# How to Design and Make Automata or Whirligig Gears Using the Power of Geometry

“…Chemistry is not an exact science” ~Mario Andrada

In this post I will do my very best to simplify the process of designing and making gears from wood and other materials. The process to build a simple Spur Gear and Pinwheel Gear will be explained.

Thanks to my background in 3d animation I have a rudimentary understanding of geometry and mathematics. I would love to be a math magician but like many people I get lost with anything beyond algebra. Thank goodness for the internet and calculators!

As my math magician friend Charlie reminded me, “To get the teeth to mesh, the spacing BETWEEN the teeth need to be the SAME on all gears.” With this in mind, using the n-gon is ideal to design a gear, the spacing between each vertex is uniform. Simply stated, an n-gon is a polygon with “n” amount of edges. The image to the left is of an eight sided n-gon. The n-gon has two radius measurements: circumcircle (rc)and incircle (ri). If you need a refresher, the radius is the distance from the center to the outer edge of the n-gon, the diameter is the complete distance from side to side (through the center). Vertices are the angular points where each edge meets (the white “edge” arrows point to vertices).

When designing gears we will focus mostly on the circumcircle radius (rc), the vertices are positioned along this radius.  The vertices will become the teeth of our gears.  If the desire is to use a gear to turn another gear uniformly each gear will be identical resulting with a 1:1 ratio. To use a drive gear to rotate a second gear at half speed the second gear needs twice as many teeth as the drive gear, a 2:1 ratio.

Below I have included a calculator to do all the hard stuff for us.

Say you want to make a pair of gears with a 2:1 ratio, the drive gear turning twice for each turn of the second. You also want the drive gear to have a 1″ radius (2″ diameter). You also want the teeth to be separated by 0.5″. This is easily accomplished with the use of the above calculator. The calculator’s default settings are Edge Length (a): 0.5 and Number of Vertices (n): 8 resulting with radius (rc)  of 0.6535. This radius is just over half of what we desire. We can’t change the Edge Length because in this example we want the tooth spacing to be .5″. Instead, increase the Number of vertices to 12. Now radius (rc) is 0.9664 just under the 1″ radius we were looking for. Perfect!

The 2:1 ratio requires the second gear to have twice as many teeth. This doesn’t mean twice the teeth makes the gear twice as large. Let’s see. In the calculator change the Number of Vertices to 16, doubling the amount of the drive gear. Radius (rc) is 1.9162.

This is important! When I started designing gears I was under the impression that to double the ratio, the radius simply needed to be doubled. This is NOT the case (thanks Charlie)! Let’s examine our calculated radius values:

• 12 vertices Drive Gear (rc): 0.9664
• 24 vertices Second Gear (rc): 1.9162

That’s double, right? No. It’s not double. By doubling the drive gear radius (rc), 2 x 0.9664 the product is 1.9328, a difference of 0.0166. Doesn’t seem like a huge deal, but a .0166 error can, in fact, impede the smooth operation of the gears. To emphasize this point let’s examine a more extreme 10:1 ratio example.

• 12 vertices Drive Gear (rc): 0.9664
• 120 vertices Second Gear (rc): 9.5552

Multiplying the 12 vertices Drive Gear (rc): 0.9664 by 10 (0.9664 x 10) results with a product of  9.664. That’s 0.1088, or a tenth of an inch, larger than calculated (rc) value.

# Making Spur Gears

Right about now you’re probably thinking, “Hey John. I thought you were going to show me how to make gears, not bore me to death with math.” Well, you’re in for a treat, let make some gears! We’ll start by making a pair of spur gears: one 1:1 and another 1:2. A spur gear is a gearwheel with teeth projecting parallel to the wheel’s axis, this is the sort of gear everyone is familiar with. For this example we’ll be making wood gears. You’ll need paper, wood, glue, drill (or drill press), saw, an accurate caliper gauge and a quality pencil compass.  If you don’t own these instruments you can find them at any hardware store – or you can be like me and score vintage beauties at flea markets and estate sales. Cheap tools may work, Harbor Freight – cough, cough, but I often find cheap tools more frustrating than productive.

### Step 1: Layout the Gear

Laying out the gear is the most important task of making your own gears. I own a few sets of old drafting tools I picked up estate sales for a few dollars. The compasses in these sets are fantastic quality and several of them have an adjustment lock. I use several compasses, and once their settings are perfect, I don’t change a thing until every gear is marked on on wood.

First, calibrate the compasses by drawing on paper. To layout the drive gear use a pencil to draw a small dot on paper, this is the center of the first gear. Set your caliper gauge set (rc): 0.9664 (or as close to this value as the gauge allows) match the pencil compass to this value. Place the compass needle on the pencil center mark and draw the circle. Reference your caliper gauge from the center of the circle to ensure the drawn circle is correct.

Set the caliper gauge to the Edge Length (a): 0.5 and adjust a second pencil compass (preferably locking) to match. Using the circle as reference, draw ticks across the circle (rc) at .5 intervals. When you’ve gone all the way around the circumference your last tick should match the first tick exactly. Refer to the Step 1 image to see my terrible first result (red circle). If it’s not perfect, something went amiss in your settings. You’ll need to start again. This requires patience and practice. The width of the pencil line complicates creating accurate marks. You’ll need to get a feel for the process.

Once you’re comfortable laying out the gear, layout the pattern for each gear on the wood you’re using. This also may require a few tries. Working on this example required about two hours to layout eleven gears from start to finish.

### Step 2: Cut Out the Gear

Now you’ll need to cut the round gear from the block of wood. Generally I use the band saw or jigsaw for the task. You can use whatever works best for you: hand jigsaw, Dremel, router, etc. Cut to the outside of the radius (rc) line you created with the compass. Try not to remove the line! Once the gear is roughed out, use a disc sander to shape the circle precisely to the line (bottom left Steps 2 &3 image).

### Step 3: Drill a Hole

I generally use 1/8″ wire to mount the gear to the project. The wire serves as the shaft for the gear to rotate about. I use an 1/8″ drill bit in my drill press for the task. Drill an appropriate sized hole centered on the depression you make with an awl. This is the middle of the gear.

After the gear is complete I use a small round file to enlarge the hole to make it rotate more easily on the wire shaft. But that’s the last step!

### Step 4: Add the Teeth

This is where personal preference, practice and experience comes into play. For this example I will be using poplar that I’ve planed to .125″ thickness. The strip of .125″ poplar is ripped on the table saw to .75″ width. Individual teeth are crosscut from the strip to a .75″ length. Each tooth is .125″ x .75″ x .75″.

I’ve constructed a miter bar jig for the table saw to hold the gear while cutting a dado for each tooth around the the gear. The dado I cut is .25″ deep and .125″ wide.  With the table saw jig I am able to center the vertex ticks drawn in Step 1 spaced at .5″ around the gear. I center the tick to the blade, cut the dado. The gear is rotated to center the next tick and the next dado is cut. This process continues until each required dado is cut.

I squirt out a puddle of wood glue on a scrap. I dip the point of a wood skewer (the grocery store kind) into the glue and spread glue into a gear dado. Then, using the skewer, add a little glue to the end of a tooth square. It is important to insert the tooth square into the dado so the wood grain is perpendicular to the dado. If the tooth is attached with the grain parallel to the dado you run the risk of the tooth breaking with the grain.

Continue this process until you’ve completed the gear.

### Step 5: You’ve Made a Spur Gear!

Congratulations on making your first gear! Repeat these steps for the second gear (keeping in mind the second gear is larger: 24 vertices Second Gear (rc): 1.9162).

# Making Pinwheel Gears

Let’s say your project requires the drive shaft to power another element or shaft at a ninety degree angle. Enter the Pinwheel Gear. You’ll need wood, drill (or drill press), saw, an accurate caliper gauge and a quality pencil compass.

### Step 1: Layout the Gear

The layout for differential gears is the same as spur gears above. Use an awl to mark center. Then draw the circle with radius (rc) using a compass. Use the compass again to draw evenly spaced vertex ticks around the circle. Because we’ll be using nails as the teeth on these gears we’ll need to draw another larger circle outside radius (rc). In this case radius (rc) is 0.9664, I generally add an eighth of an inch (0.125) resulting with a radius of 1.0914.

### Step 2: Cut Out the Gear

Cut the gear to the outside of the largest circle. Then sand precisely to the line.

### Step 3: Drill a Hole

This is exactly at Step 3 for the spur gears. I drill a 0.125″ hole centered on the awl mark.

### Step 4: Add the Teeth

I use the drill press to create an appropriately sized pilot hole at each vertex cross tick. The pilot hole should not be completely through the gear, only as deep as the nail will be driven into the wood. Here, I’m using three penny nails. Start the nail in partway then place a scrap of wood against the nail as a depth gauge. Then hammer the nail until you’re hammering the wood scrap. Continue adding nails in this fashion until your pinwheel gear is complete.

### Step 5: You’ve Made a Pinwheel Gear!

You’re an expert gear maker now. Let your imagination run wild! I’d love to see the mechanical creations you’ve built.

I started the post with a quote that originated from the 2016 Rio Summer Olympics, “…Chemistry is not an exact science…” This was an Olympics official’s response to questions pertaining to why the pool smelled rotten and the water was green. I’m here to say Chemistry is and exact science. What does this have to do with making gears? Well, making gears is an exact science also. This post, however, is the groundwork to understand how to construct gears, not exact science.

Earlier I posted about building a Pegasus whirligig kit. Assembling the kit was a fun distraction, but I honestly didn’t learn much from the task. I’m a tinkerer. I enjoy spending time considering how to make things, and how things work. I find little satisfaction in following a detailed design – robots do that. I like to build the plane while it’s in the air, as they say. It’s fun to start something, and troubleshoot and modify along the way. This is how I gain a full understanding of the project. I often build many test projects before I tackle the actual build.

Creating mechanical machines is challenging. There is a lot of trial and error involved for the novice (myself included). There is more to designing precision gears than I’ve mentioned in the post. I’ll be honest, I don’t understand most of the technical mumbo jumbo, big words like dedendum, addendum, clearance and working depth versus whole depth. If things don’t work – that’s normal. It’s an entertaining learning experience. I personally find as much enjoyment in the flops as in the successes. When the project is complete, the challenge is over – and that can be a bummer.

Making gears using this method will require trial and error. The space between the gear positions will be an issue. The heads of the nails and the lack of a taper on the ends of the spur gear will likely cause these gears to jam. Consider using a metal cutting wheel to cut the heads off the nails – and taper the metal end. Also consider sanding a taper on each tooth before assembling the spur gear.

For those makers that want a detailed, guaranteed plan you can visit http://geargenerator.com/ to design and print precise gears. This post will get you started making functioning gears. Please take what you learned here, build on it and make it your own. There’s more than one way to make a gear.

I am planning a follow up post regarding making wooden gears. There will be more information and project ideas to be found in the follow up post. In the meantime, be creative and have fun.

# Mize Mini-Mize Whirligig Automata Pegasus

As you probably figured by now, I can’t sit still. Yes, I have a zillion started projects in my workshop and plans for more in my mind and hard drive. I may get around to finishing some of these projects but I have such little time! I’m not a humongous fan of 3d printing and laser/cnc cut stuff, but every once in a while I scratch the creative itch and dabble with this sort of thing.

I decided to purchase a mini automata whirligig kit manufactured by Mize, based in South Korea, online for \$21.00 including shipping. There wasn’t a whole lot of information about this item in the description, but judging from the single image of the item it looked like it was going to be small. The package arrived from South Korea and I thought it was a thick holiday card, roughly 6″ x 9″ x .75″.

I opened the package and looked at the instructions. Yup, as assumed all the instructions are in Korean. Not a problem though because the images tend to explain everything clearly (enough). Curious, however, I photographed the instructions and uploaded them to i2ocr to translate. I don’t think it translated too well. Here’s a few selections from the translation:

• The city is divided into cities
• Excessive stress on the stomach can damage it
• You have to do the complexion
• I want to be a transit agent, too.
• Sennepusa Seeking Confession | Do not be sick
• The lungs are soaring
• Even if I left you, I would like you to be my best friend

For real. I can’t make this stuff up.

Lucky for me I work with Heeman, a talented Korean designer. Heeman was kind enough to translate the pertinent information in the image above. Thanks!

Above are the three panels of parts that create the project.  Along with the instructions this is everything in the package. Excited, I retrieved my Loctite Go2 Glue, a toothpick and a paper towel. I reviewed the assembly instructions for the first few parts. I carefully removed the necessary parts from the panels by first scoring each sprue (the little piece of material holding the part in pace on the panel) with an Exacto, then carefully nudging the part free.  After test fitting the parts together I squeezed a small puddle of glue on a scrap of paper, applied a small amount of glue to the joints with a toothpick and reassembled.

The image at the top of the post displays the assembled crank box and completed project. This was an enjoyable and easy project to build. It’s important to be patient and clamp the pieces together (when possible) as you wait for the glue to cure between steps. I’ll admit, when I was attaching the pegasus to the gearbox, pretty much the last step, I carelessly broke the propeller off the gearbox. Luckily a dab of super glue came to the rescue and worked flawlessly.

I’d be remiss if I didn’t say the pegasus whirligig is not suitable for prolonged outdoor use. The material is MDF or something similar. I am surprised at how smoothly the mechanics operate because the drive shafts are simply square cut MDF material positioned in round holes. Birthday cake candles are provided in the kit. These are used to lubricate the moving parts. The lubrication the candles provide also works much better than I anticipated. The video clip below is of the complete kit operating outside in relatively gentle gusts of wind. Most likely I will be purchasing more of these Mize kits in the near future.

# Wooden Pendulum Drawing Machine

I’ve received confirmation on my application for the 2017 Greater Newark, NJ Maker Faire Saturday May 6, 2017. This year my exhibit is titled Art Through Motion and I’ll be building various mechanisms to create Spirograph like drawings. The Wooden Pendulum Drawing Machine is the first prototype I’ve created.

This simple mechanism suspends a canvas from wires over a stationary Sharpie marker. The artist urges the canvas into a swinging motion then drops the marker into position. When the swinging of the canvas ceases the artist removes the marker and decides if more drawing is required. If so, the artist starts the process over again and may choose a different color marker.

The drawing above was created on this mechanism. I’ve titled it “Galaxies” and it’s available for \$2000.00, only kidding, it’s not for sale. It’s priceless. I couldn’t make another one just like it if I tried.

But seriously, stay tuned for more news about my projects for the Greater Newark Maker Faire. I have many more cool drawing machines in the works.

# Wooden Toy Train Whistle Made With Table Saw

“I believe don’t start if you’re gonna quit”~Eric Church

I became inspired after building the Mini-14 Street Organ to learn about making wooden whistles for musical gizmos. I figured a good place to start was to build an old fashioned wooden train whistle toy. Ya know, the kind of thing kids buy at a gift shop to drive everyone around them crazy for days. A quick internet search revealed plans for the project on The Woodcrafter Page.

The Woodcrafter whistle required drilling four 7/16″ holes into a block of wood: at lengths 4 1/4″, 4 3/4″,  6 1/4″ and 7 1/4″ and  and plugging up the whistle end with 1/2″ length of dowel. Well, I don’t own a 7/16″ drill bit that’s 7 1/4″ long – and I don’t feel like buying one. I also didn’t feel like rigging something up to drill a straight hole to that length. I turned my attention to figuring out a way to convert that design to something that can be made with a table saw. I started by calculating the spacial volume of each whistle.

7/16" Hole Length My Volume Calculation Correct Volume
3.75" .52 inch^3 .56 inch^3
4.25" .59 inch^3 .64 inch^3
5.75" .82 inch^3 .86 inch^3
6.75" .97 inch^3 1.01 inch^3

The length of the hole in the table above has subtracted 1/2″ from each depth because of the inserted whistle dowel plug. For starters, my calculations are incorrect because I subtracted 3/4″ from the length, plus I made a few extra errors. For my train whistle I used my volume calculations.

The first whistle design fixed the length of the whistle to 2″ and height to .5″. The width varied based on matching the spacial volume. Fun fact: Confirmed in hindsight, the length sets the pitch of the whistle. In the case of this whistle there are four whistles and each one simultaneously sings out a D6 note, or 1174.66 Hz. You can hear this whistle by playing the sound below.

So the first whistle wasn’t so great. I learned the length of the whistle determines the pitch. The second whistle I built fixed the height and width to .5″ and the length was adjusted to maintain my spacial volume calculations. The lengths are as follows: 2.08″ 2.36″, 3.28″, 3.88″. This whistle sounds more like a train whistle clearly making three frequencies (show in image at top).

Since I was in the zone I also built a third version fixing the height and width to .5″ and the lengths provided from the Woodcrafter Page. This whistle basically sounds two frequencies, but mostly sounds like one low note. Listen to whistles #2 and #3 here:

What did I learn? This lesson taught me I have a lot to learn about whistles. The Woodcrafter design suggests there should be four frequencies produced, in pairs of two close frequencies. I know why the first whistle only sounds one note – because all for whistles are the same depth. The second whistle may actually produce four frequencies, the two lower frequencies close to each other. I’m uncertain why the fourth one appears to only sound two frequencies. At least two of the whistles constitute the strong lower frequency because it’s wide and strong.

I have more ideas to explore when I revisit the project. And I think I’m going to consult with Charlie, my engineering and math magician friend, before diving in.