Pericyclic Gears

Pericyclic gears offer the possibility of obtaining high ratios for 3D printed gear boxes for use in DIY telescope drives, at least for the smaller models, say, up to 12 inch apertures.  There are working examples that make me want to try one.

That center, yellow, gear 'wobbles' or 'nutates' and only engages a few of the stator teeth (left most gear) at a time and does the same with the output gear (right most gear).  But there are different numbers of teeth between each of the interacting surfaces.  As such, the nutating gear cannot fit exactly into the fixed stator gear, so must rotate slightly to fit the few teeth it engages.  On the other side of the nutating gear, the number of teeth are again different which means the output gear must move a slight amount to fit the few teeth it engages.  The overall gear ratio from input to output is given by a strange looking formula  where n1 is the number of teeth on the first gear, n2 on the second, etc.

I am interested to use NEMA 17 stepper motors to drive ALT/AZ (or RA/DEC) axes on a small telescope.  They are inexpensive and easy to control but they need torque amplification and resolution enhancement that gear boxes can provide.

Consider they usually come in 200 step/rev models.  That is 1.8 deg per step - too coarse for telescope pointing and tracking.  They can be controlled to at least 16 microsteps per step (32/step is possible, too).  At 16 microsteps we are down to 6.75 arcmin per step - pretty good for pointing small telescopes with wide fovs but a bit coarse for tracking, plus we could use some torque amplification.

Consider a 50:1 gear ratio.  With microstepping we can get 9  arcsec/step which is excellent for pointing and requires one step every 0.6 seconds for tracking.

On the other hand we would like to slew much faster.  To get, say, 8 deg/sec slew (about what a C-14 does, the 32-in is at 4 deg/sec).  8 deg/sec * 60 sec/min  480 deg/min / 360 deg/rev = 1.33 rpm.  If we drive that at  66.66 rpm with a NEMA motor at the 50:1 ratio. We will get the 1.33 rpm for fast slewing.  67 rpm is doable with the motor.

The NEMA 17 motors are capable of 50 N-cm of torque.  At 50:1 (minus inefficiencies) we can hope to get about ~25 N-m which should be sufficient to move small telescopes on roller bearings.

Bevel Gears

It may, or not, be evident in the image above that the gear teeth are arranged on cones, so-called bevel gears (both inner and outer) that are widely used, primarily, to change the shaft direction.  But in the pericyclic gear box the shaft direction is unaltered - straight through.  There are excellent bevel gear design tools available, and I rely on the OpenScad tool with its bevel-gear.scad library.  In the pericyclic gear box the nutating gear 'wobbles' because it is mounted on a 'tilted' section of the drive shaft so we need to take that into account when designing our gears.  

Let us first realize that bevel gears are cones 'cut out of' spheres.  In cross- section spheres are circles and cones are triangles.  Below is an illustration of how this works for standard gears. 

The red area represents the desired outer bevel gear.  It is characterized by the cone angle (in this example, the 30 deg angle) and the radius of the sphere which is largely determined by the number of teeth desired on the gear which determines the width of the red area.  (It's a bit more involved than that, but for now it works.)

For the pericyclic gear box we are also interested in the inner bevel represented by the green area.  But as shown above we need to introduce  clearance between the two complimentary gears, and we do that by using a different face angle as illustrated below.  The bevel-gear.scad uses the supplement of the cone angle used for the outer bevel for the inner cone angle (in this example 150 - 23).  The 23 deg is the amount of tilt in the drive shaft to get the two gears to interact with everything revolving around that apex.  (VERY important)

In practice, we would like to have the two diameters of the gears to be much closer than the simple example above shows and, given that the tooth difference between the two gears is usually limited to one or two, we can accomplish this by making the 30 deg angle for the red gear much larger.  In the working example below, I chose 65 deg for the outer gear angle and a 3 deg tilt angle which leads to a 112 deg inner gear angle.

Below is a screen shot of the OpenScad script that produces a 3D printable nutator gear.  Note the number of teeth, the cone angle and the bevel-gear.scad function that produces the inner cone.  The bore hole, alignment holes and bearing recess are easily added to the bevel-gear.scad output.

This is a similar illustration for the 49 tooth outer gear (stator).  Note the number of teeth, the cone angle and the bevel-gear.scad function to produce the 3d printable model

Here is the finished gear box.