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.

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