Dynamic stability in the hover
If the heading mode is ignored as it has little effect on the other motions a quartic equation results. In the longitudinal case, not only is forward speed zero, but some of the derivatives are also zero. This is not strictly true in the lateral/directional case as the rolling and yawing motions are cross-coupled, being represented by the derivatives Lr and Np. However, if the tail rotor is assumed to be located on the roll axis then Lr can be taken as negligible and a situation analogous to that for longitudinal motion occurs.
One of the real roots can be shown to be (s — Nr) = 0 so that s = Nr. This root confirms that the yawing motion is independent of the rolling and sideways motions, and represents a damped subsidence in yaw. Any disturbance generating a yaw rate will be damped out by the moment Nr. r and the aircraft will be left pointing in a new direction This root also characterizes the yaw control response in the hover. The other real root is usually a large negative root representing a heavily damped roll subsidence.
The complex root represents a divergent oscillation involving changes in bank angle, heading and sideways velocity and is often referred to as the falling leaf mode. A disturbance in bank angle causes the helicopter to move sideways; this motion causes the rotor to flap back and eventually stop the sideways velocity but the aircraft is left
with some bank angle and the motion reverses direction. At the same time, the sideways velocity causes a change in tail rotor thrust and sideforce which then produces a yawing motion. The falling leaf motion can be considered as an undamped oscillation and the frequency of the mode is shown to be given by:
4.11.3 Dynamic stability in forward flight
The characteristic equation in forward flight again resolves into two real roots and a complex pair. The large, negative, real root again represents a heavily damped roll subsidence. The small real root now corresponds to the spiral mode, and the complex pair represent an oscillation, the lateral/directional oscillation (LDO), analogous to the Dutch roll mode of a fixed wing aircraft.
184.108.40.206 The spiral mode
As forward speed increases, the directional static stability, Nv, increases and the yaw subsidence in the hover becomes a spiral mode. The mode is dependent on the value of the factor:
g_ [N Lv — L Nv ] .
Ue [Lp Nv — Np Lv ] ф
Usually Lr, Nv and Np are positive whilst Lv, Lp and Nr are negative. Since Np depends on the height of the tail rotor above the roll axis it is unlikely to be the dominant term therefore (LpNv — Np Lv) will usually be negative and for spiral stability NrLv > LrNv.
Now Nv is a measure of the helicopter’s tendency to ‘yaw into wind’ and Lv is its tendency to ‘roll wings level’. In order for the aircraft to display positive spiral stability it must ‘roll wings level’ rather than ‘yaw into wind’. Therefore Lv (the dihedral effect) stabilizes the spiral mode. It should also be noted that the mode depends on the trimmed forward speed, Ue. However the effect of changes in speed are difficult to determine theoretically since the values of all the important derivatives change with speed.