Tail Design

6.1 Preliminary tail design

At the present stage of the design process all the characteristics of the wing and fuselage have been determined with only the size and configuration of the horizontal and vertical tails remaining unknown. The tail surfaces are primarily trim, stability, and control appendages and methods for estimating the horizontal and vertical tail sizes based on stability considerations are presented. The two most important parameters are the tail planform areas and the moment arms through which the lifting forces generated by the tail surfaces act. Therefore, initial estimates of the size, shape, and location of the horizontal and vertical tails are needed to carry out the suggested design methods by which those estimates may be refined. Here is another situation where the detailed market survey data are valuable. It is important to determine the following wing and tail surface and location characteristics from the market survey information:

Here S is the wing area and c_MAC is the mean aerodynamic chord (MAC) of the wing. We assume that the aerodynamic centers of the wing and tail surfaces are located at the quarter-chord points of their respective mean aerodynamic chords. The properties listed above will be used to form the characteristic parameters for the tail surfaces and help guide the design process. It will become apparent in subsequent sections that the appropriate parameters for the horizontal and vertical tails are given by

(6.1)

(6.2)

These ratios are called the volume coefficients of the horizontal and vertical tails, respectively, because their numerators and denominators have the units of volume. The forces developed by the tail surfaces are proportional to their planform areas and the moments those forces produce are proportional to the distances through which they act. It should be clear that for a given wing, that is, a given S, b, and c_MAC, larger volume coefficients indicate more effective tail surfaces.

6.1.1 Tail surface characteristics

The horizontal tail surface area is obtained by reasonable extrapolation of the leading and trailing edges in to the fuselage centerline, as is done for the wing. Because the vertical tail is not a symmetric figure its area is defined somewhat more arbitrarily, but generally it involves extrapolation of leading and trailing edges in to the fuselage centerline. It should be noted that sometimes the said extrapolation is extended only to the plane of the horizontal tail, so care must be taken in collecting data on tail areas. Detailed dimensional data for tail surfaces of market survey aircraft are best obtained by working from scaled three-view drawings so that consistency for all data is maintained. Similarly, the root and tip chords of the tail surfaces should be collected and the respective taper ratios and aspect ratios calculated. In addition, the sweepback angles of the tail surfaces should be noted. Then the mean aerodynamic chords of all surfaces may be determined using the definitions provided in Section 5.1. The aerodynamic centers for the various surfaces may be extracted, or at least estimated, from the market survey aircraft as being situated at the quarter-chord point of their mean aerodynamic chords. It should be obvious from the above that scaling dimensions from three-view drawings will be necessary. Such drawings may often be found in the websites of the manufacturers under the heading of technical data for airport operations.

Turbofan-powered airliners that cruise at high subsonic Mach numbers (0.7 < M < 0.9) have horizontal and vertical tail surfaces that are swept back at angles greater than that of the wing in order to make their effective moment arms as long as possible and to maintain their critical Mach numbers higher than that of the wing. Airfoils for the tail surfaces have thickness ratios smaller than that of the wing as an additional aid in keeping the critical Mach number of the tail surfaces higher than that of the wing. The thinner sections are practical because they save weight and because of the smaller aerodynamic loads they experience. Turboprop-powered airliners cruise at lower Mach numbers (0.5 < M < 0.6) and have minimal compressibility problems so the wing and horizontal tail generally have little or no sweepback. On the other hand, the vertical tails are usually swept back in order to increase the vertical tail moment arm. The airfoils used on both the horizontal and vertical tail surfaces are generally symmetrical sections so as to produce the same force magnitude for a given angle of attack in the positive or negative direction.

Some of the features of the horizontal and vertical tail surfaces for a representative range of airliners are collected in Tables 6.1 and 6.2. The aircraft are listed in ascending weight order; the first three are regional turboprops, the next four are regional turbofans, the next three are narrow-body jetliners, the next three wide-body jetliners, and the last two jumbo jets. These tables show that horizontal tail area ratios are typically in the range of 0.2 < S_h/S < 0.35, with an average of around 0.25. The vertical tail area ratios tend to be somewhat smaller, lying in the range of 0.15 < S_v/S < 0.30, with an average of around 0.21.

Aspect ratios of horizontal and vertical tail surfaces are shown as a function of takeoff weight in Figure 6.3. Note that horizontal tail aspect ratios lie in the range 3.5 < A_h < 5.5, with an average of 4.5, while vertical tail aspect ratios are considerably smaller, lying in the range 1 < A_v < 2.5, with an average of 1.6. This difference arises because the horizontal tail continually supplies a positive or negative lifting force and therefore benefits from a higher aspect ratio which reduces its lift-induced drag. On the other hand, the vertical tail is essentially unloaded in non-maneuvering flight and therefore incurs no induced drag penalty. A reduced aspect ratio may then be employed resulting in a smaller vertical tail span and therefore a stiffer structure. The sweepback angles of the horizontal and vertical tails are shown in Figure 6.4 and illustrate that the vertical tails for the turbofan aircraft have sweepback angles about 5°–10° larger than the horizontal tails, with both larger than those of their respective wings. The turboprop aircraft tend to employ large sweepback on the vertical tail and little or none on their horizontal tail.

The taper ratios for the horizontal and vertical tail surfaces are shown in Figure 6.5 where it is seen that they tend to be the same for both. It is also clear that taper ratios are smaller for the heavier jetliners and have greater variability for the lighter aircraft.

6.1.2 Preliminary tail sizing

To initiate the design process for the tail surfaces a first approximation to the size of the horizontal and vertical tail surfaces must be made. The tail surfaces are located at the aft end of the fuselage so that the moment arms l_h,ac and l_v,ac for the horizontal and vertical tails ultimately depend upon the location of the wing on the fuselage. In the subsequent sections we will show that stability requirements require that the wing be located within some precise range of distances from the airplane’s center of gravity. However, the position of the center of gravity depends heavily upon the location of the wing. In Chapter 8 we will carry out a refined weight analysis and then proceed to present a method for locating the wing properly on the fuselage. At this point it is sufficient to rely on past practice as demonstrated by the market survey aircraft.

The approximations l_h = 0.49l_f and l_v = 0.45l_f are fairly representative of actual values for a range of commercial airliners. Because the fuselage length l_f of the design aircraft has been determined, as were the span and mean aerodynamic chord of the wing, approximate values for the moment arms l_h and l_v may be determined. One may use these approximations for the tail moment arms to define approximate volume coefficients given by

(6.3)

(6.4)

The range of values for the approximate horizontal and vertical tail volume coefficients V′_h and V′_v for the representative airliners of Tables 6.1 and 6.2 is illustrated in Figures 6.6 and 6.7. Note that there is considerable variability for the smaller aircraft and much less for the larger aircraft.

Selecting target values for V′_h and V′_v using both the market survey aircraft information and Figures 6.6 and 6.7 as a guide permits determination of reasonable values for the tail areas S_h and S_v. These values are important for laying out the preliminary shape of the tail surfaces in terms of span, sweepback angle, taper ratio, and airfoil section along the lines suggested previously in this section. With the configuration for the tail surfaces now selected they may be located on the fuselage previously designed. Using the aerodynamic centers for the wing and tail surfaces along with their approximate moment arms the wing now may be provisionally located on the fuselage. A three-view drawing of the entire aircraft, excepting the landing gear configuration, may now be drawn. The overall configuration of the aircraft provides the dimensional information necessary for carrying out the refined design of the tail surfaces discussed in subsequent sections.

Additional refinement of the tail design, taking into account some basic stability and control requirements, is described in the subsequent sections of this chapter. The inclusion of additional constraints on the design permits the final size, shape, and location of the tail surfaces to be developed through a series of iterations aimed at satisfying these constraints. If time considerations preclude further calculated refinements to the tail configuration developed thus far, then the remaining material in the chapter may be used to provide some ad hoc refinements to finalize the tail design.

6.2 Refined horizontal tail design

In the simplest analysis we may separate the aircraft into two parts: the wing-fuselage-vertical tail and the horizontal tail. We are interested in the longitudinal characteristics of the aircraft, that is, the pitching motion of the aircraft in its plane of symmetry, that is, about the y, or spanwise axis, as shown in Figure 6.8. To facilitate analysis we will consider the various components of the aircraft to act independently and will account only for the most important features of any mutual interference between them. The center of gravity location indicated is that of the complete aircraft.

6.2.1 Equilibrium conditions

The aircraft, as shown in Figure 6.8, is in equilibrium in straight and level flight. This requires that there be no net force or moment acting on it. The equilibrium of forces in the vertical direction is

(6.5)

As discussed in Section 5.7, the contribution of the fuselage to the lift is neglected with respect to that produced by the wing. The moment acting around the center of gravity of the wing-body-tail M_cg is given by

(6.6)

The lift and moment about the aerodynamic center of the wing are denoted by L_w and M_ac,w, respectively, the lift and moment about the aerodynamic center of the tail are denoted by L_h and M_ac,h while the moment contribution of the fuselage about the center of gravity is shown as M_f. In Equation (6.6) the moment of the horizontal tail about its aerodynamic center is neglected with respect to the moment produced by the lift of the tail acting through the moment arm l_h. Note that moments about the spanwise, or y-coordinate, are taken as positive in the nose-up direction. Dividing Equation (6.5) by qS and Equation (6.6) by qSc_MAC permits the equations to be cast in coefficient form as follows:

(6.7)

(6.8)

Equation (6.8) may be rearranged and written as

(6.9)

In Equation (6.9) we note the appearance of the horizontal tail volume coefficient V_h of Equation (6.1) and the introduction of the horizontal tail efficiency η_h which is given by

(6.10)

The tail efficiency η_h accounts for the possibility that the horizontal tail will see a somewhat reduced velocity V_d,h due to the downwash generated by the lifting wing, particularly at higher angles of attack of the wing. The influence of induced downwash on the velocity and the angle of attack seen by the horizontal tail will be discussed subsequently. The horizontal tail volume coefficient V_h expresses the relative strength of the moment the horizontal tail can produce.

The first trim condition, that the total lift equals the aircraft weight, as expressed in Equation (6.7), may be written as

Note that the lift of the horizontal tail is negative if the wing produces a lift greater than the aircraft weight. This can be the case when the aircraft needs to not only be trimmed but also be statically stable, as described in the next section.

The pitching moment coefficient of the wing, which appears in Equation (6.9), is shown in Appendix C to be given by

Here x_cp denotes the location of the center of pressure of the wing, where the lift could be considered to act without producing a pitching moment. For brevity we have introduced

The second trim condition, that the moment about the center of gravity equals zero, transforms Equation (6.9) to

Here, for consistency, we have introduced

Then the lift coefficient required from the wing to trim the aircraft is

The displacements of the aerodynamic center and the center of pressure from the center of gravity, and , respectively, are much smaller than the normalized tail moment arm l_h/c_MAC so we may approximate the trimmed lift coefficient of the wing as follows:

For a given weight and flight condition the total lift coefficient C_L is fixed and therefore, the sign of C_m0 and C_m,f influences the lift required of the wing. If the sum of the two is negative the lift required from the wing is increased which in turn increases the induced drag. In general the zero-lift moment coefficient of the wing is the determining factor. As pointed out in Chapter 5, C_m0 for untwisted wings is generally negative, and more so for those using supercritical airfoils. Twisted wings tend to have less negative and often even positive values for C_m0 and thereby pay less of a trim drag penalty.

Thus there is a trim penalty which is primarily associated with the zero-lift moment coefficient of the wing and the fuselage pitching moment coefficient. We have tacitly assumed that the drag and thrust pass through the center of gravity of the aircraft and therefore make no contribution to the pitching moment. The mounting of the engines and the drag associated with a large vertical T-tail, for example, may require assessment as the design progresses. Similarly, propeller slipstream effects can influence the longitudinal stability; see, for example Wolowicz and Yancey (1972).

6.2.2 Trim and longitudinal static stability

The aircraft can, in general, be trimmed, that is, put into an equilibrium state where the combined lift of the wing and the tail balances the weight while the moment about the center of gravity is zero. But the question remains as to whether the equilibrium so achieved is statically stable. The aircraft is said to be statically stable if the response of the aircraft to a disturbance in angle of attack is to tend to return to the original equilibrium position. Thus, if flying in equilibrium at one angle of attack and a disturbance increases the angle of attack, the moment produced at this angle of attack must act to reduce it, that is, tend back toward the original equilibrium state. Conversely, if a disturbance decreases the angle of attack, the moment at the new, lower, angle of attack must serve to increase that angle. In other words, the rate of change of the moment about the center of gravity must be negative for static stability: an increase in α should reduce C_m,cg and a decrease in α should increase C_m,cg.

The general trend of the moment coefficient for an airplane with and without a horizontal tail is illustrated in Figure 6.9 along with the trim points where the net moment, as well as the net force, is zero. However if we displace the aircraft from these trim points by a small positive angle of attack, that is with the nose rising, the aircraft with a tail will have a negative pitching moment which will tend to drive the nose back down to the trim point. Thus the aircraft with a horizontal tail is statically stable in pitch. Conversely, the aircraft without a horizontal tail will experience a positive pitching moment which will tend to drive the aircraft further from equilibrium. In general, a tailless aircraft is unstable in pitch.

Taking the derivative of Equation (6.9) with respect to α yields

(6.11)

The moment of the wing makes no contribution to Equation (6.11) because the moment coefficient of the wing about its aerodynamic center is independent of angle of attack. The first term on the right-hand side of Equation (6.11) will be negative and contribute to longitudinal static stability if x_ac > x_cg, that is, if the center of gravity of the aircraft is forward of the aerodynamic center of the wing. This situation requires a tail down force (L_h < 0) so that the wing lift must be greater than the weight L_w > W to trim the aircraft. Increasing the wing lift increases the induced drag and this increase is called trim drag and is considered a penalty. The contribution of the fuselage is destabilizing because as the angle of attack increases the moment produced is positive, or nose-up. However, the fuselage contribution is generally small and therefore should not upset the overall stability of the airplane.

6.2.3 The stick-fixed neutral point

It is possible now to find the aerodynamic center of the aircraft, or, as it’s usually called, the neutral point, by setting the derivative of the moment coefficient with respect to angle of attack in Equation (6.11) equal to zero. In this case we assume the control stick is fixed so that the horizontal tail acts without any deflection of the control surface, that is, without deflecting the elevator. The center of gravity location which satisfies this condition is called the neutral point, denoted by x_n. The resulting expression is

(6.12)

The coefficients of wing lift and horizontal tail lift may be expressed, respectively, as follows:

(6.13)

(6.14)

Because horizontal tail airfoils are generally symmetric sections we may set α₀ = 0 in Equation (6.13). On the other hand, the wing lift-induced downwash angle ε has been introduced in Equation (6.14) to account for a reduced angle of attack seen by the horizontal tail, as shown in Figure 6.10. The term i_h represents the geometric incidence angle of the horizontal tail with respect to the chord line of the wing. Many jet transports have the capability to adjust the geometric incidence of the horizontal tail. Then, in terms of the lift curve slopes of the aircraft and the tail, a and a_h, respectively, and the downwash angle ε, Equation (6.12) becomes

(6.15)

We may estimate how far the neutral point of the airplane is from the aerodynamic center of the wing by examining the second and third terms on the right-hand side of Equation (6.15). The second term may be written as

(6.16)

Here we have assumed that l_h ∼ b/2, used the relations for aspect ratio and mean aerodynamic chord from Section 5.1, and the definitions of wing lift curve slope from Section 5.3. It is further assumed that for jet transports A ∼ 8 and λ ∼ 1/3 in order to arrive at the final result in Equation (6.16).

Analysis of the contribution of the fuselage is complicated by the fact that at angle of attack the flow over the forward portions is reasonably well described by potential theory, but the aft portions are strongly influenced by cross-flow separation and related viscous effects. The nature of the flow over a typical fuselage shape at angle of attack is shown in Figure 6.11. The notional pressure distribution over the fuselage shape shown in Figure 6.11 suggests that the load on the fuselage is essentially a pure couple. This is the reason that the fuselage contribution to the lift is not shown in Figure 6.8 and was neglected in Equation (6.5).

The classical result for a symmetric body of revolution obtained from inviscid small perturbation theory is

(6.17)

The quantity υ_f denotes the volume of the fuselage which is approximated using a relation suggested by Torenbeek (1982). Typical values of fineness ratio for commercial jet transports are in the range 8 < l_f/d_f < 11 and therefore the coefficient in square brackets in Equation (6.17) lies between 1.2 and 1.3.

It is fairly common to see a similar form which was proposed by Gilruth and White (1941):

(6.18)

In Equation (6.18) however, the coefficient K_f depends upon the location on the fuselage of the quarter-chord point of the wing in percent l_f, rather than on the fuselage fineness ratio l_f/d_f as in Equation (6.17). A curve fit to Gilruth’s data is

(6.19)

This gives agreement to within ±3% of Gilruth’s data for quarter-chord wing locations x_c/4 from 20% to 60% of the overall fuselage length l_f. For typical jet transports where the wing position is between 40% and 45% l_f the coefficient K_f lies in the range of 1 < K_f < 1.3. This compares well with the constant given by potential flow theory in Equation (6.17).

Hopkins (1951) proposed an approach which uses potential flow theory forward and cross-flow drag considerations aft. A semi-empirical relation based on a number of experimental results determined the axial location at which the calculation would switch. The approach gives reasonably good results but typically underestimates the pitching moment obtained in experiments. Among the experiments, which dealt primarily with airship shapes, were two which closely approximated the configuration of typical jet transport fuselages, having fineness ratios of 7.9 and 10.1. In addition, these two models had moments measured about locations that also approximate those of jet transports, 0.456l_f and 0.485l_f, respectively. The experimental results for these two cases were in the range

Potential flow results for these two models overestimated the stability derivative of the fuselage by 25–30%. The experimental results put the coefficient K_f in Equation 14 in the range 0.76 < K_f < 0.9 for 8 < l_f/d_f < 10. Therefore both Equations (6.17) and (6.18) should yield similar results. Then we may write

(6.20)

Here, we made the same estimates for the typical parameter values for jet transports that were used in developing Equation (6.16). Now Equation (6.15) has the following estimated form:

(6.21)

In Equation (6.21) K_f is approximately equal to unity so the second and third terms on the right-hand side are small and therefore their difference is smaller still. Thus the location of the neutral point of a conventional jet transport is fairly close to that of the aerodynamic center of the wing. We shall see that the center of gravity will lie between these two points and therefore the moment arm l_h is approximately equal to l_h,ac.

6.2.4 The stick-fixed static margin

In the development thus far, the controls have been held fixed, that is, no elevator deflection is employed. We found the neutral point and estimated that it is approximately coincident with the aerodynamic center of the wing. The derivative of C_m,cg with respect to angle of attack in Equation (6.11) may be written as

(6.22)

When the center of gravity is at the neutral point Equation (6.22) becomes

(6.23)

Subtracting Equation (6.23) from Equation (6.22) yields

(6.24)

The quantity h is the normalized distance to the aircraft center of gravity measured from the leading edge of the mean aerodynamic chord, while h_n is the normalized distance to the neutral point measured from the leading edge of the mean aerodynamic chord. The quantity h − h_n is called the static margin. It expresses how far the center of gravity of the airplane is forward of the neutral point, expressed as a fraction of the mean aerodynamic chord. The larger h − h_n, the more stable the aircraft and the less maneuverable it is. However, with greater stability the pilot workload decreases because fewer control inputs are required to keep a particular course. The airplane is said to be stiffer as the static margin increases. Typical commercial aircraft have a static margin of around 5–10%. When the static margin is zero the airplane is neutrally stable, while if the static margin is negative the aircraft is statically unstable. Modern fighter aircraft employ relaxed static stability (RSS) in order to achieve higher maneuverability, but this requires a flight control system which senses motions and uses redundant computers to provide stabilizing control inputs thus relieving the pilot of a heavy and continuous workload.

6.2.5 Estimate of horizontal tail area based on a stability requirement

The horizontal tail of an aircraft, in conjunction with other aerodynamic components, power plant, and weight characteristics, determines the longitudinal stability and control characteristics of an aircraft. In the following analysis for determining horizontal tail area only one of several stability requirements is enforced. Thus the horizontal tail area so found may be insufficient to meet other stability and control requirements. Here we present a procedure which determines the horizontal tail area S_h needed to produce a controls-fixed neutral point at a position aft of the center of gravity (CG). The aft CG position is determined after the refined weight analysis in Chapter 8 and should be placed at about 30%c_MAC. The preliminary determination of the horizontal tail volume coefficient of Equation (6.3) was used to define an initial layout of the proposed aircraft and this layout will ultimately change somewhat once the center of gravity location is found using the methods described in Chapter 8.

The refined analysis presented below includes only the most important contributions of the airplane components to the stability of the whole. Effects of the vertical position of both the horizontal tail and the CG are neglected except in the determination of the downwash derivative at the tail. The effect of power will be handled with an empirical shift of the neutral point. Other characteristics will be found for the power-off case. In addition, aerodynamic interference effects are neglected. More detailed analyses may be found in Etkin and Reid (1995) and Stengel (2004), among others.

The basic moment equation relating center of gravity position, neutral point, and moment coefficient is Equation (6.24) which may also be written as

(6.25)

The quantity h is the CG location, h_no is the power-off neutral point location, and Δh_np is the shift in neutral point location due to power (typically around 0.03); all locations are given in fractions of the mean aerodynamic chord c_MAC. The power-off, controls-fixed longitudinal stability derivative is denoted by while a_w is the wing lift curve slope . Then with the center of gravity at its most rearward point, say 0.3c_MAC, the quantity h_no + Δh_np = 0.3 and the stability would be neutral. Any further rearward movement of the center of gravity would cause the aircraft to become statically unstable. The contribution of the wing, fuselage, and tail to the stability derivative may be written as follows:

(6.26)

Note that the effect of nacelles could be included in the fuselage derivative term if more detail is desired. The wing contribution alone is given by

(6.27)

Here the quantity h_ac is the location of the aerodynamic center of the wing given as a fraction of mean aerodynamic chord. The longitudinal stability derivative of the horizontal tail is

(6.28)

The tail efficiency η_h may be taken to be about 95% unless the tail is completely out of the wing and fuselage wake, where it can be taken as 100%. Using Equations (6.25)–(6.27) we may determine the horizontal tail area to be given by

(6.29)

Now it is necessary to describe how the various terms in Equation (6.29) may be evaluated. The lift curve slope of the wing and this may be found using Equation (5.18), which is presented in units of per radian. As discussed previously, a reasonable estimate for the fuselage contribution may be found from slender-body theory, and is given in Equation (6.17).

To find , the downwash derivative at the tail, a simplified vortex theory is used so as to permit analytic solutions to the equations involved. It accounts for the following wing characteristics: aspect ratio, sweepback angle, taper ratio, and dihedral angle. Both the vertical and longitudinal positions of the horizontal tail are also included. The simplified analysis typically yields good estimates for conventional transport aircraft configurations, but may be inadequate for unusual tail assemblies.

We assume that the vortex sheet from the wing trailing edge is completely rolled up into two trailing tip vortices but partial roll-up is accounted for in the span of the tip vortices as shown in the flight photograph in Figure 6.12. A detailed discussion of wing theory which describes the trailing vortex system appears in Appendix C.

The maximum value of the downwash variation due to fully rolled up tip vortices at large downstream distances is given by

(6.30)

This value occurs in the plane normal to the plane of symmetry of the aircraft-trailing vortex system. For typical aft tail locations the contribution to the downwash due to the bound wing vortex must also be taken into account. The contribution of the wing is then given by

(6.31)

Here the quantity l_h,2 is the distance (taken parallel to the longitudinal axis) from the aerodynamic center of the horizontal tail to the trailing edge of the wing root chord, as shown in Figure 6.13 and the term x₀ may be found from the following equation:

(6.32)

Then the total maximum downwash derivative on the centerline of the plane of the tip vortices is

(6.33)

The result given by Equation (6.33) would be the maximum value of the tail downwash derivative when the plane of the horizontal tail is coincident with the plane of the trailing vortex system. For other vertical positions y_h above the extended root chord line of the wing, the magnitude of the tail downwash derivative is smaller and is given by

(6.34)

The correction factor for tail heights above the extended root chord line involves the height of the quarter-chord point of the tail measured from the plane of the trailing vortices y′ which is given by

(6.35)

Here l_h,3 is the longitudinal distance from the quarter-chord line of the wingtip to the quarter-chord point of the tail mean aerodynamic chord and γ is the dihedral angle of the wing as shown in Figure 6.14. The dihedral angle is related to stability in rolling motion of the aircraft and will be discussed subsequently. All the angles in Equation (6.35) are measured in radians and the other quantities are depicted in Figure 6.15. The second factor in the correction term of Equation (6.31) is the tip vortex span b_v which is given by

(6.36)

For a wing with taper ratio λ, at far downstream distances where the tip vortices are completely rolled up, the span between them is given by

(6.37)

Once again, the angle Λ_c/4 in Equation (6.37) is measured in radians. Note that the tip vortices move closer to each other in the far field and the distance between them is about 75% of the wingspan. The distance from the wingtip quarter-chord point c_t/4 to full roll-up of the tip vortices is given by

(6.38)

All the previous information may be used to calculate the various terms in Equation (6.29) from which the horizontal tail area may be determined.

6.3 Refined vertical tail design

The vertical tail provides stability in sideslip, or weathervane stability, by developing a yawing moment which moves the aircraft nose back to the original heading after it has been displaced to one side by a small angle β, as indicated in Figure 6.16. In the coordinate frame of the aircraft the free stream velocity V is equal and opposite in sense to the velocity of the center of gravity of the aircraft. In this situation the vertical tail produces a side force Y_v and a moment Y_vl_v acting at the aerodynamic center of the vertical tail. The wing-body combination produces a yawing moment N_wb about the center of gravity due to the asymmetrical nature of the sideslip flow. Note that the yawing moments are considered positive in the clockwise direction shown.

The asymmetry of the flow field suggests that it may have some influence on the effectiveness of the vertical tail as indicated by the shaded areas in Figure 6.17 which denote the regions of interference of the wake of the wing-body combination with the vertical tail. Note that the sideslip may occur while the aircraft is at some angle of pitch, as it would be in takeoff or landing, and this expands the interference region in the vertical direction, affecting more of the vertical tail. The fuselage itself is destabilizing in sideslip, just as it was in pitch, while the wings, which experience different drag levels, provide some reduction in the instability of the wing-fuselage combination. A second feature of the asymmetry of the flow field is the generation of a rolling moment by the side force of the vertical tail which must be counteracted by the asymmetric deflection of the ailerons, as depicted in Figure 6.18.

6.3.1 Equilibrium conditions

Taking moments about the center of gravity for a small angular displacement β, as depicted in Figure 6.16, results in the following:

(6.39)

In coefficient form Equation (6.39) becomes

(6.40)

Note that the convention in stability and control analyses is to normalize the yawing and rolling moments by the product qSb, unlike the pitching moment normalization which uses qSc_mac. Then, taking the derivative of the moment about the center of gravity with respect to the angle of sideslip yields

(6.41)

For stability the change of the yawing moment with respect to angular displacement must be positive, that is, the yawing moment must increase as the deflection β increases. The contribution of the wing alone to the yawing moment coefficient of the wing-body combination is small compared to that of the fuselage alone because the thin wing is moving in its own plane and therefore we assume that _. The moment coefficient of the vertical tail about its aerodynamic center is independent of β so that and Equation (6.40) becomes

(6.42)

In Equation (6.42) we introduce the vertical tail efficiency η_v which is given by

(6.43)

The vertical tail volume coefficient V_v, given in Equation (6.2), expresses the relative strength of the moment the vertical tail can produce and is repeated here for convenience:

The vertical tail efficiency η_v accounts for the possibility that the vertical tail will see a somewhat reduced velocity V_s,v due to the sidewash generated by the fuselage and the wings, particularly at higher angles of attack, as discussed at the start of this section and illustrated in Figure 6.17. The influence of induced sidewash on the velocity and the angle of attack seen by the vertical tail will be discussed subsequently.

As shown in Figure 6.18, downward aileron deflection on the port (left) wing increases the lift (L_w,p) while upward aileron deflection decreases the lift (L_w,s) on the starboard (right) wing. Symmetrical placement of the ailerons makes the aileron lift moment arm l_a the same for both. The resulting rolling moment equilibrium may be expressed as follows:

The total lift L = L_w,p + L_w,s = W and putting the side force in coefficient form and dividing through by qSb and accounting for possible differences in dynamic pressure seen by the port and starboard wings q_w,p and q_w,s yields

(6.44)

The vertical tail sizing approach presented in the following section requires the aircraft to be able to provide sufficient yawing moment to restore the aircraft to its original heading after a sideslip. Equation (6.44) can provide information on the aileron effectiveness only after the vertical tail has been sized. Therefore aileron sizing will be determined by the resulting vertical tail sizing. The allowable aileron span and associated moment arm l_a are limited by the flap span and location required for landing. To make full use of the wingspan, jet transports often carry outboard ailerons for roll control at low speeds where aerodynamic loads are relatively low and inboard ailerons for roll control at high speeds where aerodynamic loads are high.

6.3.2 Trim and lateral static stability

The aircraft will tend to remain headed in the desired direction in the face of disturbances, like wind shifts, that produce small sideslip angles if the vertical tail is large enough to ensure that the moment about the center of gravity is zero. From a stability point of view, the right-hand side of Equation (6.42) must be positive, so that the vertical tail contribution must be larger than the fuselage contribution, including engine nacelles. With β increasing, as shown in Figure 6.16, we see that the moment produced by the vertical tail is positive. The fuselage response is essentially the same as for pitching motion, so using Equations (6.18) and (6.20) we obtain

(6.45)

From Figure 6.16 we note that the stability derivative for the fuselage in sideslip requires a negative sign because the moment coefficient produced by the fuselage, which corresponds to a negative yawing moment, tends to increase the angle of sideslip, rather than reduce it. The vertical tail, like the horizontal tail, is in a disturbed flow field as shown in Figure 6.17. The sidewash felt by the vertical tail is analogous in effect to the downwash experienced by the horizontal tail, and the angle of attack of the vertical tail is not simply the undisturbed sideslip angle β, but the sum of β and a small sidewash angle. This is usually represented as . Thus, the variation of the side force coefficient of the vertical tail with sideslip angle may be written as

(6.46)

Equation (6.46) is similar in form to the stability derivative for the horizontal tail given in Equation (6.28). The downwash on the horizontal tail was amenable to analytic treatment but the asymmetric sidewash is more complex so that we will have to treat the sideslip problem with a more accurate empirical approach. Assuming there is no control surface deflection, that is, the rudder is not used, Equation (6.42) may be rewritten as

(6.47)

Solving Equation (6.47) for the area of the vertical tail yields

(6.48)

Using some estimates of typical characteristics of jet transports in Equation (6.48) suggests that

(6.49)

Torenbeek (1982) indicates that for jet transports the stability derivative, in our notation, lies in the range

For the smaller value of the applied moment about the center of gravity, 0.1, the vertical tail area would need to be about 0.14S, which is near the low end of the values for jet transports. However, for the larger value of the moment about the center of gravity, 0.25, the vertical tail area would need to be about 0.23S, which is near the high end of the values for jet transports.

Although an airplane with a vertical tail is stable to small disturbances, an important criterion for the vertical tail size is its ability to handle the one-engine-out condition. This situation is depicted in Figure 6.19. This is especially important for aircraft with wing-mounted engines, as was discussed briefly in Chapter 5 with respect to locating the engines on the wing. The vertical tail must provide sufficient turning force to balance the yaw moment due to the asymmetric thrust produced when one engine is inoperative. For this situation Equation (6.40), with the assumptions made thus far, may be written as

(6.50)

Consider the port engine to be shut down so that only the starboard engine is producing thrust at the level F_1eo, as shown in Figure 6.19. This thrust, commanded by the pilot from the n_e − 1 operating engine(s), defines a new equilibrium speed where the total drag of the airplane, including the drag of the inoperative engine, matches the new total thrust level. The moment coefficient produced by one engine being inoperative, as appears in Equation (6.50), may be written as

(6.51)

Here y_eo denotes the spanwise position of the inoperative engine and F_eo represents the new thrust level produced by the engine at the same spanwise position on the other wing. In the case of one engine out on an aircraft with four wing-mounted engines the approach is the same because two engines will still be operating symmetrically. This is also true if one wishes to consider the more stressing case of two engines out on the same wing, as long as one combines the thrust of the two engines as if they were one. The total thrust is equal to the drag D developed at the new speed in the one-engine-out configuration. Setting the moment about the center of gravity in Equation (6.50) equal to zero, inserting the result from Equation (6.51) for the asymmetric thrust, and solving for the required vertical tail area yields

(6.52)

An evaluation of Equation (6.52) for typical jet transport parameters yields the results shown in Figure 6.20, which also includes data for some modern jet transport aircraft. In this analysis a low-speed stressing case is assumed: takeoff configuration with an engine failure and a corresponding drag coefficient of 0.06, while the sideslip angle is taken as 0.1 radians. The two calculated curves shown have only one difference: the rudder power characterized by the angle of zero-lift shift Δ_v. It is clear that the vertical tail itself may have insufficient control authority to keep the aircraft stable without a significantly larger surface area. However, deflecting the rudder provides additional side force so as to keep the vertical tail surface area at lower values. It can be seen that current aircraft can maintain a margin of safety with reasonable vertical tail size because of the rudder effectiveness.

6.3.3 Horizontal and vertical tail placement

In Equation (6.48) it is clear that the lift curve slope of the vertical tail a_v directly influences the required vertical tail surface area. From the study of wings in Chapter 5 it is equally clear that the lift curve slope is directly dependent on the aspect ratio of the wing. Considering the vertical tail as a wing we are struck first by the rather small aspect ratio of that surface taken as an isolated body. The aspect ratio in this case is simply defined as A_v = b_v²/S_v where these quantities are defined in Figure 6.21. As indicated in Figure 6.3, vertical tail aspect ratios for jet transports are typically in the range 1 < A_v < 2.

The DATCOM method described by Hoak et al. (1978) for estimating the side force coefficient C_Y clearly indicates that the lift curve slope is enhanced by the presence of a fuselage at the base of the vertical tail and that a T-tail also improves the lift curve slope. This is the so-called endplate effect where a surface bordering the vertical tail helps reduce three-dimensional tip effects which reduce the lift curve slope. One may think of the difference between testing a finite wing in a wind tunnel as opposed to a wing that completely spans the tunnel.

Three basic tail configurations are shown in Figure 6.22: (a) the low tail, common to most large jet transports (W_to > 100,000 lb), (b) the cruciform tail, which is rarely seen on jet transports, and (c) the T-tail which is common on smaller jet transports (W_to < 100,000 lb), particularly those with fuselage-mounted engines. The cruciform tail is rarely encountered because it provides the least enhancement of the natural lift curve slope of the vertical tail and it is structurally complex. The T-tail is common on aircraft with fuselage-mounted engines because the jet exhaust would interfere catastrophically with a low tail setting and because it provides a substantial increase in the effective aspect ratio of the vertical tail alone, developing an A_eff up to 2.8A_v. The low set tail is structurally sound and provides much the same improvement in lift curve slope as the T-tail, up to 2A_v, while avoiding the complications of a cantilevered structure highly loaded at its tip.

An empirical equation for the vertical tail side force stability derivative is given in DATCOM as

(6.53)

The quantity k is a coefficient that is a function of the ratio b_v/2r₁ where b_v is the vertical tail span and 2r₁ is the depth of the fuselage in the vicinity of the vertical tail. The definition of both these parameters is rather arbitrary but b_v is often taken as the distance from the fuselage centerline measured vertically to the tip of the tail and 2r₁ is often taken as the diameter of the fuselage near the junction of the leading edge of the vertical tail with the fuselage. The magnitude of the ratio b_v/2r₁ is indicative of the endplate effect of the fuselage on the vertical tail. In the range 0.5 < b_v/2r₁ < 3.5 the fuselage has a beneficial effect on the vertical tail to the extent that its effective aspect ratio A_v,eff is from 20% to 60% greater than the isolated tail value, defined as A_v = b_v²/S_v. At larger values of the ratio there is little or no benefit of the presence of the fuselage. The factor k in Equation (6.53) has the following behavior:

(6.54)

Typical commercial transports have vertical tails that fall in the middle range 2 < b_v/2r₁ < 3 with a representative value being about 2.4.

The combined effect of sidewash angle and local dynamic pressure is given by DATCOM as follows:

(6.55)

The quantity z_w,c/4 is the perpendicular distance from the wing root quarter-chord point to the fuselage centerline. The remaining quantity in Equation (6.53) is the lift curve slope of the vertical tail, considering it as a wing. The lift curve slope for a wing, given in Equation (5.18), depends upon the aspect ratio of the wing and we have just indicated that endplate effects due to the presence of the fuselage can alter the geometric value of the aspect ratio of the vertical tail. The position of the horizontal tail in the x, z plane can also contribute to endplate effects. In addition, the relative size of the vertical and horizontal tail influences the effective aspect ratio. In order to account for these effects DATCOM presents three graphs that permit estimation of the effective aspect ratio of the vertical tail.

The effective aspect ratio of the vertical tail is given by

(6.56)

The term A_v,f /A_v is the ratio of the aspect ratio of the vertical tail in the presence of a fuselage to that of an isolated vertical tail panel which is considered to be the extension of the vertical tail leading and trailing edges to the fuselage centerline as illustrated in Figure 6.21. The quantity A_v,f /A_v is shown in Figure 6.23 as a function of the span to local body diameter b_v/2r₁; the taper ratio of the vertical tail λ_v = c_v,r/c_v,t appears as a parameter. It is clear that the effective aspect ratio for a vertical tail can be from 20% to 60% larger than its geometric aspect ratio. Indeed, for commercial airliners where b_v/2r₁ is around 2, the effective aspect ratio A_v,f /A_v ∼ 1.6. However, this effect does not carry over to the case of conventional fuselage-wing combinations. Because the wingspan is much larger than the local fuselage diameter, the endplate effect of the fuselage on the wing is negligible.

The coefficient K_h in Equation (6.56), which accounts for the relative size of the horizontal and vertical tails, is illustrated in Figure 6.24. For commercial airliners the ratio of horizontal to vertical tail area is around 1.2 so that the coefficient K_h ∼ 1. The term A_v,fh/A_v is the ratio of the aspect ratio of the vertical tail in the presence of both a fuselage and a horizontal tail to that of the isolated vertical tail panel. This quantity is shown in Figure 6.25 as a function of z_h/b_v, where z_h is the location of the horizontal tail normal to the fuselage centerline as shown in Figure 6.21. The parameter x/c_v appearing in Figure 6.25 is the ratio of the longitudinal distance from the leading edge of the vertical tail to the aerodynamic center of horizontal tail normalized by the chord of the vertical tail measured at z_h. For aircraft with low horizontal tails z_h/b_v is relatively small, and from Figure 6.25 we see that 1 < A_v,fh/A_v < 1.3. For T-tails where z_h/b_v = 1 we see that A_v,fh/A_v = 1.7. For the low tail case we would find moderate improvements of about 10–20% in the effective aspect ratio of the vertical tail A_v,eff. For the T-tail the improvement would rise from 50% to 70%. The improvement for the T-tail comes at some cost because the aerodynamically loaded horizontal tail would place substantial loads on the vertical tail and it would require strengthening that would entail added weight. The decision to use a T-tail is usually predicated on other factors, for example, to ensure that the horizontal tail won’t be immersed in the jet exhaust from turbofan engines mounted on the aft fuselage.

6.3.4 Example calculation of vertical tail stability derivative

A wind tunnel model consisting of a fuselage, wing, horizontal tail, and vertical tail has wing area S = 576 in.², quarter-chord sweepback Λ_c/4 = 30°, and aspect ratio A = 6. The wing root quarter-chord point is located on the body centerline so that z_w,c/4 = 0. The vertical tail uses an NACA 63-006 airfoil and the effect of adding this vertical tail to the fuselage-wing-horizontal tail combination is sought. The details of the tail assembly are illustrated in Figure 6.26.

From Figure 6.26 the ratio b_v/2r₁ = 2.72 and λ_v = 0.16. Entering Figure 6.23 with these values suggests A_v,f /A_v = 1.47. Again using the information in Figure 6.26 we may calculate S_v = 153.7 ft² and A_v = b_v²/S_v = 1.51, as well as S_h = 121.5 in.², A_h = 4.14, and λ_h = 0.5. Using Equation (5.6) for the mean aerodynamic chord leads to c_MAC,v = 11.86 and c_MAC,h = 5.62. The aerodynamic centers for the horizontal and vertical tails are shown at their respective quarter-chord points in Figure 6.26.

The ratio S_h/S_v = 0.79 and using Figure 6.24 we select K_h = 0.82. From Figure 6.26 we know that x/c_v,r = 0.8 and that z_h/b_v = 0. Then, using this information in Figure 6.25 yields the approximate value A_v,fh/A_v,f = 1.28. Then Equation (6.56) may be solved for the effective aspect ratio of the vertical tail as A_v,eff = 2.71. Using this result for the effective aspect ratio of the vertical tail in Equation (5.18) for the lift curve slope gives

The experimental value for the lift curve slope of the vertical tail, which has an NACA 63-006 airfoil, is given as a = 0.112 per degree or 6.42 per radian in Table 5.1. The corresponding value of κ = a/2π = 1.021. The sweepback of the mid-chord of the vertical tail is shown in Figure 6.26 to be Λ_c/2 = 41.9° so that for incompressible flow (M = 0) the lift curve slope is found to be a_v = 2.76 per radian, while for M = 0.8 the lift curve slope is a_v = 2.93 per radian.

The ratio of vertical tail area to the given wing area is S_v/S = 0.267 and z_w,c/4 = 0 so that Equation (6.55) may be solved to yield

We may use this information in Equation (6.53) to find the change in side force coefficient due to the addition of the vertical tail to the wing-fuselage-horizontal tail combination. The constant k in that equation may be found from Equation (6.54) to be k = 0.87 so that

Then assuming η_v = 0.95 we find per radian for M = 0 and −0.69 per radian for M = 0.8.

6.4 Design summary

At this stage of the design a preliminary configuration of the design aircraft may be completed with specific dimensions, except for the landing gear, which will be treated in the next chapter. The fuselage dimensions are known from Chapter 3 and the wing dimensions and characteristics are known from Chapters 4 and 5. The horizontal and vertical tail sizes and characteristics required for basic static stability are determined either by using the simple empirical approach of Section 6.1 or, preferably, by using the more refined analyses leading to Equations (6.29) and (6.52). The sizing of all these components is necessary for carrying out the refined weight estimate described in Chapter 8. Indeed, the final placement of the wing on the fuselage which defines the location of the center of gravity for the aircraft will be accomplished in Chapter 8 after the refined weight analysis is carried out. However, in order to carry out the tail sizing a nominal center of gravity location is required. Therefore, at this point it is sufficient to follow the suggestion of Torenbeek (1982) and take the approximate center of gravity to be about 42–45% of the fuselage length for aircraft with wing-mounted engines and about 47% of the fuselage-length for aircraft with aft-fuselage-mounted engines.

6.5 Nomenclature

A aspect ratio

a lift curve slope

b wingspan

b_v span of trailing vortices

C_D drag coefficient

CG center of gravity

C_L lift coefficient

C_m pitching moment coefficient

C_n yawing moment coefficient

c_MAC mean aerodynamic chord

C_r side force coefficient

d diameter

h normalized center of gravity location, x_cg/c_MAC

h_n normalized neutral point location, x_n/c_MAC

h_no normalized power-off neutral point location

i incidence angle

K_f fuselage coefficient, Equation (6.19)

K_h horizontal tail coefficient, see Equation (6.56)

k constant

L lift

l length

l_a moment arm for lift increment due to aileron deflection, see Figure 6.13

l_h longitudinal distance between aerodynamic centers of the wing and horizontal tail

l_h,2 see Figure 6.13

l_h,3 see Figure 6.15

l_v longitudinal distance between aerodynamic centers of the wing and vertical tail

M pitching moment

N yawing moment

n_e number of engines

p pressure

q dynamic pressure

r₁ local body radius, see Figure 6.21

S projected wing area

S_h projected area of horizontal tail

S_v projected area of vertical tail

V free stream velocity

V_h horizontal tail volume coefficient = S_hl_h/Sc_MAC

V_v vertical tail volume coefficient = S_vl_v/Sb

W weight

x longitudinal distance from aircraft nose

x₀ Equation (6.30)

Y side force

y spanwise distance from centerline

y′ Equation (6.35)

y_h see Figure 6.13

z_h see Figure 6.21

z_v moment arm for vertical tail side force, see Figure 6.18

α angle of attack

β sideslip angle

Δh_np shift of neutral point due to power application

ε downwash angle

γ wing dihedral angle

λ taper ratio

η_h horizontal tail efficiency q_h/q

η_v vertical tail efficiency q_v/q

Λ sweepback angle

υ volume

ς sidewash angle

6.5.1 Subscripts

ac aerodynamic center

cg center of gravity

cp center of pressure

c/4 quarter-chord

d downwash

eff effective

f fuselage

g gross

h horizontal tail

n neutral point

p port (left) side of aircraft

s starboard (right) side of aircraft

ru vortex roll-up

v vertical tail

w wing

wb wing-body combination

α derivative with respect to angle of attack α

0 zero lift

eo one engine out

References

1. Etkin B, Reid LD. Dynamics of Flight: Stability and Control. NY: Wiley; 1995.

2. Gilruth, R.R., White, M.D., 1941. Analysis and Prediction of Longitudinal Stability of Airplanes, NACA Report 711.

3. Hoak, D.E., et al., 1978. USAF Stability and Control DATCOM, Flight Control Division, Air Force Flight Dynamics Laboratory, Wright-Patterson AFB.

4. Hopkins, E.J., 1951. A Semiempirical Method for Calculating the Pitching Moment of Bodies of Revolution at Low Mach Numbers, NACA RM A51C14.

5. Stengel R. Flight Dynamics. Princeton, NJ: Princeton University Press; 2004.

6. Torenbeek E. Synthesis of Subsonic Airplane Design. Dordrecht, The Netherlands: Kluwer Academic Publishers; 1982.

7. Wolowicz, C.H., Yancey, R.B., 1972. Longitudinal Aerodynamic Characteristics of Light, Twin-Engine, Propeller-Driven Airplanes, NASA TN D-6800.