# The normal force and pitching moment derivatives due to pitching

4.7.1 (Zq)(Mq) wing contributions

Thin-aerofoil theory can be used as a convenient basis for the estimation of these important derivatives. Although the use of these derivatives is beyond the general scope of this volume, no text on thin-aerofoil theory is complete without some reference to this common use of the theory.

When an aeroplane is rotating with pitch velocity q about an axis through the centre of gravity (CG) normal to the plane of symmetry on the chord line produced (see Fig. 4.18), the aerofoil’s effective incidence is changing with time as also, as a consequence, are the aerodynamic forces and moments.

The rates of change of these forces and moments with respect to the pitching velocity q are two of the aerodynamic quasi-static derivatives that are in general commonly abbreviated to derivatives. Here the rate of change of normal force on the aircraft, i. e. resultant force in the normal or Z direction, with respect to pitching velocity is, in the conventional notation, dZ/dq. This is symbolized by Zq. Similarly the rate of change of M with respect to q is QMjdq = Mq.

In common with other aerodynamic forces and moments these are reduced to non­dimensional or coefficient form by dividing through in this case by pVI{ and [>VI~ respectively, where /, is the tail plane moment arm, to give the non-dimensional  Fig. 4.18 normal force derivative due to pitching zq, and the non-dimensional pitching moment derivative due to pitching mq.

The contributions to these two, due to the mainplanes, can be considered by replacing the wing by the equivalent thin aerofoil. In Fig. 4.19, the centre of rotation (CG) is a distance he behind the leading edge where c is the chord. At some point jc from the leading edge of the aerofoil the velocity induced by the rotation of the aerofoil about the CG is v’ = —q(hc — jc). Owing to the vorticity replacing the camber line a velocity v is induced. The incident flow velocity is V inclined at a to the chord line, and from the condition that the local velocity at jc must be tangential to the aerofoil (camber line) (see Section 4.3) Eqn (4.14) becomes for this case or

(4.59)

and with the substitution jc = — (1 — cos в)

dy v qc (, 1 cos0  S-“=r’v(A-2 + —J

but in the pitching case the loading distribution would be altered to some general form given by, say,  — = Bq + cos пв

Fig. 4.19

where the coefficients are changed because of the relative flow changes, while the camber-line shape remains constant, i. e. the form of the function remains the same but the coefficients change. Thus in the pitching case

For the theoretical estimation of rt/ and т, ґ of the complete aircraft, the contribu­tions of the tailplane must be added. These are given here for completeness. (4.75)

where the terms with dashes refer to tailplane data.