How do angles of two linked rotatable rods change in dependence on their corresponding lengths (of their ratio $\frac{r_1}{r_2}$)?

Question

Suppose a system of four rods with known dimensions $r_1, r_2, a, l$ is given (see picture).

The black rod between the points $A_1$ and $A_2$ is fixed in space, whereas the linkages at points $A_1, A_2, B_1, B_2$ can rotate. How can the change of the fist angle $\alpha_1$ be described in terms of $\alpha_2$? Is it possible to set up an ordinary differential equation? How would a plot of $\alpha_1$ and $\alpha_2$ look like? I have animated the movement in GeoGebra, but I'm still interested in the maths behind this:

This is particularly interesting in regards of a mechanism I am currently designing, I'd be thankful for any advice!

Answer 1

I reached out to my old Physics teacher and he kindly set up relationships using Pythagoras and simple trigonometry from which the following equation arises: $$r_1^2+r_2^2 - 2 r_1r_2\cos(\alpha_1-\alpha_2)-2l(r_1 \cos(\alpha_1)+r_2 \cos(\alpha_2)) = a^2$$ Solving for $\alpha_1$ analytically turned out to be rather complicated though...