# Pitch Plane

In equilibrium, J^force = 0, when drag = thrust and lift = weight (see Figure 3.9). An imbalance in drag and thrust changes the aircraft speed until equilibrium is reached. Drag and thrust act nearly collinearly; if they did not, pitch trim would be required to balance out the small amount of pitching moment it can develop. The same is true with the wing lift and weight, which are rarely collinear and gen­erate a pitching moment. This scenario also must be trimmed, with the resultant lift and weight acting collinearly.

Together in equilibrium, moment = 0. Any imbalance results in an aircraft rotating about the Y-axis. Figure 12.9 shows the generalized forces and moments (including the canard) that act in the pitch plane. The forces are shown normal and parallel to the aircraft reference lines (i. e., body axes) and abnormal to air­craft velocity. Lift and drag are obtained by resolving the forces of Figure 12.9 into

Geometry, Velocity, and Force Details

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Force and Moment Details

Figure 12.9. Generalized force and moment in the pitch plane

perpendicular and parallel directions to the free-stream velocity vector (i. e., aircraft velocity). The forces can be expressed as lift and drag coefficients, dividing by qSw, where q is the dynamic head and Sw is the wing reference area. Subscripts identify the contribution made by the respective components. The arrowhead directions of component moments are arbitrary – they must be assessed properly for the com­ponents. With its analysis, Figure 12.9 gives a good idea of where to place aircraft components relative to the aircraft CG and NP. The static margin is the distance between the NP and the CG.

The generalized expression for the moment equation can be written as in Equa­tion 12.1, which sums up all the moments about the aircraft CG. In the trimmed condition, the aircraft moment about the aircraft CG must be zero (Maccg = 0):

Mac_cg = (Nc X lc + Cc X Zc + Mc)canard + (Nw X lw + Cw X Zw + Mw)wing

+ (Nt X lt + Ct X zt + Mt )tail

+ Mfus + Mnac + (thrust X zth + nacdrag x zth) + any other item (12.1)

In the conceptual design stage, the forces and moments of each component are estimated semi-empirically (i. e., US DATCOM and RAE data sheets [now ESDU]) from the drawings. When assembled together as an aircraft, each component is influ­enced by the flow field of the others (e. g., the flow over the H-tail is affected by the wing flow). Therefore, a correction factor, n, is applied. This is shown in Equa­tion 12.2 written in coefficient form, dividing by qSwc, where q is the dynamic head,

c is the wing MAC, and Sw is the wing reference area. Subscripts identify the contri­bution of the respective components. The moment coefficients of the components are computed initially as isolated bodies and then converted to the reference wing area:

Cmcg — CNc(Sc/Sw Wc / c) + CCc(Sc/Sw )(zc/c) + Cmc (Sc/Sw )]forcanard

+ CNw(1a/c) nw + CCw (za/c)nw + Cmw nw]forwmg

+ CN( (St/Sw)(1t/c)nt + CCt (St/Sw)(zt/c)nt + Cmt (St/Sw)nt]fortail

+ Cmfus + Cnac + (thrust x zth + nacudrag X zth)/qSwc (12.2)

where:

1. n(— q /q<x>) represents the wake effect of lifting surfaces behind another lifting surface producing downwash, qt is the incident dynamic head, and is the free-stream dynamic head.

2. The vertical distances (z) of each component can be above or below the CG, depending on the configuration, described as follows:

(a) For fuselage-mounted engines, zth is likely to be above the aircraft CG and its thrust generates a nose-down moment. In underslung wings, engines have the zth below the CG, generating a nose-up moment. For most mili­tary aircraft, the thrust line is very close to the CG; therefore, for a prelimi­nary analysis, the zth term can be ignored (i. e., no moment is generated with thrust unless it is vectored).

(b) The drag of a low wing below the CG (za) has a nose-down moment and vice versa for a high wing. For midwing positions, which side of the CG must be noted; the (za) may be small enough to be ignored in the preliminary analysis.

(c) The position of the H-tail shows the same effect as for the wing but is invari­ably above the CG. For a low H-tail, zt can be ignored. In general, the drag generated by the H-tail is small and can be ignored; this is also true for a T-tail design.

(d) For the same reason, the contribution of the canard vertical distance, zc, also can be ignored.

In summary, Equation 12.2 can be further simplified by comparing the order of mag­nitude of the contributions of the various terms. Initially, the following simplifica­tions are suggested:

1. The vertical z distance of the canard and the wing from the CG is small. There­fore, the terms with zc and za can be omitted.

2. The canard and H-tail reference areas are much smaller that the wing reference area and their C («drag) component force is less than a tenth of their lifting forces. Therefore, the terms with CCc (Sc/Sw) and CCt (St/Sw) can be omitted – even for a T-tail, but it is best to check its overall contribution.

3. A high or low wing has za with opposite signs. For a midwing, za may be small enough to be ignored.

Equation 12.2 can be simplified as follows:

Cmcg — [CNc(Sc/Sw)(lc/c) + Cmc(Sc/Sw)] + [CNw(la/c)^w + Cmw ^w]

+ [CNt (St/Sw)(lt/c) nt + Cmt (St/Sw) nt]

+ Cmfus + Cnac + (thrust x zth + na^drag X zth)/qSwc (12.3)

A conventional aircraft does not have a canard. Then, Equation 12.3 can be fur­ther simplified to Equation 12.4. The conventional aircraft CG is possibly ahead of the wing MAC. In this case, the H-tail must have negative lift to trim the moment generated by the wing and the body:

Cmcg — [CNw (la /c) + Cmw ] + [Cm (St/Sw )(lt/c)nt + Cmt (St/Sw )nt ]

+ Cmfus + Cnac + (thrust x zth + nac-drag X zth)/qSwc (12.4)

Normal forces now can be resolved in terms of lift and drag; for small angles of a, the cosine of the angle is 1. The drag components of all the CN are very small and can be neglected.

Then, the first term:

CNw(la/ c) + Cmw ^ CLw(la/c) + Cm

and the second term:

Cm (St/Sw)(lt/c)nt + Cmt(St/Sw)nt ^ Cl (St/Sw)(lt/c)nt + Cmt (St/Sw)nt

where

Cmt (St/Sw)nt < Cl (St/Sw)(lt /c)nt

Hence, the moment contribution by the H-tail is represented as Cm_HT — Cut (Sh/Sw )(lt /c)nHT.

Then, Equation 12.4 is rewritten (note the sign) as:

Cmcg — [CLw (la /c) + Cmw] + Cm_HT + Cmfus + Cm

+ (thrust x zth + na^drag x zth)/qSwc

where

Cm_HT — —ClHT X [(lt/Sht)/(Swc)]nHT – — — Vh VHtClHT (12.6)

For conventional aircraft, CLHT has a downward direction to keep the nose up; therefore, it has a negative sign. Here,

VH — H-tail volume coefficient — (lt /SHT)/(Swc) (12.7)

(introduced in Section 3.20, derived here).

Then, Equation 12.5 without engines becomes:

Cmcg — [Cl w (la /c) + Cmw] Cm_HT + Cmfus

A convenient method is to analyze the effects on aircraft pitching moments of isolated aircraft components. Next, the airplane less the empennage is esti­mated, thereby determining the appropriate H-tail moment required to balance the moments at the cruise condition. Figure 12.10 shows the pitching moment contribu­tion by components of a conventional aircraft (considered mass-less to examine only

continent pitch stability ptch dab Ity van at on vrth CG variation

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.