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Magnetic delta and print angles

11 Oct 2013

I'm in the middle of designing my delta style 3D printer. Having reviewed a few designs, the magnetic joints looks to be the easiest way to go. I've noticed that many people angle the ball mount 45 degrees by either bending the plate or inserting an angle spacer. I wanted to avoid this if possible, so I calculated the minimum inclination possible for a given design.

The math was pretty simple. I use the cosine law to calculate the angle occupied by a given chord. There are two chords of importance; the diameter of the mount for the ball, and the diameter of the joint. In Xnaron's project above; we have a 3/8" ball (\(a\)), the joint is 3/8" (\(b\)), and the mount is a 6mm screw (\(c\)). This gives the joint occupying 180° (\(\alpha\)) and the mount 78° (\(\beta\)), when we are at max range, these two angles are adjacent. $$\alpha = \arccos(1 - {2b^2 \over a^2})$$ $$\beta = \arccos(1 - {2c^2 \over a^2})$$

We are looking for the minimum inclination (\(\gamma\)). If we start with the mount, it is vertical giving -90° inclination. We add half of the angle to get -51° inclination, then half of the joint angle to get 39°. This means the minimum inclination possible before the joint runs out of range is 39°. If he were using a 1/2" ball it would extend to -13°, well below horizontal. $$\gamma = {\alpha \over 2} + {\beta \over 2} - 90°$$

To calculate the minimum inclination I'll probably end up using 1/2" balls in my design, 1/4" dimple press for the mount, and then experiment with the joint design to give just enough hold. With a 3/8" joint this will give -11°, or well below horizontal.