This Is a real world engineering problem I have had trouble solving

[obiwan]

I am not sure if it is some simultaneous equation or geometry
You have two circles
The outer stationary circle has THREE outlets positioned anywhere between 90 and 120 degrees apart
The inner circle has from two to six outlets
The inner circle turns a fixed number of degrees on each iteration
The object is to align the outer circle outlets sequentially with one of the inner circle outlets with only one of them aligning on each iteration.
This part is not too hard but now it gets tricky

Keeping the degrees of rotation fixed and maintaining THE SAME position of the outer circle outlets
Blocking off one of the fixed position outer circle outlets and on each iteration of the same degree of rotation align one of the remaining two outer circle outlets with an inner circle outlet. It doe not matter if the blocked off outlet gets aligned or not.
So the variables are the angle of the outer circle outlets
The number of degrees of rotation for each iteration.
The inner circle can change and have a different number of outlets that are used to align with the outer circle outlets but the degrees of rotation cannot change between the two problems.

Answer would be in the form of
outer circle outlet angle is:
Degrees of rotation is:
Inner circle ONE number of outlets is:
Inner circle TWO number of outlets is :

 

[obiwan]

In the attached is one solution to one of the problems - its does not solve both.
Turning the inner circle 45 degrees will sequentially align one of the inner circle outlets with an outer circle outlet.

 

[Cubist]

I don't currently understand the problem. I've added some labels to your image...



Lets discuss the first scenario. This is my understanding, please correct me if I'm wrong...

The inner circle rotates clockwise. The outer circle is stationary.

Initially EC are aligned. The sequence of alignment for a whole revolution of the inner circle is:-
{ EC, DB, EA, DC, EB, DA, EC }. Are you calling each of these steps "an iteration"?

I understand that there may be more, or fewer, points on the inner/ outer circles. The points may not always be equally spaced.

Is there a requirement that the outer circle letters are "triggered" in order:- C,B,A,C,B,A ?

Is it a requirement that the angle of rotation between every successive alignment is equal?

Let's get the above correct before we move on to the "blocked" outer letter scenario.

 

[obiwan]

Yes each step is an iteration and the number of degrees rotated is always the same in all cases.
the number of points on the outer circle is three and does not change. the angle between the three outer circle points can vary but one set it remain the same for both problems.

 

[Cubist]

I think I understand the whole problem now. I can only think of a solution that violates the requirement #inner circle outlets < 6. But I'll show it anyway in case it gives you, or others, any ideas!

Divide the circle into (6+12n) equal angles, n is an integer. The following images show the n=1 & n=2 cases.


The circles filled in red are where the outlets are placed. As the inner circle rotates clockwise then outlets A,B,C are aligned in turn.

When outlet C is blocked, then change the inner circle for the following (you could also block A, but could not block B)...