oldhornet,
You described, "a cyclical (in a time sense) vertical for 3 to 4 beats, every 5 seconds or so."
That could be many things, and also by being intermittent, will drive the maintenance guys to try and smother you in your sleep, so I share your frustration.
If you turn all sas/AP's off, does it still happen? I ask that because there is a phenomena where sometimes the rate gyros can create a slight but perceptible low frequency vibration close to 4P that wiggles the cyclic and the rotor, and that the pilots can usually feel. Was fixed a while back when the vertical gyros were redesigned (a 4 bearing vice 3 bearing design, my deep memory wants to say) to eliminate a gyro output mode.
If it occurs when the sas/AP's are all off, forget that theory, and try this (which is close to Jack's explanation of a beat frequency): Sometimes one blade can have a bit more 3P than you'd like, and it can mix with the 4P to create an intermittent odd harmonic that does not show up on the vibe gear, except as a somewhat higher 3P level on the meter. This can many times be cured by tabbing that blade (get a Sikorsky rep to help). This is rare, but possible. It is the kind of vibe that is not awful, but just makes you want someone to identify it and fix it. If you made the 4P come down with bifilar maintenance, this type of beat frequency will also come down, but not go away.
PM me and I will get you in touch with Sikorsky's best vibe engineer, who can usually solve everything, even Santa's sled with a bent runner.