The lateral/directional oscillation (Dutch roll mode)
In forward flight the directional stability of the helicopter increases and the oscillation found in the hover becomes more like the Dutch roll of a fixed wing aircraft as the helicopter weathercocks with little sideways translation. It is worth noting, however, that in addition to roll and yaw, helicopters exhibit a significant amount of pitching motion during this oscillation. The LDO frequency and damping are given approximately by:
ш=vYNr+UN
Г — (Yv + N)
^ 2VYv Nr + Ue Nv
Now Nv is positive and Yv, and Nr are negative, therefore the LDO will always be stable provided there is no sign change in these derivatives. The value and sign of Nv is subject to change due to modifications in the contributions from the various parts of the helicopter. For example, changes in CG position can result in the fuselage contribution becoming negative, also main rotor wake interference may reduce or increase the value of Nv for the tailplane. Note also that if the overall value of Nv were negative the destabilizing effect would increase with forward speed, Ue. Interestingly Prouty [4.3], adopts a slightly different approach which ultimately shows that the lateral stability, Lv, can destabilize the LDO if it is excessive.
Prouty treats the LDO as an oscillation in roll and yaw but with the aircraft CG maintaining a straight flight path. In other words the LDO and spiral mode are completely decoupled. In mathematical terms this assumption means that the sideforce aeroderivatives are zero and that the bank angle and gravity effects are ignored. These assumptions change the 5 x 5 matrix presented above to the following 3 x 3:
v 
– 0 0 — U 
v 

p 
= 
Lv Lp Lr 
p 

_r_ 
r 
Generating a characteristic equation for the LDO, using det(sI — A) = 0:
s 
0 
Ue 

CE = 
— Lv 
s— Lp 
— Lr 
= 0 
— Nv 
— Np 
s — N 
= s[(s — Lp )(s — N) — L Np ] + Ue [Lv Np + Nv (s — Lp)] = 0 CE = s3 — sLp + Nr) + s(Lp Nr — L Np + Ue Nv) + Ue (Lv Np — Nv Lp) = 0 (4.38)
Assuming that Nr <§ Lp and making use of the Bairstow assumption, which states that for lightly damped systems represented by the cubic c3s3 + c2s2 + cts + c0 = 0, then c2s2 + c0 = 0 and the cubic becomes:
c2s2 + ( cj — c3 — ) s + c0 = 0
V c2
Thus Equation (4.38) becomes:
Consequently:










Using typical values for the constituent aeroderivatives it can be seen that if Lv is excessive it is possible for ^LDO to become negative, indicating an unstable dynamic mode.
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