momentum per unit time than when dropping from 1 rad/s; this is surprising, to me at least, if not particularly useful at this point.. i honestly expected it to be a constant value throughout.



Just to draw a line under it, tommorow i may redo the sim using a spooling weight, rather than rotating it down, so that gravity's force is no longer subtended by the angle of rotation. This will presumably straighten out the data somewhat, just as a cleaner point of comparison..

After that, i'll add a vMoI and start delineating momentum gain as a function of velocity versus MoI increase. This will collect some of the output GPE in the form of CF PE instead of KE, and could be set up in a variety of ways; for instance the negative inertial torque could be applied to perfectly cancel gravity's acceleration, thus causing the weight to fall at constant speed: would this cause all 19.613 J to be collected as CF PE? RotKE being a linear function of MoI, while a squaring function of speed, the reality may not be so clear-cut..

Enough for now tho, me head's spinning already..

[MrVibrating]

@Fletch - my overriding problem with 'energy gain from OB' as a concept is the time-invariance of G, M and H - if we can't lift a weight any cheaper, then we have to get more from its drop..

..yet, all else being equal, per-cycle momentum yields naturally decrease with RPM! (is it the inverse square of RPM? Or mebe inverse sqrt? Who knows, Marcello will prolly figure it out. You can see the trend tho)..

And then you have this simple axiomatic truth, that if we could only achieve a constant momentum yield per lift & drop, then input work done scales as that fixed cost times the number of elapsed cycles, while output KE squares with every doubling of velocity.. so if MoI is constant over each cycle (ie. the wheel's not changing size or mass), yet KE is > GPE, B. must've been fixing momentum yields.. this just seems an inescapable conclusion. Hence this thread's objective..

His one-way wheels evidently were gaining momentum from over-balancing, no room for doubt there.. but they couldn't've been scrimping on the input costs, and must've been subject to these diminishing L/rpm returns.. which can only mean that the momentum gained from OB must've been augmented with additional momentum - from gravity and time - yet which wasn't subject to these same constraints of rising RPM. Hence why i'm fixated on the notion of an additional GPE output dropping radially, rather than being rotated downwards on the descending side with the regular OB weight. And that's where i'm currently snookering myself, since i can't see how to gain system AM from a radial drop..

So i do believe OB is a key part of the solution.. but for the reasons above (and throughout this thread), i can't see how that alone can deliver the goods..

[MrVibrating]

OK here's the linearised version:




..and here's the data:




..so pretty much as expected; the variance is indeed due to only working with 3 sig digits, at sub-optimal sim frequency. If i had the patience to max out the freq, and maybe go up to six digits or so, it'd obviously resolve perfectly to KE = GPE, and dp+dL/dt = G.

Likewise, the difference in dL/dt in the previous purely-angular version is a function of the effective force of gravity as a function of changing angle of rotation; eliminated when the weight's constantly aligned to the G vector.

So next up, i guess i need to take a closer look at what's happening when MoI varies during the drop. For this, i'll probably just add a vMoI to this spooling rig, since it produces more familiar dp/dt figures - there seems little point proceeding with angular OB until i find some kind of design objective..


Edit: meters re-labelled correctly

[MrVibrating]

Just to provide a quick focal point to keep eyes on the prize here, let's take the worst momentum yield from the last two examples, and multiply it up over many cycles:

• this'll be the 20 rad/s yield from yesterday; 0.969 L for 19.613 J

• let's suppose we can buy at that rate for 100 consecutive cycles..

• ..so that's a total of 96.9 L, for 1,961.3 J down..

• MoI on that rig was 2 kg-m²

• 96.9 kg-m²-rad/s into 2 kg-m² = 48.45 rad/s

• 2 kg-m² at 48.45 rad/s = 2,347.403 J

• 2347.403 / 1961.3 = 119.6% of unity

Thus demonstrating that no matter how expensive a fixed momentum yield is, there's always some threshold speed above which we're OU, and below which, we're under.. all that matters is that the cost is fixed, invariant of rising RPM.


For a more optimistic example, let's take the base rate at 1 rad/s for the same rig, of 7.080 kg-m²-rad/s for 19.613 J a pop:

• let's try just 5 such cycles this time:

• 5 * 7.08 = 35.4 kg-m²-rad/s, and 5 * 19.613 J = 98.605 J total in

• 35.4 kg-m²-rad/s into 2 kg-m² = 17.7 rad/s

• 2 kg-m² at 17.7 rad/s = 313.29 J

• 313.29 J out / 98.605 J in = 317.7% of unity


Over 300% in just 5 cycs! Appetising eh? Let's tot it up again, 1 cycle at a time:

• 7.08 L on 2 I = 3.54 rad/s, and 2 I @ 3.54 rad/s = 12.532 J / 19.613 J = 63.8% of unity (ie. 36.2% under-unity)

• 14.16 L on 2 I = 7.08 rad/s = 50.126 J / (2 * 19.613 J) = 39.226 J = 127.7% of unity..

So in fact we'd only need to be able to buy at that base rate twice in succession to clear 127.7%.. but maintain it for another 3 cycs and we're over 300%.


So peering behind the scenes of these perfectly-boring tables of unity results, if we were to pick any one of them, from 'best' to 'worst' - no matter how bad - there's always a threshold number of cycles beyond which I/O efficiency is OU!

All we need to do is fix the unit energy cost of momentum over n cycles - the better that fixed rate, the lower the value of n.


This is not 'a physics problem', people.. Bessler has proven that it's merely an engineering problem; fix the unit energy cost of momentum from gravity and time, and the energy efficiency takes care of itself..

If anyone still doesn't get the maths, keep going over these examples till it clicks - just take any single result from the past two day's measurements and multiply it up while comparing net input PE to net output KE - the cheaper the yield, the less we need to accumulate to punch through the unity threshold..

[MrVibrating]

Here's summink fer yuz all to mull over w' me:

• instead of fixing the weight to the wheel to cause OB, suppose it's attached to a radial or diametric lever

• if it thus drops on the descending side, as before, then it'll under-balance that side whilst dropping; furthermore, the available G-time per cycle, and thus momentum yield, is obvs going to decrease with rising RPM

• ..but what if it drop's on the ascending side instead..? So now it'll underbalance the 'correct' side..

• .. but furthermore, is available G-time per cycle still decreasing with RPM..? Or else, is it increasing, eh? Eh?

Must be a simple way to measure this, if correct.. mebe i was too het up about the prospect of radial drops - perhaps, rather than just insulating the drop time from the descending side of the wheel by dropping thru the center instead, we might actually invert the dynamic entirely and actually raise effective G-times as a function of rising RPM's, simply be dropping levered weights on the ascending side..?

Or mebe i'm talking complete shite, again..

[MrVibrating]

Well, spent another evening checking what i'm sure i've checked a dozen times already - here a vMoI is used to apply negative inertial torque to counter gravity's acceleration, causing the weight to drop at constant speed:




• CF PE = output GPE minus the KE rise

• momentum gain is 19.613 L, over two seconds = 9.80665 L/s

..not much else to add, besides the observation that i'm clearly wasting my finite time on God's green earth with this shit.. bottom line is that gravity's a time-constant rate of change of momentum, invariant of MoI variations, and if I/O energies are in the same FoR then they're symmetrical; whatever it was i was hoping to find, it obvs ain't here.


Back to brooding over how to fix momentum yields in an accelerating system..

[MrVibrating]

Whatever B's 'epiphany' was - in that moment when he realised what worked, and why everything else didn't - whatever the mechanism was, the maths can't lie; it had the effect of stabilising momentum yields, in spite of rising RPM.

That's the goal, however it is we get there. I (we all) need to focus on that singular objective..


ETA: lemme put a finer point on that; since L per G-time is obvs constant, OU is only gonna arise if we can stabilise G-times invariant of RPM.

So a weight or weights from which system angular momentum is gained, needs to spend the same time falling / in the gravitating condition, regardless of some range of rising RPM. There ya go, that's the 'bullseye'. Everything else is just details / excuses..

[Georg Künstler]

Fletcher Wrote:

The Bessler answer is simple !

Somewhere in the basic mechanical interactions we explore is a chink in the Newtonian armour.

There is no chink and no violation of Newton's Laws.

The problem what is not seen is the indrect impact. I tried to explain it several times.
The indirect impact is producing torque in the Besslerwheel.
Mr. Vibrating is looking for different accelerations of the masses during one rotation.
An indirect impact does exactly that, different acceleration of the masses.

Bessler said that the force has to be applied in the correct angle, so it is an other construction detail what he offered.

[Fletcher]

There is no chink and no violation of Newton's Laws.

I wholeheartedly agree that there will be no violation of Newtonian Laws of Motion in the mechanics of a Bessler 'runner'.

Once those mechanics are known then this will become absurdly apparent.

So where is the 'chink' (narrow opening) that you don't believe is there ?

It relates to what Bessler said, as highlighted last post by MrV ..

Whatever B's 'epiphany' was - in that moment when he realised what worked, and why everything else didn't - whatever the mechanism was, the maths can't lie; it had the effect of stabilising momentum yields, in spite of rising RPM.

"Why everything else didn't" : Bessler wrote the following about his experiment in motion. n.b. imo he didn't build his prototype and functioning complete 'runner' until after this episode/experiment that so excited him. (I allow for translation nuances that seem to suggest a complete runner at this early stage, however B. says later when he built his first wheel, so this was an experiment (the chrysallis idea) that gave him extreme confidence that he had found a narrow gap to drive his cart through).

"AP Chapter XX11 Pg 269

For I put together the very first device which could [spontaneously] revolve a little. I saw that I had finally made the right choice, and why the earlier ones had been wrong. My heart leapt for joy at the sight of this genuine Mobile."

Then you inevitably ask yourself (at least I did) what were the common denominators to his earlier attempts at PM gravity wheels (see MT) ? Were there any significant mechanical departures, earth-shattering leaps of logic evident, in attempting to generate motion and accumulate momentum, in MT ? My answer is NO, nothing is overtly shown as that, imo. They are all caterpillars, some hanging to form chrysalises, and no butterflies lol.

It is my opinion that Bessler, for wont of a better simple descriptive, radically changed the mechanical DNA (approach) for generation of movement and the gain of momentum within his gravity wheel. And the morph was not at all similar to what he had (and many of us) endlessly experimented with beforehand. Such as the ubiquitous Bhaskara wheel and his MT13 adaptation (tho I believe it is suggested it can morph into a butterfly, as could a few others imo).

So whether you believe a chink exists or not is of no consequence to me. It is semantics. We all, including yourself and MrV, want to find that elusive mechanical morph/condition that allows an innate gain in momentum. And I would consider that a hole I would happily drive my jalopy thru, whatever you call it ;7)

[MrVibrating]

We can only gain momentum in an otherwise-closed system by manipulating up vs down swing times / G-times. This is what basic OB techniques and swinging / kiiking share in common.

MT starts with the former, investigating potential GPE I/O asymmetries, but not for long - look closer, bearing in mind this distinction that inflected 'A' is used to denote drivers versus things driven:

• MT 30 - 42 are investigating CF - GPE interactions and vice versa, culminating in MT 42 which applies both together on alternate sides

Or else radial vs angular. 'Squares' vs 'circles'. The anvils of the hammer toys on the Toys page also observing this convention.

Or MT 83 - one's sides obvs 'inputting' work, the other 'outputting it - were the picture animated, the inflected 'A' would be alternating sides each stroke.

MT 84 - a scissorjack 'swings' a cube weight. MT 85: the human figure is the driver, swinging / kiiking / gaining momentum from manipulating up / down G-times. MT 86: the action's inverted, the 'swinger' now driven by the pumping buoyant floats.

Point is, the method to MT's madness is investigation of inertial / GPE interactions, in every I/O permutation. That's the theme being developed, and culminating in the Toys page.

The torque sequence (straight 'A') on the right side is an output / something driven by the interplay of the hammer toys 'C' and 'D'. 'B' is the axis / axle, and the left / right notches running down its length are 180° rotations, aligned to the upper hammer toy 'C' with its circular anvil; so, a GPE / OB interaction, then.

The lower toy 'D' must be some kind of inertial interaction, with its square anvil, and furthermore is powered, the handles indicating an input of energy is required: output GPE (G-KE) from toy 'C' is insufficient to drive 'D'; so there must be an onboard supply of PE (presumably from spring/s) to drive it. Its action is 90° out of phase to toy 'C' (so, a 'cross-shaped' arrangement), yet not synced by its own set of 'notches' on the axle 'B'; hence activated / synced by means other than angle of system rotation.

An output of CF work / radial extension can lift a weight, or conversely, dropping a weight can retract mass against CF force, either way generating negative or positive inertial torque. Either way these interactions can only be work / energy-symmetrical.

The upturned whistling top seems to signify CF force (rotation) performing work against gravity.

Neither GPE nor inertial interactions alone can break unity; the KE you're left with afterwards is precisely whatever you had before, plus or minus whatever work you've just input or output in either field.

This last point is the real party-pooper, here - what can break unity, then?

The answer - the only one mathematically possible - is between the lines of those tabled momentum yields from the other day; pay the same energy / work done, to buy the same quantity of momentum - ie. just repeating the exact-same deal - multiple times in succession, and after n cycles, KE > PE. Prior to n, KE < PE, and bang-on that number, they're at unity.

So in summary, we're only getting any momentum at all from G & t. From GPE to KE outputs, specifically. To do so, we need to manipulate I/O G-times, and the only way to do that is via inertial interactions. Thus to break symmetry between them, input energy to the inertial workload has to be in a different (accelerated) reference frame, relative to the output GPE workload.

The maths say that if we can buy momentum at its nominal velocity rate per the KE equations, then we reach OU at the 2nd cycle.

As V increases, so does the energy cost of momentum, and thus the number of elapsed cycles of fixed unit-energy purchases of momentum required to reach unity and above.

The Toys page seems to depict 'something extraordinary' at five cycles. This means each individual cycle is only 25% efficient, each wasting fully 75% of all input energy per cycle. Those losses must either be dissipative - ie. as from inelastic collisions / braking, or else correspond to momentum sunk to gravity (such as ie. counter-momentum).

This also means that the Toys page interaction must carry a store of PE - presumably elastic / sprung PE - in order to work those handles five times on the trot, before there's any chance of hitting pay-dirt. Gaining energy in just one or two cycles is thus apparently not a likely possibility. IOW, forget about a GPE-CF interaction gaining energy on the first swoop; the only rational objective is to accumulate momentum at this fixed rate cost, dissipating most of the input energy on the way up.

So whatever the mechanical solution is, it has to be led by these inescapable mathematical constraints. One way or another, Bessler's epiphany had to involve 'speculate to accumulate' with regards to buying momentum..

[MrVibrating]

OK in the interests of just doing something other than nothing (or just vapidly gassing), i'm setting meself a short-term goal:

• one doofusly-too-literal interpretation of the Toys page interaction is that the hammer toys, due to each pair of characters facing each other, both rotate their levers in the same angular direction each shift; ie. on each toy, both levers are rotated clockwise, then both rotate anticlockwise back the other way

• the output torque train 'A' features a pair of torques both going in one angular direction for 180°, but only one going back the other way for the second 180°, capiche?

• IOW, it doesn't seem too far a stretch to suppose that the torque train 'A' might be depicting the counter-torques being applied back to the wheel axis by these lever rotations, right?

• each hammer toy is depicted in opposite phase - so one's tilting left while the other tilts right..

• ..furthermore their alignments to the axle 'B' imply they're 90° out of phase..

• ..so for instance the lower hammer toy begins each shift just as the upper one is mid-way thru its shift..

• ..thus the second pair of counter-torques it's applying interrupt the peak of the first pair..

• ..could the resulting interplay / interference thus reproduce the basic 2 - 1 - 2 -1 sequence of torques (counter-torques?) depicted by 'A'..?

It'd need to exploit gravity and time too of course, elsewise it'd just be a wobbler per N1. Dunno how that's gonna tie in, besides trying to cause a consistent over-balance..

Like i say, better to be getting on with something other than dicing word-salads..

Designing the system with data acquisition being the priority, i'm gonna start out with just a long lever attached to the background by a motor, driven by code that smoothly swings it up and down - without any sudden blips in the torque profile.

If i can work out how to do that cleanly, then i can attach two pairs of 'em to a wheel and see what happens..

[MrVibrating]

Step 1: Program motor:



..so a weight lever, bob on one end, motor on the other.. i want it to smoothly swing thru mebe a 60° arc - just as a pendulum would; gradually accelerating, reaching peak velocity mid-swing, then progressively decelerating into each apex.

Obvs. if B. was using levers, their motion would be regulated by whatever powered them - maybe a drop-weight or spring, whatevs - in sim-world however we can shortcut all that faff and just motivate 'em directly with active constraints for hinges, since all i'm really interested in for now is the interplay of counter-torques 90° out of phase..

So what are the best options for a clean control logic?

My usual go-to resort of referencing ie. the +/- x or y positions of the motor itself as a function of the wheel's rotation relative to the background is not gonna be ideal here, since i want the wheel rotation to be a result of the counter-torques being applied back to it.. IOW i want the levers to swing in their own time, and any rotation of the wheel, consistent or otherwise, as a resulting reaction..

The first option that thus comes to mind is taking a signal from the clock: to somehow convert the 'time' signal into a value that oscillates between say -30° and +30°, for a 60° arc relative to a wheel body, rather than the background.

Subtracting the motor angle from the wheel angle is easy enough; the real challenge is converting the 'time' into this smoothly-graduating 60° range.

From bitter experience, the more complex the controller logic - however effective it appears to be - the less reliable the torque * angle or power * time data will be. So whatever solution arrived at, it'll first need to be checked to ensure it reliably solves to unity before proceeding further..

Any suggestions welcome at this point.. i'm sure i'll eventually come up with something, but with my shite maths it'll prolly be summink kludgy & blippy..

So for starters it's a problem of engineering maths rather than physics per se..

OK here's a first silly suggestion:

• i could just make a basic pendulum as a reference input - ie. hide it from view or place it 10 meters off to the side - and then simply plug the output from its angle reading into that of the motor here..?

That's kludgy tho, right? And i'd then have to subtract its energy from any later usage of the 'kinetic()' function to just get the net energy of the actual system of interest..

..anyone got a better idea? Maybe using cos/sin functions or summink?