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Author Topic: offset top  (Read 46 times)

ortwin

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offset top
« on: May 13, 2021, 10:29:56 AM »

It would be a surprise if none of you would show a picture of an really existing  top of this type, but I was not able to find any such thing, so here we go:
The general appearance of the top should be like this:



But the flywheel should be made of two different materials, like this:




The brown part should be some lightweight material, for example wood.
The golden part, some high density metal (brass, copper, tungsten).
The center of mass will move away from the geometric center (that is the whole point actually).
How far off center will CM be for a given pair of densities? Let's say 1g/cm³ and 10 g/cm³ ?
Where do I need to draw that interface line to maximize the offset?
The "beer can Ansatz" we discussed earlier will hold here as well. That is why I tried to put the black circle, symbolizing the stem, on the line that separates the two materials.  The top would have a nice smooth movement that one would not expect at first glance with that offset stem.
That paper Jeremy was pointing me to, for a different reason, the one with the spinning elephant, gave me this idea.
So although there might already be such a top, I think two of you will enjoy the mathematical challenge (that I was not willing to take on).



« Last Edit: May 13, 2021, 12:08:27 PM by ortwin »
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ta0

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Re: offset top
« Reply #1 on: May 13, 2021, 10:47:57 PM »

I don't think I have such a top in my collection. Perhaps I should 3-D print one  ;)

Damn. Even using the Ansatz shortcut I get pretty cumbersome expressions for the position of the line.

r0= 1/3 m/(M+m) sin3φ /(φ - sinφ cosφ) r
with:
M = π r2 ρ1
m = (φ-sinφ cosφ) r221)

where r0 is the distance from the center to the line, r is the circle radius, φ is the half-angle of the circle sector and ρ1 and ρ2 the densities of the light and heavy sides.

I'm not going to try take the derivative of that thing. I might plot it.
« Last Edit: May 13, 2021, 10:51:37 PM by ta0 »
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ortwin

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Re: offset top
« Reply #2 on: Today at 01:58:51 AM »

I don't think I have such a top in my collection. Perhaps I should 3-D print one  ;)

...
I'm not going to try take the derivative of that thing. I might plot it.


I expect you need a big density difference in the two materials to have an impressive offset. Not so easy with 3-D print?
You could simulate a different density by making one side thinner or by leaving the inner part hollow.


If you plot it, please tell us where the minimum lies for a desity ratio of two, ratio of 10 and a ratio of 20.
Things become very easy if you go from a circle to a square. A specific solution:




The ratio of the width of the different rectangles is just the inverse of the ratio of the densities.
But how to get to the circular disk from this? "Piece of pi(e)!"- I hear Jeremy shout. Or did he say tau?


« Last Edit: Today at 04:24:48 AM by ortwin »
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