Mathematical question

[willray]

By rotating a 10T planet gear about a 60T sun gear (3:1) ratio in 180° (10 tooth) increments you can see that after being turned though 3 half turns the datum dot is now in its starting position and thus after travelling all the way around it has only traveled through 2 revolutions.
It loses a rev not gained a rev as I had (wrongly) guessed.

Umm... I think you've miscounted...

The datum dot also returns to its starting point midway between your "Rotate 180 (10T)" and "Rotate 360 (20T)" positions...

 

[Ken I]

Willray - I see your point - and got confused all over again so I drew in the other positions (rotation degrees indicated are for the 20T planet wheel so rotating it 180° equates to 10 Teeth) :-

As you can see at the North South East & West positions the datum tooth (now highlighted in red) is indeed at the south position 4 times.
However if you look at the East position the gear has only turned 3/4 of a revolution but the datum dot appears to have turned through a full revolution because it was carried on the Sun gear the additional 90°.

So the planetary gear only rotated three times relevant to the sun gear but 4 times relevant to its "compass" orientation. Hoo boy!

So who's right - It depends on your point of view ???

Regards, Ken

 

[Charles Lamont]

On the matter of Ken's triangle area conundrum, a hint for anyone still puzzled: similarity.

 

[Mike Ginn]

Guys
What we are talking about is the Coin Rotation Paradox which has similarities with the famous Asistotles Wheel which mathematician puzzled about for centuries. Looking at the coin Paradox and simplifying it and making use of Stan's tooth idea we can have a linear gear of 60 teeth and a cog of 20 teeth. I think we can all agree the the cog rotates 3 times. Stan's last diagram is also correct and shows that the cog rotated 4 times. How can this be? It's because the cog is rolling around a circle as can be seen in Stan's diagram. The result is that if you work out the number of times the coin rotates based on the circumference ratio you then need to add 1 to take account of having to go around the centre of the inner gear. This is true for any size wheel/gear/disc system.

If you really want to blow your mind then consider that the sum of all natural numbers :- 1+2+3+4....to infinity = -1/12

Its controversial but backed up by very eminent mathemations. The proof is not easy but well documented!

Maybe for a wet afternoon!

Have fun

Mike

 

[Foozer]

This old man is not to bright with math - But would the pattern be Circum A / Circum B ?

 

[Ken I]

Foozer,If you are looking for the locus of motion - here it is - red dot on the pitch circle diameter so ignore the teeth and consider it two wheels running on each other (at their PCDs). A red dot indicates its position per tooth of rotation.

It traces three epicyclic paths - demonstrating that the gear indeed rotated three revolutions against the sun wheel.

I have drawn the planet gear at the North, South, East & West positions, the reference red dot (timing mark) is at the South position indicating the outer wheel has rotated four revolutions relative to its compass orientation.

Regards, Ken

 

[mklotz]

This old man is not to bright with math - But would the pattern be Circum A / Circum B ?View attachment 138892

What you suggest is exactly what I used in my earlier post (post number 29 in this thread). However, one must do the math to understand why the number of rotations has that '1' added to the ratio of the radii of the two circles.

Go back and read #29. The math is trivial, involving nothing more than the formula for the circumference of a circle and simple division.

 

[Foozer]

View attachment 138902
It traces three epicyclic paths - demonstrating that the gear indeed rotated three revolutions against the sun wheel.



Regards, Ken

Where Grandas head is at with this circle thing .
I have grandkids 7-15 years. I like to use and encourage them to find the pattern in things - Simple Pattern - So in circle query what is actually moving around the center point of the fixed circle is the center point of the moving circle. If the unwrap distance was from the point of contact the circle center point would have traveled farther - be great for gas mileage as the wheel is just once circle rotating around another, but it don't work that way does it. [mayhaps it does but time differentials? No Thanks]

So the pattern I try to get the grandkids to see is that the number of rotations of X is the circumference of a circle r1+r2/ circumference of the circle r2 [ Older one can do Circumference].
So a 4 in fixed and a 2 in mover gives 18.849/6.283 - - 3 times.

Finding a pattern one thing, trying to explain it is another.






Asked the grandkids once to add up 1+2+3...+10. Their facial expressions were cute. Grandpa is crazy . Told them to hold up their hands fingers spread - Do you see the pattern I ask them?
10 fingers numbered.
1-10 - First [1] and Last [10] add up to 11. 10 fingers divided by 2 is 5, 5x11 is 55 so 1+2+3...+10=55 - Oh Look, you were born with the answer, Open hands before you says 55 ... 1+2+3+4...+100 same pattern -5050

If I went off into the weeds, hey I have grandkids to entertain
Robert

 

[Foozer]

However, one must do the math to understand why the number of rotations has that '1' added to the ratio of the radii of the two circles.

Have always liked your post - Missed you while you were away. So a plus 1 when the mover is outside the fixed and a -1 for when the mover is inside the fixed?

 

[Mike Ginn]

Guys
The coin rotation paradox has caused many minds to melt! It can easily be shown physically that with 2 coins of equal size - (I always favor 2 silver dollars pieces!) then the moving coin rotates twice. David Ding has produced an elegant proof for those those who like to tidy up loose ends.

His proof is found here:- www.davidyding.com/navPages/coinRotation

In any discussion it is important to define the initial conditions and this is what Ding says:-

"How many revolutions does a circular coin make while rolling around another circular coin of the exact same size without slipping?"

"At first glance, the intuitive answer might be, well, one. After all, the rolling coin has a circumference, say c, and it is rolled, without slipping, against another coin with circumference c. This is basically saying how many revolutions does a circle with circumference c make while rolled against a path with length c. Surely, the answer is one?

Well, the actual answer is two. And this seemingly strange result is called the “coin rotation paradox” and that Wikipedia has a short article describing it. Here, I will use math to solve this paradox in the general case and dish out some insights for the general result."

Ding's final equation is as follows:-

Where rf and rr are the radii of the coins so if the coins are identical then N=2. To fully understand this you will need to work through the other 4 equations.

Note that Ding's proof covers coins/disks of any size provided the ratio of the disk radii is an integer.

Mike