It seems as though my claim to fame is my collection of power transmission tutorials. I was recently asked for some help on designing lantern gears, which are a special type of gear which are generally easier to fabricate than involute gears, but which aren’t a huge loss on strength and efficiency.
Outline
The idea behind a lantern gear is probably older than involute gears: simply let your pinion be a circle of pins, and then have that turn a plate with cutouts for those pins. However, this means that every pinion/gear pair must be custom-fit to each other. In this post, I’ll only describe how to design circular-pair lantern gears, but there also a way to make a gear to match a “rack” made of pins, which I will explain if anyone is interested.
For the circular pair described below, I’ll describe the part with pins as the pinion, and the part with slots for the pins as the gear.
The Math
To my knowledge, there is no standard way of quantifying lantern gear sizes and pitches. This means you have the freedom to pick sizes by their linear pitch p (as in pulleys), diametral pitch DP (as in imperial gears), or by their module m (as in metric gears). You only need to select one of these quantities to derive the rest. They’re related by the following equations:
DP = \frac{1}{m}
m = \frac{p}{\pi}
I personally recommend dimensioning your lantern gears by their diametral pitch, as it results in round-number pitch diameters on your lantern gears.
Beyond the pitch, there are three other quantities will drive a circular pair of lantern gears. Two of these quantities are n_p, the number of pins on the pinion, and n_g, the number of slots on the driven gear. The pitch diameters of the pinion and gear, D_p and D_g respectively, are given by the same equations as an ordinary gear or pulley:
D_p = \frac{n_p}{DP} = n_p m = \frac{n_p p}{\pi}
D_g = \frac{n_g}{DP} = n_g m = \frac{n_g p}{\pi}
The last, and perhaps least interesting, of the driving quantities is the pin diameter, D_r. It has no particularly special relation to the other quantities, other than that your pins can’t interfere with each other, so you must pick a D_r which satisfies the equations below:
D_r < 2 D_p \cos (\frac{180^\circ}{n_p})
D_r < 2 D_g \cos (\frac{180^\circ}{n_g})
For the remaining examples, I will use a DP of 8 pins/inch, n_p of 8 pins, n_g of 60 pins, and D_r of 1/8" (to match a McMaster dowel pin).
Pinions
Pinions for lantern gears are very easy to draw. All we need is n_p circles, of diameter D_r, patterned evenly around the pinion pitch circle with diameter D_p:
I took the liberty of drawing an outer diameter as well, but that was more or less arbitrary. The 8 small circles here are where the driving pins will live.
I made a plate out of the sketch I drew earlier, then cut out holes for the pins, as well as a hex bore for a shaft. For the pins, I just used .125" OD, .625" long steel dowel pins from McMaster.
The Gear
On to the hard part!
I start by drawing an arc, centered on the origin, with radius \frac{D_g}{2}. I mark one of the endpoints of that arc vertical to the origin. Next, I draw a second arc, tangent to our first one, with radius \frac{D_p}{2}, and merge their endpoints. The first arc represents the pitch circle of the gear, and the second is the pitch circle of the pinion.
After constraining these two arcs to have the same arc-length, we notice that the free endpoint of the second arc follows an interesting path when we try to drag it around. That’s the path followed by the center of a single pin on the pinion, from the point of view of the gear. This path is an epicycloid, and can be drawn parametrically with equations, or by sampling points. I’ll show the sampling method, but the linked Wikipedia page can show you how to draw that path using equation-driven curves.
To sample points along that path, we make a copy of the same geometry. Two copies, to be precise! The arcs I added have the same radii as the ones I made earlier, but they all have “free” endpoints at some point along that red path from before.
Since we only need to draw half of a tooth, we can draw a line from the center to the outermost “free” point, and dimension its angle from the vertical to be \frac{180^\circ}{n_g}. To define the location of the other two samples, I drew some construction lines between the free points and their neighbors, and dimensioned them all to be equal. This practice isn’t necessarily the most effective, but it results in a “roughly” even sampling in terms of linear spacing.
I now play connect-the-dots by drawing a spline between the four points you just saw to create a section of the cycloid path.
To create the outside of the tooth (which must clear the pin, as well as the center) we offset the path taken by the center of the pin by its radius, \frac{D_r}{2}. To keep the system from binding, I add a small gap of 0.001" to that offset as well. This number is arbitrary - the larger you make it, the less likely you are to get a bind, but you also get more backlash.
I now start up a new sketch, and copy over the offset cycloid we just created. I trim down the end of the cycloid to fit within the angular size of half a tooth, mirror the path. and then draw a circle around where the center of a pin would lie to seat the pin. This is now the cutout required to make the gear from a flat, circular piece.
After creating a blank with a large enough diameter for the cutout we just created, I cut out one tooth and then pattern it around the rotational axis n_g times. The lantern gear is now done!
I decided to throw my new pinion and gear together. I noticed that the tips of the teeth of the gear interfered with the hex on the pinion, so I cut down the length of the tips of the gear teeth.