To be quiet, you need to cancel out all vibrating forces. That means making the thing perfectly balanced in all axes.
With rotary it is rather simple - the imbalance is of an excentric nature around the shaft, so a simple counterweight and zou are done.
With a linearly moving single cylinder zou have a problem: The piston assembly is "shaking" only in one axis, the crank is an eccentric imbalance. Now if you add just a simple counterweight, you may balance out either one direction (along the piston movement) or the other (perpendicular to it), but not both. The seemingly best you may do is to balance them so you get an eccentric imbalance again. But there is a problem: It is rotating in the opposite direction towards the shaft. So you would need extra counter rotating balancer shaft. Acceptavble for things like combustion engines, but not for such compressors.
But if you add an extra cylinder perpendicular to the first one, it's inertia will add the extra component on the perpendicular axis, so the total imbalance becomes again an eccentric one and rotating along the crank. So just simple counterweight and you are perfectly balanced again.
But with this what remains is the rotating load on the shaft (when cylinders are compressing) is present at about 120deg of the shaft rotation, the res 240deg there is no load. So we still have a bit of torsion vibration.
But to counter that what is sufficient is to make the compressor core (the motor + compressor assembly within the "egg") mount very light in holding the torsion, so it lets the whole motor assembly mass to counter that torsion vibration and do not propagate them to the case.
Or (and there the inverter technology with a synchronous permanent magnet motor becomes handy) design the motor in a such way so it generates the torque exactly when and how much the compression just needs it, so even the rotation vibration gets compensated.
And then your compressor becomes silent even if reciprocating.
And then other option is to use more cylinders in a regular star pattern around the axis. This has the advantage to cancel out not only the linear imbalanc components and the only what remains is an eccentric rotating along the shaft (so a simple counterweights balance that out), also the major higher order ones coming from the connecting rod arrangement distortion, plus also distributes the rotating load evenly along the shaft rotation, so making it to vibrate less even in the torsion mode. Plus this is easy to isolate from the case.
Of course this works when the cam to piston is done by a sliding mechanism, which yields exact sinewave motion. Using connecting rods (as in common combustion engines) distort it a bit, so there remains some higher harmonic imbalances (on a multiple of the crank rotation speed). But even that, because it becomes in way higher frequency, it becomes easier to "filter it out" by the core assembly mass and the suspension springs, so it could be made still quite smooth.
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