The Two Axles of MT 55.

[shap-O-vert]

Thanks all for the “Gravitates� input. Really good stuff.

To recap:

My take has been that each spherical weight, when on the ramp, “gravitates� down the ramp on its pegs towards the centre of the wheel, where it leaves the ramp and runs onto a Bowed arm that pivots off the Effort arm of one of nine Type One levers with their fulcrums at the rim of the wheel (see The Anvils of MT 138 thread).

The problem I’ve encountered is the relatively slow velocity of each weight down the ramp (at an inclination of 4°); i.e. 0.074 secs needed; best velocity achieved on straight pegs was 4.55 secs – way to slooow.

Well, with further experimentation I have been able to increase the velocity such that time taken down the ramp is less than 1.0 sec.

Test A:

Forget gravitating the weight down the ramp on its pegs.

Gravitate it (let it roll) on its spherical surface down the ramp with the rails separated by a distance equal to the radius of the spherical weight.

Result: the mean time taken to travel the 35 cm is reduced to 1.09 sec.

This is an improvement, but is still too slow to get 56 rpm.

Test B

Reduce the distance between rails of the ramp such that they equal 0.5 radius of spherical weight.

Result: Average time taken reduced to 0.95 secs. This is still too slow – but better

Test C

As in Test B, with the angle of inclination of the ramp increased to 9°.

Result: average time taken is further reduced to 0.88 secs.

Hmmmm.

OK, you can see the problem.

1. When you increase the inclination of the ramp you further increase the velocity but reduce the saving in degrees of rotation on the –ve side (because the weight has to be lifted higher).

2. When you let the weight roll on its spherical surface it also has to be lifted higher to get the surface (as opposed to the pegs) onto the ramp. It’s all a trade-off.

I’ll come back when I’ve thought a bit more about this little hitch!

In the meantime, any thoughts you might have would be appreciated.

Ciao

shap-O-vert.

PS: I understand how Grimer feels. Bang, Bang, Bang!

[path_finder]

Dear shap-O-vert,
Remember the brachistochrone.
Let's fall your weight vertically, it will reach the center much more quicker.
and for the fun again: http://www.youtube.com/watch?v=jlSv_IlXmBg

[shap-O-vert]

Yes. Those slot cars. I'll give it a shot and let you know.
The bow will twang big-time. The Bow.

Thanks path_finder for the reminder.

It'll fit with the design nicely, too.

Regards

shap-O-vert

[shap-O-vert]

This reference of path_finder’s to the brachistochrone is interesting because Bessler consistently claimed to be a mathematician.

For example he said in AP:

…I had the help of good scholars to show me the difference between good and bad learning, and so was eventually able to become successively a Cleric, a Doctor and finally a Mathematician. (p 256)

And:

All knowledge was of use to me. I particularly liked algebra…(p 257 AP).

Clearly, he did have “the help of good scholars�.

Gottfried Leibnitz, who claimed Bessler as a friend,
http://www.besslerwheel.com/accounts.html

developed the infinitesimal calculus independently of Isaac Newton, and was one of five mathematicians ( Isaac Newton, Jakob Bernoulli (Johann's brother), Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital) who solved the brachistochrone problem.
http://mathworld.wolfram.com/Brachistoc ... oblem.html

http://en.wikipedia.org/wiki/Brachistochrone_curve

Appropo another thread (On the Number 55) he (Leibniz) also noted how the I Ching hexagrams correspond to the binary numbers from 0 to 111111.
http://en.wikipedia.org/wiki/Gottfried_Leibniz

so maybe there is a link there?

And, Willem Jacob 'sGravesande (1688 - 1742), was also known to Bessler and was a colleague of Leibniz.

So did Bessler use the brachistochrone? He certainly would have been aware of its existence, IMO - given his contacts.

Regards

shap-O-vert.

[path_finder]

Dear shap-O-vert,
Many times I had to regret here on this forum the tendance for forget so many perhaps very efficient ideas. The brachistochrone has been evaluated here several times.
I do not pretend to give the solution, and I apologize do not have save the time enough for a conclusive evaluation (by a building p.e.), see here for memory:
http://www.besslerwheel.com/forum/viewt ... 6504#66504
As explained this design supposes the use of TWO counter-rotating wheels at different speeds, the both wheels exchanging their weights at the end of the path.
Mechanically it should not be an enormous job.

[shap-O-vert]

Thanks path_finder.

I viewed those posts at the time. However, I didn’t recognize their relevance to the design I was working on until your more recent post – notwithstanding my incorporation of a pivoted “brachistochrone� style arm at the Effort end of each of the nine, class one, levers that I’m using.

See: page 1 this thread: Fri Oct 08, 2010 3:17 pm Post subject: re: The Two Axles of MT 55. The Peacock’s Tail; and the post immediately above it. (How do I give a direct link??)

The application and testing of the brachistochrone to the ramp element that I’m currently working on is going to be very interesting (to me at least!!)

Regards

shap-O-vert.

[path_finder]

How do I give a direct link?

right click/copy on the small upper left icon of the desired post.
NB: should be in the 'technical support' section, but here for efficient linkage.

[path_finder]

Dear shap-O-vert,
After careful review of your above referenced posts I'm still not convinced by your design.
The peacock drawing - like mentioned in your restrictive conditions note - is only a static evaluation, useful for the potential energy: the energy resume can be obtained by the surface calculation (integration of the curve by calculus), but what is much more interesting is the source of energy (the kinetic energy) gained by the motion of your bows.
Unfortunately this is the most difficult for calculate, because everything is moving/rotating and we are not sure where and at what time the shift of the bow occurs (too many parameters).
Also I don't understand well why NINE bows.
At least the brachistochrone allows an earlier arrival, but will NEVER allow to arrive at an higher level than the start level.
But I'm curious to see the way you can link all these parts.

edited (for the fun)
How to use the bow: the 'cesta punta', see here:
http://www.dailymotion.com/video/x55qzv ... ritz_sport

[Grimer]

How do I give a direct link?

right click/copy on the small upper left icon of the desired post.
NB: should be in the 'technical support' section, but here for efficient linkage.

Applause for your attachment and very clear explanation. I only discovered that feature myself by chance.

[DrWhat]

Just trying it out!

http://www.besslerwheel.com/forum/viewt ... 0134#80134

Nice to know how to do it!

Click on the link and see what happens!

[shap-O-vert]

path_finder says:

Also I don't understand well why NINE bows.

Let me see if I can explain this a little more clearly.

There are nine bows because there are nine class one levers, each with their fulcrum on the rim of the wheel. Each of these levers has a bowed arm pivoted on the top of the effort end of the lever. The leading edge of each bow (the end nearest the centre of the wheel) as it moves down the +ve side of the wheel, lines up with the down end of the brachistochrone ramp such that the weight coming down the ramp rolls onto the bow.

The timing of the transfer of each weight from the ramp to the bow is critical to the relative torque on the +ve side compared to the -ve side. (the +ve side has torque applying for 120° of the downward rotation while the –ve side has torque applying for 94° of the upward rotation with the difference (26°) being taken on the ramp).

The transfer of the weight to the bow in this way is important because it is that shift that applies +ve torque to the downside at the earliest possible time, even though the lever doesn’t start to swing towards the wheel’s rim until a little later in the rotation. The bowed arm allows the momentum of the weight to move the weight as far out on the bow (from the centre of the wheel) as possible, consistent with the dynamic of the downward rotation.

OK. That’s the complex bit. When you get it you’ll see how simple it really is!

Why nine levers?

I tried to cover this here:

http://www.besslerwheel.com/forum/viewt ... 9404#79404
(Thanks P_F for that heads up, too)

But, in summary, let me have another crack at it.

I think nine levers are necessary because there needs to be 80° between the position of the effort arm of each lever when it's swung to the rim of the wheel and the load arm of that lever when it's at the rim. This gives the necessary weight distribution around the wheel to ensure aggregate OOB (i.e. over a full 360° of rotation).

Now, 4 x 80° only equals 320°, so we’re 40° short of a full rotation*. But, if we have another series of levers offset to the first series, we can make up that difference: 5 x 80° = 400°. And, as you can see, 4 + 5 = 9; and 320° + 400° = 720°/2 = 360°.

I hope this helps.

Finally, pathfinder says

I'm still not convinced by your design

Well, I'm not saying it works either. But I'm building it - ever so sloooowly. And with the help of this Forum's input I've already made adjustments. We'll see what happens and I'll keep you posted.

Any critique is greatly appreciated.

Regards

shap-O-vert

*But hopefully, not one brick short of a load! :-)

[shap-O-vert]

path_finder said:

Remember the brachistochrone.
Let's fall your weight vertically, it will reach the center much more quicker.

OK. I'm just back from the workshop to report that testing the first model of the brachistochrone ramp has increased the velocity of the weight to 0.56 secs down the ramp (35.0 cm). Still a fair way away from what I need to match Bessler's 56 rpm, but still a great improvement.

A further (deeper) model ramp build is underway to see if I can get even greater speed.

One technical issue is starting to arise and that is my own reaction time on the stop-watch start/stop button. There will come a point where the timing will become very unreliable. If I get to that point I will be very pleased. Incongruous I know, but true.

Regards

shap-O-vert.

PS: Do you think you can count 8 contacts of the weights with the side of the wheel in 1.07 secs with certainty?

[shap-O-vert]

A further report for those interested in my test-bed model of the brachistochrone ramp.

The best average time I can seem to get is 0.48 secs down a 35 cm ramp with rails 3 cm apart and a 5° slope (mandated by the design) measured between the take- off point and the bottom of the ramp. It needs to be recognized that there is an error of measurement of unknown magnitude (possibly as much as 0.15 sec) in this due to my own reaction time on the stop watch.

However, this is a massive improvement over the 5.4 secs that I was initially stuck with.

I think this is as good as I can get using the somewhat rudimentary time measurement technology that I have available. So I intend going with it.

Thanks again path_finder for your brachistochrone suggestion.

Regards

shap-O-vert

[path_finder]

Dear shap-O-vert,
A simple and easy way to measure the short time is the video camera.
You have to paint your ball with an equatorial white line, and just record the motion.
Then with a video editor (like kino or caffeine) you will be able to explore the movie image after image with the time displayed at the bottom in 1/100th of second.

edited: this could be useful for pequaide too.

[daxwc]

http://en.wikipedia.org/wiki/Tautochrone_curve

A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity.


After determining the correct path, Christiaan Huygens attempted to create pendulum clocks that used a string to suspend the bob and curb cheeks near the top of the string to change the path to the tautochrone curve.

Quote:
http://en.wikipedia.org/wiki/Brachistochrone_curve

The curve does not depend on the body's mass or on the strength of the gravitational constant.

According to Fermat’s principle: The actual path between two points taken by a beam of light is the one which is traversed in the least time. Hence, the brachistochrone curve is simply the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g).

Quote:
http://en.wikipedia.org/wiki/Cycloid
The cycloid is the solution to the brachistochrone problem (i.e. it is the curve of fastest descent under gravity) and the related tautochrone problem (i.e. the period of an object in descent without friction inside this curve does not depend on the ball's starting position).

The cycloid was first studied by Nicholas of Cusa

In 1634 G.P. de Roberval showed that the area under a cycloid is three times the area of its generating circle.

tks