# The derivatives Xq, Yp

These derivatives are dominated by the main rotor contributions. For teetering rotors and low flap hinge-offset rotors, the changes in rotor hub X and Y forces are the primary contribution to the pitch and roll moments about the aircraft centre of mass. Hence the derivatives Xq and Yp can contribute significantly to aircraft pitch and roll damping. The basic physical effects for the two derivatives are the same and can be understood from an analysis of a teetering rotor in hovering flight. If we assume that the thrust acts normal to the disc in manoeuvres, and ignoring the small drag forces, then the rotor X force can be written as the tilt of the thrust vector:

X = TPic (4.73)

The pitch rate derivative is then simply related to the derivative of flapping with respect to pitch rate. As the aircraft pitches, the rotor disc lags behind the shaft by an amount proportional to the pitch rate. This effect was modelled in Chapter 3 and the relation­ships were set down ineqns 3.71 and 3.72. For a centrally hinged rotor with zero spring
stiffness, the disc lags behind the shaft by an amount given by the expression двіс 26

д q yil   The Lock number у is the ratio of aerodynamic to inertia forces acting on the rotor blade; hence the disc will flap more with heavy blades of low aspect ratio. Equation 4.74 implies that the force during pitching produces a pitch damping moment about the centre of mass that opposes the pitch rate. However, the assumption that the thrust remains normal to the disc has ignored the effect of the in-plane lift forces due to the inclination of the lift vectors on individual blade sections. To examine this effect in more detail we need to recall the expressions for the rotor hub forces from Chapter 3, eqns 3.88-3.99. Considering longitudinal motion only, thus dropping the hub/wind dressings, the normalized X rotor force can be written as

The first term in eqn. 4.75 represents the contribution from the fore and aft blades to the X force when the disc is tilted and is related to the rotor thrust coefficient by the expression

F01) = -(—) (4.76)

0 aos )

This effect accounts for only half of the approximation given by eqn 4.73. Additional effects come from the rotor blades in the lateral positions and here the contributions are from the in-plane tilt of the lift force, i. e.,

Fi(2) = (у – у) віс – у9is (4.77)

During a pitch manoeuvre from the hover, the cyclic pitch can be written as (see eqn 3.72) 16

01s — ~в1с + ггq

yil

Hence, substituting eqn 4.78 into eqn 4.77 and then into eqn 4.75, the force derivative can be written in the form  ^ = Ct (1.5________________ ^ = Ct (1.5_________________________ 0L-

дq 12Ct /aos) дq 12Ct /aos     We can see that the thrust is inclined relative to the disc during pitching manoeuvres due to the in-plane loads when the blades are in the lateral position. The scaling coefficient given in eqn 4.79 reduces in the hover to

and has been described as the ‘Amer effect’ (Ref. 4.13). Further discussion can be found in Bramwell (Ref. 4.14) and in the early paper by Sissingh (Ref. 4.15). Although our analysis has been confined to hover, the approximation in eqn 4.79 is reasonably good up to moderate forward speeds. The effect is most pronounced in conditions of high collective pitch setting and low thrust, e. g., high-power climb, where the rotor damping can reduce by as much as 50%. In autorotation, the Amer effect almost disappears.