Figure 12.10. Pitch stability

the aerodynamic characteristics). The wing and fuselage have destabilizing moments (i.e., nose up), which must be compensated for by the tail to counter the wing and fuselage moments; hence, Equation 12.6 has negative sign.

The second diagram in Figure 12.10 shows the stability effects of different CG positions on a conventional aircraft. The stability margin is the distance between the aircraft CG and the NP (i.e., a point through which the resultant force of the aircraft passes). When the CG is forward of the NP, then the static margin has a positive sign and the aircraft is statically stable. The stability increases as the CG moves farther ahead of the NP.

There is a convenient range from the CG margin in which the aircraft design exhibits the most favorable situation. In Figure 12.10, the position B is where the CG coincides with the NP and shows neutral stability (i.e., at a zero stability margin) – the aircraft can still be flown with the pilot’s efforts controlling the aircraft attitude. In fact, an aircraft with relaxed stability can have a small negative margin that requires little force to make rapid maneuvers – these aircraft invariably have a FBW control architecture (see Section 12.10) in which the aircraft is flown continuously controlled by a computer.

Engine thrust has a powerful effect on stability. If it is placed above and behind the CG such as in an aft-fuselage-mounted nacelle, it causes an aircraft nose-down pitching moment with thrust application. For an underslung wing nacelle ahead of the CG, the pitching moment is with the aircraft nose up. It is advisable for the thrust line to be as close as possible to the aircraft CG (i.e., a small ze to keep the moment small). High-lift devices also affect aircraft pitching moments and it is better that these devices be a small arm’s-length from the CG.

In summary, designers must carefully consider where to place components to minimize the pitching-moment contribution, which must be balanced by the tail at the expense of some drag – this is unavoidable but can be minimized.

12.4.2 Yaw Plane

The equation of motion in the yaw plane can be set up similarly to the pitch plane. The weathercock stability of the V-tail contributes to the restoring moment.

Figure 12.4 depicts moments in the yaw plane. In the diagram, the aircraft is yawing to the left with a positive yaw angle β. This generates a destabilizing moment by the fuselage with the moment (NF = YF × l f ), where YF is the resultant side force by the fuselage and lf is the distance of Lf from the CG. Contributions by the wing, H-tail, and nacelle are small (i.e., small projected areas and/or shielded by the fuselage projected area). The restoring moment is positive when it tends to turn the nose to the right to realign with the airflow. The weathercock stability of the V-tail causes the restoring moment (NVT = YT × lt ), where YT is the resultant side force on the V-tail (for small angles of (β + σ ), it can be approximated as the lift generated by the V-tail, LVT) and lt is the distance of LT from the CG. Therefore, the total aircraft yaw moment, N (for conventional aircraft), is the summation of NF and NVT, as given in Equation 12.9:

In coefficient form, the V-tail contribution can be written as in Equation 12.11 (LVT is in the coefficient form CLVT):

12.4.3 Roll Plane

As explained previously, roll stability derives primarily from the following three aircraft features:

1.Wing Dihedral (see Figure 12.5). Sideslip angle β increases the angle of attack, α, on the windward wing, α = (V sin )/u generating Lift. For small dihedrals and perturbations, β = v/u, which approximates α = β . The restoring moment is the result of Lift generated by α. It is quite powerful – for a

Figure 12.11. Statistics of current tail-volume coefficients

low-wing , it is typically between 1 and 3 deg, depending on the wing sweep. For a straight-wing aircraft, the maximum dihedral rarely exceeds 5 deg. For a high-wing sweep, it may require an anhedral, as discussed herein.

2. Wing Position Relative to the Fuselage (see Figure 12.7). Section 12.3.3 explains the contribution to the rolling moment caused by different wing positions relative to the fuselage. Semi-empirical methods are used to determine the extent of the rolling-moment contribution.

3. Wing Sweep at Quarter-Chord, 1/ (see Figure 12.8). The lift produced by a

4

swept wing is a function of the component of velocity, Vn, normal to the c1/ line;

4

that is, in steady rectilinear flight:

Vn = V cos

When an aircraft sideslips with angle β, the component of velocity normal to

the c1/ line becomes (small β):

4

Vn = V cos( 1/4 − β) = V(cos 1/4 + β sin 1/4β)

For the leeward wing:

Vn lw = V cos( 1/4 + β) = V(cos 1/4 − β sin 1/4β)

The windward wing has V n > Vn and vice versa; therefore, it provides Lift as the restoring moment in conjunction with the lift decrease on the leeward wing.

As 1/ increases, the restoring moment becomes powerful enough that it must

4

be compensated for by the use of the wing anhedral.

12.5 Current Statistical Trends for H- and V-Tail Coefficients

During Phase 1 (conceptual design) of an aircraft design project, the initial empennage is sized using statistical data. Section 3.20 provides preliminary statistics of the empennage tail-volume coefficients. Figure 12.11 provides additional statistics for current aircraft (twenty-one civil and nine military aircraft types), plotted