Figure 12.4. Direction stability

The other requirement for static stability is that at a zero angle of attack, there should be a positive nose-up moment, providing an opportunity for equilibrium at a positive angle of attack (+αtrim), typical in any normal flight segment.

12.3.2 Directional Stability: Yaw Plane (Yaw Moment, N)

Directional stability can be compared to longitudinal stability but it occurs in the yaw plane (i.e., the XY plane about the Z-axis), as shown in Figure 12.4. By definition, the angle of sideslip, β, is positive when the free-stream velocity vector, V, relative to aircraft is from the right (i.e., the aircraft nose is to the left of the velocity). V has component aircraft velocities u along the X-axis and v along the Y-axis, subtending the sideslip angle β = tan−1(v/u).

The V-tail is subjected to an angle of incidence (β + σ ), where σ is the sidewash angle generated by the wing vortices (like the downwash angle in longitudinal stability). Static stability criteria require that an increase in the sideslip angle, β, should generate a restoring moment, Cn, that is positive when turning the nose to the right. The moment curve slope of Cn is positive for stability. At zero β, there is no yawing moment (i.e., Cn = 0).

Yaw motion invariably couples with roll motion because neither is in the plane of symmetry. In yaw, the windward wing works more to create a lift increase while the lift decreases on the other wing, thereby generating a rolling moment. Therefore, a pure yaw motion is achieved by the use of compensating, opposite ailerons. The use of an aileron is discussed in the next section.

12.3.3 Lateral Stability: Roll Plane (Roll Moment, L)

Roll stability is more difficult to analyze compared to longitudinal and lateral stabilities. A banked aircraft attitude through a pure roll keeps the aircraft motion in

Figure 12.7. Effect of wing sweep on roll stability

to β must have Clβ negative; that is, with an increase of roll angle , the sideslip angle β increases to provide the restoring moment. An increase in β generates a restoring roll moment due to the dihedral. At zero , there is no β; hence, the zero rolling moment (Cl = 0).

The wing dihedral angle, , is one way to increase roll stability, as shown in Figure 12.5. The dropped wing has an airflow component from below the wing generating lift, while at the other side, the airflow component is from the upper side of the wing that reduces the angle of attack (i.e., the lift reduction creates a restoring moment).

The position of the wing relative to the aircraft fuselage has a role in lateral stability, as shown in Figure 12.6. At yaw, the relative airflow about the low wing has a component that reduces the angle of attack; that is, the reduction of lift and the other side act in opposite ways: a destabilizing effect that must be compensated for by the dihedral, as explained previously. Conversely, a high-wing aircraft has an inherent roll stability that acts opposite to a low-wing design. If it has too much stability, then the anhedral (-ve dihedral) is required to compensate it. Many highwing aircraft have an anhedral (e.g., the Harrier and the BAe RJ series).

An interesting situation occurs with a wing sweepback on a high-speed aircraft, as explained in Figure 12.8. At sideslip, the windward wing has an effectively reduced sweep; that is, the normal component of air velocity increases, creating a lift increment, whereas the leeward wing has an effectively increased sweep with a slower normal velocity component, thereby losing lift. This effect generates a rolling moment, which can be quite powerful for high-swept wings; even for lowwing aircraft, it may require some anhedral to reduce the excessive roll stability (i.e., stiffness) – especially for military aircraft, which require a quick response in a roll. Tu-104 in Figure 12.7 is a good example of a low-wing military aircraft with a high sweep coupled with an anhedral.

The side force by the fuselage and V-tail contributes to the rolling moment, as shown in Figure 12.8. If the V-tail area is large and the fuselage has a relatively smaller side projection, then the aircraft CG is likely to be below the resultant side force, thus increasing the stability. Conversely, if the CG is above the side force, then there is a destabilizing effect.