Let the circle $d$ rotates over the circle $c$. If the angle rotation to be of 0 to 360, then how many times does the circle $d$ rotate during that process?
Since the total external angle is 360, I think the rotation on circle ( or any polygon) to be $s+1$ for real number $s$. If this is true, then we should find $s$.
The circle $d$ is rotating with an angular speed which is $\dfrac{R_c}{R_d}$ times the ones of circle $c$, with respect to their respective centers. However, the center of $d$ is moving.
So at the end the angle $\theta_d$ follows the following rule:
$\theta_d=(\dfrac{R_c}{R_c}+1)\theta_c$