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In this instructable I hope to impart unto you what I have learnt about making bevel gears. If you know more on this subject please feel free to drop a comment below, all of this knowledge has been collected through my own theoretical work and so is NOT cutting edge. But the theory is sound and the gears print and work!

I won't be going into involutes however this methodology CAN be used to create involute gears as its simply a matter of placing your chosen tooth profile instead of the one used in this example.

With that said... Onwards!

## Step 1: Basic Gear Theory

As I'm sure you know, gears are designed to transfer power and motion. The idea is to take two simple cylinders which would otherwise slip when rolling against each other and add "teeth" to better control the motion and reduce loss.

**Module, P.C.D and Tooth Count**

P.C.D or Pitch Circle Diameter is basically the diameter of the cylinder as mentioned above. This is the diameter at which the teeth mesh. The Inner and Outer diameters are related to the size of the tooth with the P.C.D being halfway between the Inner and Outer diameters. We shall call it **D.**

Tooth Count is the number of teeth around a gear. We shall call it **N. N must ALWAYS be a whole number.**

Module is a factor that determines tooth size. The larger this number is, the larger the teeth. We shall call it **M.**

The three are related like so: **D** = **N** x **M**

Usually a module of 2 will do but the larger the module the better chance the gear will have to print.

So for example if we have a Pitch Diameter of 90mm and a Module of 3, our Tooth Count will be 30.

**Speed Ratios**

The speed of a second gear can be found using the speed of the first and the tooth count. If Gear A has twice as many teeth as Gear B, it stands to reason Gear B will rotate twice for every rotation of Gear A.

**Speed**A x **Teeth**A = **Speed**B x **Teeth**B

**Speed**B = (**Speed**A x **Teeth**A)/**Teeth**B

**Choosing your Values**

In most cases you'll be choosing your values by Tooth Count and P.C.D as your limitations will be size and speed ratio. Whatever approach you take, using their relationship will keep you right!

**Involute Gears**

As previously mentioned I won't go into too much detail. The general idea is to create teeth that will mesh perfectly by using geometry and some maths to create the perfect tooth profile. It really is difficult to sum up here so I will simply mention that these following videos should help a lot and you can apply the idea to this project!

## Step 2: Bevel Gear Theory

Now, imagine the two cylinders in our basic theory had to roll perpendicularly one around the other. Naturally that's not going to happen so what we need to do is turn them into **cones**. Imagine the blue cone is rolling around the grey cone, its flat "base" will always remain at right angles to the large cone base and it will roll in a circle.

The cones in the image here are basically the extended P.C.D's of the two gears involved. Because we're working with cones the axles of the two gears can be at all sorts of angles as we simply change the "sharpness" of the cone to match. In this example we shall use two cones who's points meet but that's not necessary for the gears to work.

**Keep this cone imagery in mind, it will help with understanding what we're doing.**

## Step 3: Drawing Your Pitch Diameters

Following on from our cones we have two circle profiles here, perpendicular to each other. Feel free to change up the angle.

On the ground plane we have the large gear pitch diameter and perpendicular to it the small gear pitch diameter. These meet along the radial construction lines of the two P.C.D's.

**Note: **On the large gear we have an inner circle giving the limit of how far inwards we want the teeth to extend. This also gives us a limit as to how long the small gear will be.

We also have an angle on each that gives the width of a single tooth.

**So how did we get these?**

The P.C.D's I chose arbitrarily as 90mm and 30mm. Note they are both divisible by 3. This is because I wanted decently large teeth for visual impact.

So how many teeth each? 90 = N x 3 so **30 Teeth **on the big gear. 30 = N x 3 so **10 Teeth** on the small gear. As such we also know that the speed of the small gear will be three times that of the large gear.

**The single tooth widths** are found by dividing 360º by the number of teeth. Note that the construction line is drawn down the axis of symmetry.

For the large gear the angle is 360/30 = 12º, giving** 6º on either side of the construction line**.

For the small gear the angle is 360/10 = 36º, giving **18º on either side of the construction line.**

**In the second image **you can see the "intersection plane" showing how the two P.C.D cones meet. This will come in very handy. Also, as shown on the diagram take note of how the tooth width angle lines have been extended to meet the tangent (at gear contact point) of the larger gear's P.C.D. This is very important. We cannot draw on curved surfaces, The same will be done on the inside circle, between the intersections of said circle with the tooth width angle lines.

## Step 4: Making the Small Gear

**Image 1.**

In the first image we have the P.C.D of the small gear. I chose to make the teeth 8mm high (for smoothest motion I think you really want your teeth to be close to isosceles). This means the Inner and Outer circles mentioned in Step 1 are 4mm inside and outside of the P.C.D. Here I've simply done this by extending a line instead of drawing the separate diameters, I hope this is still clear!

The base of the tooth is found by extending a tangent out either side until it meets the tooth width line. From here the point of the tooth is found by drawing up to the Outer diameter where it intersects the axis of symmetry.

**Here is where you would implement the first involute curve theory, ensuring that at the base it made contact with the tooth width angle lines and at the tip with the axis of symmetry.**

**Image 2**

From here-on in we shall refer to the tip of the large gear's P.C.D cone as the "vanishing point". We extend along the Intersection plane sketch two lines representing the Inner and Outer diameters from the small gear P.C.D to the vanishing point. We'll call these the "Tooth Cone".

Now, note the line mentioned previously and drawn between where the inner diameter of the large gear meets the tooth width angle lines. On the Intersection plane you want to mark where this line meets it. This is where we want to draw the INNER tooth profile. Extend this line up to meet the axial line of the small gear.

Now, perpendicular to this new line we just created and parallel to the original P.C.D you want to make another P.C.D. This will be very easy, you simply start your circle centre on the axial line and extend the radius to where it meets the Tooth Cone P.C.D intersection.

From the centre draw your symmetrical tooth width angle lines down, they'll be the same angle as previously (this doesn't change). Now using where the Tooth Cone intersects this plane you can mark out the new tooth profile.

**How does this work**? The tooth cone represents the area in which both the small gear and large gear teeth exist. It represents a scaling factor where at the original outer P.C.D the scale was 1 and at the Vanishing point is is 0. The tooth size and P.C.D cones all scale with this. As such, you can also find the size of your inner tooth profile by* ((vanishing point to inner tooth plane)/(vanishing point to outer tooth plane)) *to get a scaling factor.

**Image 3. **

Simple. Loft between the two profiles.

**Image 4.**

Circle-pattern your tooth around the gear axis/P.C.D. Naturally the total number of copies is equal to your tooth count.

**Image 5.**

Finish off the basic gear shape. Here I placed a decahedron on the inner and outer tooth profile sketches and lofted, merging them. I also rounded off the tips of the teeth. Given the possible inaccuracies while printing and also how printers accelerate and idle this should help.

Also, you'll want to trim the gear to tangent of the inner tooth circle, so it doesn't overhang on the inside. We'll get to the outside in a second.

**CONGRATS! That's your basic small gear done. Moving on...**

## Step 5: Making the Large Gear

**Image 1.**

Here I've created what I call the "Tooth Wedge". Its done by using the flattened off inner and outer large gear circles lofted up, trimmed by both the Tooth Cone lines and the tooth width angle lines.

**Image 2.**

Now, the tooth wedge is as wide as the total tooth width (obviously) so your tooth profile will be as wide as this and as tall as it ** be it a simple triangle or an involute profile**. So here you can see the tooth profiles drawn on both ends from the base to the centre-top. Don't forget, this wedge basically represents a scaling factor.

**Image 3.**

The tooth will need some material to rest on. Give it some.

**Image 4. **

Here I rounded off the tops to the same radius and circle-patterned them around. Combine them to create a single body.

Next, by cutting the body with the inner and outer diameters you get a nice, smooth profile.

## Step 6: Putting It All Together

Here in preparation for a print demonstration I chewed the large gear down to save on print.

Now, the interesting part here is the small gear. You'll notice the rounded top. The reason for this is that when mounting the gear it will now fit within the same diameter as the large gear no matter what stage of rotation it is in. As such I can use a fully circular annulus ring (which in my opinion looks much nicer than a flat section cut out of an otherwise circular ring).

*You may also want to shave off or tolerance the gears a little bit depending on your printer. This may take some tinkering and be wary of overdoing this as you'll introduce backlash, where the gears move without contacting each other.*

**Now, let's move on to the print demonstration model!**

## Step 7: Print Demo and Files

So for the print demo I've made a simple assembly with 2 large gears and 3 small gears held in an annulus system.

You can find the STLs on Thingiverse here.

You can get the original 360 file with all inclusions here.

**Thank you very much for reading! If you've found this helpful or enjoyed it please contribute your vote, it means a lot to me!**

**If not, comment below and I'll see if I can address your confusion.**