The summary of my latest studies

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Here is the animation showing how to shift the axle of the inverted pendulum.
The toy of the MT138 drawing is a good analogy for a better understanding of the phenomenon.
Note the motion of the pendulum axle which is horizontal because the particular Watt linkage.
Therefore the potential energy is not concerned.
The energy for this shift can be marginal and taken from the kinetic energy of the flywheel.
This shift allows the weight to overpass the famous wall between 11:59 and 12:00.
By the way you can observe the famous 'Vesica Pisces' needed for positioning the pivots of the toy.

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I found this picture from the artist Anne Guteman.
I display it just for the spirit, very close from the 'rimless wheel' and the human 'hamster'...

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The next animation above shows how this 'human hamster' can be realized.
There is only one couple of rods for a better understanding:
- a red one, with a length of 1.922 (if the radius is taken equal to one)
- a blue one, with a length of 1.688
the both are linked by an axle (the small dot on the chain in the MT138), forming a couple wich after one turn of self-rotation is shifted delayed backward of 60 grades.
The tricky behavior is coming from the fact sometime the both fall together, sometime each one falls independently.
When the chain is completed, everything is rotating inside the dodecagonal inner rim of the wheel.
The corresponding building is in progress.

[iamneo]

Dear Howard, Still not a runner. So sad!!

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IMHO we are too much focused on the famous question of the 'one weight lifting up four weights'...
The best way to solve this is to have nothing for lift. Let's the gravity alone make its job.
In the drawing hereafter is shown a concept where the energy is used NOT for lift-up some weights, but for modify the geometry.

We suppose we have TWO wheels, on the same axis, but not on the same axle (nor shaft), rotating clockwise.
The blue one has a weight at 12:00, left free for fall.
After about a turn this weight reaches about 11:00.
At this moment the COG of the blue wheel is at the left side of the vertical plane passing through the common axis.
The idea is to shift this COG at the right side of this plane, with a mechanism in connection with the big red wheel, using a part of the whole assembly flywheel energy.
For sure we can do that with one single rod, linked at the right place and the right moment.
But if the both wheels are on the same shaft, this process has no chance to be successful.
Therefore the blue wheel center must be mobile and we need a mechanism where this wheel is suspended to.
This is exactly what the toy in MT138 is suggesting, like the double link in the drawing below.

On the practical way this means the pivots must be floating in the shot below (this is a shot BEFORE the modification, where the pivots are indicated by a violet cross, and fixed to the main wheel).

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As said many times, on my opinion Bessler did NOT use some gears with teeth:
- anyway they are noisily at hight rotation speed, and some witness would have mentioned it
- the son of the carpenter is not able to manufacture such as gears
- a famous mathematician (I don't remember who, may be one of the brothers Bernoulli, living at the time of Bessler) made the demonstration that any combined motion of several circles in rotation can be always reconstructed with some segments (the rods of the carpenter).
So far since a long time I made a collection of all mechanisms in relation with this concept.

Recently I had the need for a circle centered on the wheel main axis, but without any disk (in view to leave the space free) and for sure without any gear.
I remembered an old design used by me a long time ago, see here the principle:http://www.besslerwheel.com/forum/viewt ... 6725#56725
My experiments are summarized in the following short video.
Note in this video the main wheel is fixed, and the common point (yellow cross) of the variable lozenges rotates around the main axis(green cross), actuated by my hand for the test.
This is for a better examination of the motion, but the purpose of this design is exactly the contrary, like in the animation above.
You can show in the second picture (a shot taken from the video) all the details, in particular the position of the COG (violet cross).

The idea now is to associate four of these frames together, each one dephased of 90 grades, and to assign dynamically the yellow point at 3:00.
But in that state it is impossible to connect together the four rotating points (the four couples of crossbars are rotating in the center space).
Remove these crossbar was a big challenge. But I found a way (keeping them by a virtual link outside of each frame).

[John Collins]

A minor correction - it was not the son of a carpenter. A 'carpentor's boy' refers to a carpenter's apprentice.

JC

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Reviewing some old ideas I have been convinced the 'hamster' design could be a good principle for the Bessler wheels.
Until today like everybody I had a common behavior when looking at the wheels, considering identical the main axis and the rotation axis.
The view is entirely different if we abandon the coincidence of the both axis, and if we reverse the role of each one:
- the rotation axis is obviously the center of the elongated arms
- the main axis of the wheel is mobile
By the way the motion of the wheel is not circular, but this detail cannot be observed by the witnesses.

Now imagine a lever made of three parts:
- a pivot: located on the rotation axis (see above) and grounded
- a short arm where the whole wheel is disposed (the main axis of the wheel is located at the end of this lever)
- a long arm (more than four times the length of the previous one) where the primemover applies his force
With this constitution, a single weight is able to counterbalance the whole assembly.
The role of the primemover is to adjust the position of the four weights during the rotation with the purpose to keep the main wheel excentered.

The first drawing hereafter shows this principle.
Then the animation shows the starting phase (repeated after few seconds)
The last animation shows the effective motion of the wheel, not really far from this old animation:
http://www.besslerwheel.com/forum/viewt ... 2edf#57532

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Unbalanced went to build my new avatar.
I suggested to him the implementation of the 'geneva drive' mechanism.
For memory:
The 'Geneva stop' theory: a document from the Cornell digital library:
http://digital.library.cornell.edu/k/km ... 02_010.pdf
http://www.besslerwheel.com/forum/viewt ... 5466#55466

IMHO I know now the purpose of the two external pendulas of the first Bessler wheels: the control of the two cams
This is coming from my experiments with the 'maltese cross' design, including a 'geneva stop' multiple mechanism.
The implementation of a 'geneva drive' mechanism in a rotating wheel is much more complex.
The animation hereafter shows the motion of each of the both cams, oscillating around the main axis.

If the two parts of the 'geneva stop' are fixed we can obtain the motion like in the Wikipedia animation (see here again:)
http://en.wikipedia.org/wiki/Geneva_drive
But if the two parts are moving, and in particular if the maltese cross is rotating around the main axis, the central cam must rotate with it
In addition this central cam must recover its reset location every 90 grades (every quarter of main wheel's turn).
So far we need a complex alternative motion of the central cam, liberating then locking 'the maltese cross' four times by turn
With a clockwise rotation, the purpose of these two cams is to allow a rotation of each subassembly between 12:00 and 3:00, and also between 6:00 and 9:00.
At all other positions (between 3:00 and 6:00, and also between 9:00 and 12:00) these subassemblies must be locked.
The next animation will show the motion of the four primemovers under the control of this central cam.

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As promised the animation below shows the central cam with the four 'maltese crosses' rotating around it.
This animation is not fully acceptable: as you can see, the central cam is fixed (for a better view of the motion) and in this case the rotational motion of the crosses must be paused at 1:30 and 7:30 (because if not stopped, the rotation of the two crosses is impossible, the both shapes being incompatible).
So far you can observe a pause in the main rotation, where the both crosses can be reversed (the purpose of this design of order 'four', different from my avatar).
Now you have to consider the previous animation for the definitive motion, which is much more complex: the reason why the central cam must have an alternative motion is precisely the need for this central cam to be correctly oriented during the self-rotation of the both 'maltese crosses'.
It is a little bit difficult for explain and the final animation is also complex to build.
Note: some parts (grove, pins, etc) are not represented in the 'maltese crosses' allowing the self-rotation, only a simplified shape for the principle.

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The final motion like described above (nevertheless without the grove and the pin of the 'maltese cross/geneva stop')
You can see on this complex animation:
1. the alternative motion of the central cam (a cranked arm linked to the ground can do the job)
2. the rotation of the both maltese crosses (actuated by the pin engaged into the grove, see the wikipedia link above)
---between 12:00 and 1:30
--- and between 6:00 and 7:30
In view to remove all friction losses, the building can be made by a shaped tube (for the maltese cross) moving along a serial of small rollers (for the central cam).

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The shot below describes the building process for one of the four rotational 'maltese crosses'.
As explained above the shape are made by some curved plastic tubes and aluminium plates.
Note the four quarters of circle of the cam's shape.

In fact there is still a problem.
As indicated in the second drawing, the pin and grove couple gives well a rotation of the 'maltese cross', but in the wrong direction.
The rotation must be inverted. We can do that on the other side of the main wheel.
In addition this bi-sided process (across the bearings) will give us a better mechanical stability.

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My new avatar has the purpose to remember the principle of one of the possible designs for the 'flowerbowl', which is a device of order #3 (rotational symmetry of 120 grades).
I have no certitude but I think the Bessler's wheels were of order #4 (rotational symmetry of 90 grades).
So far the animation for the order #3 has been given earlier here: http://www.besslerwheel.com/forum/viewt ... 6355#76355
For the pleasure of all Bessler's fans I built the corresponding animation for the Bessler order #4 (just and only my opinion).

The reciprocator has been discussed earlier here so many times, I will not give any other details.
(just for the recent members, again: http://www.besslerwheel.com/forum/download.php?id=6019)

In the animation hereafter there are two reciprocators: if you observe the four bars you will see the yellow pins at each end follow a diameter of the main wheel.
Two by Two (like said Bessler) there are orthogonal ('at 90 grades of the axis' said Bessler also), assuming the typical motion of the reciprocator.
Note also each rod is tangent to a specific cycloid, the astroid.
The main wheel rotates clockwise at 'omega' speed, the four half sized circles rotates counterclockwise at 2 x omega.

But what is interesting here in this more complex design, is the complement included for each rod; a weight (in light blue) sliding on the rod because the gravity between the two ends (yellow pins)..
Surprisingly the geometry of the full assembly offers a permanent unbalance.

The question now is: how to allow the cohabitation of these four big gears? (even if acting in four different parallel planes).
The solution is in this document, see here: http://www.ias.ac.in/resonance/March200 ... p33-41.pdf

In 1725, Daniel Bernoulli (AD 1700-1782), eldest son of Johann Bernoulli discovered a nice property of the hypocycloid.
He showed that the hypocycloid generated by moving a circle of radius r on a ring of radius R is equivalent to the cycloid generated by moving a circle of radius (R-r) on the same ring of radius R.
This happens when the sum of the radii of the rotating circles is equal to the radius of the ring.
This is known as Double Generation Theorem.
Ancient Greeks played a leading role in developing the idea of an epicycle.
Ideas of epicycloid originated for explaining the movements of planets in the sky.

The second animation is a practical example applying this theorem. (There is a trouble with the small image, click on it for open a new window showing this time a correct animation)
Many thanks to Mr Poncelet for his collection of curves on this site: http://st.mengrai.ac.th/users/elearning ... chuan/zoo/

Some advices for the building very soon.

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The building of the above animation is relatively easy.
The rules are simple: the four rods must rotate synchronized together, with a rotational speed twice the main wheel's speed, and in the opposite direction.
One of the solutions: a chain and four gears (there are some others and more clever).
On the drawing below the four rod axles are located on an extra disk (in light blue) each one with a gear (in green) half size of the central gear (in red).
The chain (in dark blue) makes the link between the five gears.
Because the ratio of 2:1 between the central gear (in red) of the main wheel, and the four gears (in green) half sized, the rotation speed respects the same ratio.
The particular path of the chain allows the inversion of the rotational directions (this remembers to me some suggested links for the buzzsaw/keenie' wheels).
On a practical way a building using four wheels of bike could be the most efficient procedure.