Calculation of circles parameters embedded in 2 rotating discs?

[Richard_Steele]

The next situation is presented
2 big circles (blue ones) are rotating in different directions. The left one is rotating clockwise and the right one rotates counter clockwise. Inside the 2 big circles, 2 small circles (pink ones) are embedded.

http://img850.imageshack.us/img850/6321/pic1tu.jpg

The big circles rotate at constant speed, so the small circles also rotate at constant speed.

A straight line is "connected" from the center of one small circle to another while the circles are rotating.
http://img405.imageshack.us/img405/7163/pic2pj.jpg

Having in mind that the straight line must has a constant length..
The questions are:

1) What would be the separation (in millimeters) from the center of the big circle to the center of the small circle?
Note: I assume that in both cases (left and right) the distance of the centers is the same. If not, correct me.
http://img194.imageshack.us/img194/7290/pic3zr.jpg

2) What would be the difference of degrees between the small circles?
http://img197.imageshack.us/img197/6066/pic5k.jpg

 

[mfb]

As this looks like homework: What did you try so far? What did you get?
Can you simplify the setup (as those outer circles are not used at all, for example)?

I assume that in both cases (left and right) the distance of the centers is the same. If not, correct me.

I would expect this, too.

Edit: Hmm, that leads to strange equations. I think I made a sign error somewhere, otherwise it is impossible.

 

[Richard_Steele]

As this looks like homework: What did you try so far? What did you get?
Can you simplify the setup (as those outer circles are not used at all, for example)?


I would expect this, too.

If anyone can suggest a simplified way to get the result, will be welcomed.

At the moment I've only obtained some good result in a physics engine software doing a simulation. I've used the same degrees in left and right circles. In the X plane of the small circles I've use different values. But the movement sometimes perform small interruptions, so the parameters must to be improved.

 

[Richard_Steele]

I've thought that the simplest way to view all this stuff is to imagine a plane where a point is moving in a circular way (in this case that point would be the center of the small circles).

To calculate the distance it would be only necessary to calculate the distance between the 2 moving points. But I don't know how to express that mathematically.

 

[mfb]

What do you mean with "the X plane of the small circles"?

Those circles are not used anywhere. You just have 2 parameters to determine in your system (3 if the radii are not the same).
To get an answer in millimeters, you need additional data, like the separation between the rotation centers or the length of the line in millimeters.

 

[Richard_Steele]

If we've both circles (big and small) sharing the same center, if I move the small circle horizontally I'm moving it in the x-axis just only left and rigth.

Well, I'm starting to see the problem more simple. I think I've overcomplicated the first point of view I got.

 

[mfb]

As far as I understood the problem, those circle centers move in circles around their respective origin. If they move differently, please clarify how.

I see two trivial solutions to the problem, and if I did not make a mistake I can rule out any nontrivial solutions (with the same radius left+right).

 

[Richard_Steele]

Well, viewing the problem as simplest as possible I'll say that the problem can be seen in this way:

I've a center and a given radius. A pair of points describes an orbit around 2 given radius.
There are 2 variables:
a) The radius
b) The difference in angle between the starting point of the circle

The only thing to know is the possible configurations/combinations to maintain always the same distance between the moving points.

 

[mfb]

That looks like a good approach. You can find an expression of the distance as function of those two variables plus some rotation angle - and this distance should be independent of the rotation angle.
And if I solve that, I get a contradiction apart from the trivial solution of 0 radius.

 

[Richard_Steele]

Well, so the work will show the answers!