I wrote previously about the idea I had to make my wife Genevieve a jewelry box with the laser cutter at TechShop. In the last post, I showed a little physical mockup I made of the kind of hinges I wanted to use.  That proved it would work, but an hour with a pad an pencil and I still hadn’t unwound the math I needed to predict exactly where it would end up when it opened.

Luckily, the helpful commenters funneled through from the post on the Adafruit blog quickly answered my question. It’s called a four-bar linkage, and it’s pretty well studied in mechanical engineering. Still, I found the diagrams pretty baffling (lots of greek letters…), but ultimately I figured it out.

With the equations in hand, I used OpenSCAD to produce a set of drawings.

Clown colors for contrast only.

I’m pretty pleased with the fidelity of the four-bar calculations. Once I became friends with the Law of Cosines (and corrected all my mistakes), the numbers started working out like I expected them to, and I had good positions for my pieces. I have an angle variable that I can tweak to see where the lid would be at various places in the full motion. And it proved valuable, too: after I saw the lid angle produced by my first random set of parameters, it was clear it would have to be adjusted.

The next step is to take the panelized version to TechShop and cut out a cardboard prototype. Here’s hoping my hinging system works out!