An explanation for the 22,5 grades angle

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One of the controversial clue discussed many times in this forum, is the value of the sector angle in the famous drawing from Apologia (labeled: and you don't still understand').
If you forget the errors due to the numerous 'copies of copies' and/or parallax, it seems that a global consensus drove to the value of 22,5 grades.
Even before I took an interest in the Bessler's wheel, this value has been revealed to me as a major concern.
Hereafter I will try to explain why:

Working on the 'hamster' design (also analog with the bike pedals when doubled), I tried some different values for that hamster.
First I want to show a design wich don't give the expected relative positions and therefore don't work.
The animation below is based on this behavior, where the hamster is free to roll when in green and locked when in red.
The locking mechanism is not represented for the moment.
The size of the hamster is exactly the half of the big wheel.

With this rule the hamster is free during the first 45 grades rotation of the wheel.
Because it's size the rotation of the hamster is a combination of two rotations:
- the rotation of the big wheel
- the rolling of the hamster inside the big wheel
Therefore when arriving at this step the hamster made a rotation of 90 grades on itself.

Then the hamster is locked during the next 45 grades rotation.
This time only the rotation of the big wheel must be taken in account.
By the way when arriving at 6:00 (original starting position) the hamster cannot recover its original position.

So the question is: where we need to stop the hamster? for recovering the reset position at the 6:00 start position?

[DrWhat]

p_f, probably my first major wheel study was with two tubes at 90 degrees each side of the smaller wheel.

The attached image shows a pair of sliding shafts within which are contained heavy sliding rods. On the other side of the wheel are another pair of shafts and sliding rods but at 90 degrees to the front ones.

The futile idea was to develop a resetting movement similar to what you have shown.

Suffice to say 'no werky!'

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Dear DrWhat,
Many thanks for posting.
You are one of the rare answer I received since few weeks.

In that state, nothing to add (remember: a prime-mover alone cannot be the solution).
But you missed the half of the job.
Put your wheel (on the picture) inside another one, exactly twice bigger.
Adapt any 'anti-return' mechanism on your wheel.
Let roll your wheel on the inner rim of the big one.
If not enough torque, include another twin wheel (like Bessler said) aside the first one.
Synchronise the both.
It will run much more better.


Note:
In view do not damage your roof, or do not do that outdoor, you can use the inverted process: include another wheel, twice smaller, inside your existing wheel.
With the same remarks than above.

[Michael]

Nice though. Into the museum it goes.

[DrWhat]

Michael, thanks.

Into the skip bin it went after being perforated extensively.
You can find it at a rubbish tip somewhere in Sydney :-(

Feel free to find it and reassemble it with glue, and a lot of it!

[AB Hammer]

Well at least there is a picture. :)

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The animation above is wrong because the hamster don't recover it's reset position when returning back to 6:00 of the clock.
So far we must find the correct schedule of the both phases (free and locked).

In the reality the exact motion is really difficult to define.
The hamster is able to climb until some very different altitudes, depending of these various parameters:
- the rotation speed of the main wheel
- the size ratio between the outer rim and the diameter of the hamster rolling axle
- the ratio of the lever inside the both prime-movers
- the value of the weights, etc
In addition the diameter of the hamster has a big importance, as we will see hereafter and the fact that each part has a different rotation speed makes the calculus more complex.
Nevertheless there is only one parameter wich can be taken in account for a very good approximation: the size ratio,
and more obvious: the relative number of teeth in case where some gears are used.

In another topic (regarding MT55) mickegg said:

I count 48 and 12 teeth on the respective wheel and pinion in the MT55 image provided by ovyyus (Bill McMurtry)

I want now to explain why this choice is in accordance with the 22,5 grades value.

We will suppose hereafter that
- the wheel includes ONE primemover acting every 90 grades
- The main wheel's rotating hollow drum has 48 teeth
- the rolling axle of the prime-mover has 12 teeth

At the rest position the prime-mover is located at 6:00 of the wheels' drum (like in the previous animation).
At that position this prime-mover is active (unbalanced) and goes to the right, climbing on the inner gear of the main wheel.
By reaction the main wheel start to rotate counterwise (like the clock).
Because the number of teeth the ratio between the primemover axle and the wheel's drum is 1/4.
By the way when the main wheel rotates of one tooth (said 1/48 turn = 7,5 grades) the axle of the primemover rotates also of one tooth (said 1/12 turn = 30 grades)
And so on, by multiplying the both part of the line.
Due to the rotation of the main wheel the primemover rotates on itself like in the table below:
--- Main Wheel-------------- ---Primemover's axle angle --------------------
1 tooth 1/48 turn= 7,5° --- 1/12 turn= 30°
2 teeth 1/24 turn= 15° --- 1/6 turn= 60°
3 teeth 1/16 turn= 22,5° --- 1/4 turn= 90°
4 teeth 1/12 turn= 30° --- 1/3 turn= 120°
5 teeth 5/48 turn= 37,5° --- 5/12 turn= 150°
6 teeth 1/8 turn= 45° --- 1/2 turn= 180°

We see in the table that with this teeth ratio (12-48) the primemover recovers it's reset positions (after each 90° rotation) every 22,5° rotation of the main wheel, by the way allowing the locking ratchet to attach the primemover to the wheel. Then the primemover (attached to the wheel) returns back to the 6:00 position, where it will be unlocked. And so on.
The animation below shows this process.
The primemover is free to rotate when green and locked when red (the locking mechanism is not shown).
In this explaination there is only one primemover.
The torque is very low, but there is another solution (using the same principle) but much more efficient.

[bluesgtr44]

This concept kind of reminds me of my "springring" attempts. I thought that it would be helpful if we could come up with something that didn't maintain symmetry so I made this ring with springs and attempted to push it over.....yeah, throw another log on the fire.


Steve

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Dear bluesgtr44,
There should be no symmetry at all, this is the principle of this design (except at the 6:00 rest position where everything is dead)
The locking mechanism is not shown here. It could be either a plastic light roller sticked by a spring (like already suggested in my previous topic here:
http://www.besslerwheel.com/forum/downl ... 35678e071d),
but the simplest one is just a rod attached on the axis of the hamster wich will jump over the teeth, act like a diode and avoid the hamster to go within the reversal way.
By the way, the primemover being always shifted on the same side of the main axis, the non-symmetrical unbalance will be permanent.

Note: Don't forget that the animation shows only the gears.
The main wheel and the primemover are much more bigger around them like show previously:
http://www.besslerwheel.com/forum/downl ... c4878adff4
The size of the primemover (swinging inside the main wheel) must be sufficient for applying the 4:1 famous ratio, only able to sustend the whole assembly (like explained in my earlier post here:
http://www.besslerwheel.com/forum/downl ... 2ad04b3909)

Regarding your WM2D drawing: (thanks for sharing)
The most important is not the shape of the curved body.
It could be any (circle, ellipse, polygon, etc).
What is the key is the ability of this body to be cyclically unbalanced (like a ball filled with sand).

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I would like now to abandon a little moment the theory and to give some suggestions for the building of such as design.
The shots hereafter show one example of realization.
In a previous post I have explained how a polygonal gear can be used.
(seen here: http://www.besslerwheel.com/forum/down ... 515a448786)
The assembly shown below is a little bit different and consists in:

- two main wheel side plates with (each one) a 48 teeth gear on the internal side
On a practical way you can use some simple screwed rods fixed on a circle.
But the result will be poor, due to the position variations giving a misalignment of the teeth.
This is the reason why I used an item not specially made for this purpose: a flexible plastic bar for electrical connections.
First write a circle with a radius R. Divide it in 48 parts. Drill 48 small holes and install 48 nails (even if there are not specially well aligned). Insert the plastic bar. Tie the screws of the bar: the 48 teeth will be well self aligned at the end of the process.
Another benefits are also obtained: the teeth are in plastic and the contacts are smooth and without noise. In addition the tolerance is not so strong.

- two primemover side plates with (each one) a 24 teeth gear on the external side
This time I used some screwed rods. Take care (adjusting the distance between the two plates of the primemovers) that the fixation screws don't scratch on the 48 teeth gear.

Hoping this will help some builders...

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You should not forget an important point: the 'hamster' design needs a good synchronization between the both primemovers.
If we miss just one tooth, it cannot work well therefore we need an additional feature in view to oblige the roller (either polygonal, either with gears) to follow it's path with an excellent accuracy.
Looking at the drawing below we can observe the path of the hamster axle (the red curve): it's center follows a circular path around the center of the main wheel, with a radius of half size.
If we want to oblige the hamster axle teeth to follow the main wheel teeth, the solution is to apply an additional link with the main wheel inner gear.
The way suggested consists in two light plastic rollers, located at -120° and +120° of the hamster axis and making a triangle with the hamster axis.
Caution: this is valuable only with the design using the gears. A spring in compression is needed for the design using the polygons (because in that case the path is no more circular but epicycloidal like).
The centering mechanism of the hamster's axle is not shown on the pictures above..

But there is another mechanism able to do the job: a crank, like in the second picture where the rollers and the triangle (in grey) have been replaced by a rotating leg (in red). Due to the size of the hamster's axle, it's shape is a little bit particular if we want to avoid the collisions.

The best and almost simple and efficient way is shown in the third picture.
This method need a double side for the gear path.
An extra free roller is located at the external side of the gear and linked to the axis of the hamster's axle with a spring, By the way the axle is now sticked on the inner rim of the gear and don't miss any tooth during the rolling.

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The whole assembly (hamster design) with only one primemover including
shot #1: two 'Watt like' linkages at 180 grades (acting every 90 grades)
shot #2: three 'Watt like' linkages at 120 grades (acting every 60 grades)
shot #3: remote top view of the shot #1


Just a small remark:
the 'hamster' design I'm trying to justify since few months, was an idea already debated in 2002, like seen in this old message of Scott here:
http://www.besslerwheel.com/wwwboard/messages/494.html
and specially the lowest part (Al Bacon picture)
It's strange that since this date nothing has been happened...

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Until now the link between the hamster and the main wheel's drum was realized by the mean of some imbricated gears.

The first encountered difficulty was the missing of some teeth during the motion.
We indicated few add-on features in view to 'stick' the primemover axle's gear with the inner rim of the main wheel.
The second difficulty was coming from a desynchronization between the both primemovers (wich must be absolutely still kept in quadrature).
After several attempts and even with these corrections the motion of the primemovers was still cahotic, and they were locked or misaligned always very shortly.
The need for finding an alternative solution was critical.
I you examine the path of the axis of the hamster axle you will find easily that it's a part of circle centered on the main wheel axis.

So far a first trivial conclusion is:
instead to let roll the axle on the inner rim of the drum, why not link the center of this roller with a rod directly to the center of the wheel
A first approach of this kind has been suggested above in the second proposition of attachment (the animation centering_mechanism2_.png with the red part).
But there is a big improvement available: the full suppression of the gears.
Hereafter the second picture shows the very simple solution used for:
1. forget the gears with teeth
2. nevertheless keep the drum of the main wheel, this time with just a rack (a crown of pins)
3. provide a leg with one end centered on the main wheel axis and the other end linked to the primemover's axle.
4. add a small part acting with it's own weight like a ratchet.

Note that this new design includes 48 pins. Why?
We saw that the 22,5 grade value is a fundamental value for the effectiveness of the principle.
With 48 pins one passed pin is equivalent of 7,5 grades.
so far if the hamster is able to pass three pins, we win

The third picture shows the whole assembly (main wheel and two primemovers).
The top half is a cutted view, the bottom an external transparent view.
In the next post I will explain in details how this works.

[AB Hammer]

AH! Path_finder

Bravo on your effort.

I see you are working diligently on this. When are you going to build/finish this latest pictured design?

On the hamster cage approach, I chose to go back to the basics and what evidence there can be. This way more tracks can be covered. Any build will have to wait for a while until I finish some existing projects. But I have the design finished which would be bidirectional if it works. If it followed what you are doing I would post it, but it doesn't.

Good luck

[path_finder]

Dear AB Hammer,
Many thanks for your attention. Please be patient.
Are you alone in that desert?

Like as promised: how now the 'hamster design' works?
Hereafter we suppose the mechanical structure of the main wheel is conform with the drawing 'mainwheel_inner3.png':
http://www.besslerwheel.com/forum/download.php?id=6968
so,
1. the two main wheel's half-cylinders are supporting two identical primemovers.
On the inner rim of these two half-drums are rolling the axles of the primemovers
2. a primemover is a 'wheel inside the wheel' unbalanced every 90 grades, and with one shaft rolling inside the main wheel's half-cylinder.
Some different designs are available. One example is here:
http://www.besslerwheel.com/forum/downl ... er=user_id
3. a ratchet brings a one-way contact between the primemover's shaft and the main wheel half-cylinder.

The motion of the primemover is repetitive: there are two successive phases
- one phase of climbing: due to the unbalance of the primemover, it is trying to climb on the inner rim of the half-cylinder.
- on phase of restitution: arrived at it's maximum altitude the primemover is locked by it's ratchet and is driven back to the rest position by the main wheel itself.
Arrived at the 6:00 rest position the primemover is unlocked from the main wheel.
But the travel made by itself during the climbing phase added to the travel made back sticked to the main wheel, gave to the primemover a 90 grades rotation around itself.
Due to it's internal constitution the primemover becomes again unbalanced, and the cycle can restart.

In fact (due to the poor torque and depending of the level of frictions) two primemovers are needed, acting every 45 grades.
Hereafter three pictures of an 'hamster design' with only one primemover.
Note the elongated external bearings (see my previous post here:
http://www.besslerwheel.com/forum/viewtopic.php?t=3145
The linking mechanism has been improved also.
On this state put on the floor, it don't roll by itself.


For the fans of Bessler: I have perhaps the answer to a controversial subject discussed on this forum many times:
for what purpose Bessler did connected his wheel to a stonemill with so noisy hammers?
Some members suggested this was for justifying an useful work or showing the level of available power.
Some others asked if this was for hidden the internal noise of the wheel and avoid to obtain any useful information by hearing this noise.
On my opinion the second explanation could be the good one.

If you examine the noise made by the two ratchets, you can observe a succession of silence and noise phases.
The first primemover climbs until 22,5°: it's ratchet makes noise.
Then the following 22,5° are silencious: the first primemover slows down and stops, and during the same time the second primemover is locked (therefore no noise on the other side).
The first primemover is now locked and is fixed within the main wheel (no noise) but the second primemover on the other side climbs now (making noise).
And so on. The noise of the wheel is: scratch,silence,scratch,silence,scratch,etc. every 22,5 grades.
Now let's suppose that this tricky Bessler has triggered the hammers of the stonemill in the way they fall exactly during the silence gaps...
Nobody can detect the intermittent scratch noise. Bingo.