Category AERODYNAMICS, AERONAUTICS, AND FLIGHT MECHANICS

534 STATIC STABILITY AND CONTROL Adverse Yaw

A pilot initiates a turn from straight flight by rolling the airplane in the desired direction of the turn while at the same time applying a little rudder in order to start the airplane yawing in the desired direction. If done properly, a so-called coordinated turn is achieved where the resultant of the gravitational and inertial forces remains perpendicular to the wing. The pilot may feel a little heavier in a coordinated turn but, at least, forces tending to throw him or her to one side or the other are not experienced. A full pitcher of beer will not slosh in a coordinated turn. (Note that FAA regulations forbid the con­sumption of alcohol by a pilot eight hours prior to flying.)

A pair of aileron surfaces, which are simply plain flaps that travel symmetrically when deflected, will usually produce a motion known as adverse yaw. At an angle of attack, a flap deflected downward will produce a drag increment greater than that produced by the same deflection upward. Thus, for example, suppose the right aileron moves up and the left one moves down to initiate a turn to the right. The higher drag on the left aileron will produce a yawing moment that tends to yaw the airplane to the left, opposite to what was desired. This is known as adverse yaw due to ailerons, a characteristic that can make an airplane uncomfortable to fly.

To alleviate adverse yaw, the mechanical linkages are sometimes design­ed so that, for a given control movement, the upward movement on one aileron is greater than the downward movement on the opposite aileron. For example, on the Cherokee 180, the aileron moves up 30° but down only 15°.

Another means of alleviating the adverse yaw is in the design of the aileron. Figure 8.31 illustrates a possible way of accomplishing this. The configuration shown in Figure 8.31, known as a Frise aileron, has the hinge point below the aileron surface. As the aileron is raised, the nose projects down into the flow, thereby increasing the drag on the up aileron.

Lateral Control

In the engine-out example just presented, you may have noticed that the airplane was really not in trim. With the rudder deflected to balance the yawing moment of the engine, a side force is produced that must be coun­teracted by some means. This can be done by rolling the airplane around its x-axis to give a component of the weight along its у-axis. This is shown schematically in Figure 8.27. To carry the preceding engine-out example further, the sum of the forces along the у-axis must be zero for trim.

Thus,

+W(f> = 0

or

-i),qSvavT8r

(8.92)

Thus, to trim the engine-out situation completely, the pilot must have the capability to roll the airplane. This is accomplished by movable control surfaces that are hinged on the outer rear portion of each side of the wing, as shown in Figure 8.28. These surfaces, known as ailerons, move differentially: as one moves up, the other moves downward. Since the ailerons do not necessarily move the same amount, the aileron deflection is defined as the total angle between the two ailerons. For example, if the right aileron moves down 10° and the left aileron moves up 14°, the aileron deflection, Sa, is equal to 24°. Sa is defined to be positive when the right aileron rotates in a positive direction.

It was the incorporation of roll control that distinguished the efforts of the Wright Brothers from those of the other aviation pioneers of their day. Instead of ailerons, however, the Wright Brothers warped their box wing to provide a differential angle of attack from one wing tip to the other. This was improved on by Glenn Curtiss, who developed the hinged aileron, which was a considerable improvement over having to warp the wing. Since they had first recognized the need for complete lateral control and had devised the means to do so, the Wright Brothers brought a legal suit against Glenn Curtiss, claiming that his ailerons infringed on their patent rights. After a lengthy and bitter legal fight, the courts ruled in favor of the Wright Brothers. I have only the greatest respect and admiration for the Wright Brothers, but I question this particular legal decision, since it seems to me that Curtiss’ development of the separate movable aileron control surface was, indeed, a

control system distinct from the wing-warping system developed by the Wrights.

Leaving history and returning again to the technical aspect of roll control, observe that, when an airplane is rolled through an angle, ф, there is no mechanism present to generate an aerodynamic restoring moment. In the preceding cases involving angle of attack and sideslip angle, we encountered the derivatives CMa and CN/j, respectively, that had nonzero values. In the case of the roll angle, however, the corresponding derivative СІф is always zero. Because of this, the concept of roll stability, in the static sense, does not exist. At the most one could say that an airplane possesses neutral static stability in roll.

We define the rolling moment coefficient by reference to the wing area and wingspan, as with the yawing moment coefficient.

L = qSbC, (8.93)

Here we have a real problem with notation, since one tends to think of lift when the letters L or / are used. You will have to be careful and aware of the application of the coefficients in order to make a distinction. Generally, if the lowercase l is being used with reference to the entire airplane, it has reference to the rolling moment.

The rolling moment produced by the ailerons can be estimated by reference to Figure 8.29. As shown, the inboard end of the aileron is located a distance of у! from the wing centerline. The aileron extends from there out to a distance of y2. As a result of the aileron deflection, an increment in the section lift per unit span is produced on the right side and is given by

d(AL) = qca0T SaR dy

This incremental lift results in a differential rolling moment.

dL = – yqca0r SaR dy

Integrating this over the spanwise extent of the right aileron gives

F2

L = – qa0T SaR I су dy ■Li

Since the wing is symmetrical, it is easily shown that a similar contribution to the rolling moment is obtained from the left side, so the total rolling moment is given by

In dimensionless form, С/, becomes

Ci = —a0rSaA j (j^)xdx

where x = yl(bl2).

[3(*22 ~ *i2) – 2(1 – A)023 – де,3)]
12(1 +A)

For a linearly tapered wing this can be integrated to give

This relationship for C( is only approximate, since the increment in the section Ci is assumed to be constant over the aileron. Induced effects associated with a finite lifting surface are neglected. By comparison with experimental measurements (Ref. 8.8), it is obvious that these induced effects are significant and cannot be neglected.

If the aspect ratio of the wing is fairly high, say approximately 6 or higher, then one can use a fairly simple model to correct Equation 8.96 for induced effects. Otherwise, a set of graphs, found in References 5.5 and 8.3, should be used.

When deflected, an outboard aileron will produce an incremental lift distribution something like that sketched in Figure 8.30a. Thus, on either end of the aileron, an incremental vortex system is shed in a manner similar to that shed at the tips of a wing. This incremental vortex system will induce an additional downwash over the aileron, thereby reducing its incremental lift. It is assumed that this reduction is in proportion to the reduction in lift that one would expect from a finite wing having an aspect ratio equal to the aspect ratio of that portion of the wing spanned by the aileron. If this ratio is denoted by Aa, Equation 8.96 is corrected by multiplying it by the correction to the lift curve slope for a wing.

______ Aa______

Аа + [2(Аа + ШАа + 2)

For the case where x2 = 1.0, Aa, A, xt, and A are related by

(1-*,)(!+A) ~1A

1 -Xi +A(1 + Xt)j2

Figure 8.30b presents C(j predicted on the basis of the foregoing rela-

Figure 8.30a Simplified model of an aileron.

tionships. In preparing this figure, it was assumed that: a0~ 0.106C|/deg

7] = 0.80 (correction to t—see Equation 3.49 and Figure 3.33) jc2= 1.0

— = 0.25 c

For other aileron-wing chord ratios, this figure must be corrected for the effect on t. This correction is provided as an insert in Figure 8.30b. In order to determine C|8 for a value of cjc different from 0.25, the value of C(s read from the figure is multiplied by the factor ka. It is interesting to note that Figure 3.33 for г/ results in extreme nonlinearities in т for flap angles beyond about 15 or 20°. Yet aileron data appear linear with 5a up to angles of approximately 30°. This is further evidence that, because of induced effects, the section angles of attack are reduced over the ailerons. Indeed, according to Reference 8.8, as the aspect ratio of the portion of the wing covered by the aileron decreases, the range of Sa over which С/ is linear with Sa increases.

If x2 is less than unity, superposition can be used to calculate C, s. For example, suppose Xi = 0.5 and x2 = 0.9. Then C(s can be calculated from

Again, the subscript a is dropped on the S in writing C, s, since Sa is the primary control for producing a rolling moment.

Directional Control

Control of the yawing moment about the z-axis is provided by means of the rudder. The rudder is a movable surface that is hinged to a fixed vertical stabilizer. The rudder is the vertical counterpart to the elevator, and its effectiveness is determined in the same way. Referring to Figure 8.26, the vertical tail is shown at zero sideslip angle with the rudder deflected positively through the angle Sr. The rudder produces an increment in the vertical tail lift (or side force) that results in an increment in the yawing moment, given by

AN = -1, ALv

The increment in the tail lift is given by

A Lv = r),qSvavT S,

V

Figure 8.26 Rudder control.

so that in coefficient form, the rate of change of CN with respect to the rudder angle becomes

Сщ = ~ViVvavr (8.91)

The subscript r is dropped on S, since it should be obvious that 5 refers to the rudder when considering CN. т is the effective change in the angle of the zero lift line of the vertical tail per unit angular rotation of the rudder and is estimated in the same manner in which т was obtained for the elevator.

As an example in the use of the rudder, suppose the airplane pictured in Figure 8.25 lost power on its right engine. If each engine is located a distance of Ye from the fuselage centerline, the resulting asymmetric thrust would produce a yawing moment about the center of gravity equal to TYe. In steady

trimmed flight the thrust, T, must equal the drag and N must be zero. Therefore,

DYe + qSbCNs 8r — 0

or

Given the airplane and rudder geometry and CD as a function of V, one can then calculate the vertical tail volume necessary to keep 5r within prescribed limits, usually a linear operating range.

LATERAL AND DIRECTIONAL STATIC STABILITY AND CONTROL

We now turn our attention to the static forces and moments that tend to rotate the airplane about its x – and 2-axes or translate it along its у-axis. First consider directional stability and control about the z-axis.

Directional Static Stability

Figure 8.25 illustrates an airplane undergoing a positive sideslip. In this case, the component of its velocity vector along the у-axis is nonzero and positive. The airplane is “slipping” to the right. This results in a positive sideslip angle, /3, being defined as shown. As a result of the velocity vector no longer lying in the plane of symmetry, a yawing moment, N, is produced by the fuselage and by the side force on the vertical tail. The airplane will possess positive static directional stability (sometimes called weathercock stability, after the weathervane) if

Figure 8.25 An airplane having a positive sideslip.

The criterion for longitudinal static stability required the slope of the moment curve to be negative. In this case, the sign is opposite because the yawing moment is opposite in direction to the angle fi.

Generally, the yawing moment from the fuselage is destablizing. However, it is usually small and easily overridden by the stabilizing moment contributed by the vertical tail, so airplanes generally possess static direc­tional stability. However, the vertical tail is not normally sized by any considerations of static directional stability. Instead, the minimum tail size is determined by controlability requirements in the event of an asymmetric engine failure or flying qualities requirements related to dynamic motion.

The yawing moment is expressed in coefficient form by

N = CsqSb (8.88)

The contribution to CN from the fuselage can be estimated on the basis of Equation 8.72. However, in calculating CNfB corresponding to CMaB, the difference in signs and reference lengths must be remembered. CygB will also be of opposite sign to Cl»b.

The contribution of the vertical tail to the static directional stability is formulated in a manner similar to that followed in determining the horizontal

tail’s effect on longitudinal stability. If the aerodynamic center of the vertical tail is located a distance of lv behind the center of gravity, then

Nv = r)tqSJvav[(3( – e„)] or

CN = T,’f ьаЛ1~€^ (8 89)

The vertical tail volume, Vv9 is defined by

о /

X/ = v —

v~ S b

so that Equation 8.89 becomes

С» = v, VvavO – єр) (8.90)

P

av is the slope of the lift curve for the vertical tail. An estimate of this quantity is made more difficult by the presence of the fuselage and horizontal tail. If the vertical tail is completely above the horizontal tail, then it is recommended that an effective aspect ratio be calculated for the vertical tail equal to the geometric aspect ratio multiplied by a factor of 1.6. If the horizontal tail is mounted across the top of the vertical tail (the so-called T-tail configuration), this factor should be increased to approximately 1.9 to allow for the end-plate effect of the fuselage on the bottom of the vertical tail and the horizontal tail on the top. This latter factor of 1.9 is only typical of what one might expect for an average ratio of fuselage to tail size. For a more precise estimate of this factor, one should resort to wind tunnel tests or Reference 8.3 or 5.5.

The sidewash factor, ep, is extremely difficult to estimate with any precision. For preliminary estimates, it can be taken to be zero. However, one should be aware of its possible effects on Nv.

Ground Effect

Downwash velocities are decreased when the airplane is operating close to the ground. Since the ground is a solid boundary to which the normal component of velocity must vanish, its presence alters the streamline pattern that exists around the airplane out-of-ground effect. In order to determine ground effect, an image system representing the airplane’s vortex system is placed below the airplane a distance equal to twice the height of the airplane above the ground. The vortex strength of the image system is of opposite sign to the original vortex system. Thus, midway between the two systems, their induced velocities in the vertical direction will cancel, satisfying the boundary condition along the ground.

An additional graph to estimate the effect of the ground on downwash is not needed in light of this information. Instead, one can again make use of Figure 8.6a. For example, suppose an airplane is operating a height above the ground, h, equal to half of its span. Also, let the distance /(>c equal b and

h, lb = 0.05. Relative to b’,

ft = 0.637b’

/ac = 1.273ft’ ft, = 0.0637 ft’

Relative to the image system the tail is located at,

l, = 1.273ft’

*ac

ft, = (0.637 + 0.637 + .0637)ft’

= 1.338ft’

Thus, from Figure 8.6a, for the image system

ea = 0.06^

This downwash is subtracted from that for the airplane out-of-ground effect which, in this case, is

e„ = 0.527Aa

Thus, the downwash in this case is decreased by 11% due to the ground effect. This decrease will, of course, improve the static stability and change the trim.

Effects of Propulsion System

Propellers

Both propellers and jets can affect longitudinal static stability. Consider first the case of a propeller-driven airplane, shown schematically in Figure

8.20. As illustrated in the figure, a propeller develops not only a thrust, T, directed along its axis of rotation, but also a force, PN, directed normal to its axis. It will be shown that PN is proportional to the propeller thrust, a, and the advance ratio, J, and is directed upward for a positive angle of attack.

Since the thrust is directed along the propeller axis and rotates with the rest of the airplane, its contribution to the moment about the center of gravity does not change with a. Hence Сщ is not affected by T. On the other hand, Сщ is affected by T since, generally, the thrust line will not pass through the center of gravity. If the thrust line lies a distance of Zp above the center of gravity, then Сщ, given by Equation 8.5, is decreased by the amount

—-©(t)

(8.73)

The normal force, PN, affects both the trim, Сщ, Сщ is increased by the amount

and the stability, Сщ.

—Н’ч

II

(8.74)

while Сщ is increased by

Д CM =CN kr

ina Iypa £

(8.75)

PN

Figure 8.20 Effect of propeller forces and slipstream on longitudinal static stability.

516 STATIC STABILITY AND CONTROL where

r ЭСЧ

Cn"°‘ 9 a

In addition to contributing to the forces and moments affecting longi­tudinal motion, a propeller at an angle of attack produces a yawing moment that couples with the lateral-directional behavior of the airplane. Let us now examine the origin of both the normal force and yawing moment. Figure 8.21a shows a side view of a propeller at an angle of attack a. It is seen that a component of the free-stream velocity, Va, lies in the plane of the propeller and is directed upward. Figure 8.21 b is a view in the plane of rotation looking in the direction of flight. Measuring the displacement of a blade from the

(c)

vertical bottom line, it is seen that a blade section experiences a velocity normal to the blade given by the sum of tar and aV sin ф. In Figure 8.21c we view a section of the blade looking in toward the hub. The section lift and drag combine vectorially to produce a thrust per unit blade length of

Since all three of these force components vary with the blade position, ф, average values are obtained by integrating each with respect to ф from 0 to 2ir and then dividing by 2-tr. For example,

where В is the number of blades and R is the propeller radius.

In order to cast the equations in a tractable form, it is assumed that the angle ф is small so that

Vr — о)Г Va sin ф

With this assumption dL/dr and dDIdr become

рЦр = ip(wr – Va sin ф)2сС( (8.77)

= p(a>r – Va sin ф)сС<1 (8.78)

Ci, as well as the local velocity, varies with ф, whereas Cd is assumed to be constant. Using a quasisteady approximation (neglecting unsteady aerodynamics), Ci is written as

С, = а(/3-ф) — а(/3 — tan-1— —A

cor — Va sin Ф/

/3 the blade pitch angle defined in figure 6.9, will be chosen so as to give a constant С/ along the blade when a is zero.

Thus,

The differential, instantaneous pitching, and yawing moments about the y – and z-axes, respectively, will be given by

dMp = r cos ф dr
dNp = r sin ф dr

The foregoing relationships are combined and integrated over ф from 0 to 2ir and over r from 0 to R. The chords c and Cd are assumed constant, so they can be taken outside of the integration. The following results are obtained.

It is convenient to express PN and Np in terms of T and the product TR, respectively. Since the same assumptions are involved in calculating T, PN, and Np, the ratios of PN and Np to T are probably more accurate than the predicted absolute values of these quantities. Also, (aqA) cancels out, so that one needs only to estimate the average lift coefficient of the blades in order to obtain the force and moment ratios.

In steady, level flight, the thrust is equal to the airplane drag. Thus, knowing the drag, propeller geometry, advance ratio, and q, one can calculate the Ci from Equation 8.80. The result is relatively insensitive to Cd, so that a typical value of 0.01 can be assumed for this quantity.

Figure 8.22a and 8.22b presents the derivatives of PN and Np with respect to a in ratio form as a function of advance ratio for constant values of Ct. If the propeller geometry is not known, a reasonable value of C( would probably be around 0.8.

X

co

Figure 8.22c Correction for upwash ahead of a wing. Model of figure 8.5 assumed using an unswept elliptic wing. c0=root chord, x = distance ahead of quarter-chord line.

The direction of PN is independent of the direction of the propeller rotation. However, the sign of the moment, Np, is reversed if the propeller rotates opposite to that shown in Figure 8.21. As derived, all rotations and moments follow the right-handed coordinate system of Figure 8.1.

This method for predicting PNa and NPa is original with this text and is untested. Sufficiently complete systematic data against which to compare the predictions could not be found. An alternate method for predicting PN and Np is offered in Reference 8.13. Graphs based on this reference can be found in other sources, but they seem to be specific to particular propellers, and their general applicability appears questionable. The conclusions of Reference 8.13 may be applied, however, without these graphs. They state that a propeller behaves like a fin having the same area as the average projected side area of the propeller with an effective aspect ratio of approximately 8. The moment produced by the propeller is approximately equal to the lift on the fin multiplied by the propeller radius. “Average” side area means an average area projected in one revolution. This is given approximately by one-half the number of blades times the projected side area of one stationary blade.

In calculating the propeller normal force, one must account for the fact that there is an upwash ahead of the wing that effectively increases the angle

of attack. Thus Equation 8.75 becomes

ACMa = CNpf(l~ea)

This can also be written as

(8.83 b)

which assumes the thrust equal to the drag. Again, ea is the rate of change of downwash with a. Ahead of the wing, ea is negative so that (1 – ea) is greater than unity. This factor is presented in Figure 8.21c. This graph was calculated using the vortex system shown in Figure 8.5. Again, in the final analysis, one should resort to wind tunnel testing of a powered model to determine these effects accurately.

The effect of the propeller slipstream on the wing and tail will not be treated here in any detail. At higher speeds, these effects are small and can usually be neglected. At the lower speeds, however, the increased q in the propeller slipstream can increase both the wing and tail lift. In addition, particularly for V/STOL (vertical/short takeoff and landing) applications, the propeller-induced velocity can become appreciable relative to the free-stream velocity, resulting in significant changes in the section angles of attack of both the wing and tail. For a detailed treatment of those effects, refer to Reference 3.3

Jets

The two principal effects of a jet propulsion system entail the forces developed internally on the jet ducts and the influence of the jet exhaust on the flow field of the tail. The following is based on material in Reference 8.2.

Figure 8.23 depicts schematically the flow through a jet propulsion system. The notation is similar to that of the reference. First, it is obvious that an increment to the trim moment results from the thrust, T, which is independent of a. This increment can be written as

(8.84)

Second, as in the case of the propeller, a normal force is developed by the jet. This force, Nj, is directed normal to the exhaust velocity V, and acts at the intersection of lines parallel to V, and V, through the centers of the inlet and exhaust. This is usually close to the inlet. Applying momentum principles to the system, the magnitude of N, is found to be

Nj = mjV6

where nij is the mass flow rate through the engine. The angle в equals the

angle of attack of the thrust line a minus the downwash angle e, at the inlet. If the inlet is ahead of the wing, e, will generally be negative.

в = a — є,-

From this, it follows that the normal force, N,, contributes an increment to CMa, given by

Equation 8.85 can be evaluated given the characteristics of the engine. The quantity (1 – €jj can be estimated using Figure 8.22 or Figure 8.6.

The influence of the jet exhaust on the flow field external to the jet is illustrated in Figure 8.24. Viscous shear along the edges of the jet produces an entrainment of the external flow into the jet. To the external flow, the jet appears somewhat like a distributed line of sinks. As shown in Figure 8.24, this jet-induced flow can produce a change in the local flow direction for a lifting surface in proximity to the jet.

Again, the calculation of flow inclination and the effect on a lifting surface is a difficult task and is best determined from wind tunnel tests. The effect can be appreciable and generally should not be neglected. According to Reference 8.2, it may vary with a enough to reduce the stability significantly.

From material to be found in the appendix of Reference 8.2, an ap­proximate estimate of the inclination angle due to jet entrainment can be calculated from

(8.86)

where

CD = airplane drag coefficient n = number of engines

x = distance aft of exhaust exit plus 2.3 diameters of the jet at its exit r = radial distance from jet centerline b = wingspan A = aspect ratio

Although a does not appear in Equation 8.86, the inclination of the jet itself varies with a, which causes the distance, r, to change. For more detailed information on this, see Reference 8.2 and the original sources on this topic, which are noted in the reference.

Effect of Fuselage and Nacelles

A body, such as a fuselage or nacelle, is not a very efficient producer of lift by comparison to a wing or tail surface. Thus, except for missilelike configurations, where the body is relatively large, the contribution of the fuselage or nacelles to the total airplane lift can be neglected. However, in the case of the moment, fuselage and nacelle contributions can be fairly significant and result in a measurable shift of the neutral point. Generally, the
increments to CMa are positive, resulting in a decrease in h„. Depending on the incidence of the wing relative to the fuselage and nacelles, these components can also contribute to Сщ. Thus, including the fuselage and/or nacelles, Equation 8.7 becomes

См — Сщ + Д Сщ + (Сщ + ДСМ>

The increments Д Сщ and ДСщ result from the addition of the fuselage and/or nacelles to the basic wing-tail combination. Thus, the relationships derived thus far are unchanged if the total values for Сщ and CMa are used. An increment to CLa can also be included.

It is somewhat futile to present either data or theoretical results for predicting Д Сщ or ДСщ in view of the many possible fuselage-nacelle-wing – tail combinations. Real fluid effects and lifting surface-body interference effects severely limit any application of theoretical solutions of the body alone.

For approximate purposes, expressions for CL„ and Сщ for bodies alone are presented that are based on material from DATCOM (Ref. 5.5). The lift curve slope of an axisymmetric body, having a length of / and a diameter d, can be expressed in the form

(8.71)

Here, Cl is based on a reference area, S0, which is the body’s cross-sectional area at a station X0 back from the nose. X0 is a function of a station Xt; X, is the station along the body where ds/dx has its minimum value. Usually this means the point of maximum cross-sectional area. In terms of the body length, X0 and X, are related by

Х0 = 0.367 + 0.533Х,

The slope of the body moment curve can be estimated from

Сщв = CL. JXM — XQ+ To)

Cmg is based on a reference area of S0 and the body length, /. Xm is the station for the moment center expressed as a fraction of /. /0 is an average length that, multiplied by S0, gives the body volume ahead of the station X0. Again, T0 is relative to the total body length.

Since Cl„b and Сщв are based on the body geometry, their values must be corrected when they are applied to the total airplane. More specifically, cLaB must be multiplied by the ratio S0IS before being added to Equation 8.11. Similarly, Сщв must be multiplied by (S0l)l(Sc) before being added to equations previously derived from Сщ – It should be emphasized that the expression for the lift curve and moment curve slopes hold only for small angles of attack.

Stabilator Angle per g

The preceding developments will apply to the stabilator configuration if Д Se is replaced by Aihs, CMs by CM„ and CLs by CLi. A separate analysis of the stabilator with regard to a steady maneuver is therefore unnecessary.

Stick Force per g

Stabilizer-Elevator Configuration In order to calculate the stick force required per g of normal acceleration, we again use Equations 8.27 and 8.37.

Си — bOct + bi 8e + Ьз St

Vt ~q~s6 = bl[a(l ~ e“)- ‘a* + 21 q J + b2 8e + b3 8,

Now let 5, be adjusted to trim P to zero for unaccelerated flight. For this condition let a = a0 and Se = 8^, such that a = а0 + Да, 8e = 5^+ A<5e. It then follows that

Thus the stick force per g becomes

The position of h for which P equals zero is known as the stick-free maneuvering point and is denoted by h’m. In terms of h’m, the stick force per g becomes

where

Stabilator The stick force per g for the stabilator is obtained in a manner similar to that which was followed for the stabilizer-elevator configuration. The moment coefficient about the pivot line is once again given by Equation 8.32. Also, in the steady pull-up, Equation 8.63 again applies and 8 is related to ihs by Equation 8.33. Thus, for the stabilator,

г) GqSc = "" e“) ~ ihs + 21 <?] + b2(keihs + S0)

Now let S0 be adjusted so that P = 0 for q = 0. For this trim condition we again let а = a0 and ihs = ihso, so that, generally, а =а0+Да and ihs + ihSo + Дihs. Thus, ,

G^jsT, * – «■> ■- ЛІ. + * p(j)] + W. ii.

The similarity of this relationship to the corresponding equation for the stabilizer-elevator tail configuration is obvious. Again, we can express the stick force per g in terms of the stick-free maneuvering point.

where

An airplane loaded so that the center of gravity is close to the stick-free maneuvering point presents a dangerous situation. The pilot can impose extreme inertia loads on the airplane with the application of little or no control force. Such a situation, however, is rarely encountered if an adequate static margin is maintained, since the stick-fixed and stick-free maneuver points are aft of the corresponding neutral points.

Steady Maneuvering

A wing-tail combination is pictured in Figure 8.19 undergoing a steady pull-up maneuver. For clarity, the angle Да, is shown much larger than it

Figure 8.19 Wing-tail combination in a steady pull-up maneuver.

actually is; that is, the ratio of /, to the radius of the flight path, R, is much less than one. The acceleration, an, of the airplane toward the center of curvature requires a lift in excess of the weight given by

L=w(l+^j (8.47)

The lift-to-weight ratio is known as the load factor, n.

The relationships derived thus far that relate CM to CL (or a) hold for the maneuvering case. However, an additional increment must be added to CM as a result of the increment in a, due to the angular velocity, Q.

AM = — |pV2T/,S, a, Да,/,

But

(8.49)

Thus, in coefficient form,

Notice that we can also visualize the airplane as translating with a linear velocity of V while rotating about its center of gravity with an angular velocity of Q. The tail is then moving down with a velocity of l, Q, which leads to Equation 8.49.

Horizontal Stabilizer-Elevator Configuration: Elevator Angle per g
Including the effect of Q, we can now write Equation 8.7 as

См = Cm„ + Cm„& + Cms 5e + CmqQ (8.51)

However, neither Q nor CMq is dimensionless (although their product is). Thus, it is sometimes desirable to write Equation 8.51 as

Cm = CM<) + См„а + Cms Se + CMfl

where q is a dimensionless pitch rate defined by

The choice of cl2 as a reference length is simply according to convention and has no physical significance.

From Equation 8.50, CMq is seen to equal approximately

CM<, = – n, a,VH^ (8.54)

Cm, is referred to as “pitch damping,” since it represents a negative moment proportional to the rotational velocity about the pitch axis.

In a steady pull-up maneuver, there is no angular acceleration, so CM must be zero as it is in trimmed level flight. Assuming Сщ to remain constant, we can therefore equate the increment CM resulting from the steady maneuver to zero. Since CM is assumed linear with a, S, and q, we can write

0 = См„ Да + Cm5 AS, + Cm,<? or

/См Aa + Cm <?

a*-(–c„, ’) <855>

Remember that q now represents the dimensionless pitch rate and not the dynamic pressure. Since

V = QR

and

it follows from Equation 8.53 that

«=^(7) ,8’561

The quantity (ajg) is the acceleration expressed as the ratio to the ac­celeration due to gravity. When, for example, we speak about “pulling 4g’s,” we mean that ajg – 4.0.

Да can be obtained from Equations 8.47 and 8.48.

or

Cl + A CL 1 an CL g

Thus,

ACl=Cl —

= CLa A a + Cls A Se

so that

where CL is calculated for 1 g flight. Combining Equations 8.55, 8.56, and 8.57 results in an expression for the elevator angle per g of acceleration.

A Se Cm CL + gc/2 V2CMqCLa (ajg) CMfLa ~ CMCLs ( )

Substituting Equation 8.13 for CMq, Equation 8.58 becomes

ClCl

CmsClr — CmCl where m is the mass of the airplane. We now define a relative mass parameter, p.

2m

pcS

cl2 is the same reference length used in the definition of q (Equation 8.53). This length multiplied by the wing area gives a reference volume. The parameter p is then the ratio of the airplane mass to the air mass within this volume. Using the definition of p, equation 8.58 becomes

Similar to the neutral point, there is a particular value of h, known as the stick-fixed maneuver point, hm, for which no elevator deflection is required to produce an acceleration. Indeed, if the center of gravity moves aft of this point, the effect of the elevator is reversed. From Equation 8.60,

hm = hn-^CMq (8.61)

Since Cm, is negative, hm lies aft of the neutral point.

In terms of the neutral point,

The quantity (hm – h) is known as the stick-fixed maneuver margin. Again, it is emphasized that CL is calculated for straight and level flight. It is not the total airplane CL required during the steady pull-up maneuver.

Stabilator Configuration

The stabilator trim tab is mechanically linked to the fuselage, as shown schematically in Figure 8.18. The tail is trimmed by moving the attachment point of the link to the fuselage. With the stick free and the hinge moment trimmed to zero, the entire tail is floating around the pivot line. If the fuselage angle of attack is disturbed, the tail will tend to maintain a constant angle of attack. However, the tab, being linked to the fuselage, will deflect, thereby changing the aerodynamic moment about the pivot line. This, in turn, will cause the entire tail to float to a different position at which the moment is again zero. Thus, to determine the free elevator factor for a stabilator, the linkage to the fuselage must be taken into account. The tail cannot be treated as an entity, as was just done for the horizontal stabilizer-elevator configura­tion. We begin by setting Equation 8.32 equal to zero and substituting

(8.45)

For the stick-fixed case, where the incidence angle is fixed, the tail lift curve slope in the downwash of the wing is simply

a,( 1 – e«)

By comparison to Equation 8.45, it follows that for the stabilator configura­tion, the free elevator factor is given by

p = , (1 – тке)Ь і

bx – b2ke

Using the numbers previously estimated for the Cherokee 180, the value for Fe is estimated to be

Fe = 1.30

•Thus, the effectiveness of the stabilator in providing longitudinal static stability is not degraded by freeing the stick; indeed, it is actually improved!

Stick-Free Static Margin

The effect of freeing the elevator on the neutral point can be determined simply by multiplying a, by the free elevator factor, Fe, and using this effective tail lift curve slope in the expressions previously derived for the stick-fixed case.

If Fe is less than 1.0 the neutral point shifts forward, thereby decreasing the static margin for a given center-of-gravity position.