The zoetrope was a form of early animation, originally patented by William George Horner in 1834. The classic zoetrope consisted of a cylinder with a set of still images arrayed radially around the edges; when spun at the right speed the eye blurs the images together creating the illusion of motion.
As an engineer by training and an artist in my free time, recreating this style of zoetrope presented a fun interdisciplinary challenge. My objective was to 3D print a fluid mesh animation and create a spinning base and projector for zoetrope display and illumination.
This project was inspired by a unique take on the zoetrope-- a 3D printed sort of frameless zoetrope created by artist Akinori Goto. Her zoetrope consists of a circular seemingly abstract lattice of plastic strands, that are in fact a continuous loop of movements created by connecting the frames of a dancing figure through time. This sculpture is illuminated by a shaft of light that reveals only slices of the object at a time, isolating single frames for the viewer and creating an animation when it spins.
Internal gears are commonly used in gear reduction applications because they solve the problems of size and stress by offsetting gear axes and using multi-stage reductions to reduce shear forces. The two main types of internal gears are epicyclic, or planetary gears, and cycloidal gears.
Epicyclic gears are composed of a central “sun gear” and three or more “planetary gears” that orbit within an outer ring gear. The forces on the gear teeth in this configuration are generally lower because they are distributed amongst multiple points of contact at once, however this configuration does not easily accommodate gear reduction ratios much above 30:1.
Cycloidal curves are created by tracing a point on a circle as it rolls around a "base" circle. A curve created by a circle rolling externally around a base circle is called the epicyloidal curve, and the curve formed interior to a base circle is hypocycloidal. In a cycloidal gear, the part of the tooth flank that lies outside the pitch circle, the addendum, is epicycloidal. Conversely, the part of the tooth flank that lies inside the pitch circle, the dedendum, is hypocycloidal – because the outer rolling circle used to create the addenda tooth flanks on one gear is used as the inner rolling circle to create dedenda tooth flanks of the other gear the angular velocity remains constant despite the gear's strange offset configuration.
The hypocycloidal gear is driven by an eccentric input shaft that is mounted to the cycloidal disc. This disc has lobes rather than teeth, and so the shear forces are dramatically reduced in this configuration as forces are more completely distributed. The cycloidal disc engages with pins around the interior of a ring gear, and the disc's rotational motion is translated to the output shaft via roller pins protruding through the disc. The number of internal pins (always one more than the number of lobes on the disc) determines the gearing ratio of the output motion. Because of its eccentric mounting and the low shear forces, cycloidal gears enable gear reduction of anywhere from 30 to 300:1.
Disk design can be defined with 4 parameters:
D = outer diameter of the circle of pins
d = diameter of each pin
e = eccentricity, the distance between the geometric center and the input shaft
N = the number of pins; this value determines the gears gear ratio
Using these parameters we can draw the profile of the cycloidal disk:
→ More in-depth explanation of the math behind hypocycloidal reducers and gear profiles.
→ Quick equation generator to save time.
To create the cycloidal gear in Autodesk, I used these equations and wrote a python script that parametrically generates the gear's curve as a spline. My version of Autodesk, Autodesk Fusion 360, does not have built-in parametric modeling capabilities like “sketch equation curve” that are a part of Autodesk Inventor, so instead I leveraged the Fusion 360 API (Application Programming Interface) to wire in my own code for sketching an equation-governed curve. This tool is quite versatile and can also be used to automate repetitive tasks and to write add-ins like those found in the Autodesk app-store.
The code below can be run directly from the python compiler Anaconda, or saved and run in Fusion 360 directly using the ‘Add-ins’ dialog box. It was based on the example script by Richard Parsons (Capo01).
For my final design I decided on a stacked dual-phase cycloidal gear with an 5-node disc in the first phase and a 8-node disc in the second, compounding to produce the desired rotational ratio of 40:1 from input to output.
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The complete design of these gears and the zoetrope was executed in Autodesk Fusion 360 CAD, and all parts were created with 3D printing with the exception of some purchased hardware (screws, ball bearings). Although 3D printed plastic is not the most suitable material for high-stress gearing, it was the only manufacturing resource I had at my disposal at the time. Under these circumstances however, I did make efforts to optimize for material strength and functionality.
Optimizing the strength and precision of 3D printed parts:
Gears were printed individually rather than trying to fit multiple on the platen at the same time; this creates a cleaner finish on the teeth of the gears and reduces oozing and retraction of parts. Although slightly slower, this increases the integrity and precision of the parts which is critical for gear performance.
I also made conscious decisions on orientation of parts during printing; because of the layering process of deposition/extrusion printers, a printed part's strength is very directionally dependent. The molecular bonds forming the material extrusion itself are usually stronger than the adhesive bonds of one extruded layer of plastic laid on another, and so material deposition parts are strongest in planes parallel to the print bed. In my application this meant I had to print gears flat but print axel-type parts like the input shaft lengthwise on a curved edge with supports.
→ More on optimizing print strength: 3D Printing Settings Impacting Part Strength (markforged.com)
Even with these considerations in printing, I found it was important to reinforce some of the more delicate parts with hardware. Specifically, the smaller transitional shafts have been designed with holes for screws to strengthen these connections where the strain on the plastic alone would be too much.
To additionally strengthen 3D printed parts one can select more resilient materials such as polycarbon filament and use post-processing techniques such as annealing after printing to improve the parts firmness and tensile strength. For this project I printed my test parts with the more common and cheaper PLA filament. These test parts worked, and because they are not serving a critical purpose (this project was purely for entertainment), I decided not to reprint or process them further.
Note: The tenacity of PLA, as measured by it’s mechanical durability under impact testing, is not as good as other comparable and similarly cheap materials such as PETG or ABS, so I would not generally use this material for this sort of mechanical application. PLA however does measure up favorably in terms of tensile strength, so it has its uses. PLA is just what I had on hand, and it was sufficiently strong in this case.
→ To read more about annealing and different plastic strengths I’d recommend this blog.
Design choice of using ball bearings:
Another challenge with 3d printing gears meant to operate at high speeds is that the plastic creates a lot of friction. To prevent the gears from binding I used rollerball bearings at all critical points of contact; this added significantly to the budget of the project but was key to its success.
Assorted diagrams of design process and planning:
External involute gears are probably the most common gear type. External refers to the relationship between the two gears axles and involute describes the gear tooth profile. For example, a basic spur-pinion pair uses parallel axis configurations and can be used to change torque and RPM when two gears of different sizes are paired.
The involute gear tooth profile means that each gear tooth is the involute of a circle – the curve that would be traced if you wound a string around the perimeter of a circle.
This curve enables a standardization of gear production, and was a great advance in gear design, because the tooth profile of an involute gear depends only on the number of teeth on the gear, pressure angle, and pitch and so it can fit with any other involute gear with a given pressure angle and pitch independent of how many teeth each gear has. Contact between a pair of gear teeth occurs at a single instantaneous point where two involutes of the same directional spiral meet. The line of action or line of contact is perpendicular to the tangent of any point along the gear profile’s curve. The angular velocity ratio between two gears of a gearset must remain constant throughout the mesh for smooth transmission of power.
Another advantage of the traditional straight-cut external spur-pinion gear is that this simple design allows for a high degree of manufacturing ease and precision.
Despite their general convenience and broad applications, for my purposes using external gears would have been impractical; the high gearing ratio would require drastically mismatched gear sizes whose disparate dimensioning would be unlikely to mesh well and would make a gear box mechanism that would be unwieldy to mount on the blender base.
I divided this project into two distinct phases: designing the artistic part, the zoetrope itself, and the mechanical part, the motorized base that spins the zoetrope.
Art
I decided to start with a relatively simple animation: the organic fluctuation of ocean waves. I used still life images and marked points of motion, a similar technique to that used in animation -- marking key points and tracking them between frames.
In Autodesk I arranged these images radially and used the rail feature to trace curving connections between each point of motion.
For the final 3d print, the material’s properties and strength were not important so I prioritized the piece’s aesthetics; I ordered a special translucent blue PLA to optimize the effect of the illumination, and make the structure appear more water-like.
Motorized Base
A key component of a successful zoetrope is the motorized spinning display. After some research, I determined that it would be cheapest to repurpose an old kitchen aid blender base that I had lying around, rather than buying motors and switches. Additionally, I’d been wanting to find a use for the blender base, and it’s repurposing presented a fun gearing challenge.
The first challenge of this repurposing was the mismatch in speeds; an ideal speed for zoetrope is less than 100rpm, so I needed to find a method for gearing down the blender.
→ Ideal zoetrope speed: 25 rpm
→ Blender speed: 10000 rpm
→ Resulting Gear Ratio:
After extensive testing, fitting, tweaking, and re-printing the gear reducer works!
I installed ball bearings, sanded rough surfaces, bolted the housing for stage 1 and 2 together, and secured the gear reducer unit to the base with long bolts that fit in the blender's grooves.
For the zoetrope display I mounted a laser-leveler so that the beam intersected the zoetrope display.
The laser only had the desired effect in a dark room so the video quality is not great:
Now that I have a motorized zoetrope display base, I hope to continue this project by experimenting with more interesting zoetrope animations and forms. Stay tuned for updates!
