Figure 12.8. V-tail contribution to roll

12.3.4 Summary of Forces, Moments, and Their Sign Conventions

Given below is the summary of sign convention in the three planes.

12.4 Theory

Forces and moments affect aircraft motion. In a steady level flight (in equilibrium), the summation of all forces is zero; the same applies to the summation of moments. When not in equilibrium, the resultant forces and moments cause the aircraft to maneuver. The following sections provide the related equations for each of the three aircraft planes. A sense of these equations helps in configuring aircraft in the conceptual design phase.

12.4.1 Pitch Plane

In equilibrium, 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

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 (Mac cg = 0):

Mac cg = (Nc × lc + Cc × zc + Mc)canard + (Nw × lw + Cw × zw + Mw )wing

+ (Nt × lt + Ct × zt + Mt )tail

+ Mf us + Mnac + (thrust × zth + nac drag × 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, η, 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 )(lc/c) + CCc(Sc/Sw )(zc/c) + Cmc(Sc/Sw )]for canard

+[CNw (la /c) ηw + CCw (za /c)ηw + Cmw ηw]for wing

+[CNt (St /Sw )(lt /c)ηt + CCt (St /Sw )(zt /c)ηt + Cmt (St /Sw )ηt]for tail

where:

1.η(= qi /q∞) represents the wake effect of lifting surfaces behind another lifting surface producing downwash, qi is the incident dynamic head, and q∞ 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.

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:

Normal forces now can be resolved in terms of lift and drag; for small angles of α, 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) + Cmw

and the second term:

CNt (St /Sw )(lt /c)ηt + Cmt (St /Sw )ηt ≈ CLt (St /Sw )(lt /c)ηt + Cmt (St /Sw )ηt

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

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