Grinding Gears: Figuring Out The Ratio

Practically any combination of motor and gearbox can be mathematically arranged to fit all sorts of problems. You could lift a crane with a pager motor, it just might take a few hundred years. However, figuring out exactly what ratio you need can feel bit backwards the first time you have to do it.

A gear is nothing more than a clever way to make two circles rotate in concert with each other as if they were perfectly joined at their circumferences. Rather than relying on the friction between two rotating disks in contact, the designer instead relies on the strength of a gear tooth as the factor limiting the amount of torque that can be applied to the gear.

Everything is in gearing is neatly proportional. As long as your point of reference is correct, and some other stuff. Uh, it gets easier with practice.

\frac{radius_1}{radius_2} = \frac{velocity_1}{velocity_2} = \frac{tangentialforce_1}{tangentialforce_2} = \frac{torque_1}{torque_2}

Now as my physics professors taught me to do, let’s skip the semantics and spare ourselves some pedantics. Let us assume that all gears have a constant velocity when you’ve averaged it all out. Sure there is a perceptible difference between a perfect involute and a primitive lantern gear, but for the sake of discussion it doesn’t matter at all. Especially if you’re just going to 3D print the thing. Let’s say that they’re sitting on perfect bearings and friction isn’t a thing unless we make it so. Also we’ll go ahead and make them perfectly aligned, depthed, and toleranced.

Typically, a gearbox is used for two things. You have a smaller torque that you’d like to make into a bigger one or you have one rotational velocity that you’d like to exchange for another.  Typically torque is represented with a capital or lowercase Tau (Ττ) and rotational velocity likes to have a lowercase omega (ω). It also doesn’t matter at all; it just makes your equations look cooler.

Now a lot of tutorials like to start with the idea of rolling a smaller circle against a bigger one. If the smaller circle is a third as large as the big one, it will take three rotations of the small circle to make the big one rotate twice.  However, it is my opinion that thinking it in terms of the force applied allows a designer to think about the gearing more effectively.

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If the friction between the two surfaces of the circle is perfect, then any force applied tangentially to one of the circles will result in a perfectly perpendicular and equal force to the other circle at the point of contact between the two. Midway through writing the preceding sentence I began to understand why textbooks are so abstruse, so I also drew a picture. This results in two equations.

 \tau_1 = F * r_1 

\tau_2 = F * r_2  

Evernote Snapshot 20160831 181223Now, when you have a force perpendicular to the line drawn to describe the radius, the equation for torque becomes really simple.

Multiply the length of the “lever arm”, “radius”, etc. by the force to get the preceding equations. Make sure to include the units.

You should end up force-unit * length-unit. Since I usually work in smaller gears I like to use N * mm. American websites typically use oz-in to rate motors. It is technically ozf-in (ounce-force), but the US customary system has a fetish for obtuseness.

We can make some observations. The smaller gear always sees less torque at its center. This initially seemed a bit counter-intuitive to me. If I’m using a cheater bar to turn a bolt the longer I make the bar the more torque I can put on the bolt. So if I touch the outside of a really large gear I should be seeing a ton of torque at the center of a small gear rotating along with it. However, as we mentioned before, any torque applied on the outside of the larger gear is seen equal and tangential on the smaller. It’s as if you’re touching the outside of the small gear. The torque has to be smaller.

This is why you have to pedal so much harder when the rear sprocket on a bicycle gets smaller.  Each time you make the sprocket smaller you shrink the torque input into the wheel. If the perpendicular output where the wheel hits the ground is <input from the small gear> / <radius of the wheel> then it’s obvious why this happens.

Hopefully my diagram doesn't win a prize for awfulness. Then again, an award's an award.
Hopefully my diagram doesn’t win a prize for awfulness. Then again, an award is an award. Remember that the bicycle wheel and  its input gear are rigidly attached to each other.

It’s also important to note that any time you increase the torque, the speed of the gears slow by the same proportion. If you need 60 N*m out of a motor that can give 20 N*m and you use a 3:1 gearbox to do it. If the motor previously ran at 30 rpm it’s now running at 10 rpm.

Let’s jump right into an example. Let’s say you want to make a device that automatically lifts your window blinds. You’ve got some junk and a 3D printer.

The problem set-up.
The problem set-up.

Now you’ve taken a spring scale and pulled until the shutter moves and you know you need 10 lbs. of pull to get the blinds to pull up. To make it easy on yourself you multiply this number by two so you know you need exactly 20 lbs of force to pull the curtain up. Then to make it really easy on yourself convert it all to Newtons. It’s approximately 90 N.

Now you don’t really care how fast the blinds pull up, but you go ahead and pull them up yourself. You get the feeling that the blinds won’t appreciate being lifted faster than the whole range in two seconds. You personally don’t care if takes ten seconds to, but you’d like it not to take too long.

You also measure the length of string pulled out to raise the blinds. It’s 1.2 meters.

A classic. Cheap. Unreliable. Weak. Also Cheap.
A classic.

Lastly, you only have one spare power supply and a matching motor left in your entire laboratory after you followed the advice in a Hackaday article. Cursing the day the author was born, you sullenly write down the last specifications. You’ve got one of those cheap GM9 gear motors. 5 V, 66 rpm, and 300 N*mm. You damn him as you think fondly of your mountain of windshield washer motors and 80 lb server rack power supplies that you tossed out.

To start with, you do some experiments with a pulley. You arbitrarily pick, 3D print, and find that a 100 mm in diameter pulley seems to wind it up nicely by hand. By the end of the winding the outside diameter of the string is 110 mm. So you use the torque equations above. You find that at the end of the rotation, if you attach the motor directly, there is only 5.45 N of force being applied to the string. Not nearly enough.

Hrm..
Hrm..

So, since you know everything is more or less proportional, you divide 90 N / 5.45 N, and arrive at an answer of 17. So, at a minimum for every turn of the pulley you need 17 turns of the motor to get the torque needed.

That would be okay, but it messes with our other specification. At a 17:1 ratio, it will take our 66 rpm motor pretty close to a minute to wind the blinds up.

Damn.
Damn.

This is a moment for some pondering. Make a coffee. Maybe go write a relaxing comment to a Hackaday writer listing their various flaws, perceived and true, in excruciating detail.

What if you wound the string up on a closet rod? Those are only about 30 mm in diameter. You take a bit of rod and wind it up. It seems to work and since it’s wider the string only ends up adding 5 mm to the final diameter. You rework the calculation and find that in this case you only need a ratio of 6! Yes.

Now some of you who have done this before are likely gnashing your teeth, or more likely already down in the comments. Unfortunately it’s all proportional. While you only need a ratio of 6:1 now, nearly a third. You also need to rotate the pulley approximately three times as much to pull the same length of cord.

Sometimes you can’t win. In this case the only solution is to order a new motor. You look online for a bit and realize that one of the 12 V motors you threw away last week would work perfectly for this. You wouldn’t even need a gear box. You could attach it straight to the pulley. You look around your perfectly clean and orderly garage and feel empty.

However, just for fun you build a 6:1 gearbox anyway. It’s a hack after all.

Cover photo of the hilariously complicated Do Nothing Machine credit to the Joe Martin Foundation.

Ask Hackaday: What Are Magnetic Gears (Good For)?

Magnetic gears are surprisingly unknown and used only in a few niche applications. Yet, their popularity is on the rise, and they are one of the slickest solutions for transmitting mechanical energy, converting rotational torque and RPM. Sooner or later, you’re bound to stumble upon them somewhere, so let’s check them out to see what they are and what they are good for.

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Star Track: A Lesson in Positional Astronomy With Lasers

[gocivici] threatened us with a tutorial on positional astronomy when we started reading his tutorial on a Arduino Powered Star Pointer and he delivered. We’d pick him to help us take the One Ring to Mordor; we’d never get lost and his threat-delivery-rate makes him less likely to pull a Boromir.

As we mentioned he starts off with a really succinct and well written tutorial on celestial coordinates that antiquity would have killed to have. If we were writing a bit of code to do our own positional astronomy system, this is the tab we’d have open. Incidentally, that’s exactly what he encourages those who have followed the tutorial to do.

The star pointer itself is a high powered green laser pointer (battery powered), 3D printed parts, and an amalgam of fourteen dollars of Chinese tech cruft. The project uses two Arduino clones to process serial commands and manage two 28byj-48 stepper motors. The 2nd Arduino clone was purely to supplement the digital pins of the first; we paused a bit at that, but then we realized that import arduinos have gotten so cheap they probably are more affordable than an I2C breakout board or stepper driver these days. The body was designed with a mixture of Tinkercad and something we’d not heard of, OpenJsCAD.

Once it’s all assembled and tested the only thing left to do is go outside with your contraption. After making sure that you’ve followed all the local regulations for not pointing lasers at airplanes, point the laser at the north star. After that you can plug in any star coordinate and the laser will swing towards it and track its location in the sky. Pretty cool.

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1980s Toy Robot Arm Converted To Steam And Other Explorations

We were doing our daily harvest of YouTube for fresh hacks when we stumbled on a video that eventually led us to this conversion of a 1980s Armatron robot to steam power.

The video in question was of [The 8-bit Guy] doing a small restoration of a 1984 Radio Shack Armatron toy. Expecting a mess of wiring we were absolutely surprised to discover that the internals of the arm were all mechanical with only a single electric motor. Perhaps the motors were more expensive back then?

The resemblance is uncanny.
The resemblance is uncanny.

The arm is driven by a Sarlacc Pit of planetary gears. These in turn are driven by a clever synchronized transmission. It’s very, very cool. We, admittedly, fell down the google rabbit hole. There are some great pictures of the internals here. Whoever designed this was very clever.

The robot arm can do full 360 rotations at every joint that supports it without slip rings. The copper shafts were also interesting. It’s a sort of history lesson on the prices of metal and components at the time.

Regardless, the single motor drive was what attracted [crabfu], ten entire years ago, to attach a steam engine to the device. A quick cut through the side of the case, a tiny chain drive, and a Jensen steam engine was all it took to get the toy converted over. Potato quality video after the break.

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Scratch-Fabricated Plastic Gobbling Shredder Brings Recycling Home

[Jason Knight], an intern at FabLab RUC, has worked hard for 9 months to make a sheet plastics shredder for HDPE and LDPE from things like plastic bags, bubble wrap and air cushion packaging with the goal of recycling the shredded plastic. Why shred these things? When broken down to smaller pieces they can be melted in a consumer grade oven (like where you cook your frozen pizzas) then molded into new objects or extruded into 3D printing filament.

We especially like his big homemade 1.1 inch (30mm) thick wooden gears, for transferring the rotation from the motor to the cutting shafts while giving a step up in torque. As you can see in the video below, the gears definitely add an extra look of power to the machine.

The blades are the shape you most often see in shredders, gear-like disks side-by-side with teeth cut from them that pull the plastic in while shredding it (in contrast to this lower-throughput experimental DIY shredder made with two steel pipes). [Jason’s] multiple teeth are a bit of work to fabricate — not only were all the teeth milled from sheet metal but they then had to be individually sanded to remove burrs from the edges. It was worth it, as this has no problem chewing waste plastics to pieces.

Shredders can be dangerous machines for wandering fingers so [Jason] added a few safety features. Those include a drawer that you open to insert your plastic into the shredding area and a guard that completely surrounds the gears. And both features include transparent plastic areas so that you can still watch the impressive working parts in action.

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Gear Clock Uses Stepper Motor

[Rjeuch] liked a wooden clock he saw on the Internet, but the gears were produced with a proprietary software tool. So he built his own version. Unlike the original, however, he chose to use a stepper motor to drive the hands.

The clock’s gears aren’t just for show, and the post does a good job explaining how the gears work, how you might customize them, and how they fit together. The clock’s electronics rely on an Arduino.

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Incredible Marble Music Machine

We tried to figure out how to describe the band [Wintergatan]. It took a lot of googling, and we decided to let their really incredible music machine do it for them. The best part? Unlike some projects like this that come our way, [Wintergatan] documented the whole build process in an eight part video series.

The core of the machine is a large drum with two tracks of alternating grey and black Lego Technic beams and pins. The musician sequences out the music using these. The pins activate levers which in turn drop ball bearings on the various sound producing devices in the machine. The melody is produced by a vibraphone. At first we thought the drum kit was electronic, but it turns out the wires going to it were to amplify the sound they made when hit. At the end of their travel the bearings are brought up to the hopper again by a bucket conveyor.

The final part count for the machine sits at 3,000 not including the 2,000 ball bearings rolling around inside of it. If you’ve ever tried to make a marble machine, then you’ll be just as impressed as we were that the machine only appeared to lose a few marbles in the course of a three minute song. Aside from the smoothness of the machine, which is impressive, we also enjoyed the pure, well, hackiness of it. We can spy regular wood screws, rubber bands, plywood, bits of wire, and all sorts of on-the-spot solutions. Just to add bonus cool, the whole project appears to have been built with  just a bandsaw, a drill press, and a few hand power tools.

The machine is great, but we also really appreciate the hacker spirit behind it. When a commenter on a YouTube video told him he was a genius, he replied, “Thank you for that! But I do think, though, that it is mostly about being able to put in the time! I mean the talent of being stubborn and able to see things through are more important than the abilities you have to start with. If you work hard on anything, you will learn what you need and success! Its my idea anyway! So happy people like the machine!”. Which we think is just as cool as the machine itself. Video of the machine in action and part one of the build series after the break!

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