Category AERODYNAMICS, AERONAUTICS, AND FLIGHT MECHANICS

Up to this point we have been concerned with “stick-fixed” longitudinal static stability. The pitching moment as a function of a has been determined assuming that the necessary force was applied to the control stick to hold the elevatdr (or other types of control surfaces) at its trimmed position. Suppose, however, that the pilot does not hold the stick fixed. Sometimes, in a cross-country flight, one likes to be able to trim the airplane and sit back and relax. The degree to which one can relax depends on the airplane’s stick-free static stability.

As we will see shortly, freeing the stick has the effect of changing the slope of the tail lift curve. Thus, the results that have been obtained for the stick-fixed case will still hold if an effective a, is used.

The stabilator and the stabilizer-elevator configuration will again be considered separately, since there are basic differences in the analyses. The final results from the two analyses will, however, be similar.

Elevator-Stabilizer Configuration

This is the easiest of the two tail configurations to treat. If the stick force is zero, it follows that the hinge moment of the elevator must be zero. From

Equation 8.37,

Or Se, the floating elevator angle, is related to a, and S, by

The lift coefficient of the tail is given by

CL, = aAatt + т 8)

with the small contribution from 8, being neglected. With 8e related to a„ CL, becomes

Thus,

(8.41) where Fe is called the free elevator factor and is defined by

(8.42)

Normally, as seen by a previous example, bx and b2 are both negative for the stabilizer-elevator combination. Thus Fe is typically less than one. For the earlier example, bx = -0.33 and b2 – -0.69. CJC was given as 0.37, so that r would equal approximately 0.58. For this particular tail the value of Fe becomes 0.72.

Again, the Cherokee 180 will be used as an example, and I emphasize that the numbers are in no way endorsed by the manufacturer. The tail dimensions and angular movements were measured by students during a course on techniques of flight testing for which the Cherokee 180 was used. Being an easy and forgiving airplane to fly, it is a good vehicle for such a course.

The horizontal tail of the Cherokee, shown in Figure 8.16, is of the type pictured in Figure 8.7c. This type of tail is sometimes called a “stabilator.” The gain between the elevator and the horizontal stabilizer, ke, as well as the gearing, G, can be determined from the graphs presented in Figure 8.16. The stick position and angles shown here are arbitrary. They were obtained by moving the control wheel to a given position and measuring the distance from an arbitrary reference point on the wheel to an arbitrary reference point on the instrument panel. At each position, a protractor with a bubble level was placed on the horizontal stabilizer and then on the elevator to obtain the angle of these surfaces relative to the horizontal. Using these graphs, you should be able to determine that

ke = -1.50 G = +0.5 rad/ft.

The following quantities are estimated from the tail geometry assuming the airfoil to be symmetrical with the aerodynamic center at 0.25 c and a section lift curve slope equal to 0.106C//deg.

a, = 0.0642CJdeg (Equation 3.74)

t = 0.55 (Figure 3.32)

г) = 0.80 (Figure 3.33)

-0.185 (Figure 3.31)

AC;

For the symmetrical airfoil with 5, = 0, CMac = 0, so CM at the pivot line, which is 1.7% of the chord behind the C/4 point, will be

Cm = 0.017a, a,

Thus,

b, = 0.0625/rad

The increment in CLl due to the elevator deflection will be

A CL, = а, тт) 8e = 1.616 5,

with 5, in radians.

The increment in CM about the pivot line produced by 5, will equal the sum of the increment in CMac and the moment produced by the increment in

CL acting ahead of the pivot.

ДСМ = -0.185(1.616) Se + 1.616(0.017)5,

= 1.616 5Д-0.185+ 0.017)

Thus, it follows that

b2 = -0.271/rad

The following numbers are typical of the Cherokee 180 with four pas­sengers, full-fuel, but no baggage.

W = 2255 lb

h = 0.197 (dimensionless location of center of gravity)

For ea of 0.447, h„ = 0.442, so that

Cm„ = CLa(h — hn)

= 4.50(0.197-0.442)

= – 1.10/rad

Also,

Cm, = ViVh^i

= (1)(0.384)(3.68)

= 1.41

The constant, A, in Equation 8.34 becomes

A = 0.0476

The stick force, from Equation 8.35, becomes

Р = С5Л(|)л[і-(0]

= (0.5)(25)(2.5)(14.1)(0.0476)[l –

■2o4‘-(0]

The estimated stick force, P, is presented in Figure 8.17 as a function of airspeed for different trim speeds.

Reference 8.3, similar to Reference 5.5, is a comprehensive collection of data relating to aerodynamics, structures, and performance. If one requires an extensive treatment of the effects of airfoil thickness, aerodynamic balance, trailing edge angle, and other geometric parameters on the hinge moment coefficient, this reference should be consulted. The many geometric features that affect the aerodynamic performance of an elevator are pictured in Figure

8.10. These are all considered to some extent by the reference.

For illustrative purposes, a limited amount of data is presented here in Figures 8.11 to 8.15, based on Reference 8.2 and also Reference 8.3. Figures

© Gap

(© Seal (or lack of)

(J) Aerodynamic balance, balance ratio, BR

© Nose shape

(б) Trailing edge bevel

(6) Control horn balance

(7) Spanwise extent of elevator

(5) Total tail aspect ratio and taper ratio (¥) Ratio of ce to c

<b)

8.11 and 8.12 present factors kx and k2, from which b{ and b2 in Equation 8.37 can be obtained. These curves actually represent deviations from a norm defined by the following factors.

1. 10% thick, symmetrical 2-D airfoil.

2. The ratio of elevator chord-to-airfoil chord (cjc) equals 0.3.

3. No aerodynamic balance.

4. A round nose on the elevator.

Each curve in these figures represents the effect of varying one parameter while keeping the others at their nominal value. For the nominal case,

b = —0.55 b2=- 0.89

Thus,

b,.-OMk0y(LjuBR)kt{L)

b2 = -o. m2(^y2(ty2(BR)k2fy

For example, suppose the following is given for a horizontal tail.

b.

b

| = ».25 b

BR = 0

The notation is a little confusing, since the subscript t, denoting “tail,” has been dropped but added again to denote “tab.”

For this case, from Figure 8.11,

k2(BR) = 1.0

Ці) = 0.73

Thus, b and b2 are estimated to be

bx = -0.55(1.16X0.49)

= -0.31

b2 = -0.89(1.05X0.73)

= -0.68

The effect of the tab on the hinge moment is obtained from Figure 8.13. For a two-dimensional airfoil,

^=-0.83

к

к = 0.97 b3 = -0.81

Since most trim tabs are located near the midspan of the elevator, the correction for aspect ratio is assumed to be negligible. However, since the tab extends over only 25% of the tailspan, the value of b2 is reduced in propor­tion. Thus,

b2 = -0.20

For this example, it follows that

CH = -0.31a – 0.685, – 0.205,

The angles a, Se and 5, are in radians.

The foregoing relationships assume linear relationships between CH and the angles a, Se, and 8,. As the material relating to flaps in Chapter Three showed earlier, linear aerodynamics holds only up to some combination of flap angle and angle of attack, beyond which flow separation occurs. An estimate of these limits can be obtained from Figure 8.14 (taken from Ref. 8.4). This graph is for a NACA 0009 airfoil having a plain flap and operating at a Reynolds number of 3.41 x Ю6. This value of R is typical for the horizontal tail of a light to medium aircraft at landing speeds.

Figure 8.14 is not applicable to trim tabs. However, the reference notes on conventional control surfaces (elevator with trim tab), a satisfactory maximum for tab deflection exists between the angles of+/-15and +/—20° for

moderate flap deflections. Thus, for a constant tab chord, it is better to use a large-span tab’deflected to a small angle than a short-span tab deflected to a large angle.

The combination of the vertical and horizontal tails on an airplane, that is, its complete tail assembly, is called the empennage. On some airplanes the empennage consists of a vertical tail mounted at each tip of the horizontal tail. These “twin tails” act as end plates to the horizontal tail and effectively increase its aspect ratio. An estimate of this increase can be obtained from Figure 8.15. The effective aspect ratio equals the geometric value of A divided by the factor r. For example, a tail with A = 4.0 would have a lift curve slope of approximately 0.06CJdeg. If end plates having a height-to-span ratio of 0.4 were put on this tail, its effective aspect ratio would increase to 6.7. Thus a, would increase to approximately 0.073, an increase of 22%. Although possibly not as great, one would also expect a similar improvement in the effectiveness of the elevator.

The relationship for stick force and stick force gradient will now be developed for the horizontal tail configuration of Figure 8.7b. Such a configuration is trimmed either by changing the incidence of the horizontal stabilizer (so in a sense it is not fixed) or by deflecting a small flap, known as a trim tab, on the trailing edge of the elevator. The particular method of trim is not pertinent to the present development.

Assuming a symmetrical airfoil section, the elevator hinge moment coefficient will be written in the same form as Equation 8.32

8, refers to the trim tab, as shown in Figure 8.9. Following the same procedure as for the all-movable tail, it can be shown that

F’= (t)a + [fi2 ~ b)ihs + b}8t~ в]™у2 (838)

Cl„ (Cm6Clb — СцаСц) ЬгСіСи

If iks or 5, is adjusted to trim P to zero at a trim speed of FTR, Equation 8.28 becomes

зЬИІМ’-ШІ <8-з9>

The rate of change of stick force with speed will be

dP 2GSece(WIS)A dVra Ftr

In the case of an all-movable tail, the moment, H, can be expressed in terms of a moment coefficient, CM, about the pivot. For a control surface that is hinged to a surface ahead of it, the “hinge” moment is usually expressed in terms of a hinge moment coefficient, CV

H=pV2ScCh

S refers to the planform area of the control surface (such as the elevator), and c is the mean chord of the control surface.

Let us now consider the stick force (“stick” is used by most pilots and engineers as synonymous with control wheel) for a movable tail with a linked tab. For simplicity a symmetrical airfoil will be assumed, so that the moment coefficient can be written as

Cm, — ba, + b2 Se

The notation, bt and b2, is borrowed from Reference 8.2 and avoids the use of triple subscripts in denoting the partial derivatives. Obviously,

The constant b can be positive or negative, depending on the location of the pivot relative to the aerodynamic center. b2, on the other hand, is usually negative.

To allow the stick force to be trimmed to zero, a constant term, 50, is added to the linked elevator angle so that

Se кelhs So

Substituting Equations 8.31, 8.32, and 8.33 into Equation 8.27 gives P

GqS, c,

Replacing ihs by Equation 8.26 and using

WIS

results in

(8.35)

The gradient of the stick force with velocity at a particular trim speed is found by differentiating Equation 8.35 with respect to V and letting V = VTr.

dP 2 GS, c,(WIS)A

dVr r Vtr

For a given trim speed a positive P, that is, a pull on the control, should result in a decrease in the speed. Thus, for the proper feel, the constant, A, in Equation 8.35 should be positive. This is referred to by FAR Part 23 as a stable stick force curve and is a requirement for all conditions of flight.

For obvious reasons, in the design of an airplane’s control system, the stick or control wheel forces must lie within acceptable limits throughout the operating envelope (V-n diagram) of the airplane. In addition, the variation of these forces with airspeed about any trim point should be such as to give a proper “feel” to the pilot. Generally, this means that a push forward on the longitudinal control should be required to increase the airspeed, and a pull should be required to fly slower.

With regard to the longitudinal control forces, FAR Part 23 allows a maximum of 60 lb for a temporary application to a stick and 75 lb for a wheel.

A prolonged application is not allowed to exceed 101b for either type of control.

Gearing

The control force, P, is directly proportional to the control surface hinge moment, H.

P = GH (8.27)

G is referred to as the gearing. To determine G, refer to the sketch in Figure 8.8. Here a schematic of a control linkage is shown between a stick and an elevator. If the system is in equilibrium,

(8.28)

where the stick force, P, and the hinge moment, H, are shown in their positive directions. Now allow the stick at the point of application of P to move a distance s in the direction of P. In so doing the elevator will rotate through an angle S„ given by

s

LI,

Substituting this into Equation 8.28 gives

(8.29)

Comparing Equation 8.29 to Equation 8.27 gives

This result is general and independent of the details of the linkage. Since Se is negative for a positive stick displacement, it follows that G is positive. It can also be obtained simply by noting that if the system is in equilibrium throughout its displacement, the net work done on the system is zero. Hence,

Ps + HSe = 0

from which Equation 8.30 follows.

Equations 8.19 to 8.23 apply to any of these configurations if i, is taken to mean the incidence angle of the tail’s zero lift line. To pursue this further, we know from Chapter Three that the lift coefficient for a flapped airfoil can be

expressed as

С/ = a(a + tS)

Here, t is the actual value of the flap effectiveness value and equals т obtained from Figure 3.32 multiplied by 17 from Figure 3.33. Thus, referring to the tail configuration of Figure 8.7b, if the elevator is deflected downward through the angle 8e, the incidence angle of the zero lift line is decreased by the amount t 8e, If ihS denotes the incidence angle of the fixed horizontal stabilizer, then the incidence angle of the tail zero lift line will be

it = ihs — t 8e (8.24)

The coefficients CM, and Cl, then become

дСм _ дСм d 8e dit d 8e dit

or

Similarly,

Using Equations 8.22, 8.24, and the above expressions for Cm, and CLl results in

(8.25)

Thus, as a function of CL, Se has the same form as it, except that the slope of Se versus Cl will be negative, since the positive rotation of the elevator is opposite to that for the incidence angle.

Stabilator

Now consider the configuration in Figure 8.7c. Let 4, (which is now variable) and 8e be related by

Using Equation 8.24, i, now becomes

і, = 4,(1 – rke)

This is identical to Equation 8.22, except for the factor 1/(1 – rk3). By varying ke, one is able to alter the rate of change of the tail incidence angle with CL – Usually a negative value of ke is used to make the tail more effective for a given displacement. This increases the tail lift curve slope since, for the linked elevator,

a( 1 – тке)

Control Position as a Function of Lift Coefficient

According to Equation 8.7, the pitching moment coefficient can be written as a linear function of a.

См — Сщ + Cmjx

From Equation 8.10,

Cl — Cl„oc + Cii’t

where CLi = – r),a, S,IS. Substituting for a in the equation for CM and setting the result equal to zero gives

0 — Сщ + (Cl — Сції)

Сщ is given by Equation 8.5.

All-Movable Tail

If we now consider the tail incidence angle, it, to be a variable, we can write

Сщ = Смас + Сщі{ (8.21)

where CMi = r),VHa,.

Substitution of Equation 8.21 into Equation 8.20 results in the following equation for the trim tail incidence angle as a function of the lift coefficient.

Cm Cl + ClCm^ CmPl„ ~ СмСц

Сма is negative for a statically stable airplane and CL) is always negative. См, с is usually negative and CLa is always positive. It therefore follows, if

1,1 c > (h„ – h), that i, as a function of CL is of the form

i, — ACl + В

where A and В are positive constants. Notice that

dit _________ Сма_____

dCL (CMiCLa – CMCL) < J

This allows one to determine experimentally the neutral point for a given airplane. This is done by measuring i, as a function of Cl for different center-of-gravity locations. The slopes of the experimentally determined plots of i, versus Cl are then plotted as a function of center of gravity and are extrapolated to the value of the center of gravity that gives a zero slope. From Equation 8.23, CM„ for this center-of-gravity position is obviously zero. From Equation 8.13, it then follows that this center-of-gravity position corresponds to the neutral point.

The foregoing material applies strictly only to aircraft with all-movable tails. There are two variations on this configuration. The first is the fixed horizontal stabilizer-movable elevator configuration. The other is the movable horizontal stabilizer with a linked elevator (or trim tab). These three configurations are pictured in Figure 8.7. A variant on the configuration of Figure 8.7c is to input the control directly to the tab. Deflection of the tab pro­duces a moment that rotates the rest of the tail. This configuration is referred to as a flying tail. Some persons refer to any all-movable tail as a flying tail.

The estimation of a representative angle at the tail that allows for the downwash from the wing is a difficult task. The vortex system shed from the wing is unstable and rolls up into two trailing vortices, so the model for the trailing vortex system is not well defined. Even if one could circumvent this problem, the downwash in the region of the horizontal tail will not be uniform over the tail. Thus, to use a simple correction to the tail angle of attack leaves something to be desired.

With these reservations in mind, the simple model illustrated in Figure 8.5 is proposed to calculate e«. The wing is replaced by a single bound vortex with a vortex trailing from each tip. As the vortex sheet rolls up, the edge moves in toward the centerline, so that the span between the two trailing vortices, b’, is less than the wingspan. For an elliptic wing, it can be shown that

_ 7Г

b ~ 4

Using the model shown in Figure 8.5 and the Biot-Savart law (Equation 2.64), the graphs of Figure 8.6a and 8.6b were prepared. To obtain ea from these graphs, one first determines the distance, /ac, from the quarter-chord of the tail to the quarter-chord of the wing, b’ is calculated from Equation 8.18 to give /ac/b’. Figure 8.6a is then entered interpolating for hjb’. Note that h, is the height of the tail above the plane containing the wing and parallel to V. The value of ea obtained from Figure 8.6a is then multiplied by the factor presented in Figure 8.6b to correct for sweepback.

With the use of these graphs, an estimate of ea can now be made for the light plane of Figure 3.62. In this case,

/ac = 159 in.

b = 360 in.

S = 160 ft2

Because of the dihedral and at an angle of attack, h, is approximately zero. Using Equation 3.74 and a value for a0 of 6.07/rad leads to a value for a of 4.19/rad. From Figure 8.6a,

= 0.6

ea = 0.447

Because this material is an introduction to the subject of airplane stability and control, a certain amount of elegance will be foresaken for the sake of simplicity and clarity. Most of the derivations to follow will relate to a “basic airplane” consisting simply of a wing and a tail. Analysis of this simple configuration will disclose the basic principles involved in determining the motion of the complete airplane. Following these fundamental developments, the effects of the fuselage and propulsion system will then be considered.

Stick-Fixed Stability

The forces and moments on a wing-tail combination in the longitudinal plane are shown in Figure 8.2. The x-axis in this case is chosen to coincide with the zero lift line of the wing. Relative to this line, the tail is shown nose down at an incidence angle of i,. The positive direction of i, is an exception to the right-hand rule followed for other angles, angular velocities, and moments. Pointing one’s right thumb along the у-axis, the fingers curl in the direction of a positive rotation (but not for /,).

The angle of attack of the wing’s zero lift line is shown as a. At the tail,

however, the angle of attack is reduced by an angle e due to the downwash from the wing. The aerodynamic .center of the wing is located a distance of (h – h„w) c ahead of the center of gravity. Thus h and hnw are dimensionless fractions of the mean aerodynamic chord c and are measured from the leading edge back to the center of gravity and aerodynamic center, respec­tively.

If the “airplane” is in trim, the vector sum of the forces and the moments about the center of gravity will be zero.

2м = о

or

(8.1a)

(8.1b)

Now consider the graph in Figure 8.3, which presents the pitching moment about the center of gravity as a function of a. In trim a must be positive in order to produce lift. Thus M must be zero for a positive a. Now the question is, “Should the slope of M versus a be positive or negative for static pitch stability?” To answer this, suppose the airplane is trimmed at point A when it is disturbed, possibly by a gust, so that its angle of attack is ■ increased to a value of A’. If the slope of M versus a is positive, a positive moment, corresponding to point B, will be generated. This positive moment, being a nose-up moment, will increase a even more, thus tending to move the airplane even further from its trimmed angle of attack. This is an unstable

M

situation, as noted on the figure. On the other hand, if the slope is negative, so that point C is reached, a negative moment is produced that will reduce a to its trimmed value.

Expressing the moment in terms of a moment coefficient, it follows that the requirement for longitudinal static stability is

^<0 (8.2)

da

where M = qScCM-

Stick-Fixed Neutral Point and Static Margin

Denoting dCM/da by CMa, we note that CMa will vary with center of gravity position. There is one particular location for the center of gravity, known as the neutral point, for which CMa = 0. The neutral point can be thought of as an aerodynamic center for the ejitire airplane.

Lw=qSaa

L, = r),qSta, (a ~ ~ “) (8.3b)

Mac = qScCM, c (8.3c)

where

dCLl

dat

Mac, will be neglected as being small by comparison to the other terms in the equation. Usually Mac, will be zero anyway, since most tail surfaces employ symmetrical airfoil sections.

The term de/da is the rate of change of downwash angle at the tail with a and, for a given wing-tail configuration, is assumed to be a constant (although it can vary with the trim condition). From here on it will be denoted by e„. The determination of ea will be discussed in more detail later.

Substituting Equation 8.3 into Equation 8.1b gives

Note that for a = 0,

where Vh is called the horizontal tail volume and is defined by

VH=§4 (8.6)

о c

CM can therefore be written as

См — См0+Сма« (8.7)

where

Сма = (h — hnw)a — 7],atVH(l — ea) (8.8)

In order to find the neutral point, we set CMa equal to zero and h equal to h„. However, the term VH includes l„ which is a function of the center-of – gravity position. From Figure 8.2,

I = h, – h (8.9)

Also, we can write, for the total lift of the wing and tail,

L = qSaa + r),qS, a([a(l – ea) – /,] or

CL = [a + У, Jr at(l – e„)]a – щ, ^a, (8.10)

Thus, the combined lift curve slope is given by

CLa = e + ij,|’a,(l-0 (8.11)

Substituting Equations 8.9 and 8.11 into Equation 8.8 and equating the result to zero gives

fc. _ t Ь(8.12,

Cl JQ

If this result and Equations 8.9 and 8.11 are substituted into Equation 8.8, a surprisingly simple result is obtained for CMa.

CMa = – CLJh„-h) (8.13)

Considering Equations 8.5 and 8.13 in light of Figure 8.3, it is obvious that a trimmed, statically stable airplane must have a positive tail incidence angle and its center of gravity must be ahead of the neutral point.

The quantity (h„ – h) is known as the static margin. It represents the distance that the center of gravity is ahead of the neutral point expressed as a fraction of the mean aerodynamic chord. To assure adequate static longitudinal stability, a static margin of at least 5% is recommended. Let us briefly examine the foregoing for the light airplane pictured in Figure 3.62.

For this wing-tail combination the following quantities are calculated from the drawing.

h, = 2.78
S,/S = 0.153

For the 652-415 airfoil,

CMac = – 0.07
hnw = 0.27

Using Equation 3.74 and a Cta of 0.106/deg,

a = 0.0731/deg
a, = 0.0642/deg

t)(, the ratio of q at the tail to the free-stream q, is assumed to be 1.0. 17, can be greater or less than unity, depending on the fuselage boundary layer and any effects from the propulsion system (such as a propeller slipstream).

Using a procedure to be described shortly, e„ is estimated to be 0.447. Thus,

CLa = 0.0785 CJdeg • hn = 0.443

so that

CMa = -0.0785 (0.443 — h)

There are configurations other than the conventional wing-tail com­bination that are trimmable and are longitudinally statically stable. Since the preceding relationships regarding longitudinal static stability place no res­triction on the relative sizes of the wing and tail, it is easily seen that the smaller surface can be placed ahead of the larger. This is referred to as a Canard configuration. In this case, the rear lifting surface is called the wing, and the forward “tail” is referred to as the Canard surface. For purposes of supplemental control, Canard surfaces are occasionally employed with con­ventional wing-tail arrangements or with delta wings such as the XB-70A, pictured in Appendix A.3. In the case of a true Canard configuration, the center of gravity will be toward the rear, just ahead of the wing.

An unswept cambered wing without a tail can be made stable simply by flying it upside down. In this case, CMac becomes positive so that, with the center of gravity ahead of the aerodynamic center, the resulting aerodynamic moment curve about the center of gravity will appear like the stable curve of Figure 8.3. A swept wing with positive camber can be made stable by washing out the tips. The result will again be a positive value of the Смас for the wing, even though CMac is negative for the wing section. Similarly, the CMac for a delta wing is made positive by reflexing the trailing edges; that is, by turning them up slightly.

CM. c and Aerodynamic Center Location for a Finite Wing

The location of the aerodynamic center and the CMac for a finite wing can be determined precisely by applying the lifting surface methods described in Chapter Three. These quantities will now be determined approximately for a linearly tapered wing. The method should indicate how one would apply a more rigorous approach to the calculations and, at the same time, provide results that are readily usable for approximate calculations.

With reference to Figure 8.4, the line through point A swept back through the angle Л is the locus of the section aerodynamic centers. Therefore the pitching moment about a line through A normal to the chord line will be

(8.14)

If XA is the distance of the aerodynamic center behind the point A, then

Mac = Мд + LXA

Dividing by qSc, this becomes

Differentiating Смас with respect to a gives

0 = dCMA + ^_ da c

Substituting Equation 8.14 and recalling that

results in

1 r2

XA = 7I cClay tan Л dy J-bl2

If it is assumed that C(a is constant the preceding equation can be written in the form

where у is the spanwise distance from the centerline out to the centroid of the half-wing area. Equation 8.15 is applicable to any wing where the section aerodynamic centers lie on a straight line.

For a linearly tapered wing, Equation 8.15 becomes

(8.16)

The mean aerodynamic chord, c, is defined as the chord length that, when multiplied by the wing area, the dynamic pressure, and an average CMac, gives the total moment about the wing’s aerodynamic center. If the CMac is not constant along the span, one has a problem in defining an “average” СМж. We will therefore calculate c according to

(8.17)

This definition of c follows directly from the preceding relationships if the section Q and CMac values are assumed to be constant. For a linear tapered wing, c becomes

c =

where c0 is the midspan chord.