Dynamic stability in the hover
Some of the derivatives can be assumed to approximate to zero in the hover case (Xw, Zu, Zq, Zw, and Mw) and the characteristic equation typically solves to give two real roots and a pair of complex roots. One real root can be shown to be (s — Zw) = 0, so that s = Zw. This represents a heavily damped subsidence such that if a helicopter is disturbed, by a vertical gust for example, the subsequent heave motion is quickly damped out. The motion is a pure convergence with no oscillation and confirms that the vertical motion is completely decoupled from the pitching and fore/aft motions, a prediction arising from examination of the Z-force equation in the hover. The other real root represents the forward speed mode. In the hover the pitching oscillation or falling leaf mode masks this mode.
The physical description of the motion associated with the complex root is handled easily. Assume that the hovering helicopter experiences a small horizontal velocity disturbance, the relative airspeed change causes the rotor to tilt backwards and exert a nose-up pitching moment on the helicopter. A nose-up attitude then develops and the rearward component of the thrust vector decelerates the aircraft until its forward motion is stopped. At this point the disturbing disk tilt and rotor moment vanish but the helicopter is left in a nose-up attitude and backward motion begins. This causes the rotor to flap forwards and exert a nose-down moment. The thrust vector tilts forward and the rearward motion is stopped but the helicopter is now left in a nose – down attitude which accelerates it forward and the cycle begins again. The motion is generally unstable and its amplitude increases steadily, bearing out the analytical solution. The instability is entirely due to the characteristic backward flapping of the rotor with forward speed. However, the pitch damping derivative Mq will affect the rate of divergence. Making Mq more negative will reduce the real part of the complex root and hence increase the time to double amplitude of the motion. It can never make the motion stable, however, and at best only neutral stability can be achieved.
Bramwell [4.1], discusses the implications of positive Mu on the motion, that is the possibility of making the rotor flap forward with speed, but this only leads to a pure divergence which is even more undesirable. He also quotes the results of investigations into the effects of configuration changes on the dynamic stability of the hovering helicopter but it appears that no reasonable change will significantly improve it. In particular, CG position has little effect on the stability but simply affects the fuselage attitude adopted, even for blades with offset hinges or hingeless rotors. Although a hub moment can be exerted in these cases and it is therefore no longer necessary for the thrust vector to act through the CG, thrust changes due to forward speed and pitch rate are typically zero in the hover. The motion can be approximated to a neutral oscillation whose frequency depends on Mu and Mq:
4.9.3 Dynamic stability in forward flight
The characteristic equation in forward flight resolves into four roots but it is not so easy to generalize as in the case with a conventional fixed wing aircraft. For the latter,
the characteristic equation resolves into pairs of complex conjugate roots representing two oscillatory motions, one of short period and high damping – the SPPO – and the other of long period which is lightly damped – the phugoid. In the case of a helicopter the characteristic equation solves into four roots but for a particular helicopter two pairs of complex roots may be found at one flight condition, two real and a pair of complex roots at a second condition and four real roots at a third. The reason for this is the large variation in the value of the derivatives over the flight envelope. In the longitudinal case Mw has the greatest influence, with Mq, Mu and Zw also playing a part in the overall result. Although rotorcraft do not, strictly speaking, exhibit the SPPO and phugoid motions described for fixed wing aircraft, there are certain analogies which can be drawn.
The phugoid motion of a fixed wing aircraft is an oscillation involving changes in height and speed at approximately constant incidence. A disturbance producing, for example, an increase in lift causes the aircraft to climb slowly as lift now exceeds weight. The climb (which is at constant incidence) results in a decrease in speed and consequent loss of lift and eventually leads to a situation where lift and weight are again equal but the aircraft continues to slow down. A descent begins as weight exceeds lift and the consequent increase in speed produces an increase in lift and the climb starts again. The oscillatory motion experienced by a helicopter is influenced by the respective values of the speed stability, Mu, the angle of attack stability, Mw, the pitch damping, Mq and the pitching moment of inertia, Iyy.
Consider the motion following a disturbance that causes the helicopter to adopt a nose-down attitude and start to descend. Initially, rotorcraft with neutral angle of attack stability (Mw = 0) will be considered. The component of aircraft weight acting along the flight path accelerates the helicopter but as speed increases the rotor disk flaps back (speed stability) and a consequent nose-up pitching moment and angular acceleration occurs. This soon produces an angular velocity such that the pitch damping causes the fuselage to rotate to a greater angle of attack than the rotor, and this effectively neutralizes the thrust vector tilt due to static stability. The flight path is still downwards, however, and the component of weight acting along it ensures that the airspeed continues to increase so that the preceding steps are repeated. The angular velocity continues to increase with consequent increase in fuselage angle of attack. In turn, the thrust continues to increase until it is sufficient to level off the glide path. At this point the helicopter has reached its maximum forward speed, maximum nose-up pitch rate, and maximum fuselage angle of attack. The thrust now exceeds the weight and the aircraft begins to climb. The component of weight acting along the flight path now starts to slow down the helicopter. The rotor disk flaps forward as the rearward tilt due to speed stability is exceeded by the forward tilt due to pitch damping. The nose-down damping moment reduces the pitch rate and the fuselage angle of attack until they reach their trim values. The helicopter is still climbing, however, so the speed continues to decrease and the rotor flaps forward. The resulting nose-down pitching moment starts a similar sequence of events (with opposite signs of course) and this will be repeated until the oscillation eventually either damps out (stable) or grows worse (unstable). It is possible to characterize the long-term mode as [4.1]:
The frequency of the long-term mode is therefore inversely proportional to the trim speed, Ue. An increase in trim speed thus reduces the frequency resulting in a larger period for the oscillation. An increase in Zu (related to the lift coefficient), on the other hand, has the opposite effect. The damping of the long-term mode is affected by the same factors, but in the opposite sense, so an increase in trim speed will increase the damping in addition to reducing the frequency of the response. The drag of the helicopter (related to Xu) also affects the ‘phugoid’ frequency. A helicopter carrying large external stores (higher drag coefficient) is therefore likely to exhibit a more heavily damped long-term response. A less simplified relationship for £ shows that the pitch damping, Mq, adds to the damping of the long-term mode whereas Mu reduces it [4.3].