Dave,
With all due respect to your attempts to determine how a blade flaps, you are boggled right here:
"However, I see nothing that will cause the rotor to flap faster."
What exactly DO you know about what makes a blade flap at a given angle? Frankly, not much. Now don't be angry, I don't know much either (I am one fact ahead of you on this!) In fact, the best brains at Sikorsky knew so little about this for the S-76 that we made three sets of mixing bellcranks for the cyclic, and tried them out in flight, and settled on the best compromise for production. I know exactly how the S-76 cyclic behaves, and it is just about right, even though the cyclic phase angle is 57 degrees (33 degrees wrong by the Lu-Dave math).
The belief that the real phase angle is 90 degrees is where you and Lu are stuck. Just when you see the light that this is NOT TRUE, the blade DOES NOT FLAP at 90 DEGREES, your old, solid, comfortable belief kicks in, and you go back around the circle.
OK what can I say? The fact that you don't see why is a puzzle for you. You believe that it must flap at exactly 90 degrees, so all those poor sods at Sikorsky and Robinson who make these machines at the "wrong" angle must be on drugs.
Oh well, I guess it's time for me to give up, but I'll try one more time:
1) The required phase angle is a number that often differs from 90 degrees. The books are wrong when they call it gyroscopic precession.
2) The real phase angle depends on the flapping inertia of the blade, its chord, its rpm, its hinge offset, and several other factors. The real phase angle changes up to 5 degrees, depending on the speed of the helicopter, for pete's sake.
3) The real phase angle has been as much as 135 degrees on some helos, and as little as 57 on others, and these helos all flew nicely, and the cyclics worked like cyclics should.
4) On a robbie it is 72 degrees, on an S-76 it is 57 degrees.
5) The robbie and S-76 cyclic behave properly.
6) Lu and Dave cannot see WHY these statements are true, so they disbelieve them.
7) Nick has reached a nice place to sign off. Adios!