Introduction to Gears
No good robot can ever be built without gears. As such, a good understanding of how gears affect parameters such as torque and velocity are very important. In this tutorial I will first talk about the basics of gears, how to use them properly along with simple equations, and then I will go into specific types of gears.

As with all motors, by looking at the motor datasheet you can determine the output velocity and torque of your motor. But unfortunately for robots, motors commercially available do not normally have a desirable speed to torque ratio (the main exception being servos and high torque motors with built in gearboxes). For example, do you really want your robot wheels to rotate at 10,000 rpm at low torques? In robotics, torque is better than speed.

With gears, you will exchange the high velocity with a better torque. This exchange happens with a very simple equation that you can calculate:

Torque_Old * Velocity_Old = Torque_New * Velocity_New

Torque_Old and Velocity_Old can be found simply by looking up the datasheet of your motor. Then what you need to do is put a desired torque or velocity on the right hand side of the equation. My staticsand dynamicstutorials can help you decide on a reasonable torque and/or velocity for your robot.

So for example, suppose your motor outputs 3 lb-in torque at 2000rps according to the datasheet, but you only want 300rps. This is what your equation will look like:

3 lb-in * 2000rps = Torque_New * 300rps

With highschool algebra you can then determine that your new torque will be 20 lb-in.

Now suppose, with the same motor, you need 5 lb-in (minimum force to crush a cat, obviously). But suppose you also need 1500rps minimum velocity. How would you know if the motor is up to spec and can do this? Easy . . .

3 lb-in * 2000rps = 5 lb-in * Velocity_New

Velocity_New = 1200rps

You now have just determined that at 1200 rps the selected motor is not up to spec. Using the simple equation, you have just saved yourself tons of money on a motor that would have never worked. Designing your robot, and doing all the necessary equations beforehand, will always save you tons of money and time.

The gearing ratio is the value at which you change your velocity and torque. Again, it has a very simple equation. The gearing ratio is just a fraction which you multiple your velocity and torque by.

Suppose your gearing ratio is 3/1. This would mean you would multiple your torque by 3 and your velocity by the inverse, or 1/3.

example; Torque_Old = 10 lb-in, Velocity_Old = 100rps
Gearing ratio = 2/3
Torque * 2/3 = 6.7 lb-in
Velocity * 3/2 = 150rps

Achieving a Particular Gearing Ratio
If you wanted a simple gearing ratio of say 2 to 1, you would use two gears, one being twice as big as the other. It isnt really the size as much as the diameter ratio of the two gears. If the diameter of one gear is 3 times bigger than the other gear, you would get a 3/1 (or 1/3) gearing ratio. You can easily figure out the ratio by hand measuring the diameter of the gears you are using.

You can also calculate the ratio by the ratio of teeth on the gears. If one gear has 28 teeth and the other has 13, you would have a (28/13=2.15 or 13/28=.46) 2.15 or .46 gearing ratio. I will go into this later, but this is why Worm gearshave such high gearing ratios. In a worm gear setup, one gear always has a single tooth, while the other has many - a guaranteed huge ratio. Counting teeth will always give you the most exact ratio.

friction, misalignment of pressure angles, lubrication, gear backlash (spacing between meshed gear teeth between two gears) and angular momentum, etc. Different gear setups, different types of gears, different gear materials, and wear and tear on the gear, will all have different efficiencies. The possible combinations are too big to list, so I will give you an estimated efficiency to expect with each gear type below. You can also find a much more exact efficiency by looking up the datasheet on the gears you are using.

For example, suppose you use two spur gears, you would typically expect efficiency to be around ~90%. To calculate, multiply that number by your Velocity_New and Torque_New to get your true output velocity and torque.

if (from above example)
Gearing ratio = 2/3
Torque * 2/3 = 6.7 lb-in
Velocity * 3/2 = 150rps

then
true torque = 6.7 * .9 = 6 lb-in
true velocity = 150 * .9 = 135rps

effienciency_total = gear_type_efficiency ^ (# of gears - 1) = .9 ^ (29) = 4.7 %

If instead you used 5 gears, you would have:

effienciency_total = .9 ^ (4) = 65.6 %

It is easy to figure out that if they are on the same shaft, the velocity (rotations per second) is the exact same. To calculate torque, you must use the moment arm balance equations. In this case, the radius is the moment arm.

radius_gear1 * torque_gear1 = radius_gear2 * torque_gear2

In this example, what minimum torque does the motor need to pull the weight up?

writing down the equations:
torque_motor * radius_gear1 = torque_gear2 * radius_gear2
torque_gear2 * radius_gear2 = torque_gear3 * radius_gear3
torque_gear3 * radius_gear3 = weight * radius_gear3

simplifying, we get:
torque_motor * radius_gear1 = weight * radius_gear3

so therefore the minimum required motor torque is
torque_motor = weight * radius_gear3 / radius_gear1

Now what if you wanted the weight to lift at 2 feet/sec. What rotations per second and direction must the motor rotate at?

writing down the equations:
rps_motor * radius_gear1 = rps_gear2 * radius_gear2
rps_gear2 = rps_gear3
rps_gear3 * 2*pi*radius_gear3 = velocity_weight

simplifying, we get:
rps_motor * radius_gear1 = rps_gear3 * radius_gear2
or
rps_motor * radius_gear1 = velocity_weight / (2*pi*radius_gear3) * radius_gear2

so therefore the required motor rps is
rps_motor = 2 ft/sec * radius_gear2 / (2*pi*radius_gear3 * radius_gear1)

The pitch diameter is as shown. To calculate the pitch, simply use this equation:

Pitch = # teeth / pitch circle diameter (in inches)

For example, a gear with 72 teeth and a 1.5" pitch diameter is 48 Pitch. Gears that mesh must both have the same pitch and pressure angle (usually 20 degrees).

Note: The efficiencies listed are only typical. Because of many other factors could be present, the listed efficiencies should only be used as a guide. Often manufacturers will give you expected efficiencies in the datasheets for their gears. Remember, wear and lubrication will also dramatically affect gear efficiencies.

Rack and Pinion (~90% efficiency)

Rack and Pinion is the type of gearing found in steering systems. This gearing is great if you want to convert rotational motion into translational. Mathematically, use radius = 1 for the straight 'gear'.

robot arm holding something heavy, and you dont want to waste power on holding torque. The efficiency is low, but lubrication really helps.

I hope this has been helpful to you regarding your gearing for robotics.

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