Basic Gear train calculation

Consider the simplest of gears - spur gears (straight cut gears). There are a plenty of special literature covering the topic in details, I just want to provide the basic params that will be sufficient for you to design gears. For example, it can be used for 3D printing.

When two gears are in the mesh, the first one is a pinion gear - has number of teeth z₁, and the second one is a driven one - has number of teeth z₂. Thus, gear ratio is r = z₂/z₁

There are at least four circles describing the gear: outer circle (d_a), inner circle (d_f), pitch circle (d), and basic circle (d_b). They all are shown for the driven wheel on Fig.1. The entire tooth is bounded by outer and inner circles. If we consider two cylindrical surfaces equivalently replacing gears so that they are in contact with each other and there is no slide between them while rotating, then they would have diameters equal to the gear pitch diameters. Thus the ratio of their diameters is the ratio of the gear train itself.

In order to avoid vibrations and make the gears to run smooth, the tooth has a special profile shape, usually involute. If you consider a straight rail being rolled around a cylindrical surface, the tip of the rail will follow some trajectory. This will be the shape of involute. That’s why we need the basic circle - it's the circle to roll our "rail" around to provide the involute shape for our teeth. Having involute shape of the teeth, the pressure line between the two teeth being in contact will remain constant (red line 1 on Fig.1), providing the constant gear ratio and smooth operation.

The pressure angle α between the pressure line 1 and the pitch circle tangent line 2 is usually taken equal to 20° as a standard. The pressure line is always tangent to both basic circles.

The distance between the two adjacent teeth measured along the pitch circle is a gear pitch p. However, in practice the gear module m is used:

You can imagine the following to understand the module better. Take the pitch circle and expand it to the straight line preserving the full circle length. Then if you compress it to the pitch diameter d (shrinking π times), the gear pitch p will also be shrinked to the module m. That's why this also is correct:

The values of module are standardized and should be chosen from the tables.

The tooth height above the pitch circle is equal to module m. The tooth height below the pitch circle is equal to 1.25·m. Thus, the full tooth height is 2.25·m.

Given the module, z₁, and z₂, the gear circle diameters are expressed as:

The distance between the gear axis:

If the gear diameters are given, we can express z₁, z₂.

It's better to avoid designing gears with z less that 16. The shape of the teeth is being distorted so much so it causes the problems with meshing. If z₁ or z₂ appears to be less that 16 for a given diameters, choose the smaller module.

Next I will share some approach to build a tooth in CAD software.

Important! The major of the modern CAD software gives you a plugins out of the box to automate gear train design, so you should definetelly use them. However, if you still need to design something very custom, you can build the tooth shape manually. There are a plenty of approaches known for that purpose, I will share the one I personally use sometimes. This was proven to work fine for design and 3D print of a plastic gears.

So, let's start with defining a base circle diameter:

Let's introduce angles φ₁ and φ₂:

Draw a straight line 1 through the gear axis (Fig.2). Draw a straight line 2 having an angle φ2 with line 1. Mark the point 5 on the intersection of the line 2 and the pitch circle. Here we've got the first point of the tooth shape. Draw a line 3 thorough the point 5, which is tangent to the basic circle. Mark the point 4. The distance L between points 4 and 5 is:

If we now roll the line 3 around the basic circle towards the base of the tooth, the point 5 will move to the point 6. The point 6 lies on the inner circle, the line O₂-6 makes an angle φ₃ with the line O₂-4:

Draw a line 7 that makes an angle φ₁ with the line 3 and is also tangent to the base circle. We obtain the point 8. The distance L₁ between the points 8 and 9:

Draw a line 10 that makes an angle 2·φ₁ with the line 3 and is also tangent to the base circle. We obtain the point 11. The distance L₂ between the points 11 and 12:

Now we can finally draw the spline through the points 6, 5, 9, 12. We have d_b < d_f (which is not always the case), so we can add a straight radial line 6-13 to the shape of the tooth. We can also add an arc 14-15 that lies on the outer circle and will be the tip of the tooth. Finally, mirror the contour 13-6-5-9-14-15 about the symmetry line O₁_0₂. Now we have the full tooth shape.

If we need to build a shape for the rack, use the layout from Fig.3. The rack is basically the gear with an infinite large diameter, so the involute shape transforms into a straight line, and the entire tooth shape is a trapezium.