Yawing Moment with Boll Rate
Referring again to Figure 8.35, it is seen that on the right wing, which is moving down, the lift vector is inclined forward through the angle PylV. On the left wing, the inclination is to the rear. The components of these tilted lift vectors in the x direction give rise to a differential increment in the yawing moment equal to
dN = -2 yqcCL^ydy
Integrating from 0 to b/2, the total yawing moment increment in coefficient form for a linearly tapered wing becomes
Rolling Moment with Sideslip Angle—Dihedral Effect
The rate of change of rolling moment with sideslip angle, С((,, is important to the handling qualities of an airplane. Generally, a small negative value of Cip is desirable, but too much dihedral effect makes an airplane uncomfortable to fly. The principal factors affecting С/э are sweepback, placement of the wing on the fuselage, and the dihedral angle of the wing, Г.
The primary control over Clf) is exercised through the dihedral angle, Г, shown in Figure 8.38. Although the sweepback angle also affects Clf) significantly, Л is normally determined by considerations other than C/r From Figure 8.38, it can be seen that a positive sideslip results in an upward velocity component along the right wing and a downward component along
the left wing. Added to the free-stream velocity, this results in an increase in the angle of attack over the right wing equal to
Да = (ЗГ
Over the left wing, an opposite change in a occurs. This differential increment in a results in a differential rolling moment, given by
dl = 2qcajiYy dy
For a linearly tapered wing, this reduces to
The derivation of Equation 8.108 has neglected induced effects that are appreciable for low aspect ratios. Its primary value lies in disclosing the linear relationship between Г and C/0.
It is recommended that the contribution to C[fS from Г be estimated using Figure 8.39. This figure represents the departure of Q0 from the following normal case and is based on graphs presented in Reference 8.11 or 5.5.
A =0.5 A = 6.0
Д1/2 = 0
C, f = -0.00021 r/deg(T in deg)
Figure 8.39 Effect of wing dihedral angle on CNote that factors represent relative effects of varying each parameter independently from the normal case. Л = 0.5, Л = 6.0, Лі/2 = 0.
where the functions kA, kA, and kA are obtained from Figure 8.39 as functions of A, A, and Лід, respectively.
The effect of sweepback on Cis determined with the help of Figure 8.40. A swept wing is shown operating at a positive sideslip angle /3. From the geometry, the velocity component normal to the leading edge of the right wing is given by
V cos (Л – /3)
The corresponding velocity on the left wing is
V cos (Л + /3)
If Cin is the section lift coefficient based on the normal velocity and “normal chord,” then the differential lift on the right and left wings will be
dLR = q cos2 (Л – /3)c cos ЛС(, ds
dLL = q cos2 (Л + /3)c cos ACln ds
The differential rolling moment is therefore
dl = qcCiny [cos2(A + j8) – cos2 (A – /3)] cos A ds But
у = s cos A
dy = ds cos A
l = qCln[cos2(A + /3) – cos2 (A – /3)] І су dy
The total lift (for /3 = 0) is given by
L = 2q cos2 АС/ I c dy
= qSCtn cos2 A
Thus the wing CL and the normal section C, n are related by
Cj = Cin cos2A
and the rolling moment coefficient becomes
If this equation is differentiated with respect to /3 and evaluated at /3 = 0, the following results.
For a linearly tapered wing, Equation 8.110 reduces to
Again, this result is only qualitatively correct. Generally,
Ci0 = -/(A, X)CL tan Л (8.112)
Figure 8.41 (based on Ref. 5.5) presents CtJCL as a function of Л for a range of aspect ratios. The variation with tan Л is seen to hold only for the higher aspect ratios. This figure can be used with Equation 8.111 to estimate Ci0 for other taper ratios.
Observe that wing sweep can contribute significantly to dihedral effect. In order to avoid an excessive dihedral effect on aircraft with highly swept wings, it is frequently necessary to employ a negative dihedral angle on the wing, particularly if the wing is mounted high on the fuselage.
The effect that the wing placement on the fuselage has on Ctj) is seen by reference to Figure 8.42. In a plane across the top of the fuselage, the cross-flow around the fuselage is seen to go up on the right side and down on the left. Thus, for a high wing, this flow increases the angle of attack of the right wing while decreasing a on the left wing. This results in a negative rolling moment comparable to a positive dihedral effect. For a low wing, the effect is just the opposite. This is the reason, as you may have observed, that
Figure 8.42 Rear view showing the cross-flow around a fuselage.
the dihedral angle for unswept low wings is generally greater than for high wings. Many high-wing airplanes do not have any dihedral angle at all.
As a rule, it is recommended that the following be added to ClfS to account for the fuselage cross-flow (Ref. 8.6).
High-wing Д Clfj = -0.00016/deg
Midwing AC/f) = 0
Low-wing AC/fj = 0.00016/deg
This information on C/0 and the other stability derivatives is intended only as an introduction to the subject. For more complete information on these quantities, see Reference 5.5 and 8.3.