AERO-DERIVATIVES FOR ROTORCRAFT
During the different phases of a helicopter’s flight it is subject to a number of forces and moments that must be balanced if the aircraft is to remain in trim. Alternatively should the pilot wish to control the helicopter by modifying its flight path he must generate an out-of-balance condition by moving the flight controls from their trim position. On the other hand atmospheric disturbances may upset the force and moment balance and cause the flight path to change unintentionally. The nature of the out-ofbalance condition following such a gust affects the stability characteristics of the rotorcraft by dictating whether the helicopter will return to trim without pilot intervention. The amount by which flight controls and atmospheric disturbances modify the forces and moments acting on a rotorcraft are the key to determining its stability and control characteristics. Typically, these forces and moments can be changed by any of the following, alone or in combination:
(1) Disturbances in linear speeds or angular rates;
(2) Changes in main rotor and tail rotor blade pitch;
(3) Movement of the centre of gravity position.
Measuring or predicting how a given force or moment will change as a result of variations in the above parameters is fundamental to determining the handling qualities of a particular air vehicle. A shorthand based on the derivative has therefore been developed to simplify discussion of these effects.
4.2.1 The derivative
The concept of the derivative is best understood by means of a simple example. Consider an aircraft in trimmed straight and level flight at an angle of attack (AOA) a0. If the variation of the lift and drag can be simply portrayed, as in Fig. 4.1, it is possible to determine their trimmed values (L0 and D0).
Suppose as a consequence of a gust the AOA is increased by da to a1. From Fig. 4.1 it is a simple matter to determine L1 and D1. If, however, only small changes in AOA occurred it would be possible to calculate changes in lift and drag using the local slope around the trim point. Thus:
L – L = dL = dL da and D – D = dD = dDda da da
Since these slopes represent the rate of change of lift and drag with AOA around the trim point, they can be used directly to calculate the change in these forces. As shall be seen later, all equations of motion assume the aircraft is in equilibrium prior to being disturbed by a gust or pilot action. Thus in our example, a, L and D should represent the change in AOA, lift and drag from their equilibrium values. Therefore, written more formally:
r dL dD
L = —— a and D = —— a da da
Alternatively, taking note of the symbols used for vertical and horizontal forces and standard aero-derivative notation:
L = — Z = La. a= —Za. a and D = — X = Da. a= — Xa. a
In fact, modifications to the AOA can result from changes in forward speed (u) and from changes in vertical speed (w). Thus more generally:
X = Xu. u + Xw. w and Z = Zu. u + Zw. w