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The Fundamental Law of Gearing and 3D Printing Good Gears

Five Tooth Gear

As the spurs come into contact, rotate, and disengage, it is important that the rotation speed of the gears remains constant. This is needed for smooth transmission of power. The fundamental law of gearing essentially states:

The angular velocity ratio between two gears of a gear set must remain constant throughout the mesh.

 Norton, R.L., 2006, Machine Design: An Integrated Approach, 3rd Ed, 148190-8

To achieve this, an “involute gear” profile is generally used to design gear teeth. An involute curve is created as you grab the end of a string on a cylinder and unwind it. The end of the string traces out an involute curve. The edge of the gears teeth will have this same type of shape resulting in a kind of budging, rounded tooth.



The involute curve on the spurs also results in the force between the two gears being at a constant angle, regardless of where the teeth are found as they continuously mesh and re-mesh. The angle can be engineered into the gear but is almost always at twenty degrees along a “line of action”.

This angle is called the pressure angle. Decreasing the pressure angle reduces the force on the gear bearings. However, teeth with a smaller pressure angle are not going to be as strong as teeth on a gear with a higher pressure angle. Higher pressure angles allow for stronger teeth but greater force on the bearings and often, more noise.

It is important to match the pressure angles in a gear train. Gears working together should have the same pressure angle.

Teeth Per Gear and Gear Ratio

There can be some practical limits to the number of teeth on a gear. A small number of teeth may require that the teeth be undercut to allow meshing of the spurs. A small number of teeth also will not run as smoothly.

In the five tooth gear below, you will notice the spurs get narrow near the base, using ten teeth or more may be better for your design.

Five Tooth Gear


One of the difficulties in using more teeth per gear is that many 3D printers may not be able to print at the resolution needed for accurate teeth.  Consider the two gears below that are roughly the same sizes.


Adding more teeth requires the teeth to get smaller. As the radius of the gear is reduced at a size that is dependent on the resolution of the 3D printer, it will not be possible to print a useable thirty tooth gear, while the larger five tooth gear is still workable.

A second benefit to using larger teeth is that the distance at which the gears interact is also larger. This can sometimes give valuable tolerance when actually assembling the part. Slightly misaligning two 30 spur gears could have a more serious effect (gears not engaging)  than slightly misaligning two five-pin gears.

This post is up to about 500 words so it looks like a good time to stop.  In my next post I’ll try to add a bit more useful information.