# Normalized and non-dimensional derivatives

Although the full set of derivatives introduced above may be needed for an accurate mathematical representation of rotorcraft dynamic characteristics they are not all required when discussing typical helicopter handling qualities. Since the pilot is most interested in the attitude changes that occur as a result of control input and atmospheric disturbances mainly moment derivatives will be retained in the reduced set. Due to their widespread use, most of these derivatives have acquired a common descriptor based on their effect on the stability and control characteristics of a typical helicopter, see Table 4.4.

All the derivatives considered so far are affected by the aerodynamic characteristics of the aircraft. They will therefore be modified by changes in flight condition (density altitude, airspeed, rotor speed) and rotorcraft design (rotor radius, blade area). In order to make appropriate comparison it is common practice to make aero-derivatives non-dimensional in a similar manner to that used for lift and drag. In addition, as shall be seen later, when the equations of motion governing aircraft behaviour are manipulated it is sometimes convenient to normalize the derivatives using the mass and inertia properties of the helicopter. There is unfortunately no internationally agreed set of standard symbols used to distinguish between these various classes of aero-derivatives. A set appropriate to rotorcraft that has been developed from those used by Babister [4.2] is presented in Table 4.5.

Aero-derivatives help us understand the stability and control characteristics of

Table 4.4 Most commonly used aero-derivatives.

 Common descriptor Common descriptor Xu Drag damping L Roll damping Yv Side force Mq Pitch damping Zw Heave damping Nr Yaw damping Lv Lateral static stability LA Roll control power Mu Speed stability MB Pitch control power Mw Angle of attack stability NeTR Yaw control power Nv Directional static stability Ze0 Heave control power Tail rotor roll Ye eTR Tail rotor drift Mec Pitch change with power Nec Torque reaction

Table 4.5 Symbol set for various classes of aero-derivative.

Non­Dimensional Normalized dimensional

 Derivative type derivatives Divisors derivatives Divisors derivatives Force/linear velocity Xu Y Zw mass Xu Yv Zw psAQR xu yv zw Force/angular velocity Xp Y Zr mass Xp Yq Zr psAQR2 xP yq z Moment/linear velocity Lu Mv Nw inertia Lu Mv Nw psAQR2 lu mv nw Moment/angular velocity LP Mq Nr inertia Lp Mq Nr psAQR3 lp mq n. Force/control deflection XAl YBl Z0c mass XAl YB Z0c psAQ2R2 xA yBl z0c Moment/control deflection L A MB N% inertia LA mb n% psAQ2R3 lA mB n0c

rotorcraft by providing a convenient manner through which to describe the factors that affect these characteristics. Equations of motion can be developed that tie these derivatives directly to the dynamic behaviour of rotorcraft. It is possible, therefore, to determine the particular derivatives that are key to shaping the pilot’s perception of his aircraft. Likewise, the contribution made by the components of a typical helicopter to these derivatives can be identified and how the design choices made by rotorcraft manufacturers may affect the suitability of a helicopter for a particular role can be understood. Aero-derivatives are, therefore, fundamental to any study of aircraft stability and control.