AERODYNAMICS OF THE HORIZONTAL TAIL
7 21 Contribution of the Horizontal Tail to the
Aerodynamics of the Whole Airplane
Airplane in straight flight The lift acting on the horizontal tail adds considerably to the pitching moment of the whole airplane because of its large lever arm compared
to the wing chord (see Fig. 71). Let Lя be the lift of the horizontal tail and rH the distance of this lift force from the moment reference axis (usually the lateral axis through the wing center of gravity). Then, from Fig. 75, the contribution of the horizontal tail to the pitching moment of the whole airplane is
MH = – r’„LH (71)
where the noseup pitching moment is taken as positive. Here the contribution of the tangential force of the horizontal tail to the pitching moment has been disregarded because of the small high position of the tail surface relative to the fuselage axis. For the contributions of the horizontal tail to the lift LH and to the pitching moment MH, dimensionless coefficients are introduced through
Lh = сшАнЧн (72я)*
Мн = cMHAc^qm (72Z?)
Here qH is the dynamic pressure at the location of the tail surface. It is, in general, smaller than the dynamic pressure of the undisturbed flow qx because of the effect of the wing on the tail surface. The moment coefficient of the tail surface referred to the wing quantities is obtained from Eqs. (7l)(72h) as
cmh — 
SmAmLk ClH Qoo A cM 
(73e) 

with 
II 
(1сш ( Эая ioBV* ЫнЧ 
(73b) 
The lift coefficient of the horizontal tail сщ depends on, in addition to the geometric data, its angle of attack ccH and the elevator deflection 1?Я (see Fig. 76a). The term dclH/daH represents the lift slope of the horizontal tail without interference, and (0ая/Эря)т? я the change in the direction of the horizontal tail for zero lift caused by the elevator deflection. For the plane problem of the airfoil with control surface (flap), this coefficient has been given as a function of the controlsurface chord ratio; for additional information see Chap. 8.
Generally, the incident flow direction of the horizontal tail is considerably different from that of the wing because the tail surface is strongly influenced by the wing and fuselage and lies in the wing downwash (interference). The incident flow directions of the wing and horizontal tail differ, as shown in Fig. 76я, by the downwash angle aw = w/U^, induced by the wing and fuselage at the location of
*Note that the index l has been chosen for the lift coefficient with reference to the tailsurface quantities.
Figure 75 Contribution of the horizontal tail to the pitching moment (schematic). C. G. = center of gravity of the airplane. W = weight of the airplane.
Figure 76 Aerodynamics of a horizontal tail in straight flight. WJ5 = geometric neutral point, N = aerodynamic neutral point, (N2S )д = geometric neutral point of horizontal tail, (a) Incident flow direction of the horizontal tail, ад = а + Єд + а у/, (b) Aerodynamic forces on the wing and horizontal tail.
the tail surface. Here, w< 0 means downwash and w> 0 means upwash. The angle of attack of the horizontal tail thus becomes
&h — + єн + №w
where Єд is the setting angle of the horizontal tail relative to the wing chord and a is the angle of attack of the wing. Hence the contribution of the horizontal tail to the pitching moment at zero elevator deflection becomes
where r’H is the distance of the neutral point of the horizontal tail from the
(76)[33] 
moment reference axis of the airplane. The change of the moment with the angle of attack at fixed setting angle of the horizontal tail (stability coefficient) is then obtained as
The quantity
is termed the efficiency factor of the horizontal tail. For the moment change with setting angle of the horizontal tail at constant angle of attack, Eq. (75) yields
(78)
Comparison of Eqs. (76) and (78) shows that the moment change with angle of attack (stability contribution of the horizontal tail) depends on the interference between the wing and the horizontal tail. It is proportional to the efficiency factor of the horizontal tail, дан/да = (1 + 9aw/3a). The efficiency factor of the horizontal tail is generally considerably less than unity, as will be shown more accurately later. The moment change with setting angle of the horizontal tail (control), however, is not affected by the interference if the ratio ЯнІЯ<*> is disregarded.
To establish the contribution of the horizontal tail to the lift of the whole airplane, it is advantageous to define the lift coefficient of the horizontal tail, in analogy to Eq. (72b), as
Lh – clhM
In analogy to Eq. (75),
(710)
Here the comments made in connection with Eq. (75) apply also to the derivatives of сїн with respect to a and єн.
In the investigations made so far of the contribution of the horizontal tail to the pitching moment and the lift of the whole airplane, the respective coefficients have been established as functions of the angle of attack of the airplane and the setting angle of the horizontal tail. For some problems it is more favorable, however, to establish the contribution of the tail surface to the angle of attack and to the pitching moment as a function of the lift coefficient of the whole airplane and of the setting angle of the horizontal tail.
The lift coefficient of the whole airplane is composed of that of the airplane without the horizontal tail Ci0H, and the contribution of the horizontal tail cLH, that is, Ci = cLoH + cLff. Hence the lift slope of the whole airplane, without consideration of the effect of the tail surface on the wing at fixed setting angle of the horizontal tail, is obtained from Eq. (710) as
The sought change of the angle of attack with the lift coefficient of the whole
The change of the angle of attack with the setting angle of the horizontal tail zH at constant lift coefficient of the whole airplane becomes
(712)*
Here the second factor is given by Eq. (710).
Like the wing alone, the whole airplane has a neutral point, that is, a point on which that portion of the lift force of the whole airplane acts that is proportional to the angle of attack (compare Sec. 133). As shown in Fig. 76b, let the distance of the neutral point of the whole airplane from the neutral point of the airplane without the tail unit be designated as xNH. This distance is identical to the neutralpoint displacement caused by the horizontal tail. This neutralpoint displacement xNH, as shown in Fig. 76b, can be determined from the moment equilibrium about the neutral point N0H of the airplane without the tail unit xjyHL — rHN^H – Here rHN is the distance of the neutral point of the horizontal tail from the neutral point of the airplane without a horizontal tail. The result is
Introducing Eqs. (710) and (711) into this equation finally leads to the neutralpoint displacement caused by the horizontal tail,
(713)
In this equation, the first fraction on the righthand side determines the percentage of the tailsurface lever arm by which the airplane neutral point is shifted rearward relative to the neutral point of the airplane without a horizontal tail. In many cases, the second term of the sum of the denominator can be disregarded in comparison with the first term. The neutralpoint position of the whole airplane is obtained from the neutralpoint position of the wing alone (Chaps. 3 and 4), from the neutralpoint displacement caused by the fuselage (including the wingfuselage interference, Chap. 6), and from the neutralpoint displacement caused by the tail surface as given above.
The change of the moment coefficient of the horizontal tail as a function of the lift coefficient from Eq. (129) is obtained immediately from the neutralpoint displacement caused by the horizontal tail in the form
with bcJbEfj – ЭсЕЯ/Э£Я
Finally, the change of the moment with the setting angle of the horizontal tail may be determined for a fixed lift coefficient of the whole airplane. This is a free moment because the total lift remains constant. As shown in Fig. 766, MH = —t’hnLh, where LH is the lift of the horizontal tail caused by the change of the setting angle of the horizontal tail, and r’HN is the distance between the neutral points of the horizontal tail and of the whole airplane. Thus, observing Eq. (710) and with rHN = rHN — xNH, the following relationship is obtained:
(715)
This relationship is also valid for (dcMldeH)CLconst. The two coefficients of Eqs. (714) and (715) can be taken from Fig. 726, the first as the difference of the slopes of the curves cM(c£) with and without tail surface and the second from the curves with different setting angles zH of the horizontal tail.
To evaluate the above equations for the contribution of the horizontal tail to the lift and the moment, attention must be paid to the ratio qHlq„. Special attention must be paid, however, to the angle of incidence aH of the horizontal tail, because it depends strongly on the interference between the wing and the tail surface. The angle of incidence of the horizontal tail aH is decisively affected by the induced downwash angle aw < 0 caused by the wing at the location of the horizontal tail [see Eq. (74)]. In Fig. 1la, the change of aH with the rearward position of the tail surface is shown under the assumption of a horizontal tail chord parallel to the wing chord (eh = 0). At the wing trailing edge, aH = 0 because here the kinematic flow condition requires that a + aw = 0. With increasing distance
from the wing trailing edge, aH increases and assumes a constant value at a large distance that is considerably smaller than a. The distribution of aw and thus of aH behind the wing can be computed with wing theory. This matter will be discussed below.
The lift slope of the horizontal tail dc[HjdaH for a horizontal tail free of interference may be determined with wing theory.
Airplane in pitching motion So far only the influence of the airplane angle of attack on the aerodynamics of the horizontal tail has been considered. In addition, however, the rotational motion of the airplane about the lateral axis is particularly important for the aerodynamics of the horizontal tail. During rotation of the airplane about the lateral axis with angular velocity ojy, an angleofincidence distribution aH of the horizontal tail is created as shown in Fig. 1lb that increases linearly with distance from the axis of rotation. This angle of incidence at the location of the horizontal tail (threequarter point) becomes, with the distance from the axis of rotation rH,
(716a)
into Eq. (736) and the resulting formula into Eq. (73a), the change of the moment coefficient with the pitching angular velocity is obtained, with as
(717)
This coefficient is termed the contribution of the horizontal tail to the pitch damping. Comparison of this formula with Eq. (76) shows that the contribution of the horizontal tail to the stability is proportional to (Ая/А)(^/см), and that to the damping is proportional to {AHjA){r’HICfj)2.