Ok, five years later and I think this thread might need resurrecting.. again! I have read through a lot of it. I'm well aware that what they teach you in flight school is never the full story - cf Bernoulli and Venturi effect when explaining lift - reasonable methods of understanding but not entirely correct as the real answer involves Navier-Stokes equations, lots of nasty vector calculus and computational solutions (as I seem to recall from fluid dynamics lectures many yonks ago). In trying to understand this, my thoughts on the matter thus far are as follows.
"Gyroscopic properties" of a rotor disc
heedm provided the best explanation of conservation of angular momentum that I've seen in a while. All of what he said accords exactly with my understanding of this concept. Conservation of angular momentum is a principle that applies to any rotating body, therefore, to my mind, it applies to a helicopter rotor system. But the question is - a) how does it apply and b) how important is this effect in understanding rotor disc dynamics? I imagine it is small but present, since (pure intuitively) I would have thought that the angular momentum of the rotor disc is on the small side compared to the lift generated by the blades. I agree that a helicopter rotor system is not a gyro, however, the system must obey conservation of angular momentum.
Lift and blade flapping
I'm sure my view on this is incomplete, so here goes. Lets consider at a zero wind hover to start with, and lets not worry to start with about cyclic inputs vs the direction you're trying to fly in.
At some point on the circle blade pitch angle is increased, increasing AOA. This increases lift on that blade, and as the pitch angle increases, the lift on that blade increases. This causes an overall increase in lift at the position of maximum blade pitch angle.
But blade flap acts to counteract that increase in lift, because as the pitch angle increases, the AOA increases, and lift increases, causing the blade to flap up, and the AOA to decrease (somewhat) due to the motion of the blade upwards. However, according to Newton's Second Law, the increase in lift will take a finite time to accelerate the blade upwards (we are now considering linear momentum and elasticity of the blade in the plane perpendicular to the plane of rotation and taking into account the centripetal/tension force on the blade which will depend on Nr amongst other things). So the relative decrease in lift (the flap) will lag the increase in lift caused by the increased AOA, causing a point of net maximum lift at some point between the max blade pitch point and the max blade flap point. I imagine the lag is greater in rigid rotor systems, where the elasticity of the blade is what allows flap.
To change direction of flight, we must tilt the disc in that direction. So, what forces cause the disc to tilt, and in what direction relative to min and max blade pitch angles? This is what I have surmised:
1. If you spun a rigid gyro in zero gravity and applied a force to it, it would demonstrate precession. If you spun a rotor system in zero gravity (but in air) it would still obey conservation of angular momentum, so it would display some precression-like effects, although it would be a 'sloppy' gyro as its spokes can move out of the axis of rotation to a certain degree. So, obviously, the more rigid the rotor system, the more gyro-like effects come into play. Any time a blade is able to exert a force on the mast, and thus tilt the whole disc, the disc will exhibit gyro like properties.
2. Blade position caused by flapping will cause the rotor disc to tilt. Increase the pitch angle of the blade and the blade will flap up, and at the point of maximum flap, the disc is tilted most up. There is a lag between max blade pitch angle and max blade flap up. I think this lag must be designed to be on the order of 90 degrees (but at least between 45 and 135) in order to induce simple harmonic motion (i.e. a sinusoidal path) of the blades round the circle. I imagine on a hinged rotor system the angle is always 90 degrees since the hinge stop (I assume the degree of hinging or tilt is limited?!) could "damp out" the top of the sinusoid curve, or else it has to be designed to be 90 degrees or greater to stop the blade from hitting the hinge stop. In a rigid rotor system I would imagine that the lag angle would vary depending on rotor speed, blade coning (i.e. how much vertical elastic tension the blade is already under due to lift requirements, i.e. variation in gross mass) and I'm sure other things I haven't thought of. Therefore I would assume the designers pick an advance angle for the swash plate somewhere in the middle of this variation and let the pilots do the rest with the cyclic.
Therefore, the total tilt of the rotor disc will be a combination of blade flap angle and a force transmitted to the mast either through the elasticity/tension of a rigid rotor system or the pressure of the blade against the hinge/tilt stop in a hinged/tilted system. The first is a mechanical 'effect' - the disc is being 'tilted' by an alteration in the position of the blades relative to the axis. Gyroscopic like effects don't apply to this type of movement because the individual blades are not acting on the whole system. Incidentally, conservation of angular momentum is conserved on each individual blade by hunting - as the blades centre of mass moves closer to the point of rotation (the mast) as it flaps up, the blade's speed will need to increase to conserve angular momentum. The second only applies if the blade is in a position to affect the movement of the whole disc, because it is constrained in some way, either by the inherent elasticity/tension of the blade or if it hits its hinge or tilt stop. This proportion of the force will act "gyroscopically". This force will generate a vector that will tend to tilt the rotor in a direction at 90 degrees to the max blade flap angle (or at 90 degrees to the max net lift position, I'm not sure which and it might depend on.. things), but in my estimation will be much smaller than the blade flap induced tilt. The resultant of the two tilting forces (I think this is called your phase lag angle?) will therefore be somewhere between the max blade flap up and 90 degrees ahead of this, but it will vary depending on your flight conditions. This might explain why I think I heard that rigid rotor systems are more likely to have an advance angle (I think this is the right terminology - the number of degrees the swashplate input leads the cyclic input direction) less than 90 degrees.
Ok go on then, pick this apart!