Ok, after taking literally a couple months of working time mostly working on various revisions of this roots blower mechanism, for the openERV.org project, I had to abandon the approach.  Ultimately the mechanism is just too expensive, probably.

However, it works marvelously well and might be useful for something else.  Maybe a silent pc, or a super quiet ventilation fan.

The point of using it rather than a fan is that it can give a silent airflow even at significant pressure differential.  I haven't tested it exactly, but I think you could get it to operate at less than 0.3 sone at about 500 rpm, which it gives 0.65 liters per rotation, so that would be like 4.5 liters per second or something, 10 cfm, and maybe 20 mm of h2o air pressure.  It is actually possible to make a centrifugal fan that can do that, but meh no one has published a design for it yet.

If you wanted a truly silent flow at more than about a mm of h20 pressure, you could reduce the rpm still further, and then this would be the only way to get that.  Any fan will produce noise because of turbulence at the blade tips, at the rpm needed to produce that much pressure.

So it does have a small window of operating parameters where it can do something nothing else can do. 

It is made to last a very long time, just like a brushless fan.

However, you still need a silent motor to make this happen.  I have found some, they are a certain variety of 32 mm diameter gimbal motor, used in camera gimbals.  They can be had for about $3 USD each on ali express (including shipping).  There are several manufacturer's.  They have a hollow shaft, are black and have 3 wires.  Not all gimbal motors will work, and these ones are bigger and more powerful than most of the little gimbal motors out there, and they are non-cogging which helps a lot to make them silent.

Then, you can get a so called sine wave motor controller, or a Field oriented control (FOC) driver for it.  Sine wave controllers are reasonably easy to get although they are at least $15 each, they are available for drones.  Make sure it says sine wave control.  A square wave one might actually work ok, though, I don't know.

For my experiments I used a servo motor that is of a type that is common on the used market, but they make noise on their own so there isn't much point any more.  However, if search 12 volt servo motor with encoder on ebay or ali express you will find several suppliers.  You can match them to the picture.  They are old and used and some of them are siezed.  They are available on Amazon, too, and not much more expensive there.

I implemented a PID controller to control motor speed, but that's really not necessary.  You can just give the motors a PWM signal and they will exert a certain torque, and thus end up going a certain speed.  The pwm regulates torque, not speed. 

The roots blower has several rubber parts that need to be printed in TPU. I used 85(A?) shore durometer ninjaflex, the softest TPU I could get.  Harder TPU might not work very well to isolate the vibrations.

The cushions on the shaft just make it easier to print and assemble, because you can get a tight fit without precise printing, they don't help much with the noise.

The cushions that the bearings go into help a bit, because they prevent the bearings from creaking around, mostly.  Also rolling element bearings produce a bit of noise naturally, which you can't hear unless it's coupled into a nearby object.  The cushion prevents that coupling.  The cushions are super glued or epoxied into place.

The motor shaft cushion and rubber mounting washers are the most useful.

I used 3 mm hex cap screws for everything, they are easy to get.

The cad file is a bit of a mess, but I tried to make it show how things work.  The bigger of the two motor mounting plates isn't as good.  The smaller one has smaller bolt holes and stuff that lets the bolt grab the plastic, which makes things easier than using a nut.

You should check  using the model to make sure things fit ok before printing.

The hardest part was making the vanes and vane profiles. You can see there is significant clearance between them, which will imply some air leakage.  You might be able to decrease that, but there is a harmonic oscillation that occurs which causes vane clattering at one particular RPM, and the clearance prevents that.  It is I think mostly harmless.

The vanes are made by using an epicycloidal/hypocycloidal profile, and then rotating and trimming them in a laborious way to give a constant gap between the closest surfaces, throughout their rotation.  This is not trivial and helps optimize the ratio of the size of the gap mentioned above to the resistance to clattering.  This is because if you just take the epicycloidal/hypocycloidal profile and offset it to shrink it slightly, the maximum angular misalignment between vanes before they clatter varies depending on the rotational position of the mechanism, i.e. the position in it's cycle it is at.  They are much more prone to touching in some regions than others.  I used fusion to manually rotate and trim them so the tolerable angular misalignment is about the same throughout the rotation of them.

They use magnetic gears, with backup gears, for sychronization.  The backup gear has a large backlash, which causes the teeth to engage only if the magnetic gears are overwhelmed, because they can only stand so much torque.  This keeps the vanes sychnronized no matter what.  This is important for practical operation for many reasons, including the clattering that may occur, which can overwhelm the magnetic gears.

To make the magnetic gears, you use 6 mm diameter, 3 mm thick NIB magnets, I used N42 grade magnets, N52 would of course be better, they are stronger magnets, further reducing the tendency to clatter.

You simply apply 30 minute epoxy, make sure you mix it right, and put the magnets in their holders, with alternating polarity.  This alternating polarity also helps them hold themselves in position, making assembly easier, but it is a bit finicky.  You should practice without the epoxy first probably.  You can do it by putting one in, rotating the gear so the one put in is covered by your finger, then add another one, rotate again, so the nearby previous ones are always held down by your finger.  Also, you can hold the magnets in a stack, and put the end of the stack in, then slide sideways to deposit a magnet.   I recommend having a magnet taped to the table, which you leave there and use as a reference for polarity.  It critical that the magnets always have the right polarity, and that both gears are identical in their pattern.  The magnets must always have the N side exposed in the areas between the teeth of the backup gears, for instance, for the gears to mesh properly.  Or the S side, but it has to be the same for both gears, that's all.

Also, make sure the magnets always alternate in polarity. I had a could reject gears because I only discovered after epoxying them that one of the magnets was the wrong way.  I had to pry the magnet out, clean things laboriously and re-epoxy it, and it was a close call, it might break the plastic, depending how good your epoxy is. 

Don't let the magnets get close to any other magnets or ferromagnetic objects before the epoxy is set, of course.

You might be able to use 5 minute epoxy, which actually takes longer than 5 minutes to set, but IDK.

Don't use PVA glue, I can tell you that, because when the mechanism is at rest the magnets pull pretty hard on each other, and the PVA glue fails and the magnets come out of their holes at that position.  Very annoying.

For printing, I think I made every thing for 0.333 mm layer height, 0.45 mm wall thickness.  Tell it to print thin walls for the vanes.  There is some minor error in the vane I couldn't get rid of, this large triangular face or whatever, it's mostly harmless as it doesn't print anyway.

You need to calibrate your printer for perpendicularity, most printers don't need to be checked for linearity as the linearity is determined by the pulley diameter, etc.  which is reliable and quantified.

 There is software called goskew which does a slightly crude job, but works to correct the gocode for skew.

Use the tangents method when getting the data for skew, that is the standard. Make sure you do it carefully and write it down so you don't loose the data, and don't mix the data points up or anything. 

Also the plugins for cura to correct skew don't work, neither does any other software out there, I tried it all.

There is a feature in the later versions of Marlin which allows for the same tangent figures to be entered, which is a better way to correct skew, but I didn't use it yet because you have to recompile Marlin and re flash your printer, which is not the easiest thing, and might lead to bricking the printer.

Goskew does cause some artifacts on the prints, though, so it is definitely better to use the firmware if you can, plus then you can't forget to compensate a file, which would ruin a print.

As a closing comment, if you search for "roots blower noise convel blower" there is a good document that discusses the subject in the context of turbo chargers, and they describe a mechanism that is probably even better, more quiet and stuff, but also more complicated.  To bake that into a printable mechanism with magnetic gears to synch the vanes would make a lot of sense.  It would probably cost about the same and be quieter still and allow higher airflow and pressure levels for the same noise level.