# Math and lasers

Published on .

Back to index

In this article series, I'm trying to build a working equatorial mount for a 10" Dobsonian telescope.

As with any project, there's going to be some backtracking, and this one is definitely not an exception to the rule. My original plan of using a set of variable gear ratios to achieve the rotation speed of 86016 seconds for the mount (assuming 1 RPS for the motor) was good, but in the end I managed to achieve a precise 86164s rotation period (1-to-1 to Earth's rotation period) using some subtle, but well-placed delays in the motors rotation.

I figured the simplest way to achieve a rotation period of 86164s is to rotate the motor once every 8.6164 seconds, and then just have 4 x 1:10 gear pairs to "dilute" the motor's rotation period 10000-fold to 86164s. In this scenario the gear ratios wouldn't be a problem, but getting the motor to rotate once in exactly 8.6164 seconds would require some tricks with Arduino's `delay()` method, as it only takes an integer (milliseconds) as argument.

As floating point numbers were out of the question in setting the delay, I had to average the rotation period to 8.6164s in 10-rotation intervals, instead of having each rotation be exactly that 8.6164s. Here's the way to do it:

`((43ms * 200 + 16ms) * 10 + 4ms) / 10rotations = 86164ms/10r = 86.164s/10r = 8.6164s/r`

We assume that the stepper motor has 200 steps. By having each step take 43ms, 10 rotations will take `43ms * 200 * 10 = 86000ms` to complete, so 1 rotation averages to 8600ms. That's already very close to our target average of 8616.4ms (8.6164s). All we need now is some calibration. By adding 16ms to each rotation (so to every 200th step), we get `86000ms + 16ms * 10 = 86160ms`. 16ms is of course also multiplied by 10 here as we are averaging to 10 rotations. Then we just need to add a 4ms delay after every 10 rotations, and we are at 86164ms per 10 rotations!

The base delay of each step (43ms) was found via trial and error, and could be virtually anything, as long as you adjust your calibrations accordingly. I just found this to be the easiest way for me. You should also divide all calibration delays as evenly as possible per the 10 rotations to not get any jankyness to your mount's rotation. You can find my complete solution here. The relevant rotation delays are in the `loop()` method in the end of the file.

## Designing the gears

With the precise 8.6164s motor rotation period we just need 4 pairs of 1:10 gears to achieve a 86164s rotation period for the mount itself. The problem for me was that I don't own any CAD program, and don't indend to own one in the future, so designing the gears without actually involving pen and paper was going to be the next problem to solve.

After a bit of googling and "overflowing the stack", I found a promising web tool for creating gear templates. The tool provided by woodgears.ca allowed me to play around with different gear ratios and see what kind of practical problems different ratios would cause. I knew I had to get a ratio of 1:10 for the 2 gears in each pairs, so my options were basically 4:40, 5:50, 6:60 etc. You can actually input a 2:20 ratio to the website, but that kind of gear pair just looks plain silly.

Eventually I settled on a 6:60 ratio. 5:50 would've been okay too, but with a 8mm shaft diameter the structure of the smaller gear in a 5:50 pair would have been very weak. I didn't touch tooth spacing, contact angle or any of the more advanced options, as I didn't understand their repercussions and trusted the default config was fine. The only things I set were "Gear 1 teeth" (6), "Gear 2 teeth" (60) and "Shaft hole dia." (8mm).

I was surprised how easy it had been to get to this point in the gear design process. Before finding woodgears.ca, I had a brief dead end as I was figuring out a way to install AutoCAD Inventor on a VirtualBox Windows image (I don't currently own any real Windows devices), but luckily I got to avoid a hasty crash-course into CAD design. Maybe another time, when I can actually afford the license. Now, as I had the final gear design ready, I was able to move on to realising the gears into flesh.

## Printing the gears

Laser-cutting was the obvious choice for me. Acrylic, which is a number one choice in many laser cut studios, is hard and fairly strong, and suited my gear design perfectly. I'd heard of a good laser-cutting studio in the Kallio area of Helsinki, simply called Laser Cut Studio, so finding a location wasn't a problem either. And best of all: for this one, integral part of the project I could just sit back and trust the professionals.

Sadly I can't submit a website for lasercutting, so I still had to figure out a way to get the design from woodgears.ca into a cutter-readable, Adobe Illustrator file (.ai). Woodgears.ca has an option to print the gear design, but unfortunately that didn't cut it for me. Another problem was that the bigger gear wasn't completely visible in the design, as it was overflowing from the <canvas> element it was drawn on. So I had to somehow enlarge the canvas size and then get the drawn paths out of the canvas into a format Illustrator understands.

Eventually I was able to plug the awesome canvas2svg JS library into the woodgears.ca gear.js. I extracted the SVG markup through the dev console and placed it into its own SVG file. You can find this woodgears.ca SVG support rewrite and the final gear SVGs on my GitHub.

Now I just needed to paste the generated SVG files into Illustrator and I was good to go. Luckily I hadn't used my 1-month Illustrator trial period, so I was actually able to do this part for free. I checked the acrylic sheet size from the laser cut studio's website, and created a new Illustrator artboard accordingly (1000mm x 600mm). Most of the artboard I filled with the large gears, which have a 294mm diameter. The rest of the space I simply filled with as many of the smaller gears (36mm) as possible. I've made the resulting Illustrator file available here.

Working with the Kallio Laser Cut Studio was a breeze. After some back-and-forth questions, my cut job was completed in a week's time, with a price tag of around 100€. Below you can see the final result.

Building an equatorial mount for a Dobsonian telescope - Grinding gears (Part 5)

Back to index