I'm afraid there's another dimension here.
The vector of the weight at any given moment. This differs from that of the wheels at the weight's current location, because the weight travels a different path. Good luck with that math, I can't do it.
At any point between its ultimate slot positions, the weight are administering a differnt force on the wheel than its current location would suggest. Let alone when hitting the slot's stop.
The longer the relative length of the slots, and the free-er the movement of the weight, the bigger the effect I try to explain.

Have you tried using the 'Golden Ratio' in some of your 'hypotheticals'? Funny numbers like PHI (and powers of PHI...) have a way of showing up a lot in nature (natural math!). Something to play around with anyways.
Have you tried the same idea using curved slots?
PC

   @All,
 Here is a quick math break down that shows the over balanced side would travel
about 2 inches more to rotate 180 degrees.
 The design allows for 10 degrees of rotation in addition to the 10 degrees of radial shift. This is why the movement upward allows for 190 degrees of movement. When the weight moves towards center, if the wheel does not rotate, the weight will tarvel and extra 10 degrees of rotation. And if the weight moves inward with 10 degrees of upward travel, then it will travel about 1 inch less moving upwards.
 edited to add; the math referenced if for an inner position 10cm's from center. Over balanced position is 10cm's from center and 3cm's @45 degrees (paralell with the plane of the axle when it can move outwards).
                                                                         Jim
 
 degrees % of rotation
115 = 63.8%
 20 = 11.1%
 45 = 25.0%
10.07 = 63.8 %
8.885 = 11.1 %
7.07  = 25 %
1R Phi x % = length
10.07 * 3.14 * 63.8% = 20.17
8.885 * 3.14 * 11.1% = 3.096
7.07  * 3.14 * 25.0% = 5.549
----------------------------
                      28.815
 
degrees % of rotation
55   30.55%
20   11.1%
115  58.33%
10.07 * 3.14 * 30.55% = 9.659
8.885 * 3.14 * 11.1%  = 3.096
7.07  * 3.14 * 63.88% = 14.18
------------------------------
                        26.9

The math is extremely simple.


If the weights will end up in the same position after one complete revolution - regardless of its initial position, the result is already given.
No output.


Vidar.

Jim,
What I wanted to express, and you may well be onto that for long, is that with the weight moving inside their slots, their actual radial load on the wheel is different from a static one. You could say the wheel is exerting energy to throw the weight into a wider orbit. It's costing energy, because the radial speed of the weight is supposed to be increased at the same time.
There should be a gain on first glance from overbalance, but there isn't because you can't get the weight into the wider orbit AND at increased radial speed. There's not the gravity available to do that. It's just 9.8m/s² on tap like the rest of the wheel is bound to.
We need the weight to behave in a certain way. And it it would, we'd have OU right there. Unfortunately, the weight just uses up all the potential energy it's got, for the vertical loss available. The outer orbit has a kinetic energy requirement gravity cannot account for. Due to the angle of the slots, whichever angle or curve you choose.
 
To understand why this wheel doesn't work, might set you on a path to find the ultimate clue, or cheat it.

Until, the search is for a rock that has yet to fall, to use for gravity extraction. Or, a hole that has already been dug.
Anyone ever calculate how much energy it would give to let the surround height slide into the Grand Canyon in a controlled fashion?

   cloxxki,
 Remember when you mentioned that the weight A has unused energy ?
I have thought of something. If when it stops, it's stop is at an angle, it's
force can be directed in the direction the wheel is rotating.
 Why this can help is when weight B moves to it's start position, it will lose
momentum. This will help to over come that. Also, when weight B moves inward,
it's momentum can be deflected upwards and downwards. This would keep it from
trying to rotate the wheel backwards. That is something I have always over looked.
 If you look at it's position, it rolls inward in a direction that stays below the level
of the axle. But if the end of it's slot has 2 angles, then it's momentum could effectively
be cancelled. Then it might have a chance of working.
 This would be because while weight B is moving to it's start position, the other 3 weights
would be accelerating, basically acting as a flywheel. Then it might be able to coast in a balanced position. While it is balanced, it would use a minimum amounbt of energy to continue rotating. And it seems a minimum of about 40% overbalance is required. If it's inner CoG is 30 cm's from center, then 30*4= 12cm's of over balance.
 There are a few tricks that would/could help to time and increase the shift of the 2 weights.
 
                                                                                 Jim             
 
edited to add; first pic shows how weight B can have it's force redirected in 2 different directions. This will allow the momentum fromweight A to
have more potential. The 2nd pic shows how a small drop will get the weights rolling. With weight B, this can help to over come inertia. And because of inertia, the drop off for weight B might need to be a mm or 2 higher than weight A.
 in the end, if the wheel can accelerate for 10 degrees out of every 90 degrees of rotation, it might have a chance. Also, if the over balance is a little more than 40% it should help. One thing to remeber though, the more over balance there is, the less time for acceleration.
 What might be an odd thought is this, while weight B is moving towards it's start position, if it isn't creating much resistence, then the distance it is from center for the period of rotation is a non factor. This means that for the extra 10 degrees or so that I added to the under balanced side would be putting limits on it that might not exist to that degree. Of course, if all things are equal, then the wheel being able to accelerate is what is important. It is acceleration that is converted into momentum.
 Happy New Year All
                     Jim

The problem with B moving inward, is that it's immediate falling behind radially and vertically. You can temporarily save some inertial cost from it by letting it roll "back". the wheel would only support B, not lift it for a moment. Then though, it leaves fewer degrees for the wheel to complete B's lift afterwards.
In the various Abeling discussions, we found that even a weight path along the axle doesn't help. Not with all the leverage in the world, because leverage goes at the equal cost of angle.
Abeling, as far as we could tell, made the B side lighter by letting the weight roll up a ramp not part of the rotating wheel. But as a weight rolls up a ramp, it loses potential energy it might as well have invested in the wheel.
 
Off-topic for a moment because I think it might be the key to all gravity wheels that were or weren't.
If Abeling is for real, I expect weights to be spinning magnets, of which the orientation varies the load on the wheel up and down. Using the inertial differences at free moving speeds rather than the mass which is equal when stationary.
Let's take for a fact that there's a is difference between a single magnet falling off a roof, versus 2 magnets pressed together in opposing mode, or spinning relative to earth in a certain direction. A scale won't be able to tell, but experients I've read on, suggest that there is a difference in flight path and duration.
If earth cannot pull as well on a magnet in a certain spin or situation, it may also be cheaper to smack up, to reach height Y. I can think of several way to then change orientation to earth, or other magnets, at no or little cost, possible smaller than the gains to be had in the vertical part.
What if Bessler and Abeling had pretty basic machine exploiting an anomalous characteric of the chosen weight? The various designs are the smoke screen, the trick is what you're lifting, not as much how. that's how I'd patent something like that. Describe, vaguely, all ways that will work for the anomaly, without mentioning it.
Bessler kept his weights a secret right? They could have been magnets. Oriented differently up than down. Or being spun up by the wheel, either by rolling on a smaller stumb than its outer diameter acting as axle or cog to get good rpms. The inertia being recovered at the end of the lift or drop, and used to repeat. A clock maker would know how. A simple spring would collect most of the spin from the weight, and pass it on to the next weight that needs it.

   Cloxxki,
 Last night, I was looking at 2 drawings I had printed. I realized they do go together.
They also have a detail that is in a 3rd drawing. They do support what I am building.
 Thse 3 drawings would be the proof of Besler's wheel. I think this is where now I can relax and
enjoy doing the build.
 Because I am building, I am going to focus on that. I hope you don't mind.

@replaced, there is still hope for you. Of course, I could be wrong.

                                                                                                                Jim