LONGITUDINAL STATIC STABILITY

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,

Figure 8.2 Forces and moments acting on a wing-tail combination.

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.