# Creating a 3D Printed Spirograph

### The Challenge

The best part about having a 3D printer is that instead of buying toys, you can just make them yourself!  Recently, I was walking through a game store and was surprised to find that they were selling really fancy and complex spirograph kits.  You know, just like the ones YOU used to play with as a kid. It always amazes me how a simple application, with weirdly shaped hole-filled gears, could be capable of such intricate patterns. That got me thinking—why not 3D print my own set and get all the shapes imaginable?

Armed with some simple governing equations of motion, I was able to recreate my own personalized version of the ever-popular spirograph. In order to verify that I can get the shapes and patterns I want, I’ve used a motion study trace plot to simulate the results of various hole patterns. The end result is as entertaining to play with as it is technically challenging to configure and analyze.

### Equations of Motion

For this design, I’m creating a spirograph kit that generates hypotrochoid curves. These curves are created by tracing the path of a point P located on a circle of radius r rolling on the inside of a much larger circle with radius R. This point P is at a distance a from the center of the smaller circle

#### After doing some research, I found the parametric equation for the trace of point P to be below:Assumptions:

• No-slip condition where both circles contact due to gear teeth meshing
• Point P is somewhere inside of the smaller circle
• 0 ≤ t ≤ 2π

With these equations and conditions, you can generate the x and y coordinates of the point P at any given angle t. As expected, governing parameters for the size and shape of the curve are the ratio R:r (i.e. the gear ratio) and the length of the distance a (i.e. hole placement on the smaller gear).

#### Modeling the Spirograph in SOLIDWORKS:

I began by copying the desired size and type of spur gear profiles from the examples available in the SOLIDWORKS Toolbox add-in and saving them into separate parts. Next, I created sketch construction geometry that described the pitch diameter of each component and the profile center point for use in creating assembly mates. Additionally, in the case of the ring gear that becomes my template, I added an additional sketch circle that describes the path of the center of the gear as it moves along the inside of the template for use in a path mate. Finally, I made a sample array of holes in each gear to use as references for a motion study trace plot.

Moving on to the assembly, I used a path mate and gear mate to set each gear’s orientation and position correctly relative to the template and animate the proper meshing of gear teeth during movement. This constrained each gear and template pair for analysis in a SOLIDWORKS Motion Study.

#### 3D printed prototype and Motion Study comparison:

It works! Side by side are the comparisons of trace plot from a SOLIDWORKS Motion study. The drawn result you see is using the same hole position and gear/template pairing. I printed this model (120mm X 120mm) on my home printer, a modded Monoprice Mini.

Note: Above ^ play a loop of “Circular Spirograph Assembly.avi”

In this example, my smaller gear completed the hypocycloid curve in five rotations and continued to overlap itself with continued drawing. However, by simply altering the starting orientation of the gear, by even as much as one tooth clockwise and maintaining the drawing location, I can stack slightly out of phase patterns like those shown in my actual drawing!

### Conclusions

The examples shown above are just a few of the many ways this concept could be executed using customizable 3D printed templates. The number of shapes and patterns expand when you consider that you could also use rectangular-shaped templates and curves of constant width for gear shapes. As long as you have creativity and enough filament, the possibilities are endless.