Thank you, Xavier, for copying this article from "Airgyro Aviation"
My two cents. The author writes:
"
With an optimum undersling the 2/rev vibration caused by this moment is minimised but there is always some. Incorrect undersling (either too large or too small) increases the moment of inertia around the spanwise axis and increases the resulting 2/rev vibration. The best solution is to optimise undersling"
I calculated the evolution of the torque on the plate, according to the underslung
. It gives
Γ[SUB]h[/SUB] = 8 Ω[SUP]2 [/SUP]M sin a[SUB]1 [/SUB](sin² a[SUB]0[/SUB]. R[SUP]2[/SUP]/3 – sin a[SUB]0[/SUB] . R.h + h²) sin (2Ω t)
With a[SUB]0 = [/SUB]coning, a[SUB]1[/SUB] = longitudinal flapping, M = masse of one blade (kg), Ω = rotation speed (rd/s), R = rotor radius (m) and h = undersling (m)
It shows well that the minimum of this vibratory torque is obtained when theteeter bolt is on the line which encloses the centers of gravity of each blade. This was generally accepted
The amplitude of the movement of the masses dm, parallel to teeter bolt, during a flapping angle α1 is worth:
sin a[SUB]1[/SUB]. (r sin a[SUB]0[/SUB] -– h) sin(2Ω t)
Their speed v along this axis is :
v =-sin a[SUB]1[/SUB] .2 Ω . (r sin a[SUB]0[/SUB] - –h) cos(2Ω t)
Their acceleration is :
a = -
sin a[SUB]1[/SUB]. 4Ω[SUP] 2[/SUP]. (r sin a[SUB]0[/SUB] -– h) sin(2Ω t)
Therefore it requires an elementary force:
dF = -4dm Ω[SUP]2[/SUP] sin a[SUB]1[/SUB] (r. sin a[SUB]0[/SUB] - h). sin(Ω t). Hence, with dm= (M/R). dr : dF = -4sin(Ω t).(M/R) .
Ω[SUP] 2[/SUP]. sin a[SUB]1.[/SUB] (r. sin a[SUB]0[/SUB] - h) . dr
So, for two blades, the sum of these dF between 0 and R is
F= -8M.Ω[SUP]2[/SUP] sin a[SUB]1[/SUB]. (½R sin a[SUB]0[/SUB] - h) sin(2Ω t).
This force also acts around the pitch pivot to z meter below. and adds his torque on the control plate
So, the total torque vibration 2/rev would totally disappears when the undersling lower that "optimum"
Here, If z = 0.2 m, the required force for canceled the stick vibration is 68 mm, instead of "optimum" 82 mm