Stability Coefficients of Longitudinal Motion

Straight flight For longitudinal motion, the resultant aerodynamic force may be represented by lift, drag, and pitching moment. Their dependence on the angle of attack (see Fig. 3-61a) has been discussed in the previous section. The two most important coefficients are lift slope dcLjda and pitching-moment slope dcMjdcL. The latter determines the position of the aerodynamic neutral point of the wing by Eq. (1-29). The lift slope dcLjda is presented for various wing, shapes in Figs. 3-25,3-38, 3-42, 3-44, 346, and Table 3-5. The neutral-point positions of various wing forms can be obtained from Figs. 3-37, 3-38, 3-43, 344, and Table 3-5. The flight mechanical computations for the neutral-point position require great accuracy. The neutral-point position depends strongly on the individual planform. In general, therefore, it is required that for its determination the lift distribution should be computed by using the lifting-surface theory (see Sec. 3-3-5).

Pitching motion Pitching motion is actually a nonsteady motion. In general it proceeds slowly enough, however, that it can be treated as “quasi-steady.” When the wing

Figure 3-61 The motion modes of the wing, (a) Straight flight, (b) Yawed flight, (c)-(e) Rotary motions: rolling, pitching, yawing.

(Fig. 3-62) performs a rotary motion with angular velocity coy about the lateral axis through xs (Fig. 3-6Id), a vertical additive velocity Vz = озу(х — х$) is produced that varies linearly over the wing chord. Together with the incident flow velocity V, the rotary motion in chord direction produces an additive angle-of-attack distribu­tion a(x) = VzjV of magnitude

«(®) = ^ (s — XS) (3-147)

This angle-of-attack distribution produces an additive lift distribution, the integra­tion of which leads to an additive lift and an additive pitching moment. These quantities are designated lift due to pitch rate and pitch damping. Both depend linearly on cOy. It is expedient, therefore, to introduce the coefficients dcLjdQy as lift due to pitch rate and bcMjbQy as pitch damping, where Qy = tuyC^jV is the dimensionless pitching angular velocity and is the wing reference chord, introduced earlier by Eq. (3-5b). These coefficients depend only on the wing geometry and the position of the axis of rotation.

Now it wfll be explained how these two quantities can be determined and, in

Figure 3-62 Explanatory sketch for aerodynamic coefficients of the pitching wing.

particular, how their values change with the position of the axis of rotation xs[20] It is evident that there is an axis of rotation xQ for which the lift due to pitch rate is zero. For a rectangular wing, this axis of rotation lies at a distance § c from the leading edge, according to the Pistolesi’s theorem (see Sec. 2-4-5). The pitch damping, however, cannot be zero for any position of the axis of rotation. For the computation of the lift due to pitch rate, the angle-of-attack distribution of Eq. (3-147) is rewritten in the form

oc{x) = ^ (x – xQ) – F ^ (a?0 – x8) (3-148)

By setting xs = x0 in this equation, the second term becomes zero, whereas, by definition, the first term produces zero lift due to pitch rate, (dcL/d£>y) = 0. The contribution of the first term to the pitch damping will be expressed by (dcM/dQy). The second term in Eq. (3-148) represents a constant angle of attack and gives the total lift due to pitch rate as

bcL _ dcL х0 – xs

9 Qy J s d? os cy.

Here dcLjda. is the lift slope of the wing. Equation (3-149) shows that the lift due to pitch rate is a linear function of the position of the axis of rotation (see Fig. 3-63a). The moment of the pitching motion is obtained from Eq. (1-28) as

which leads with Eq. (3-149) to the pitch damping:

&cm /деm жіу — xs xo — :cs

dQjs 8®Jo Сц Cm doc

This equation shows that the pitch damping depends parabolically on xs. In particular, it is immediately obvious that for Xg = xN and for x$ = x0 the pitch damping has the same value, namely, (bcM/dQy)0, as in Fig. 3-633.

To be able to compute the pitch damping from Eq. (3-150) for an arbitrary position of the axis of rotation xs, the determination of (bcMjbQy)0 and of x0/cM is required, whereas the coefficients dcLjda and are known from earlier

discussions.

For xs – xN, Eqs. (3-149) and (3-150) yield

(3-1513)

Thus the problem of determining the lift due to pitch rate and the pitch damping for an arbitrary position of the axis of rotation has been reduced to the computation of the two coefficients (3cijdDy)N and {bcMjbQy)N for the position of the axis of rotation in the neutral point. These latter two coefficients are obtained from the lift and pitching moments as determined from lifting-surface wing theory for the angle-of-attack distribution corresponding to Eq. (3-147):

(3-152)

In Table 3-6, numerical data on the positions of the axis of rotation for zero lift due to pitch rate and. of the corresponding pitch damping are compiled for a trapezoidal wing, a swept-back wing, and a delta wing (Table 3-5). Compare also Garner [62] and Gothert and Otto [24].

In the case of airplanes with a separate horizontal tail, the contribution of the wing to the lift due to pitch rate is small compared with that of the tail surface. An

4

Figure 3-63 Lift due to pitch rate (a) and pitch damping (b) vs. position of axis of rotation Ху. хдг = neutral-point

Table 3-6 Position of the axis of rotation for zero-lift due to pitch rate and corresponding pitch damping for a trapezoidal wing, a swept-back wing, and a delta wing* і Trapezoidal 1 wing Swept-back wing Delta wing Zla-0 0.533 0.485 0.604 C/JL – 0.358 – 0.498 – 0.285 d&Jy)0 *The distance Ax0 is measured relative to the geometric neutral point NiS, the position of which is given in Table 3-5. Table 3-6 is based on data from Table 3-5.

accurate computation is therefore not required. On the other hand, in the case of all-wing airplanes, whose total pitch damping is almost completely produced by the wing, a more accurate computation may be required, depending on the specific case.