Precession of teetered autogyro rotor

[llwindy]

"...the blades will will NOT AT ALL change their plane of rotation, they will only change their angle wrt. the plane of rotation and so no resultant force will act upon the air frame in the vacuum..."

I disagree with this statement, and your drawing of the centrifugal force component explains why. Whenever I change my axis, the flapping angle beta appears. I think a more worldly example is attaching a rope with a small mass at the end and attach it to a rotating axis. In that situation, it forms a disk plane perpendicular to the rotational axis. This is a very very un-aerodynamic object thus any cyclic pitch change will not be effective. This is essentially the same as in a vacuum. However, when I change the axis of rotation by tilting, the object will follow and the new disk plane will be perpendicular to the rotational axis.


EDIT: after posting, I immediately had to get back to this statement. On the one hand the system is basically one where by changing the rotational axis, potential energy is stored in the spring. Thus, without any forcing term (since we're in a vacuum), and no damping, there will be a continuous exchange between inertia and potential energy, as plotted. The solution solely depends on the initial conditions which are flap angle and flap velocity. Due to the tilting of the axis, these initial conditions are non-zero thus a sinusoidally varying solution is retrieved.




However, this brings me to the next problem. Below I have a snippet of table 7.1 of Prouty's book. Above, I described the case with no hinge offset. And the computation of the differential equation shows what is in agreement with the table. But, why then in vaccuum a hinge offset causes the tip path plane to align perpendicular to the rotor shaft?!

 

[XXavier]

(...)


However, this brings me to the next problem. Below I have a snippet of table 7.1 of Prouty's book. Above, I described the case with no hinge offset. And the computation of the differential equation shows what is in agreement with the table. But, why then in vaccuum a hinge offset causes the tip path plane to align perpendicular to the rotor shaft?!
View attachment 1157648

Inertial forces play a rôle, and they don't depend on the presence of air. This fragment from page 259 pf 'Cyclic and Collective' by Ray Prouty may be of help:

 

[llwindy]

Inertial forces play a rôle, and they don't depend on the presence of air.

Inertial forces are represented by the " I \times \ddot{\beta}". Hence, when constructing the equations of motion around the hinge of a rotor with hinge offset, I do not see where the damping term comes into play. That is, one needs damping to dissipate the energy conversion from the accelartion term and stiffness term. Otherwise, it will just behave as I showed in the plot of a previous reply.

Nevertheless, the example with the rope is an intuitive way of explaining the hinge offset and its ability to align the rotor..

 

[kolibri282]

The drawing in my post clearly showed that we are discussing the case of the currently used, two blade, recreational gyro rotor, which, obviously, has no hinge offset.