Flapping. Part 1. Natural or Resonant Frequency.
Remember pendulums in school physics? Take a piece of string with a weight on the end, swing it, time 20 swings then work out the period of oscillation. Lengthen the piece of string and repeat the exercise. The result is a longer period, or to put it another way a lower natural frequency. Shorten the piece of string the natural frequency increases.
Now imagine a rotor system. It rotates at a given RPM, which we could equally describe in terms of a frequency. The rotor systems’ natural frequency is therefore equal to the Nr. If you then hinge a blade at axis of rotation (teetering head) it is free to flap in resonance with the rotation and because it is hinged at the point of rotation it will have the same natural frequency (the distance from hub to Centre of Inertia is the same as the distance from the hinge to the Centre of Inertia). Even if the system is damped, by air density, by dampers, springs or whatever the system will remain in resonance.
Now hinge the blade out board of the hub, the natural frequency of the blade will change – just like our pendulum the shorter distance between the point of flapping and the Centre of Inertia will increase the natural frequency of the blade. BUT the NR has stayed the same so the two are no longer in resonance. In aerodynamic speak (and control theory for that matter) one now has reduced the frequency ratio (excitation frequency [NR] / undamped natural frequency). This means if the blade is disturbed instead of returning to its’ original flapping position 360 degrees later it will return less than 360 degrees later. This is without aerodynamic effects it is pure inertia (the same stuff that drive gyros).
Who ever heard of a gyro with flapping hinges in it? No one, because to be a true gyro the mass muss be rigid, if it can flap out of resonance it will react differently and out of phase and won’t be a gyro.
Coming soon – Phase angle as a function of damping and frequency ratio. (Sorry run out of time for now)