Effects of asymmetry on the dynamic stability of aircraft

rotation has been found and its derivation is described; the induced drag aleo introduces a side component and its magnitude is computed.

The new derivatives due to the oblique wing B,nd the general methodol- ogy for evaluating stability derivatives are also described in this Chapter.

Chapter IV deals with the derivation of the equation of motion.

The numerical results are given in Chapter V. First, the span- wise distribution of lift, downwash, and induced drag, a.s computed by strip and lifting line theory, are reported and compared. Then, the results for rigid and flexible wing, as computed by lifting line theory, are described. Special emphasis is given to the spanwise distribution of the increments of the aerodynamic forces during simulated perturbed conditions, and the resulting stability derivatives. Root loci vs.

skew angle along with the mode shapes are shown for all the natural modes.

Chapter VI contains the final conclusions and remarks.

II AERODYNAMICS 2. 1 Genera li ties The lift generated by a wing can be divided into two contributions: lift due to twist and camber, and lift due to the flat wing at angle of attack.

According to linear theory these two contributions are independent of ea,ch other, and the lift is simply equal to their superposition. In addition, the twist contribution does not depend on the angle of attack and therefore on its variations, but it does depend on sweep and speed changes (or, more exactly, on dynamic pressure variations). In a stability analysis we must know the spanwise distribution of the aero- dynamic force and how such distribution varies when the wing undergoes perturbations. Let us now limit ourselves to the lift component of the aerodynamic force since dra,g and side forces can be derived from the knowledge of the lift.

For a symmetric aircraft it is possible to obtain the lift distri- bution for the perturbed condition by applying strip theory to the cruise spanwise lift distribution [Ref. 7]. Though an approximate one, this method produces satisfactory results in the linear range.

The perturbations considered in a sta,bility analysis can be divided into two groups: 1) perturbations which affect the total lift distribution (flat wing plus twist contribution) 2) perturbations which affect only the flat wing.

To the first group belong the u, r ,and p perturbations; to

the second group the perturbations involving change in the local angle of attack (ex , p ,and q) .

For a symmetric aircraft the flat wing spanwise lift distribution for the cruise condition is symmetric.

Let us now consider the flat wing lift distributiop (FWLD) of a skewed wing. For such a wing at no sweep, the lift distribution is symmetric and elliptic (because of the chord distribution). When the wing is swept, the new spanwise lift distribution will now show an increase in lift on the a.ft part of the wing and a decrease on the forward one, because of a.n upwash generation in the first case and a downwash increase in the second one. The need 'for twist in this wing will be based now not on a 'tip stall last' requirement, but primarily on the need to obtain a symmetric lift distribution from the asymmetric flat wing lift. Fig. 2.1 shows the asymmetric lift, lift due to twist, and total lift versus span. The accurate evaluation of the change in aerodynamic loading will lead to good precision in computing the wing contribution to stability derivatives. We shall now follow two approaches in determining the spanwise lift distribution during the perturbed motion.

The first, based on strip theory, can be used for all perturbations, but in the case of sideslip it fails to give any acceptable result. In order to improve this situation an empirical method, ba.sed on a. modifica.- tion of the Pope-Schrenk method [Ref. 6J which can compute the spanwise lift distribution for a flat oblique wing was obtained and will be briefly discussed in section 2.5 • The second approach is based on the eva.1ua.tion of the spanwise lift

fa

i$

;:i5~

§t--.

~~

q~

~ FLAT WING LOAD DISTlIBUTION, (L)F

DESIGN "lG" LOAD DISTRIBUTION, (L)Tot \ ·h.5

' -_~-~:\

C C ';' -- --- --

-- -

C C // .....

.... ave L ... " ~ .n.o

....

."

."

, " " " ~ " "'-J ,,- " LOAD DISTRIBUT!ON DUE TO TWIST (L)B

" .-

/

'

\ I 0.5 \ I

'\ . - \

~ _ 5 - .. ',\

.- '\

. -'

_.-

.- l'"

-"

_ .. -,,- ,-'"

'- .. -- .. - - ... - - _ ..

'- FIGURE 2.1 - Total Lift and its Components distribution of an oblique wing by means of linear aerodynamic theory.

In this case, the spa.nwise lift distribution is computed for both cruise and perturbed conditions. This second method also allows the static effects of a flexible wing to be included. The load distributions ob- tained by means of these two methods will then be used in determining the sta.bility derivB.tives B.nd the results compared. Once we have com- puted the stability derivatives we shall use these va.1ues in the equations of motion and determine the na.tura1 behavior and the time response to control disturbances.

2.2 Reference Axes To discuss the problem of stability, it is necessary to set up a system of reference axes which form the basis of a system of notation used to describe the motions of an airplane. Two basic body fixed axes systems, each consisting of three mutually perpendicular axes passing through the center of gravity of the airplane, adequately cover most of the a.erodynamic problems in stability considerations. These are the "body axes" and the "wind axes", which will be referred to as "stability.

axes" • 2.2.1 Body Axes (x ' Yb ' zb)' b The body a.xes system is rigidly fixed in the airplane and is the system ofmutua11y perpendicular axes pa.ssing through the airplane's center of gravity and whose X-axis is parallel to the thrust axis, the wing mean aerodynamic chord, or some other longitudinal reference and is positive in the direction of the nose of the airplane. Herein it is taken along the body centerline reference. Figure 2.2 shows this system together with all forces, moments, displacements, and velocities.

The XZ plane is the plane of symmetry for the airplane.

2.2.2 Stability Axes (x ,y ,z).

s s s The sta,bility axes system differs from the body axes system in that the X-axis is parallel to the relative wind, positive forward. The Z-axis is positive down (Fig. 2.3). The Y-a,xis is positive to the right.

The moment, angle, and angular velocity conventions are given by the right hand rule.

The sta.bility axes system is the one used for basic aerodynamic performance work, and it will be used in our ana,lysis. In our study we shall also a,ssume that for zero angle of atta,ck stability and body axes wi 11 coincide.

2.3 Strip Theory The strip hypothesis asserts that we may calculate the aerodynamic force on each strip as if it were an isolated airfoil moving with the resultant velocity which it has because of its local position on the aircra.ft [Ref. 7].

The strip hypothesis is a first approximation to the actual case where the trailing vortices from each strip possibly interfere with the others, but this first approximation is confirmed by experiments to give results which are in excellent agreement with the facts as observed, even at incidences above the stall, for a symmetric aircraft. Unfortu- N,r Z,W T: Zb Notation for body axes.

T, = rolling moment p = rate of roll M= pitching momont q = rate of pitch N = yawing moment r ~~ rate of yaw [X, Y, Z] = components of rcsunant aerodynamic force [n, 1!, w] = components of velocity of C relative to atmosphere Figure 2.2 - Body Axes [Ref. 21J x s Z B Figure 2.3 - Stability Axes nate1y, when dealing with motions involving rate of pitch or ra.te of roll, the strip theory fails to give a good approximation for the ca.se of an oblique wing, since it predicts no lift change at the wing root chord for the case of a roll, or at the wing station. whose quarter chord is crossing the Ys axis, for the case of pitch.

The advantage of this approach is that it allows us to evaluate the sta,bility derivatives using different sources of data, such as wind tunnel or flight tests, and computer results.

2.3.1 Variations in Aerodynamic Forces due to a Perturbation a.

Since the twist (or dihedral) contributions are independent of changes in angle of a,ttack, only the flat wing lift distribution (FWLD) needs to be considered. (Subscript F will indica.te flat wing quantity).

We can now introduce the following approximations.

2.3.1.1 Section Lift Slope.

Assuming the FWLD is given, the local lift slope can be computed as C (y) = (2.1) L a where local lift slope at station y (ljrad) local lift coefficient of the flat wing at station y angle of a,ttack for cruising condition (rad).

2.3.1.2 Section Downwash.

The downwash is a consequence of the wing not having an infinite span. For a 2-D wing, the section lift coefficient is given by (2.2) where m = 2-D section lift slope

o

e(y) section twist.

The presence of trailing vortices in a finite wing introduces local downwash velocities whose effect is to reduce the local angle of attack and, therefore, the lift produced by the wing. The two terms of (Eq • 2.2) will become, for a 3-D wing (2.3) where

= flat wing downwash angle

downwash angle due to twist.

Therefore, the section total lift coefficient for a 3-D wing is (2.4 ) The previous expression applies to a straight a.s well as to a skewed or swept back wing; the only change, a.ssuming all quantities are measured

in the flight direction, would occur in the value of so' the 2-D section lift slope. We refer to R.T. Jones [Ref. 7J for a detailed discussion of the derivation of

a •

o

We shall now evaluate the downwash angles by computing the difference in the local lift distribution from the 2-D case. This is a "crude" approximation, but in a strip theory analysis it is the only way to evaluate the downwash velocity and, therefore, the spanwise induced drag distribution.

In section 5.2.1 we shall compare the downwash results obtained with this method against the correspondin~ results obtained applying the method based on linear theory described in the next section.

Let us define (2.5) where [CL(y)]F = flat wing section lift coefficient

[CL(y)J = section basic-lift coefficient. It represents the

B a-independent contribution to lift due to twist.

(2.6) [CL(y)JF = aO[a - EO(Y)] O [CL(y)]B = aO[e(y) -l'.E(y)J' (2.7) We can now assume, according to linear theory, that the two down- washes are independent from each other, and furthermore, that the down- wash due to twist does not depend on the angle of attack. With these assumptions we have (2.8) [e (y)J L a 1 - (2.9) OE (y) O (2.10) EO (y) = da a O The tota.l downwash is therefore given by (2.11) ~E(y) = (2. 12) 1 -

oa

2.3.1.3 Section Induced Drag.

The aerodynamic drag in a finite wing has two components: the first is due to skin friction and pressure distributions on the boundary, the second is the one induced by the lift because of the presence of trailing vortices.

Both components are normally of the same order, and dependent on the aircraft speed [Ref.lSJ. The analytic spanwise evaluation of these two components is a difficult task.

According to Multhopp [Ref. 9J we may write the induced drag coef- ficient for the wing as b(2 (2. 13)

f C CL(y) ex dy

i -bj2 where the so-called induced incidence is bf2 1 1 d (2.14 ) y - 71 d71

f

-b(2 and the section induced drag coefficient is g'iven by (2. 15 ) Cn. (y) = CL(y)a i ~ Garner, [Ref. 10J who has discussed induced drag and its spanwise distribution in incompressible flow, has shown that, for swept back wings, the quantity C CL(y) a does not give an acceptable spanwise i distribution of induced drag as suggested by Robinson and Laurmann [Ref. 11J. This conclusion can be expected since the induced incidence a ' as computed in Eqn. (2.14) implies that all the bound vortices of i the horseshoe vortex system, used as a model for the wing, lie on a straight line perpendicu1a.r to the velocity. Therefore, the downwash angles correspond to the ones of a straight wing and the induced drag obtained is the product of the lift distribution of a swept wing times the downwash angle of a straight wing having the same wing span and the same spanwise lift distribution.

We shall return to this subject when we evaluate the induced drag distributions by means of linear theory. For the sake of clarity we w shall recall that the downwash angle is the ratio where V

w = downwash velocity measured at the lifting line

v = free stream velocity.

In our strip analysis the downwash angles are obtained directly from the actual spanwise lift distribution; we assume that the downwash angle is the cause of the difference in lift coefficient from the corresponding 2-D one, and no assumptions are made on the wing geometry.

We may therefore expect a. better accuracy in the estimate of the spanwise drag distribution.

l.et us now see how to relate the previous discussion to the evalua- tion of stability derivatives by means of strip theory.

For the lift ca.se we have one term, the basic lift, which is "0: independent" and the other, the flat wing, which does depend on 0: ; the same considerations can be applied to the drag.

In the "0: indepen- dent" drag contribution we can group skin friction, pressure distorsions, and, if we extend to the drag the same assumptions made for the lift, also the induced drag due to ba.sic lift. This approxima.tion is quite accurate for skin friction and pressure distortion, but it becomes less accurate when considering the induced drag of the basic lift, as is shown next.

The total section induced drag is obtained by substituting into Eqn.(2.1S) the total values derived in the previous sections (2. 16) We can see that the induced dra.g produced by the basic lift distribution (2.17) [CD.(Y)]B = [CL(Y)]B [EO(Y) + L.E(Y)] ~ has the term [CL(Y)]B EO(Y) that is a dependent since EO(Y) , as discussed in the previous section, is a dependent.

Therefore, only the quantity [CL(Y)]B L.E(y) can be assumed, within the range of linear theory, as a independent, whereas (2. 18) is the a dependent component of the drag. The change in the section drag for perturbations introducing a local change in the angle of attack can now be computed as the rate of change of Eqn. (2.18) with a (2.19) and by substituting Eqns. (2.5), (2.11) and (2.12) into (2.19) we obtain (2.20) 2.3.2 Evaluation of Stability Derivatives by Means of Strip Theory.

Assuming the spanwise lift distribution is given for the flat wing as well as for the wing with the nominal twist and dihedral, it is pos- sible to evaluate the stability derivatives by using the approximations used in section 2.3.1 and the expressions given in sections 3.3.2 and 3.3.4 • For the stability derivatives due to side-slip it is necessary to know the spanwise lift distribution corresponding to the new sweep angle differing from the nominal one by A = - ~. Because of the peculiarity of the shape of the FWLD and its dependence on the sweep angle, an ap- proxima,tion assuming cos A (2.21)

cos Au

where A=J\-~ would fail to give an acceptable result. The knowledge of the spanwise lift distribution for the new sweep angle would therefore be required.

For the case when such lift would not be available, an empirical correction to the Pope-Schrenk' s method [Ref. 6,12] was derived and it is described in section 2.5 • 2.4 Lifting Line Theory 2.4.1 Introduction A more accurate way of evaluating the stability derivatives, spe~ cially roll and pitch derivatives, is to use linear aerodynamic theory.

The literature offers a wide variety of methods which can be used; many of these are very complex and allow the user to eva.luate both the span- wise and chord-wise load distribution. The simplest three-dimensional wing theory is that based on the concept of the lifting line. In this theory the wing is replaced by a straight line [Ref. 13J. The circula- tion about the wing associated with the lift is replaced by a vortex filament. This vortex filament lies along the straight line; and at each spanwise station, the strength of the vortex is proportional to the local intensity of the lift. According to Helmholtz's theorem, a vortex filament cannot terminate in the fluid. The variation of vortex strength along the straight line is therefore assumed to result from superposition of a number of horseshoe-shaped vortices, as shown in Figure 2.4. The portions of the vortices lying a.long the span are called the "bound vortices". The portions of the vortices extending downstream indefinitely are called the "trailing vortices".

The effect of trailing vortices corresponding to a positive lift is to induce a downward component of velocity at and behind the wing. This downward component is called the "downwash". The magnitude of the down- wash at any section along the span is equal to the sum of the effects of all the tra.iling vortices along the entire span. The effect of the downwash is to change the relative direction of the air stream over the

section.

/'

Line of Direction of "-... ~rodynamic airstream Bound ~ - centers vortices Figure 2.4 - Vortex Pattern Representing a Lifting Wing [Ref. 13 ] The section is assumed to have the same aerodynamic characteristics with respect to the rotated air stream as it has in normal two-dimen- siona1 flow. The rotation of the flow effectively reduces the angle of attack. Inasmuch as the downwash is proportional to the lift coef- ficient, the effect of the trailing vortices is to reduce the slope of the lift curve. The rotation of the flow also causes a corresponding rotation of the lift vector to produce a drag component in the direction of motion.

The methods using discrete vortices to represent the continuous distribution of circulation of the vortex sheet are attempts· to simplify the cmnputations. In the methods employing discrete vortices, two-dimen- siona.1 theory is used to determine the most representative locations of the vortices a,s well as of the control points. If only one vortex line is used, it is placed along the center-of-pressure line in two-dimensional flows, which for a flat plate at an angle of attack and at subsonic speeds is the quarter chord line.

For subsonic speeds, the downwash va.ries inversely with the distance behind the quarter chord line; at the position of the three- quarter chord line it just equals in magnitude the vertical component of the flow tangential to the flat pla.te having the same circula.tion.

Conversely, if ,the condition of tangential flow is satisfied at the three-quarter chord line, the strength of the concentrated vortex will indicate the lift on one wing due to angle of attack. Of course, these methods of obtaining correspondence between the lifting lines and the vortex sheets loose their validity near the corners of the wing, where the flow differssha.rply from the two-dimensional.

In a stability analysis it is not required to have great accuracy in the chordwise load distribution. Therefore, it is sufficient to use only one vortex line pla.ced, for subsonic cruise condition, at the quarter-chord point. In doing so, little is lost in accuracy and a lot is gained in simplicity.

The supersonic case has to be approached in a different way accord- ing to the properties of the supersonic flow. The method outlined in this chapter applies to subsonic speeds only, but it can be extended to the supersonic case. It can be shown that the horseshoe-vortex system of Figure 2.4 is equivalent to the one where each horseshoe vortex has a constant strength equal to the sum of the strength of the bound vortices contained at the corresponding section of Figure 2.5 • When the wing is skewed, two models can be used. Figure 2.6 shows the first one where the bound vortices a.re aligned with the wing span, -r-;'-~~~ I-- f-- I-- i-

<:e

10סo.

I- ,

f

Figure 2.5 - Horseshoe Vortex Pattern (unyawed wing) Figure 2.6 - Bound Vortex Normal to Flight Stream ORIGINAL PAGE OF POOR QU IS ALITY

I

I

i Figure 2.7 - Bound Vortex Parallel to Wing Span and are at a skew angle with. respect to the free stream velocity.

The second model, shown in Figure 2.7, assumes that the bound vortices are perpendicular to the flight direction. This model corre- sponds to the case where the wing at a skew angle is actually replaced by a finite number of straight wings, and will be used in this study.

The first model is expected to give a better accuracy when the number of horseshoe vortices is small, but the two models coincide when the number of horseshoe vortices goes to infinity.

The method used in this study is based on a modification of the Weissinger-L-Method and applies at subcritical Mach Number. This method is derived from excellent work done by Gray and Schenk in 1953 [Ref. 14] and has been modified for the oblique wing case. Among the advantages of such an a.pproach to the evaluation of the spanwise dis- tribution of the loading, is the possibility of evaluating the loading increments· due to aileron or flap deflection, effects of wing flexibility, and accelerations on the wing. These features are fully described in Appendix D.

2.4.2 Steady State Loading on an Airplane with an Oblique Wing.

The fundamental problem involved is the development of a series of equations which relate the spanwise lift distribution for an arbitrary wing plan form in a given flight condition to the properties and a.tti- tudes of the individual sections that form the wing.

For a 2-D wing, the following relationships can be found in any standard textbook on aerodynamics [Ref. 7,15, l7J: (2.22)

r

w =- (2.23) r 21Tr where

m = local lift slope

o

C = local chord length

t = section lift

ex = total angle of attack (see Fig. 2.8) f 1 V d .

q 2 p = ynam~c pressure

circulation

r =

At a specific distance r behind the lifting line, the resultant of the downwash velocity Wand the flight velocity V is parallel to r the section zero lift line. Then, (2.24 ) from Eqn. (2.22) we obta.in (2.25 ) Substituting (2.25) into (2.23) results in W (2.26 ) r Equating (2.26) and (2.24) or (2.27) Since the theoretical section 2-D lift curve slope is 27T, r must equal C/2, which is the distance between the lifting line and the three-quarter-chord point.

Therefore, for the 2-D (unswept wing) ca..se (2.28) corresponds to the control point where no flow exists normal to the zero lift line. Whenever the local lift slope differs from 27T, expression (2.28) becomes (2.29) The essential difference between a 2-D wing and a wing of finite aspect ratio arises from non uniform spanwise loading which produces the trailing vortices. The equations presented so far are considered to apply to a finite wing when the effects of all the vortices, both bound and trailing, have been taken into account.

Equation (2.29) can be written in matrix form \WI

~~o]

(2.30)

l,vl3C/4

lUi, This matrix relation represents a series of equations, each appli- cable to a. particular station on the span of the wing.

The elements of 1~}3/4c' everyone of which is affected by the

- bound and trailing vortices of the wing stations can be evaluated from l

l~} - _1_ [SlJ lr

(2.31) Iv 3/4 C- 47TV ;, I I a.nd by expressing 1 r l in terms of I.e I.

The [SlJ matrix is the downwash matrix and is derived in Appen- dlx D.

Combining equations (2.30) we obtain

_1_ [SlJ I.e I = [,~] la I

(2.32) 81Tq I I 2Th I f l or (2.33) 2.4.3 Section Final Angle of Attack lafl.

The final B,ngle of attack across the span ! a j can be considered

f to be composed of three essenti.al parts (see Fig. 2.8).

= la I + la 1+ la I (2.34 )

I rl g I I sl

where la 1= angle of attack caused by structural deflection of a

I s\

flexible wing la 1= angle of attack caused by built-in twist, apparent or I gl aerodynamic twists, control deflection, angular velocities, iari= flat (and rigid) wing angle of attack (measured w.r.t.

root-section zero-lift line).

I I

The angle of attack la ' caused by structural deflection of a sl flexible wing due to the section lift at the section a,erodynamic centers is linearly related to the matrix as

· "EquHibrium position of

,.

,. section zero-lift line

, , Or

Root-section zero'-

lift ~ine

...... Undisturbed

wind direction

Figure 2,8 - Final Angle of Attack 10: I = [S ] j t 1 (2.35 ) I sl 2 I I where [S2 = structural deflection matrix (described in Appendix D).

J The contributions to the angle of attack 0: are described in g Appendix D.

2.4.4 Section Induced Drag.

As discussed in section 2.3.1, only the induced drag varies with 0:.

The induced dra,g arises from the rotation of the aerodynamic forces due to the downwash velocities induced by the trailing vortices.

In our model, where the wing has been replaced by a lifting line placed, for the subsonic analysis, at the quarter-chord line, the section induced drag can be evaluated by computing the downwa,sh angle (O:i)i at the station bound vortex. This is similar to what is done

for (!i). , except that downwash induced by a bound vortex on itself

V 3/4 C is zero.

Therefore

L: (K ) Cf4 £ j

j ij (2.36) where (Kij)C/4 is the same as computed in Appendix D with the following two exceptions: c.

1.

1) the term 2 ' distance of the control point from the lifting line, must be dropped since the downwash is now evaluated at the lifting line; 2) the contribution to the downwash sould be disregarded for the case when the control point is within the horseshoe vortex, as discussed in Appendix D.

A more accurate result would be obtained using the vortex la,ttice method. The results obtained in this analysis are discussed in Refer- ence 16.

2.5 An Empirical Correction to Schrenk's Method Another approach to the problem of spanwise loa,d distribution having much less theoretical foundation is presented by Flatt [Ref. l8J.

It follows a method first presented by Schrenk [Ref. 19J.

Schrenk's method makes allowance for the effect of the varying down- wash along the spa,n of a nonelliptic wing by assuming that the final span load distribution for an untwisted wing is ha,lfway between the actual planform shape and a semi-ellipse of the same area. However, Schrenk and Flatt did not consider a swept back wing; Alan Pope and William R. Haney, assuming that the effect of sweepback on the non- dimensional span loading is linear, proposed an empirical correction [Ref. 6, 12] to take care of the effect of sweep back. It has been successfully employed in preliminary design of subsonic aircraft.

The following empirical formula which was obtained during our stUdy of the oblique wing, can be applied to Schrenk's method.

~ 2'

1 - ~ (2.37)

where = skew angle A c IcosA: root chord measured in the flight direction c = R R b cosA: actual wing spa.n b = O wing.

b = wing span for the unskewed O Since the goa.! of this method is to provide, in a fast way, the shape of the spanwise lift distribution, the wing lift coefficient is assumed to be known. In addition, the new spanwise lift distribution obtained with this correction must be norma,lized by multiplying it times the ratio of the wing lift coefficient to the lift coefficient obtained by integration of the new lift distribution.

The results of this correction, forthe case ofa flat wing, were checked aga.inst the lift distribution as computed by the numerical program of Reference 20.

Figures 2.9a, b, and c show the comparison between the two methods for three different cases.

To date, no check has been done for the ca.se of a wing having twist. The major difficulty for this case is to estimate the downwash velocities produced by the lift due to twist and, therefore, to determine the effective twist.

Though for symmetric wings experience suggests that an effec- tiveness of 50% is an acceptable assumption, we do not expect that the same can be applied to a, skewed wing.

As pointed out before, the effect of a positive sideslip corresponds, = , = , for an oblique wing, to s, negative change in sweep and the perturbs,tion will therefore affect both the FWLD and the basic lift distribution.

To a first order approxims,tion it is possible to a,ssume that the chs,nge in the shape of the lift distribution for the total lift will be proportiona,lto'the change with sweep of the FWLD.

Therefore, the simulation of a, sideslip can be dorie by 2.6 Summary.

The reference axes system used in this work, the stability axes, has first been defined. The method for evaluating variations in the aerodynamic forces due to perturbations was then outlined. A simplified method, based on strip theory, for computing spanwise distribution of the induced drag was also proposed.

A second and more systematic way of computing spanwise lift distribution based on lifting line and including effects of wing flexi- bility was the described. The spanwise induced drag distribution was then evaluated by using lifting line theory.

In the last section of this Chapter, an empirical correction to Schrenk' method for the oblique wing case was proposed.

", III STABILITY DERIVATIVES 3.1 Basic Features 3.1.1 Flat Wing Lift Distribution (FWLD).

In section 2.1 we have described how the FWLD behaves when the wing is skewed. This behavior is undoubtedly one of the most important elements of difference from the sYmmetric case. The upwa,rd field gener- ated by the forward wing, causes the aft wing to "feel" a higher angle of attack. This, as we mentioned before, is the cause of the loss of sYmmetry in WLD, but, because of the higher angle of a.tta.ck actually experienced by the aft wing, the stall condition will be reached in this region first. This is simila,r to the case of a tip stall for a symmetric wing, except that now only one tip would stall and the loss in balance would produce not only rolling but also pitching moments. The recovery from such a stall is very difficult.

This can be expla,ined if we analyze in detail the motion of the aircraft following the stall of the aft wing.

As we mentioned before, the loss in the lift sYmmetry introduces, in the case of the left wing forward, positive pitching and rolling moments. The nose-up motion deriving from the pitching moment introduces an increase in the angle of. attack of the whole wing as well as an angular velocity; whereas the rolling moment produces only an angular velocity.

The effect of these two angular velocities results in a linear va.riation CluE the loca.l angle of attack for the wing, increasing it on the a.ft part and decrea.sing it on the forward one.

Therefore, the aft wing will experience a further increase in the angle of attack which will worsen the stall condition and extend it inboard.

On the forward wing, instead, the increase in angle of attack of the whole wing is counteracted by the decrease deriving from the angular velocities and the final trend is consequently toward a delay in the forwa.rd wing stall, which makes the nose-down attitude, required for a recovery, difficult to achieve.

The center of pressure of the flat wing lift distribution (FWLD) lies on the aft- portion of the wing and produces both pitching and rolling moments as the angle of attack varies, introducing new important derivatives. In fact, a.ny cha.nge in the angle of attack affects the FWLD only and destroys the synnnetry in the tota.l lift distribution obtained by twisting the wing. The consequence of this is a loss in the moments' balance. Figure 3.1 shows the variation in spanwise lift distribution, as given by linear theory, for a case where, for the sake of clarity, the "perturbation" 0: is assumed to be equal to 50% of 0: , It will be seen that a positive change in 0: produces an increase in the lift whose aerodynamic center is displaced toward a point in the aft part of the wing, thus producing a rolling moment as well as a pitching moment. By considering the corresponding change in induced drag, it is possible to evaluate the yawing moment.

%c

':b~

0$

~f1;.

.o/:-Y

~~

t;Q

Steady State Load Distribution ~:

I

t f

/

/

Increment in Load Distribution

I

due to a change in ex

!

t

I

, w

I CO

j

--l-- ---

-

.,

-

...- ...

'" ..- ....

./ .......

./ ,- " '" /' ,-

'"

"

'"

"

Forward Tip Aft Tip Figure 3.1 - Increment in Load Distribution due to a Change in ex.

3.1.2 Side Force A wing at a skewed angle experiences a. side force deriving from the induced drag component in the y direction. Let us now derive its magnitude.

Figure 3.2 shows a section of a wing B,t a. skew angle A.

The principle of independence [Ref. 8J a,ssumes that, in a friction- less flow, all the results of the two-dimensional flow theory can be applied immedia,tely to an infinite oblique wing simply by subtracting the axial component of velocity.

No matter whether we consider the free stream direction or the direction normal to the leading edge, the lift generated by the wing section is obviously the same. Therefore, the lift force is given by 1 2

[ C • 1 ] = 1 P v c:.£...:..l

(3.1)

t = '2 P V C

o L o 2 L cosA

o

where the subscript 0 defines the quantity in the flight direction.

Since v = V0 cosA (3.2) and (3.3)

C = Co cosA

from 3.1 it is possible to derive the well known relationship C = C cos A (3.4 ) L L

o

Figure 3.2a shows the decomposition of the velocity vector into two components; one normal and one parallel to the leading edge. The two elementary strips have the same area.

ORIGINAL PAGE IS OF POOR QUALITY

r

I

A i

I I I I , [ , I I I

a)

I cosA

c

i i ~i 4-

c

r

:, ~O

b) ih

.. ~. 1

•

Figure 3.2 - Side Force due to Induced Drag.

If we now introduce a plane defined by the velocity vector V

o

and by the trailing edge segment where the two strips coincide, we can derive the relationship between the angles of attack measured in the two directions. This relationship is useful when evaluating the angle of attack during sideslip. This can be done by simply noticing that the height from the leading edge to the previously defined plane is the same at the point k as well as at k. This, of course, implies O that the leading edge is a straight line. The effects of twist or dihedral would then be superimposed.

The distance h being a constant, by pure trigonometric consider- ations applied to Figure 3.2b we obtain Co sina = h O (3.5) C sina= h and from (3.3) we obtain sina = sina cosA (3.6)

o

Therefore

rv = i -1 (S ina O )

(3.7) VI, s n A cos and for small angle of attack (3.8) The induced drag measured in the flight direction and the corres- ponding one measured in the direction normal to the leading edge are not the same.

The lift is the same in both·cases;but the downwash angles are not.

The downwash velocity at the lifting line [see 2.3.1 and 2.4.4J must be computed with respect to the flight direction, since it is the wing span measured with respect to such direction that will determine the downwash velocity.

An easy mistake would be to extend the use of the well known expression for the downwash angle (for an elliptic lift distribution) (3.,9) to the case normal to the leading edge W C L - = V rriR In fact, using equation 3.4 and since iR = iR / cos 11. , we find O that (3.10) This implies that the induced drag measured in the two directions is ider.tical, since a.nd where

d. = section induced drag, flight direction

~O

d. = section induced drag, direction normal to the leading

~ edge.

The downwash velocity at the lifting line is induced by trailing vortices aligned with the flight direction.

We shall call the downwash measured this way W.

~ The downwash angles in the desired direction can bow be computed Thus, the section by dividing W. by the corresponding velocity.

~ induced drag measured in the flight direction is given by and, in a direction normal to the leading edge w. W.

d = 1,...1:.= 1, 1 (3.11) i V V0 cos!\.

or d.

d = (3.12 ) i cos!\.

Equation 3.12 shows that the induced drag measured normally to the leading edge is greater than the component in the flight direction.

In a symmetric swept back (or swept forward) wing, the two side- components would cancel each other so that the total induced drag actu- ally experienced by the wing coincides with the component in the flight direction. For the oblique wing, the lack of symmetry introduces a side component of the induced drag whose section magnitude is given by (d.) = d. sin!\. = d tan!\. (3.13) i 1 y 1 0 Therefore, for the case of the left wing forward, the side force contribution due to the induced drag of the wing is (3.14 ) In addition to the side force (Fy)D due to the drag component in the y direction, a second term must be introduced when the wing is swept and at an angle of attack. The wing rotates perpendicularly to x and, therefore, parallel to zb (Fig. 3.3).

b Let us now consider the wing position w.r.t. the stability axes for

the case a 1 0 and A1 0 (Fig. 3.3). This position can be better

o

visualized by considering these two steps: 1) a rotation of the aircraft about its Ys == Yb axis by an angle a ; 2) a rotation of the wing O about zb by an angle A. The final position of the wing is no longer parallel to the Ys axis, the left (forward) tip being higher than the aft one. By idealizing the wing with a straight line and considering its projection Y3 onto the plane, we define the angle YO (Fig. 3.3b). The lift vector being by definition perpendicular to the wing axis and to the velocity (parallel to x ), will therefore be s banked by an angle producing a contribution (positive in this case) to the side force. This feature would not be present in a flying wing since, by banking the wing by YO ' it would be possible to realign lift and gravity, whereas in a conventional configuration the lift produced by the tail and the different inertia properties produce a somewhat more complex picture.

We can now quantify this side force for the case 6f a straight rigid wing, by introducing some geometric considerations.

Let us consider the wing for and assume for simplicity a = A= 0

o

that our stability axes have their origin at the wing pivot; in this case the axes would be aligned with the wing span.

Let us now rotate a about Ys ; we obtain the set of axes x ' O b Yb ' zb ,where Yb = Ys is still aligned with the wing span.

The rotation matrix for such transformation is r cosa 0 - sina x O O 's

~l

= 0 I 0 Ys 0 cosa Z

L sioa

:: J

O O s -~···X S === Wing position for 11.=0 a) s

====== Wing position for a: = 0

o and 11.:1 0 (left wing forward) L Yo .'"" .

Figure 3.3 - Side Force due to Wing Rotation cosA Y(Y) ORIGINAL PAGE IS -------.;;::~::=-------- y s OF POOR QUALITY Figure 3.4 - Effects of Wing Bending on Side Force If we now rota.te the wing by about its pivot, the wing axes A be defined by the following matrix transformation will cosA sinA 0 cosA cosA x cosob sinA - sina ~

r x -I o s

- sinA cosA 0 - sinA cosa sina = cosA sinA Yb Ys O O

y: r

0 0 1 sina 0 cosa z zb L zw.

O s O x x w s z z w s The lift vector is perpendicular to the plane defined by V and

o

Yw and in terms of the stability axes (A~ unit vector) sinA cosa

o)

cosA

S'w (in stability axes) = [R]-l [~] = [R]T [~] =(-

sinA sina O Let us define L=-L'2'

where '2' is a. unit vector .J.. to 'X and

3 s

'2' .'X

or o

3 s '2'·9 ;::0 3 w z e x 'Z' = we have

(~) and since 'X' (in stability axes) =

s 3 0/ z3 z = 0 z3 x sinA sino: = 0 cosA z3 + z3 O y z therefore

'Z' 3 =(~l tanA S inCl \ -.t= 1==j;~-':!:"'2 ...... i

O

1 "'I 1 + tan A sin %

to 'X' and 'Z' We can now define s 3

Y' • 'i? =-0 =

3 s • 'Z' = 0 tanA sino: + y3 = 0 O z

= ( ~ - ) -'=.1 =====1:;;r=2 =. ~2 i

tanA s ino: "'I 1 + tan A s ~n 0: O The aerodynamic force in the x ' Y3 ,z3 reference axes has the following components We can now express these in terms of our stability axes 1 0 0 - D F x - tanA sina O F = '0 y 1 2 · 2 1 2 · 2 + tan s~n a + tan s~n a A A O O tanA sina O 1 F 0 - L 2 2 z 1 2 · 2 + tan s~n a 1 + tan A sin a A O O The value of YO w.r.t. the stability axes X ' Ys ,zs is found s to be tanA sina O (3. 15) sin YO = 2 · 2 ) (1 + tan s~n a A O and for small angles (3. 16) Yo = a tanA O The corresponding lift contribution to the side force is (3.17) All the previous analyses can be extended to the case of a flexible wing having a built-in dihedral (and neglecting the twist contribution) by superposing a correction to the angle (Fig. 3.4) (3.18) y(y) = YO + cp(y) cosA (+ bending modes) where cp(y) • geometric dihedral a,t A='O In this case the side force hlils to be evaluated by means of an integration along y • The equilibrium condition may be reached by s banking the aircraft by an angle cPO which, in terms of stability axes, corresponds to introducing a gravity component in the ys direction.

For a straight rigid wing with no twist, considering total side force (contribution of the oblique wing itself and contribution of the geome- try of the wing rotation) and gravity, the equilibrium equation is

Y = (Fy)n + (Fy)L + mg sin ~O ~ (L)Tot a tanA+ (Fy)n + mg sin ~O

O

• •

SiXlce for the trimmed condition (L)Tot = mg

3.1.3 Wing Rotation.

Figure 3.5 shows the geometry of the wing rota,tion about the pivot and the moment arms of the aerodynamic forces with respect to the center of mass (C. G.) of the aircra,ft.

The aerodynamic force, as assumed in this study, is applied at the quarter chord. Since the wing is stra,ight we can also assume that the line joining all quarter-chord points is straight. Such line is repre- sented by line "m" in the unswept condition, and by line "n" in the . ,~.

swept one. The moment arm x about x 11 used when computing yawing

•

a.nd rolling moments as well as when cOlllPutinl the section perturbation velocity due to rate of pitchtnd/or rate of yaw.

The moment arm y about the. Ys axis, is used when computing angular velocities due to rate of pitch and the pitching moments.

In determining the pitch stiffness C [Ref. 21J it is convenient -0: to refer to the quarter-chord line for the unswept case, the moment arm about such line is indicated by YMA.C' Since the wing rotation affect.s the magnitude of th~ above moment arms, it will also modify the magnitude of all aerodynamic moments .

•

The wing rotation also affects the position of the ailerons with respect to the aircraft centerline. Figure 3.6 shows the ailerons and section of wing which is affected by their deflection. Figure 3.6a shows the case for zero sweep. The shadowed area is defined by two straight lines a.ligned with the free stream velocity V passing TO through the aileron's outer and inner stations. When the wing is swept, (Fig. 3.6b) we can notice that the two straight lines defining the area affected by the aileron a.re noW passing totRe left of the corresponding point on the quarter chord line for the unswept case.

The result of this is an increase in the moment arm about the longitudinal axis for the left aileron, and a decrease in the moment arm for the right wing. The position of the pivot behind the quarter chord line will reduce and eventually eliminate such an effect.

By simple geometric considerations, we can compute the quantities x s <: fJl o (,) 't:l

-_ .. ---- -----J

i x I X Ys = stability axes s X Yo = axes pa.ra.l1el to stability axes centered at wing root o quarter-chord.

c = pivot-C.G. distance p d = pivot quarter-chord distance x = moment arm about x s = moment arm about y YMAC = moment arm about m (quarter-chord line for A= 0) Geometric expressions for the moment arms: x = yO tanA + d sinA Y = c + d cosA - yo tanA p Y = c + d(cosA- 1) - yo tan.t\ MAC p Figure 3.5 - Geometry of Wing Rotation and Moment Arms

-1-----

o :> .

--J-

~ H

t

c,).

. >',: L __

C,) _ ............ _-- .

---.

<a : :

I

i

I

• I I

~OO~N3 ~ _ ~~

'I I::> J.ND I ORIGINAL PAGE IS OF POOR QUALITy V A/C

o

"

~

V

o

C2. ~-~_ .. ~_

i

YCNTIL I Cl ~-t>4~---'-'"' .... - ....... -------- '.

costl. '~-, YCNTOL "- ....

V1 W ~, ,-------i----l.--- C4 .YCNTIR • coati.: I !

-- ... - :

i YC~·--·--costl. ---- --r---- C3

II)

Figure 3.6 Influence of Wing Rotation on Ailerons Geometry

•

and being the same as and C for the case when the ailerons are symmetric with respect to the wing centerline).

3.2 Derivatives 3.2.1 General Methodology.

The perturbation quantities considered in a stability analysis are: - velocity in the x direction: u - velocity (sideslip) in the y direction: v - velocity in the z direction: w - rate of roll (or angular velocity about the x axis): p - rate of pitch (or angular velocity about the y axis): q - rate of yaw (or angula.r velocity about the z axis): r Of these quantities, the second and third are generally normalized with respect to the free stream velocity V • In doing so the new TO perturbation quantities are (3.20) which correspond respectively to a cha.nge in the angle of attack and a change in the sideslip angle as can be seen in Figure 3.7.

While the effects of a perturbation u, ~ , or r must be consid- ered separately, the perturbations p and q ca.n be dealt with in the same fashion as for a, since in fact they all affect the local angle of attack, as will be shown in the next section.

We shall now describe the general methodology for evaluating the

~~

~ts

~~

V TO

.o~

w "" 0; =- 0; - tan V

r==::=::= 1 0; • I - x s

!2~

TO w

t::§]

v

~&

~~ 'z 2 2 '" s V = V TO + W - V

~

TO VI VI V ~ TO v "" .

i C::::::::::, .... 1 0= X S (3 - ta~ V v TO ~ 2 2~ ""

VI = V + V - V

s TO TO .,-.( Figure 3.7 - Perturbation Angles stability derivatives in a systematic way according to the previous subdivision; s. much more complete discu~.ion can be found in References 15, 21, and 22. Such methodology applies to dimensional stability derivatives.

We refer to Appendix A for the relationship between non-dimensional and dimensional derivatives, and to Appendix B for detailed calculations of the non-dimensional stability deriva,tives.

3.2.2 ex, p ,q Derivatives.

Let us now concentrate on a section of our wing. Whenever we consider a, positive perturbation ex in the local angle of attack, the

•

wing a.ctually experiences a new perturbation velocity, w. The result of this velocity w is that the free stream velocity becomes V having a new direction which differs by an angle ex from the previous one.

Consequently, the lift and drag components of the aerodynamic force must now be referred to this new direction. This result is shown in Figure 3.8.

L V __ c....::-+~ + ~_ ... _. X , 0 S

'1r- w

z s • Figure 3.8 Section lift and drag force for an ex perturbation.

Let us now evaluate the forces with respect to the stability axes system for this perturbed case.

d cosO: '":: J- d x J- sina - 0: - == d (3.218) Y tanA == J- "I - d sinO: ':::

z -J- cosO: - [J- + d a]

--

and corresponding moments are: the

z x

== £S (3.21b) M ==

-z YMAC

S N -x x + Yy == S For the equilibrium case, the equations are , ~

X

== -

dO

o

tan (A) == YO dO J-O"lO J- ==

-

Zo

(3.22 )

o

o£

== Zo x

S The changes due to the perturbation 0: are therefore given by ==X-X == J- a - !::>.d !::X !::>.Y - [j. tanA == J-"I - J- 0"1 0 (3.23) /:§..

t:J,+ do: == 6M == - S

I

Because of the small perturbation auumption these equations can be linearized by expanding them in a Taylor .eries about the equilibrium condition (or a= 0) and retaining only the first-order terms. For the first equation, this approach leads to

/§. = .2- [J.a - t:. d"] a (3.24)

oa a= 0

and the same can be done for the other equations • . This is the classical assumption of linear a.erodynamic theory which is to a.ccept the following approximation for stability derivatives: (3.25) where

A = a.ny aerodynamic force (or moment)

b = any perturbation quantity.

Consequently, the stability derivatives at our section for a per- turbation a are given by (3.26) where

X = [o~ (t a-&1) ] a- 0 = [t+~~ a- ~~ ]a= 0 = to - (~) a= 0 (3.27)

a (3.28) (3.29) (3.30)

~ = o~ [0: tanA + ccilnst termsJo:= 0 = tanA

The same approach can be used when studying 0: and 0: perturba- p q tions. In both cases the perturbation will produce a change in the angle of attack. Such change will no longer be constant along the span, but will be (J.3l) for the pitch, and (3.32 ) for roll.

Although the distribution in the angle of attack is a.ntisymmetric, the corresponding increment in lift, and consequently drag,because of the shape of the FWLD will not be asymmetric. The asymmetry in lift, case of a symmetric wing in roll, does not produce any variation to the lift vector. Loss of asymmetry introduces a change in the total vertical force Z as well as in X.

The same approach used for the 0: derivatives can be applied to the evaluation of roll and pitch derivatives.

x s • / 81. • des lip Figure 3.9 - The only differences are: angle of attack which is no longer a a) the section perturbation but varies according to Eqns.

constant as in the previous case, (3.3l) and (3.32); b) the linear operator must be replaced, for the pitch by

oa

(3.33) and for the roll by (3.34 ) The angle y remains unchanged.

The wing aerodynamic derivatives with respect to the stability axes can be obtained by integrating the section values over the span.

3.2.3 ~ Derivatives.

The effect of introducing a sideslip angle ~ corresponds to a negative 6A for the wing (Fig. 3.9). Consequently, for the wing

o "

(3.35 )

,,~= - "A

The evaluation of the ~ derivatives is not trivial when consider- ing the aerodynamic moments. In this case, because both FWLD as well as the distribution of lift due to twist will be affected, it is not as simple to derive a,nalytic expressions as for the previous cases. A numerical calculation would be possible by considering results at two different wing sweeps and then comparing them. For the cases where no moments are involved, the total aerodynamic forces considered are (3.36) In addition, the following a.pproximation ca.n be made.

(3.37) (L)Tot = [(L)Tot] A= 0 cos A = D + I (L) (C) .1 cos A

o Tot L Tot TT lR

(3.38 ) and, since lR = (lR) A= 0 cos A , it is possible to evaluate the ~ derivatives of these forces.

3.2.4 r Derivatives.

The new yawing derivatives introduced by the oblique wing are due to two factors: a) side force and b) aerodynamic coupling between rolling and pitching moment. A rate of yaw increases the local stream velocity according to the relation L.u= - x • r (3.39) Therefore, for the dynamic pressure the variation is H.O.T.

q(y)

~2)

(3.40) ( Substituting (3.39) into (3.40), we obtain (3.41) The linear antisymmetric change in q(y) due to the perturbation r will now affect the aerodynamic forces due to both the flat wing and twist (including effects of dihedral and camber) contributions. Since the antisymmetric variation in q(y) is applied to a symmetric (cruise condition) distribution, the variation in the loading distribution will also be antisymmetric, like for the case of a symmetric aircraft. There- fore, when first order variations only are considered, no changes will occur in the wing total aerodynamic forces, but all three aerodynamic moments will be affected.

Since q(y) is the only term which changes in the expressions for

x Y, and Z, the evaluation of the yaw derivatives is reduced

to computing (3.42) and similarly for y, and Z

oY x

.. Y 2 (3.43)

dr = 0 DO

(3.44) A simple integration over the span of the last three expressions will determine the ya~ derivatives of the aerodynamic forces. For the ~s ,M and N moments, the expressions to be integrated are S S x - f = - Z 2 - x

s o U

o

x (3.45) M = + Zo 2 - y S U

o

3.3 Summary.

In this Chapter we have introduced and evaluated the two side force contributions, one due to induced drag and one due to wing rotation, that an oblique wing aircraft would experience. The effects of wing rotation were then analyzed. The general methodology for computing stability derivatives was also miscussed.

IV EQUATIONS OF MOTION 4.1 Rigid Body The equations of motion for the aircraft can be derived from Newton 's Second Law of Motion, which states that the summation of all external forces acting on a body must be equal to the time rate of change of the momentum of the bbdy, and the summation of the external moments acting on a body must be equal to the time rate of change of the moment of momentum (angular momentum). The time rates of change are all taken with respect to inertial space. These laws can be expressed by two vector equations.

I

V = .! Ei

(4.1) m (4.2) where I indicates the time rate of change with respect to inertial space.

Now, the external forces and moments consist of equilibrium values plus a perturbation value which stems from a, difference from this equi- librium condition. Thus,

F=F +iF

o

(4.3)

T=T +Er

o

In the dynamic analyses to follow, the aircraft is always considered to be in equilibrium before a disturbance is introduced. Thus, F and O TO are identically zero.

The equilibrium forces consist of lift, drag, thrust, and gravity, and the equilibrium moments consist of moments resulting from the lift and drag generated by the various portions of the aircra.ft a.nd the thrust.

The following observations are made to a.chieve a linear analysis.

First, the aircraft is in an equilibrium condition before perturbation.

Second, the mass of the aircraft remains constant during any particular dynamic analysis. Third, it is assumed that the aircraft is a rigid body. Fourth, it is assumed that the earth is an inertial reference and unless otherwise stated, the atmosphere is assumed to be fixed with respect to the earth. The time rate of change of the velocity vector with respect to the earth, in the assumed reference axes, is given by I S V=V+Wxv (4.4 ) where S indicates time rate of change with respect to the assumed reference axes system (non inertial); w is the angular velocity of the aircraft with respect to the earth; and x signifies the cross product.

Similarly I S H=H+wxH (4.5 ) V and W can be written as (4.6) where and represent the steady state conditions and and /::.V are the perturbation quantities.

Their components in stability axes are: The angular momentum is given by the dot product of the inertia tensor, I., and the angular velocity vector, W I'J H = I • W (4.8) I'J and Equation (2.5), I being time invariant because of assumption number four, can be rewritten a.s I

H = I . .§. + W x (I . ill)

(4.9) W The inertia matrix of an oblique wing aircraft differs from the one of a synnnetric aircraft because of the non-zero product of inertia. I xy For an oblique wing aircraft, the inertia matrix in body axes is I I I xz xx xy I = I I 0 b xy yy I 0 I xz zz Because our stability ana.lysis is based on the stability axes system previously introduced, it is necessary to transform the inertia matrix from body to stability axes. This transforma.tion is carried out in deta.il in Appendix E.

Because of the transformation the inertia matrix in stability axes will no longer have a zero In fac t, a,s shown in Appendix C, it I yz is given by I' = - I sina: (4.10) yz xy where the prime denotes stability axes.

The equilibrium condition we are going to consider in our analysis is that of ~*raight level flight, therefore

P = Q = R = V = W = 0 (4.11)

o 0

For this condition, after substituting (2.4) into (2.1), and after algebra, obtain some we F ' qw - rv x 1 - 1 F = ru - pw (4.12 )

F = v + u r

+

m m y o

vp - qu F Til - q u > z O linear terms non linear terms Similarly, Equation (2.2) becomes • + I I I' p+ I' r (I' q-I' r) q+(I' -I' )qr xz xx xy q xz xy zz yy f • + I I I' (r _ 2)+1' = I' · + I' i: qr+(I' -II )pr T= M + yy q yz xz p . xy xy p xx zz

-

1,(2_2)_1' • + I I

• + I' r qr+(I' -I' )pq I'

N q zz xy p q xz P yz xz yy xx , , linear terms non linear terms (4.13) It is now necessary to expand the applied forces and moments and to express them in terms of the changes in the forces and moments that cause or result from these perturbations. These latter forces are usually of an aerodynamic a.nd gravitational origin.

If it is assumed that, as the disturbances are small, the partial derivatives are linear, the differentia,ls can be replaced by the actua.l increments, and F and T can be written as

d~ \ b

( i

i Ib.=o

1.

(4.14 )

T = 6T = 1: (o~~ \ b i

i 1.)b .=0 1.

OF)

o~T) represent the aerodynamic stability

where ~b. and

(

(

1. b.= 0 i b = 0 1. i derivatives eva.luated at the steady state condition when the perturbation variable b is zero.

i The linearized set of equations therefore is: r

v + u = E (::~bo

O i 1.

(4. 15)

I' '+1' 4+1' r=

xx p xy xz

II p+II 4+1 r=

I: (Q~). b

. xy yY yz i i 1. b .=0 1.

(4.15 ) 1 1 II 1>+1 4+1 r= I:

I)b

xz yz zz i i

(

1. b.=O 1.

In the set of Equations (4.15) the terms II 4 and II p, not

xy xy present in the case of a synnnetric aircraft, introduce the inertia

coupling between pitch and roll motions. The terms II 4 and II r

yz yz have also a coupling effect and seem to make the coupling between longitudinal and lateral dynamics even stronger, but their coupling is only apparent since it depends on our choice of axes system.

In addition to the inertia, coupling existing in synnnetric wing air- craft between roll and yaw, oblique wing 8.ircraft experience an inertia coupling of pitch and yaw motions. In table 4.1 we report the matrix representation of the set.of Equations (4.15) in terms of Laplace trans- form.

4.2 Effects of Flexibility.

The analysis done so far applies to a rigid aircraft.

Now, it is known that the stability and control characteristics of flight vehicles may be profoundly influenced by the elastic distortions of the structure under a.erodynamic load. Many of the important effects of distortion can be accounted for simply by altering the aerodynamic derivatives. The assumption is made that the changes in aerodynamic

e

~c

~$

~~~~

£:'"t:t

/:""ftA.

$~

t;; S - F - (F + sF ) - (F +SF .)

- S F - (F + SF ) I I u - F x x X x 1' x x x x a tx u p a q II II r F Y - F - (F + SF ) -[5(W +F ) +F ] I I a

[S(UO·F )-F J - SF U -F + ~

O Y Y Y Y Y a Yp Yep. o Y 5 u a Y ll ll q r - F [5(U -F .> -F ] - (F- + SF ) 5(V -F ) - S(U + F ) II o - F E E E o z z o z 1: E~ a a u p 11 q ·r

I I I

b.

• D Fx(s) I - f - (f + sf pIS - (f +5~) 5(1'5-£)

I I 5(I'5-f)

a I' 5 - f u a x p ll xY q xz r -....J N

I

a - M I I - <ii + SMa> - (~+S~) 5 (I' 5 - M ) 5(1'5 -N') I' 5-N u a yx p yz Y q r - N - (N + 5Ntr> I I r - (Nil +5N~) S(I'S-N) I' s2 -SN I' S - N u a xz p yz q z r TABLE 4.1 Lapla.ce Transform of the Equation of Motion.

loading take place so slowly that the structure is at a.ll times in static equilibrium. This is equivalent to assuming that the natural frequencies of vibration of the structure are much higher than the natural frequen- cies of the rigid-body motions. Thus, a change in load produces a proportional change in the shape of the vehicle, which in turn influences the load.

When the separation in frequency between the elastic degrees of freedom and the rigid-body motions is not large, then significant inertial coupling can occur between the two. In that case, a dynamic analysis is required, which takes account of the time dependence of the elastic motions.

The elastic motions have no inertial coupling with the rigid-body .

motions except through I. However, it has already been assumed, in the

previous paragraph, that such time rate is second-order and negligible in the small-perturbation theory.

The only coupling existing between elastic and rigid body motions is due to the ~erodynamics.

The aerodynamic derivatives associated with the deformations of the airplane are of two kinds: those that appear in the rigid body equations, and those that appear in the added equations of the elastic degrees of freedom.

In our analysis we shall use a. quasi-steady approa.ch neglecting unsteady aerodynamic effects and assuming that only the wing is flexible.

Therefore, we shall also neglect the added equations of the elastic degrees of freedom and refer to Ashley, Etkin, and Blakelock [15, 21, 22J for the complete study.

In a quasi-steady analysis the effects of flexibility on the wing are basically reduced to static bending and torsion.

Because of this bending, the aerodynamic center sh~fts forward changing the value of all the aerodynamic moments and their related deriva.tives. The side force due to lift is also affected. These static effects have been included in our numerical analysis and will be described in that section.

I

V NUMERICAL ANALYS IS 5.1 Model Description In the previous chapters we have discussed qualitatively the influ- ence of an oblique wing on the aerodynamics and dynamics of aircra.ft.

In this chapter we shall attempt to quantify these new properties and evaluate their influence on the stability of an oblique wing aircraft.

The aircraft configuration that has been studied is the Boeing single-body yawed wing aircraft model 5.3 (Fig. 5.1), that wa.s studied for NASA [Ref. 24J. Unlike what is shown in Figure 5.1, our analysis, as well as the one done by Boeing, was carried OlJ,t for the "left wing forward" case. The nominal configuration and flight condition for the simulation were: Mach mnnber = 0.8 Altitude = 20,000 ft (6096 m)

Gross weight = 400,000 lb (181,440 Kg.)

Wing sweep = 45 degrees

C.G. location = .355 (M.A.C.)~O (body station 57.8 m) The inertia properties being referred to body axes were transformed to the equivalent in stability axes according to the transformation described in Appendix C.

The .355 M.A.e. center of gravity location in the nominal con- figuration produced a longitudinally unstable vehicle. This con- dition was selected by Boeing in order to achieve a more satisfactory overall vehicle design; since the design included a longitudinal S.A.S. this was also the case in the Boeing SST design in the subsonic regime.

In the unstable (without SAS) configuration, the normally complex short period roots migrate significantly; one moving to the positive real axis and one combining with the "rolling convergence" root to form a new complex pair in the LHP.

In order to study the effect of the oblique wing under more "normal" conditions, we chose to modify the nominal configuration . so as to achieve a stable configuration for the zero sweep ( A = 0 ) condition. This was accomplished by moving the e.G. forward.

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FIGURE 5.1 NASA-BOEING DESIGN FOR OBLIQUE-WINGED TRANSPORT (Courtesy of NASA) 5.2 Aerodynamic Results In chapter II we described two different methods for evaluating the spanwise lift distribution for the oblique wing and how to obtain the spanwise induced drag distribution from the knowledge of the lift.

Two numerical programs, based on these two methods, have been written: OWSD/ST computes the stability derivatives for a rigid oblique wing aircraft by means of strip theory; OWSD!LT , instead, applies the linear theory approach and can a.1so include the static effects of a flexible wing.

We shall now report the numerical results obtained with the two methods, for a wing at a skew angle A of 45°.

In our strip theory analysis, the wing was assumed to be rigid; therefore, the results obtained with that analysis will be compared against the rigid wing case of linear theory. The, we shall compa.re the results given by linear theory for the rigid and flexible wing.

5.2.1 Comparison between Strip Theory and Linear Theory (Rigid Case).

Figure 5.2 shows the spanwise lift distribution at A = 45° as

computed by means of the lifting line method, for the flat wing as well as for the case including built-in twist. The twist is linear, and the tip values were chosen so that the resultant lift was acting at the wing centerline. In a real design the airfoil camber distribution would be so that the cruise lift is acting at 50% chord. The assumed twist distribution certainly is not the optimal one, leading to a minimum induced drag, but, at least, it produces no rolling moment .,~ and, for the purposes of our analysis, is a satisfactory cruise condition.

The twist values, for the unswept wing, are Left tip

= + 2.6 degrees

Right tip = - 3.35 degrees

The downwash angles for the flat wing computed by the two methods are shown in Figure 5.3. The lifting line approach (Fig. 5.3a) shows a large variation in the spanwise distribution of the downwash; it also indicates that the right wing, beyond the 60% half span, experiences an upwash which, in terms of horizontal force, results in a thrust.

The induced drag distribution is shown in Figure 5.5 where the nega.tive value corresponds to the thrust mentioned before.

The results obtained by means of strip theory show a slight varia- tion in the downwash angle distribution and the corresponding induced drag distribution appears to be almost symmetric about the wing center- line. The drag distribution, as obtained with the lifting line theory, produces a yawing moment which tends to unskew the wing. This effect is not shown in the results obtained with strip theory.

The tendency to unskew the wing has been experienced also during wind tunnel tests run at NASA-Ames Research Center. Since the model was not a pressure one, it is not possible to tell whether this was due only to a higher drag an the forward wing, or also to a thrust force in the aft one.

The side force distribution (Fig. 5.4) predicted by the two methods differs since the drag terms differ. The side force predicted by the ORIGINAL PAGE IS OF P{)()D ATT A T T ....... _

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ORIGfN'AL PAGE IS lifting line theory increases the unyawing moment produced by the drag, instead, the strip theory result shows the opposite effect.

We can therefore conclude that the spanwise induced drag distribu- tion, as computed by means of strip theory, is a poor approximation.

We shall now continue in the comparison between the two methods for the cases when perturbations are considered. Unfortunately, the numerical program computing stability derivatives according to strip theory computes only the nondimensional lift distribution. Thus it is only possible to compare the results obtained with the two methods qualitatively.

The first perturbation quantity considered is a.

(positive forward) Figures 5.6 and 5.7 show respectively the & x For the sake of clarity we recall that, for the perturbed and !:iF y case !:iF is given by x Figure 5.8 shows the change in lift due to a roll perturbation.

According to strip theory, no change occurs in the lift at the air- craft centerline, whereas the lifting line s~lution shows, for positive rate of roll, a negative !Sl at the aircraft centerline.

A greater difference shows up in the change in lift due to a pitch perturba.tion. This greater difference is also due to the fact that, in the case of strip theory, the vertical velocity introducing the aero- dynamic twist is computed at the quarter-chord, whereas, in the lifting line method, it is evaluated at the 3[4-chord point.

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ORIGINAL PAGE IS

OF POOR OTT 11 T TTV Figure 5.10 shows the total lift distribution for a negative side- slip of 5 degrees.

We shall return on the sideslip case in the next section, where we discuss in detail the behavior of the oblique wing in sideslip. For the moment, we shall only point out how good the agreement is between the lifting line theory results and those obtained with the empirical method described in section 2.5.

5.2.2 Comparison between Rigid and Elastic Wing (Linear Theory).

The lifting line method described in section 2.4 implies the know- ledge of the station 2-D lift slope. For this reason a value for this is considered an input data for the numerical program OWSD/LT. In our analysis the section liftslope has been assumed to be constant along the span.

This is not true in most cases and specially for our model, since the thickness to chord ratio varies along the span. The value of the 2-D lift slope has been adjusted so that the wing lift slope coincides with the value computed by Boeing [Ref. 24J in its study. No built-in dihedral was assumed, but only a linear twist was introduced, satisfying (as mentioned in the previous section) the zero rolling moment condition.

In our model, the pivot location (50% root chord) does not coincide with the quarter-chord line (where the lift force is assumed to be acting); therefore, when the wing is skewed, the wing center line does not coincide with the aircraft centerline. For that reason the right wing has a span larger than the left one. Therefore, the lift distribution for the cruise condition, satisfying the equilibrium condition about the roll axis, can no longer be symmetric in order to balance out this asymmetry in the two semiwings. This asymmetry can be noticed in Figure 5.11 which shows the

spanwise lift distribution at A = 45 (rigid and flexible wing)for this twist

distribution and for an angle of attack of 3.75 degrees (.0652 rad.).

Since the twist was computed for the rigid wing case, the lift distri- bution of the elastic wing will produce a strong positive rolling moment.

The flat wing case is shown in the next figure (Fig. 5.12) for the same angle of attack.

It is interesting to notice that the flexible Wing case produces almost no rolling moment about the aircraft centerline (The resultant lift vector is applied at the quarter-chord station having a distance to the right of the longitudinal axis equal to 0.72 ft).

Such a small difference does not even require a built-in twist, but only a little aileron correction. At this point it should be pointed out that the built-in twist is, for an oblique wing aircraft, a poor design solution. In fact, since the twist measured in the flight di- rection varies with the cosine of the sweep angle, the twist correction is maximal when the wing is unskewed ( and then we do not need any twist correction since the FWLD is symmetric) and decreases when we skew the wing.

A built-in dihedral is, therefore, a far better solution, producing no twist. for the unskewed case, and increasing the twist correction when the skew angle increases.

We shall complete the analysis of the cruise condition, as from our ORIGINAL PAGE IS nF pnnR OTT AT J'l'V Ii i. : d: : I 'i :\

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: " itf :[ '",: i':': j 1 .~ J ~ ii ,II" 1: , I C +flf' numerical model, with Figure 5.13. Figure S.13a shows the downwash distribution for the flat wing, where the negative values correspond to the upwash. The flat wing drag distribution corresponding to this downwa.sh is shown in Figure S.13b. The flexible wing shows a much higher drag on the forward (left) wing and a thrust over a larger portion of the aft wing. This is expected; in fact the forward wing, by bending, in- creases the angle of attack and consequently the lift (this behavior is well known as the divergence problem of a swept forward wing), where- as the opposite happens on the aft one.

The direct consequence of the higher lift of the forward wing is an increase in the upwash velocities induced on the aft one. Instead, for the aft wing, the decrease in lift will result in a decrease in the up- wash field induced on the forward one and, consequently, lead toward a further increase in the downwash of the aft wing. These are the rea- sons why the flexible wing experiences a higher downwash value on the forward wing, and a larger portion of the right one is affected by the upwash velocities.

Then, when we multiply the lift (a positive value) times the corre- sponding downwash, in order to obtain the spanwise induced drag distri- bution, we multiply the already greater downwash times a larger value of lift for the left wing. For the right wing, instead, the lift magni- tude is smaller and this kJ the reason why, though affecting a larger portion of the aft wing, the thrust reaches a lower maximum value than the corresponding one for the rigid case.

We shall now continue our comparison analyzing the results for the AWl 1 V ltV (lU\}U \!to SI GIBVd 1VNIBIHO "F""

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cases simulating perturbed states as are assumed in our computation of the stability derivatives.

The numerical program OWSD/LT evaluates the stability derivatives by first computing the distribution of the aerodynamic forces (lift, drag, and side force) for the perturbed condition, then evaluating the increment from the corresponding cruise distribution, and finally inte- grating these increments as well as their product with the corresponding moment arms in order to obtain the incremental moments over the entire wing span.

The desired values of the stability derivatives are then obtained by dividing the results of the integrations by the perturbation quantity and by the appropriate nondimensionalizing factors [See Appendices A and B]. Then, derivatives are computed with standard expressions [Appendix B and/or Ref. 15, 21] and the yaw derivatives can be evaluated according to strip theory as described in section 3.2.4. Therefore, for both of them it is necessary to simulate the perturbed condition. Consequently, the conditions to be simulated have been reduced to: angle of attack, roll, pitch, and sideslip.

The a-perturbation has been simulated by increasing the cruise angle of attack by 3 degrees (.05236 rad.). The new lift distribution, including twist contribution, is shown in Figure 5.14.

The rigid wing shows a visible increase in the aft winglif~,due to the increase in the flat wing 1ifto Eor the elastic one, this effect is much less pronounced. These two behaviors can be predicted by looking at the shape of the FWLD (Fig. 5.12); in fact, a change in a will introduce a c> , ORIGINAL PAGE IS OF POOR OTT AT ,lIT first order variation in the lift, proportional to these shapes. The variation, 6F ,in the horizontal force (Fig. 5.l5a) is strongly af- x fected by the lift contribution in that direction during the perturba- tion, as we have discussed in section 3.2.2; in fact The increase in lift for the rigid wing is more pronounced in the aft part where it contributes to the thrust already experienced because of the upwash field, but on the forward wing its increase is not suffi- cient to offset the drag increase.

For the elastic wing, the FWLD has ~ more symmetric shape than for the rigid one; we would therefore expect the transition point from negative (drag) to positive (thrust) X-force to occur further outboard on the left wing, instead, the two shapes behave similarly, except for the magnitude. This is not the ca.se, according to our results, a.nd the reason is probably because of the downwash distribution: stronger down- wash on the forward wing, and consequently a higher induced drag to counteract the higher increase in lift; but also stronger upwash on the aft one with higher resultant thrust where the lift increment is lower than the rigid case. The same reasoning can be applied to the variation in the side force (Fig. 5.l5b), where the two shapes look similar, al- though they differ in magnitude.

The perturbations due to roll and pitch a.re simulated by intro- ducing an aerodynamic twist cor:;:'esponding to a linear va.riation in the free stream vertical velocity [See section 3.2.2J. The roll angular

velocity is chosen to be such that the twist is equal to -5 degre~s· at the

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ORIGINAL PAGE IS ,",-T.\ nAAU ATT AT .TTY left tip and zero at the aircraft centerline [see Eq. 3.31].

For the pitch velocity, the left tip condition remains - 5 degrees and zero twist now coincides with the wing station corresponding to the intersection of the quarter-chord line with the lateral axis y • s The three different twists used in our study are shown in Figure 5.16.

Figure 5.l6a shows the built-in twist; the aerodynamic twist corres- ponding to our roll simulation is shown in Figure 5.l6b. The aerodynamic twist due to the pitch simulation (Fig. 5.16c) is not perfectly linear because the downwash velocities have been measured at the 3/4-chord points, which do not lie on a stra.ight line. As we would expect, the aerodynamic twist introduced by a positive rolling motion increases the lift distribution on the aft wing and decreases it on the forward one.

Figure 5.17 shows the spanwise lift distribution corresponding to the cruise angle of attack plus a twist given by the superposition of twists a) and b) of Figure 5.16. The increment in the vertical force is of greatest interest. Although such an increment has a contribution due to the drag, according to [see section 3.3.2J where such a contribution is negligible compared to the actual lift changes, therefore the 6. Z shown in Figure 5.18 can be thought of as the actual change in lift without any serious loss in accuracy.

(Y) .; ORIGINAL PAGE IS OF POOR QUALITY !

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. + ORIGINAL PAGE IS OF POOR QUALITY From those distributions we notice that the rigid wing shows a positive variation on the aft wing, larger, in magnitude, than the negative one. The opposite is true for the flexible wing. Moreover, if we look at the shape of the corresponding case, as simulated by strip theory, we can observe how erroneous the strip theory approximation is for the flexible wing case.

The variations I::.x and I::.y look quite complex (Fig. 5.19 a) and b».

The most important derivative computed from these two distributions is the yawing moment. In a symmetric aircraft the side force is always negligible. This, as we have proposed in section 3.1.2, is not true for an oblique wing aircraft, in fact the side force has a magnitude compa- rable with the horizontal one even in a perturbed condition. By inspec~ tion, we can also notice that the yawing moment produced by the I::.x and I::.y distributions is, in this case, negative, and the rigid wing shows a larger magnitude.

The same analysis can be extended to the pitch perturbation; there is only one difference: the way the aerodynamic twist is measured in our linear theory model.

Eqn. 3.31 implies a linear variation in the aerodynamic theory; this is a common assumption and was used also in our strip theory analysis.

The linear theory, as we mentioned earlier, implies the evaluation of the vertical velocities, induced by the angular velocity q, at 3f4-chord.

Figures 5.20 and 5.21 respectively show the spanwise lift distribution corresponding to the cruise angle of attack plus the superposition of l"ITt",j ,#1ljl,. iil,l.e':.f"- I-t- j: + t _

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Figure 5.22 shows the ~Z distribution a,nd 5.22b the ~ Y distri- bution.

In our model, the sideslip is simulated by increasing the skew angle by 5 degrees. This, as we have already mentioned, corresponds to a nega- tive sideslip according to the reference axes system chosen. Thus, when looking at the variations in the aerodynamic forces, we must change the sign when considering the sideslip case. Figure 5.23 shows the total lift distribution for the new skew angle.

Increasing the skew angle, the angle of attack measured in the flight direction decreases (Eq. 3.8). In our case, the relationship between perturbed and cruise angle of attack is: cos 45

CX = CX ° 0

cos 50 The results shown in Figure 5.24 are of great interest.

Increasing the skew angle (which corresponds to a negative slip), the total lift decreases and the spanwise distribution of such variation is asymmetric (Fig. 5.24). This asymmetry will result in negative pitching and rolling moments; therefore, a positive sideslip introduces positive rolling and pitching moments. This is an unsta.ble behavior and can be visua.lized by considering the trend of the spanwise lift distribution when the wing is skewed.

The upwash field generate~ by the forward wing introduces the build-up in lift on the aft wing.

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~~ lEi , j 7 :): As we increase the skew angle, this build-up will shift more and more towards the aft wing tip. Thus, this shift will introduce negative pitching and rolling moments (left wing forward case). When we unskew the wing, the trend will be the opposite; consequently the lift build-up will shift towards the center decreasing the magnitude of the negative rolling and pitching moments. These are positive increments in the moments and the reduction in skew angle corresponds to the sideslip situation. The effects of flexibility worsen this undesired behavior.

In our model, we have used linear built-in twist to stabilize the rolling moment. In this case also,we see how poor such a solution would be as compared to the case of a built-in dihedral.

The effectiveness of the linear twist decreases with the cosine of o the skew angle; thus, the amount of twist required at 45 to trim the aircraft would introduce a strong positive rolling moment when the wing is unskewed.

Since a sideslip perturbation affects the total lift distribution, the contribution due to twist in a positive sideslip situation is an increase in the undesired positive rolling and pitching moments. Since the equivalent twist due to dihedral varies with the sine of the skew angle, the previous unfavorable situation now becomes a favorable one; in fact, a decrease in sweep will introduce a variation in the equivalent twist which results in a decrease in the forward wing lift and an in- crease in the aft wing lift, which is a stabilizing trend. The yawing moment due to the variation in the longitudinal force, ~ , is positive for our negative sideslip simulation (Fig. 5.25a.), and this is a favor- = , ~ a.ble result.

From the shape of the side force (fig. 5.25b) we can observe that the contributions of flat and rigid wings to the yawing moment can differ even in sign. In fact, for the negative sideslip case, the rigid wing shows a decrease in side force whose resultant force is applied to some point on the aft (right) wing. The rigid wing, instead, shows the resultant side force acting on the forward (left) wing. Therefore, the rigid wing, in sideslip, experiences a side force that produces a desta- bi liz ing yawing moment (pos i ti ve for the nega.ti ve sides lip case shown in Figure 5.25b), whereas the elastic wing shows a tendency to produce a favorable rawii1g~moment.

The effect of the wing rotation on the ailerons geometry has already been discussed in section 3.1.3.

Table 5-Ia shows the ailerons dimensions for the unswept case, and Table 5-Ib shows the corresponding ones assumed in the computer simula- tion for the 45 skew angle; the quantities used in this Table are defined in Figure 3.6.

Three aileron deflections are considered: 1) left aileron deflected 5 degrees 2) right aileron deflected 5 degrees 3) both ailerons deflected; left 5 degrees, right - 5 degrees.

The elastic wing case considered so far assumed the elastic axis (E.A.) to coincide with the wing quarter-chord line. In addition to this elastic case and to the rigid one, we also considered, for the ailerons only, the case when the E.A. is at a distance from the quarter

YCNTOL = 93.63 ft CNTCOL = 2.2 ft

Left Aileron

YCNTIL = 73.93 ft CNTCIL = 3.66 ft

YCNTOR = 93.63 ft CNTCOR = 2. 2 ft

Right Ai leron

YCNTIR = 73.93 ft CNTCTR = 3. 66 ft

a) Ailerons Nomina.l Dimensions at !I. = 0

Wing Station Distance from (non dimen.)

ArC Centerline [ft] Outboard Inboard Outboard Inboard Left Aileron .937 .837 64.45 55.31 Right Ai leron .862 .662 66.03 51. 75 b) Computer Approximation at A = 45 TABLE 5-1 Ailerons Dimensions o The latter represents the real situation, but, since at 45 skew chord.

angle the bending contribution to the wing twist is by far more important .

than the contribution due to torsion, the differences between the two elastic cases are small, at least as fa,r a,s our analysis is concerned.

We shall see how the same is true for the ailerons also.

Figures 5.26 through 5.28 show the total lift distribution corre- sponding to the three aileron deflections considered, for both the rigid and elastic wing cases, having the E.A. coinciding with the quarter-chord line.

The aileron effectiveness for these three cases is shown in Figures 5 • 2 9a , b, and c.

Once more, we can observe the peculiar behavior of the oblique wi~g.

In fact, the forward aileron, affected by a stronger downwash, is less effective than the aft one, which is instea.d influenced by a strong upwash. The variations in horizontal force, Dx" are shown in Figures 5 • 3 Oa , b, B,nd c.

Because of the lack of points in the areas of interest, the shape of the curves looks unusual; nevertheless it is interesting to notice the Dx variation introduced by the aft (right) aileron.

The positive right aileron deflection produces an adverse yaw moment much smaller than the cor::esponding one produced by the left a.ileron (fig. 5.29 and 5.30). We have alrea.dy noticed how the favorable upwash field decreases the drag on the aft wing; in case of right aileron de- flection the induced dra,g rise is not only lower than the corresponding one for the forward aileron, but the increment in lift, because of the = = ORIGINAL PAGE IS 1'\l:1 1),V,\"o OTT I! T I'l'V , ,~ ,-, .. ; l l . tj:~;, 1: 1 : n!I;;': '.!, .'; 't! Ili,lril: 'ii: r',!" :;i.: I;! I,i[ il l :>;11 :"lllli; "il 1:':11 1 ,I'!: : i'li': .::!, : ~::.,; '.'.;,:, ':::: ..

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The same effect can be noticed in the case of antisymmetric deflec- tion (Fig. 5.30c).

The negative deflection in the right (aft) aileron produces a decrease in drag, or equivalent thrust, followed by a drag increase on the outboard part of the wing. This behavior suggests the possibility of using the aft aileron only for control during the cruise condition; but this solution implies, first, the experimental confirmation of the analytic results and, second, that there are no adverse aeroelastic effects related to that solution.

Figures 5.31 through 5.34 are referred to the elastic wing case having the E.A. passing through the wing pivot (5ifroroot chord) and parallel to the quarter-chord line~ Very little difference can be noticed between these results and the corresponding ones for the other elastic wing case considered.

In fact, the twist contribution due to bending is much larger than the contribution due to torque.

5.3 Stability Derivatives and their Influence on the Natural Modes.

At this point, the logic flow of our ana.lysis would require the discussion of the numerical results of the stability derivatives. We shall postpone it to the next section and carryon, instead, a qualitative analysis of the influence of the stability derivatives on the natural mode of an oblique wing aircraft.

= , ORIGINAL PAGE IS .f\1<' 1){){)D f\TT A T T'f'V = , = = ORIGINAL PAGE IS ........... "'''''A-n, ATT A T Trt"V In Chapter III, the new stability derivatives due to the skewed wing were introduced without any attempt to define their importance to the dynamic stability and natural modes of the aircraft. This analysis had been carried out only on the basis of understanding the behavior of an oblique wing aircraft and its differences from a symmetric aircraft. We shall now discuss the result of a numerical investigation to determine the influence of the new stability derivatives, as well as of the usual ones, on the dynamic stability of an oblique wing aircraft.

The set of six linear differential equations dervied in Chapter IV was numerically solved using an available computer program (GSA), which wa.s originally developed by Lockheed Missiles and Space Company. The program solves a set of linear differential equations using the Laplace transform method and gives root locus, bode or time domain plots for the system. The characteristic polynomial is of the 8th order.

The influence on the roots of the characteristic equation is now analyzed for each nonzero derivative by means of a root locus. In this way, it is possible to find the derivatives having influence on the natural modes. This study has been carried out for the new as well as the conventional derivatives, by varying one derivative at a. time, start- ing from zero to a value double that of the corresponding one reported in Table 5-111, column 1.

It is obvious that this is actually not possible in a realistic analysis, since the parameters affecting one derivative may affect other derivatives also; the range itself does not reflect a real case except for some of the derivatives depending on the skew angle. The purpose of

this analysis is only to localize those derivatives whose contribution to the root location is negligible and to compare them with each other.

For the sake of clarity, we have labeled the roots according to the classical definitions though, in the case of a skewed wing, they may lose part of their meaning. Table 5-11 reports the results of this investigation and shows the influence on each natural mode. They were evaluated according to the percent change in the natural modes while the value of the derivative ranged from zero to the maximum value assumed. The symbols used are: less than 5% 5 to 20~ * 20 to 50% ** ,~** 50 to 80 % *,~** over 80 % 5.4 Stability Derivatives.

In section 5.2 we have analyzed the spanwise lift distribution of the aerodynamic forces as computed with our numerical models. The stability derivatives are the direct consequences of those spanwise distributions.

To date, neither program computes the stability derivativee with

respect to rate of change of a. These derivatives take into account

the time required for the effect of downwash produced by the wing to reach the horizontal tail. For symmetric aircraft, the methods described in References 27 and 28 are usually used.

The effect of downwash is to reduce the angle of attack of the stabilizer. Because the wing is at a skew angle, the distance from the wing to a conventional stabilizer varies from tip to tip and complicates the computation of such derivatives.

Since the analysis done in section 5.3 showed that the influence of these derivatives on the dynamic of the aircraft is negligible and because of lack of time, we decided not to further investiga.te the analytic and numerical evaluation of such derivatives.

We recall that the nondimensionalizing quantities used in our analysis [Appendix A] are: cruise speed, wing span, and mean aerodynamic chord, both for the unskewed configuration.

The results obtained, together with the Boeing ones, are shown in Table 5-111. Tables 5-IV through 5-IX are the computer printouts of the first six cases of table 5~III. The symbols in these printouts are explained in Appendix E.

In addition to the stability derivatives, the wing contributions are reported separately.

Whenever a stability derivative differs from its corresponding wing contribution, this is because of the tail contribution.

We shall now make a few remarks concerning the most influential derivatives (Table 5-11) as computed with lifting line theory.

The comparison between rigid and flexible wing, in terms of stability derivatives, can be done by comparing columns 3 and 4 in Table 5-111.

TABLE 5-n - INFLUENCE OF THE STABILITY DERIVATIVES ON THE NATURAL MODES.

-- ~piral Short Period Dutch Roll Rolling Phugoid

- Mode_ ~amp. __ tlh _ Eamp. __ tlh Mode

tlh

__ Damp ...... _ ~~*~'d~ C xu C ~ .. * xa *.,'(,'\.,,, C *~'~** zu C ya ,,\ "i": .,,\"k *,,\ C * *** za .. /\ .,'(*.,'\"';'\ 7('1(,;,\'1\ "k-k ,'\*,'Cj'( *'k*~~ *~~** CjJ:i, ok ok *"1("1\-/\ .,'\"'(i,\"k i'n':: "'\'1\,'\';'\ C * nx 7(i( 'Ok C * na C xp *",\"k"k C yp C zp "k.,'("ki'.: '"/\,'(;'(./\ "k*,,\.,'( i'e ,'c"k.,,,:.,,,: C .ep C mp "k,'("ki( ,'(i'("ki( "k*i'(Ok "1\"k7("/\ ~~**~'~ C * np C Xp C yp C Zp .. ,'(,'(,'(,'( .. /\,'(,'(./\ 7(i'( ·'/(i'(** "k"ki'(* *i'(*i'\ C ..ep i'c"k "/\7(*";'( *.,'( i'( **7(* C *** mp ok C * * np C X~ C ytf C zlf Dutch Roll Rolling Phugoid Spiral Short Period

'% Mode

Mode Damp. Ub. Damp.

Damp. Uh

~b"

--

** * -- ~" -- ~"

c~ ;'(-;'( "k-k "k·k"l\i'( C

-- --

***~" -- ~"

m'l C

--

-- -- --

-- -- -- --

nlf - C

-- -- -- -- -- -- -- --

yr ok "k*~("i'(

***'k -- -- -- --

C J,r

**

C

-- ~"**~" --

** -- -- --

~" mr ~"ok*~'(

C ~"* -- ~"**~" -- -- *~b"'k

--

nr C z~

-- -- --

-- -- -- -- --

~'(

C -- -- --

-- -- -- --

ma Influence of the Stability Derivatives on the Natural Modes

TABLE 5-111 - COMPARISON BETWEEN STABILITY DERIVATIVES FOR A = 45°

4 5

1 2 3 , , 6 , 7

, , I ..

N.A. N.A.

.123 .133 .131 .108 .108 C x a

.568 .537 -- --

.283 .563 .378 C Y

a

..

-4.11 -4.24 -4.53 -4.24 -4.11 -4.24 -4.24 C za -.0133 -.297 -.0573 -.0127 -.203 -.164 - .17 C,t

a

.281 .292 -1.6J' -1. 10 -1.16 .297 -1.55 C m

a

.0084 .0075 -.0647 -.0673 -.00235 -.0913 -.0359 C n a --

-.00929 -.00895 --

-.00657 -.00857 -.00657 C x~ -.252 -.252 -.283 -.283 -.282 -.283 -.282 C Y~ -.546 -.535 -0.228 - 0.258 -.459 -.525 - .459 C z~ -.0881 - .0705 -.0493 -.0146 -.0237 -.0172 -.023~ C,e ~ .108 .510 .282 -.0378 .004 .195 .00223 C m~ .034 .125 .131 .035 .131 .125 .131 C n~ -1. 03 -1. 03 -1.03 -L03 -1. 03 -L03 -1. 03 C** z"

a

-4.78 -4.78 -4.78 -4.78 -4.78 -4.78 -4.78 c** m"

a

-.044 -.044 -.0937 -.109 -.0279 -.06 .0105 C Y p .29Q .298 0 .61 -.0685 -.406 -.328

c

z p -.444 -.19..: -.193 - .4 -.251 -.278 - .212 C,t p -3.12 -1.85 -1.84 -2.66 -2.23 -2.50 -1. 94 C m p , .0175 .0085 .0119 .0064 .0146 .0184 C -.0153 n J cont'd p .059, -.0305 -.378 .0202 -.55 -- -- -4~·23 -3.17 -6.73 -3.08 -3.09 -2.02 1.58 -2.23 -2.50 -2.14 -1.84 -1.84 -2.66 -2.79 -26.7 -26.6 -27.1 -28.7 C -30.1 -32.8 -28.6 m q -.354 -.163 -.133 -.0908 -.0831 -.163 -.159 .260 .263 .258 .261 .26 .229 .229

c -- -- -.13

z r .0525 .0549 .0544 .0541 .055 .0885 .095 .266 .244 .292' .298 .296 .379 .436 C m r -.130 -.1650 -.1654 C -.129 -.13 - .13 - .131 n r * No Stability Augmentation (SAS) included.

** These data were taken from Boeing study [Ref. 5J since our program

does not have the capability, to date, of computing thesederiv&tive~ The eight columns represent: 1) Strip Theory: spanwise flat and total lift distributions as from Reference 5 (Fig. 5.35). Sideslip evaluated according to empiri- cal method described in section 2.5.

2) Strip Theory: spanwise flat and total lift distributions as com- puted by lifting line theory (Fig. 5.2). Sideslip evaluated as for case 1 • 3) Lifting line theory, rigid wing.

4) Lifting line theory, elastic wing (E.A. coinciding with quarter- chord axis.

5) Same as No~ 4, but no built-in twist.

6) Boeing results rigid wing.

7) Boeing results elastic wing.

~~

t .... E t-ulLL",it .. , .\,.f.

lI,t ~INC r.Cf\1~ItUTIWS Tli STAflLllrv IJr:RIVATIVFSISTAeILITY AXESI '

."s

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~~

el~h =-G. 21E lL eMk", = (I.?I,/:f 00 0:1'01 = O.14lE-J2 ClPW "'-0.406E 00 crow. 0.371E 00 It"" = ,).2>!H-Cl e~tA = C. o~C-Ol

LY Il" =-.'.' 1'1i:- 02

D."

~g; STA~LlITY O£~IVATIVESISTAAILITY AXES I (IU =-0.553£-C' (Ill =-0 ... 43E-04 ~: (VA = oj. nBE cu C ~A .-0.155E 01 (llA = O.l(&E )\' CIA =-(1.424E 01 eLA =-0.203£ 00 tNA --0.359£-01 (LB =-!).239E-Ol CMS a 0.223E-02 eN8 .. O. HIE 00 oe =-0.651E-02 CYB =-0.282E CO CIB =-(l.459E 00 CYAO = 0.( CLAD" 0.0 CMAO .-0.418E 01 (XAC = G.(; CIAO =-0.10~f 01 CNAO - 0.0

cvac = O.C Clee = c.o ClBO = 0.0 eMaD .. 0.0 IPee = 0.0

CNBO - 0.0 CVP= 1l.105E-Ol (II' =-(l.40~E 00 CLP =-0.251E 00 eMP -0.223E 01 CNP --0.155E-Ol (.)f = 0.0 CYU E 0.202 e-Ol Cle ",-0.423E 01 CLC =-0.223E 01 CHO --0.301E 02 CNI; --O.3Sfte 03 exc E 0.0 CYi< = 0.20CE CC. eli< = 0.0 ClR .. 0.525E-Cl CMR - 0.266E 00 eNA --O.l29E 00 (X'; = G.G AE~(DV~AMIC CENTER BUILDUP A("e ~.250~A({dZEPC SWEEPI ACT 1.122~A((oIFRC S~EEPI ACfS }.34q~~(I.IERC S~EEPI S AS e.c ~At{iZcRO S~EEPI t) H(I~ETI O.121M.CI.IERO SweEP) ~ F(R A tE~Tc~ CF G~AVITV ~ ~.355~~CI.ZERC S~EEPI THE STATIC ~APGIN IS 0.366MACI.ZERO sweEP.

CONTFCl'OERIVATIVES elEvATC~ C~FL~CTICN Of = 0.0 CfGREEISI CleE --O.13&E-OZ (MOE =-C.E90E-C2 RlLCeq CEFLECTICN DR = 0.0 OEGREEISI CY~R E C.191E-02 CLOR = O.11BE-C3 CNOR·: 0.215E-05 AllE~C~~ OtflECTICNS OA-LEFT - 0.0 [EGREEISI OA-RICHT - 0.0 OECREErS) CYC~ - 0.0 CIDA • 0.0 CLOA - 0.0 CHDA : 0.0 . C~OA • 0.0 TABLE 5-1V Computer Printout for Case 1 of Table 5-111.

~g I-t ""OQ hiE FCLLCl,,;liG ARE TH~ wWG eU',fRIBUTIO'jS TO SH51L1TY oeRIVATIVESl STABILITY AxeSI

°z

CLew. O.ZI8E-03 eMew z 0.39ge-02 1)1 CN~W z-0.235E-02 CYhW • 0.283E 00 0>-

CLA" =-0. 164E 00 CMAW "-0.103E ~t-l CLPw .-0.173E 00 c~~w .-O.2SQF 01 CSP~ .-0.174~-OJ ClOW .-0.250E CM~W .-O.233E 02 C~CW a-O.163F 00 CL uw ClQ~ "-C.116f 01 • C.312E-Ol C~Qw = 0.~9tE 00 CNP,W = 0.<'98f-[13 CIPW :-0.328': 00 CYow - ~.58QF-Cl ~ .....

~ ·v CYPW = 0.1~9E-Ol Cyp," : C.IJ~E-Ol

l;l>>-

t""Q I-ttsl STABILITY DF~IVATIVESlST4BILITV ~XES I

~(jj

CXu :-0.4 .3~-04 ClU =-0.553F-03 ex~ : C.I08E 00 eYA : 0.263E 00

Clio "-0.424E 01 CLA =-0.164E 00 C/olA =-O.llOE 01 CNt. --c. 235!:-02

eXB =-0.t57E-02 C ~B =-0,282E 00 CiP =-0. 't59E 00 CLB =-0 237E-Ol O"B .. 0,399E-02 ("lll .. O. nle 00 eXAO .. a.OuOE 00 ey tC = O.OOOE 00 CLAD =-0.103E 01 CLAD - O.OOOE 00 CM~O --0.478E 01 OlhD - O.OOOE 00 eX50 .. O.OODE 00 eyeD = C.OOOE 00 ClBO = O.OOOE 00 C"~O = O.OOOE 00 c~eo'- O.OOOE 00 CL!lO - O. COOE 00 CXP = O.OOOE 00 C ~P --00279E-01 Cl P =-0, 328E 00 CLP z-0,'278E 00 CMP --0.250E 01 trjP • 0.6391:'-02 CXU " O.OJOE 00 CVI; '" 0.21l2E-01 Cl(; :-0.317E 01 CLQ "-0.250E 01 CMQ a-O.328E 02 CNQ --0. 163E 00 CXR .. C.OOCE 00 CYIl .. 0.263E 00. Clll - O.OOOE 00 CLII '" 0.549E-01 CNR --0.139E 00 CMIl. - 0.29"E 00 AERCOV~AMIC CE~TER 8UILDuP ACW!l C.250~AC(iZE~Q S"EEPI 4CT C.122MAC(ilE'l~ S~EEPI AepS O.242MACl.lERC SwEEPI SAS O.OOJMAt(~lERO S"eEPI XAClNETI C.614~AC(ilERC SwEEPI t"" CJJ CJJ FeR A CENTER OF GRAVITY i 0.355~AC(.lERO SWEEPI THE STATIC MARGI~ IS O.Z59MACcaZEIl.O SklEEP) CONTROL UERIVATIVES ELEVATOR CEFLECTION OE z G.OO eEGREEISI CluE "-0.188E-02 CMCE --Co890E-02 RUDDER DEFLECTION OR & O.OOOEGREEISt eYOR - 0.197E-02 CLCR a 0.176E-03 CNOR .-O.OOOE 00 AILERONS (;EFLECTlONS tA-LEFT - 0.00 tEGREEI SI OA-RIGHT. 0.00 DEGREE.CSt CYOA • O.OOOE 00 CleA - O.OOOE 00 CLOA - O"OOOE 00 CMOA .. O. OOOE 00 CNOA - O·,OOOE 00 TABLE 5-V Computer Printout for Ca.se 2 of Table 5-III.

THE FeLLC~Ir;G AKE TI1E WING eGNTRIBUTIt:JNS TO SUBILlTY DERIvl>,TIVESISTABILITY AXES) ~iGIO ~1~G ANALISY~ cyaw • 0.563E 00 CNAII =-0.91:E-Ol CLaw =-0.110~ 00 CMAW --O~109E 01 CMEn = C.195E 00 eLe~ = C.630E-02 Cl4QW s-0.191E 02 eLPw =-v.2C8E vu (NOW --0.133E 00 C~P~ =-0 816c-02 CLew =-0 214E 01 CHP~ :-0.194E 01 ClPw --0.b85E-01 C~~w : 0.292E 00 CNPW = J.858E-03 ell,;w =-0.'111E 01 CYOW --0-30SE-Ol CLRh • C.310E-Ol Cl8W --0,600~ 00CN8W =-0.673F.-02 CY~W =-0.505E-02 CYP~ =-O.129E-01 ~ b:l ~ STABILITy DE11VATIVESISTABILITY AXeS) V1 ell,. =-0,632E-03 exl,. =-0.461E-04 I ClA s-0.424£' 01 CLA =-0.110E 00 CMt> '-0.116!: 01 CNA =-0,9Ue-01 eXA : 0.123E 00 e 'A = 0.503E 00 <: H cve =-C.263E 00 Cle =-0.525E 00 CLa --O.I12E-Ol CMS - 0.19~E 00 eNS - 0.125': 00 ex B :-0.859E-02 ClAC =-0,103E 01 CMAC --0.418E 01 CXAD : 0.0 CYAD - CeO CLt>O - 0.0 ClOUD - 0.0 o,ao .. 0,0 eveo = C.O elflC = 0.0 CLeO = 0.0 Ctl~O - 0.0 CXBD : 0.0 ('") exp : 0.0 GYP --0.539E-01 CZP --0.665E-Ol CLP :-0.212E 00 CHP .-0.194; 01 CNP - 0.1106e'':01 C YQ :-0.305 E-Ol eNQ --0.1331: 00 CHI =-0.613E 01 CLO --0.214E 01 CHO --0.286E 02 CXQ '" 0.0

~

C '!l.': 0.25810 00 C LR : 0.544E-Ol CMR • Oo292E 00 CNR --O~ 130E 00 CXR : 0.0 elq - 0.0 != rt (1) AERGCYNAMIC CENTER BUILDUP Ii Acwe '"d Co250~AC( lERO ShEEP) Ii C·.122~t>CI lERC ShEEP) ACT .....

ACP S C.256~ACI lERe ShEEPI ::l SJI S rt C.O ~ACI ZERO SWEEPI --------------------------------------------- != XACIIlET) • C.627~ACCiZERC ShEEP) rt t-h FeR A CE~TER OF GRAVITY a 0.355~Ael.lERc S~EEPI ThE STATIC ~ARGIN IS 0.272M.CCilERO SIIEEP,) ....

W Ii ~ ('") eeNTROL CERIVATIVES Q) CIl ELEV~TOR CEFLECTION DE = 0.0 CEGREEIS) (1) ClOE =-0.213E 00 C~CE --O.lOlE 01 W RucceR CEFLECTIGN OR: O.C DEGREECSI 0 t-h CYuH : 0.113E 00 CLCR = 0.10110-01 C~DR --0.563E-01 1-3 AILEP,C~S CEFLECTIONS CA-LEFT = 0.0 DEGREEIS) Oil-RIGHT. 0..0 DEGP,EEIS) III 0" J-' AILERCNS GECMETRY W.R.T. hING CENTERLINE INON-OI~. STAT) (1) V1 LEFT WING RIGHT W!t.lG I ()UTBCARD INBOARC OUT80ARD INBOARD H CESIG~ eC~CITIO~S -0. ~21 -0.732 0.732 0.921 H COMPUTER APPRnxIMATIO~ -0.':;31 -0.831 0,862 0.. 662 H .

LEfT hiNG AILERON OEFLECTEC ONLY e~DA • 0.253E-02 ClCA :-0.128E 00 CLOA = 0.242E-Ol eMOA = o~ 251E 00 eNOA =-0,372F.-02 RIG~T ftI~G AILERON OEFLECTEC ONLY Cyot> = 0.564E-02 eleA =-0.20210 00 CLOA =-0. 521E-Ol CMOA =-0,445E 00 CNOA --0.420E-03 ANTISY~~EIRIC AILERC~ CEFLECTIO~ CYOA --0.896E-02 ClCA • 0.745E-Ol CLOA • 0~163E-01 CMOA = 0.702E 00 CNOA --0.457E-03

~~

T~E fGLLC~lNG \~E T~E .iNG (UNTQI8UTICNS TO ST~~'L'TY OEQIVATIVESISTABILITY AXES'

-

~LtSTlC .r',G A<,ALI SYS

8~

CLb~ = 0.381E-Ol CMP~ = C,S10E 00 ClA~ =-0.127 -01 Ci'Hw = 0,382E (0 C'IAW =-0.1>47E-Ol CY~W .. 0.568 00 ~~ ClF. =-O.l&SE JO C~F~ =-C.18SE 01 C~Pw =-0, 502 -02 Cl CW =-0, 184E C1 C~QW =-0.171E 02 ('4Qw =-0 909 -01 ClQ. =-J.ll7E Dt-e; 01 ~LPW = C.301E-Ol C"lRW = 0.298 00 CNr.W = 0.15I>E-03 CZPw = 0.2.,<;1;' 00 C YOW --0.378 00 cyp~ =-0.4~1E-Ol CY-w =-C.203E-02 CZFlIo =-0,664 OOCNBW =-0,5851'-02

s;:>

t-t~ -t:9 ~ STABILITY OE~IVATIVESISTABILITY AXESI b:I

~m

~ CXl, =-0.476E-04 CZU =-0.058E-03 ':;XA = 0.133E 00 CH = 0.568E Oll cza "'':'0.411E 01 C l~ =-0. 127E- 01 CMA .. 0.297E 00 C~~ --0.6'011'-01 VI CX~ "-0 .• 929E-02 CyI} --0.,283E 00 I CZ 8 "'-0·. 5465 00 CLB '" 0,146E-Ol (Mil '" 0.510E 00 0''1 '" 0.1251' 00 CXAU = 0.0 <: (.YAO ., C.O CIAC "'-0.103E 01 CL,.O '" 0.0 C ... ,.O =-0.4781' 01 ('41,0 " a a H cyec = C.O CXBC ., C.C CIBC = 0.0 (MBO .. 0.0 CLBO '" 0.0 CIi'!O - 0.0 H CXP ., 0.0 C yp "'-0,937 E-Ol . CZP = 0,2995 00 ClP =-0.1<;3E CO (MP "'-0.18~E 01 O!P s 0.18'0"-01 CYQ s-C.37BE 00 CX~ = 0.0 CH: =-0.308E 01 CLO =-0.18ttE 01 (I4Q "'-0,267E 02 CNO "'-0,9081;'-01 (") CYR = C.261E 00 (XII, " 0.0 eZR = 0.0 C LR '" 0.5'oIE-01 CI4R .. 0.298E 00 (Nll. --0.131E 00

~

AERCDY~AMIC CENTER BUILDUP (:: rt ACIooB C.250"'CI ZERe S~EEPI (1) ACT ti C.12bl'.tl ZE"O SIoEEP, ACP S s -e.093"ACI ZERe SIIEEP' SAS c.o "ACI ZEP.C SIIEFPI ti

'"

J-'.

--------------------------------------------- ::l xAC"'IETI C.283"ACI.IERC SIIEEPI rt o (:: fO~ A CE~TER OF GRAVITY • C.355~ACIGIEll.C SwEEP) TH~ ST~TIC MARGIN IS -0.072MACI.ZERO SIIEEPI ....

rt w U1 t-h ceNTROL DERIVATIVES o ti ELEVATOR CEFLECTICN DE '" 0.0 CEG~EEISI (") ClOE ~-C.213E 00 CHeE =-0.101E 01 s:u CIl (1) RUDCER CEfLECTICN O~ = o.e DEGREE lSI .p- (YDR s 0.113E 00 CLC~" O,lOIE-Ol CNOR .-0,Sb3E-OI o AIlERCNS CEfLECTIONS CA-LEfT = 0.0 OEGREEISI DA-RIGHT" 0.0 DEGREElSI t-h t-3 AILERCNS GECMETRY h.ll..T. III~G CENTERLINE I~CN-DIM. STATI s:u 0" LEFT WING RI GHT WING t-' (1) CUTec.a~o INBOARD GUTROt,RO IN80ARC CESIG~ CC~OITICNS -Oo~27 0,732 0 921 -0.732 VI -0.837 CCMPUTEP APPPC~I"ATIC~ -C.~37 0.662 0.81>2 I H H LEFT IoING AILEPC~ (EFLECTEC O~LY H CLOA s 0.312E-Ol [YOA .. 0.103E-Ol ClCA =-0.177E 00 CNDA "-0.tt4 3E-0 2 0.33'0£ 00 CMeA RIG~T III~~ AILEP.C~ CEfLECTEC CNLY [YDA ~-0.15JE-02 CZCA =-C.BB8E-Ol CLOA =-0.279F-01 CMeA =-0.243£ 00 C~!OA " 0.520E-03 4NTISYM~ETRIC AiLE~C~ CEfLECTIC~ CYOA s 0.7~IE-J2 CleA s-C.B87E-Ol CLDA s O.591F-Ol (1'104 0.577E 00 CNOA --0.300E-02 TH~ FLLlCWlhG •• ~ T~~ wlNC CO~TRIBUlIONS TO ST~8ILITY DERIVATIvESISTARILITY AX~S) ~g 1-1 ELASTIC ~I~G A~ALI~YS

~~

: O. ,HbE ')0 ~-.). t :Br -;)1 cu,,~ : ,).147(-.11 CYA ... O.531E 00 C'U.W ~-0.671E-Ol C/o' fW C.2'l2f Dol U~'~W CLil~

0>-

CNew "-0.831E-Ol C~Qw "-0. 17 IE 02 =-0.IB4E Cl eL~W =-O.~93E-t)2 CNPW :-0.1!l4E 01 eM FW :- 0.1 'jdf OJ CLI'.'j ~t:-4 CYQW "-O.550E 00 CIPW .. 0.298E 00 = O. 858E -C3 CNRW = O.29tE 00 CMRw : 0.31bE-01 CLRw =-0.1"6:: oll C Hi ~ ,0 I-d =-0.337E-02 CNew =-0.631E 00 Clew a-C.228E-02 CYR .. :-0. b~5E-Cl C YP " ~ t::d

§2>

Q ~ r STABILITY DERIVATIVESISTABILITY AXESI p.-;t?j \Jl I

CXU =-0.412E-04 Cll. =-0.645E-03 ~c;j

<: OU ., 0.292E 00 coa "-0.613E-Ol CX" = (l.DIE 00 C YA = 0.531E 00 Cla =-0.411E 01 CLA =-0.133E-Ol H eYB =-C.211JE 0:) eMB = 0.l82E 00 C~8 .. 0.lZ8E 00 CX~ '-0. eS5E-02 CIB =-u.535E 00 CLa =-0.882E-02 H H cxaD = 0.0 CNAt) .. 0.0 CY t C = 0.0 CLAD =-0.103E 01 CLAD" 0.0 C"'AO =-0.478E 01 exec = 0.0 CYBO = 0.0 Cleo = 0.0 cLlm = 0.0 cr~BO - 0.0 CNBD - 0.0 cXP ~ 0.0 (YP =-C.l09E 00. ClP " 0.ZgeE 00 CLP =-0.193E 00 04P --0.184E 01 CNP - 0.175E-Ol (') CXC = O.C cya =- C. 5 5 0 E 00 ClC =-O.309E 01 ClO "-O.IME 01 (MO =-0.26fE 02 C~Q --0.831~-01 (XR .. 0.0 CYR .. C.2&OE 00 CZR = 0.0 ClR .. 0.550E-01 CMR - 0.296E 00 C"R --O.UOE 00

~

c:: rt AEROC'~AMIC CENTER BUILDUP (I) Ii e.250~AClilERC S~EFP) Aewe C.12b~~CI@lERC S~EEPI ACT I'd Ii .. -C.092MACCiIERO S~EEP) ACPS 1-" C.O ~ACC~lERO SwEEP) SAS ~ rt o C.28~MACI.IERO SwEEPI UC CIIET I c:: rt t-' FOR A CENTER OF GRAVITY ; 0.355~AClilERO ShEEP) THE STATIC ~ARGtN IS -O.071MAt UlERO SWEEP» CJJ Hl Cj') o Ii CCNTRCL DERIVATIVES (') III ElEvATO~ CEFLECTIQN DE" C.O CEGREEISt {Il CIOE =-0.213E 00 CMOE =-0.101E 01 (I) \Jl Rl.CCER CEFLECTIC~ DR • 0.0 CEGREEISI ~YDR ~ 0.113E 00 CLOR., C.I0IE-01 C~DR =-0.563~-Ol o Hl AILE~CNS CEFLECTIONS CA-LEFT = 0.0 CEGREEISI DA-RIGhT .. 0.0 DEGREE lSI 1-3 III c" AILEP,eNS GHMErRY H.R.T. ~ING CENTERLINE INON-DIM. STATI t-' (I) qIGHT LEF T WING WING \Jl IN80ARD OUT9(1ARO INBOARD OUTeOARD I 0.732 0.9Z1 -O.73Z OESIG~ CCNCITIO~S -0.~21 H 0.662 0.862 -0.837 COMPUTER tPPROXI~ATIO~ -0.937 H H LEFT .. INC ~ILERON DEFLECTEC Olft y eYDA .. 0.110E-Ol CICA =-0.177E 00 tNDA -,-0.303E-02 eLOA - 0.312E-1)1 tMOA - 0.334E '00 P.IG~T IoING AILERON CEFLE~TEC ONLY CYOJ =-0.57~E-Ol CICA =-a.8BBE-01 CLOA "-0.279E-Ol C~OA --0.243E 00 CNOA .. 0.211E-02 ANTISYMMETRIC AILERON OEFlECTIO~ CLOA .. O.591F.-OI CYOA - 0.123E-Ol CleA --0.881E-Ol CHDA - 0.571E 00 tNOA --0.319E-02 The elastic wing, unlike the rigid one, shows a positive C rnCl!

or pitch-up moment.

There are no major discrepancies between our results and

Boeing's except in the C # derivative. Since the major contribution

n to the latter is from the fin, the difference may result from our O!O assumption (;J3 = O.

It is useful to consider stability derivatives as functions of skew angle before discussing dynamic response.

The effect of skew is seen in the accompanying graphs (Figures 5.35 through 5.41) on a configuration that was stabilized by moving the C.G. .15 M.A.C. further forward.

The behavior of these derivatives is explainable in terms of two important changes in the lift distribution resulting from an angle of attack perturbation.

The first of these is the predominance of lift on the trailing wing. The second is the effective reduction of the lift forces arising from roll and pitch rates or changes in pitch attitude, that is, C is reduced.

La The program approximates CLO! as the calculated lift divided by dynamic pressure, wing area, and cruise angle of attack. From simple two dimensional sweep theory, the lift on a wing at constant angle of attack and free stream velocity varies as the cube of the cosine of the sweep angle. The effective dynamic pressure is reduced by a factor cos A and the effective section angle of attack by a further factor of CosA. The lift distrubition calculated by the program closely matches this Cos A variation. C is proportional to C La L and hence behaves as Cos A as well.

It is not possible to easily follow the details of a change of skew angle through the lifting line analysis to verify the cos A dependence which results.

The trailing vortices move closer to the downwash control points, and bound vortex segments decrease in length in proportion to CosA , which accounts for a factor Cos fl. The change in the geometry relating the vortex segments to the control points is complex however and gives rise to three dimensional effects such as the lop-sided lift distribution, which are totally unaccounted for in two dimensional simple sweep theory.

In addition to decreasing C ' the closer lateral proximity La of the skewed wing to the body reduces the roll induced velocities.

However the extension of the skewed wing fore and aft of the C.G.

causes increased pitch induced velocities.

In pitch, roll, and yaw manouvers the lift perturbation distribution is unsYmmetrical with respect to the center line so that consideration must be given to the changing moment arms in interpreting the behavior of the derivatives.

The moment arms are decreased for rolling moment derivatives, increased for pitching moment derivatives and relatively unchanged for yawing moment derivatives.

These observations lead to the following understandings of the effect of skew on individual derivatives: (1) C ' C experience a decrease as explained earlier. A L La cos A dependence applies only to small variations about an equilibrium position. The airplane C would be maintained at L a constant cruise velocity by increasing cruise angle of attack as Cos -3 A..

The tendency of lift to shift to the rear wing leads (2) C~: to a rearwand shift of the neutral point as shown in the graph of vehicle aerodynamic center, or an increase in the static

margin. However the decrease in C~ dominates, and C = C x

ma La

static margin is reduced.

(3) C : displays slight increase until skew reaches 20 decreasing xa

thereafter. From Appendix B, C = CL(l-2kC~) C and C are

xa L La both decreasing which results in the calculated reversal.

("Reversal" will be used to mean a reversal in the sign of a curve's slope, and "reversal in sign" a zero crossing of the curve). The second term in the parenthesis is the contribution of induced drag, while the first results from the rotation in the perturbed body frame of a lift vector fixed in stability reference axes.

(4) C : increases with the appearance of a sideways rotation of ya the lift vector and a sideways component of induced drag. The reversal is caused by the dramatic decrease of C and C~ at L high skew angles.

(5) C =-C za. ~ (6 ) C~, where £ is rolling moment, increases with the shift of lift to the trailing wing, but eventually reverses with decreasing C and moment arm La (7 ) C~ is the result of X ' the X force due to an a perturbation, a acting through a moment arm proportional to CosA and Y through a

an arm proportional to Sin A: initially side force generated

Decline sets yawing moment predominates and C~ increases.

in as the longitudinal force generated moment becomes in- creasingly smaller and C itself begins to decline. Again ya the reduction of C and C with skew has an important role.

L La (8) C Monotonicly increases for the same reasons as C yp ya The increase is less dramatic because of the decreasing distance of the wing tip from the centerline, and reduced roll induced angles of attack. The non-zero value at ~ero skew is due to the rudder.

(9) Normally zero for symmetric aircraft which have an C zp antisymmetric roll generated lift distribution, the C zp curve has two startling reversals and a reversal in sign.

The derivative is very sensitive to slight departures of the lift distribution from antisymmetry. In the range 0 to 10 skew, lift decreases more on the leading wing in a roll than it increaseB on the trailing wing. After 10 the trend is reversed until the lift increment on the right trailing o wing dominates as in figure 5-18 for A = 45 .

The final reversal reflects decreasing lift curve slope and effective roll induced angles of attack.

(10) C : decreases monotonicly with declining roll induced tp velocities, effective section angle of attack, and lift curve slope, causing a reduction in roll damping

(11) 9n : increases with skew because of an antisymmetric roll

p distribution acting through an increasing moment arm.

Declining roll induced effective angles of attack and lift curve slope cause a reversal.

NO (12) C : Decreasing X- force yawing moment arms are offset by np increasing Y- force arms. The longitudinal position of the center of gravity will influence the shape. Figures 5-19 (a) and (b) show distortion of the expected antisymmetric X and Y force distribution. Since the incremental lift distribution

in roll shown in figure 5-18 for A = 45 is almost normal, the

distortion is related to an abnormal downwash distribution, figure 5.13 a). The departure from an antisymmetric force 0 0 distribution decreases C from 0 to 30 • nP (13) C: Figures 5-22 (a) and (b), showing pitch generated yq X and Y- force distributions, are similar to those for roll generated horizontal force distributions. fm figure 5-21 the corresponding lift increment is noticeable greater on the trailing wing. Skewing the wing has a less detractive effect on pitching stability derivatives because induced velocities and aerodynamic twist increase. The detractive effects of diminishing effective angle of attack,and dynamic pressure remain however. The horizontal force distributions are symmetric at 0 skew but become less so as portions of the leading wing extend in front of the center of gravity.

The distortion mentioned earlier opposes this tendency.

(14) C : declines steadily with an increasingly antisymmetric zq lift distribution (Figure 5-21) (15) C : increases for the same reasons that C decreases, n ;:,q zq until the familiar high skew angle deterioration takes place.

is initially non-zero for a center of gravity not on the (16) C mq The introduction of antisymmetry in the incremental quarter chord.

pitching lift distribution, and increased induced velocities, and pitching moment arms increase C rapidly at first.

mq

(17) C : The rapid increase after A = 15 , and the tapering

hq

off above A = 30 correspond to the varying antisymmetry

in C The initial decrease may be because the wing quarter yq

chord lies behind the center of gravity at A = 0 , so that

when side force components of induced drag first appear their contribution to yawing moment is positive. As the skew angle increases, and the forward wing extends ahead of the center of gravity, its contribution to yawing moment becomes negative. The non-zero initial value of C could nq not be explained, but may result from an unsymmetrical downwash distribution at zero sweep.

.8 C .6 L .4

~

.2 0 A 0° 4.

3.

2.

1.

A o 0° 10° 20° 30° 40° 50° .8 Vehicle aerodynamic center (Neutral point) in terms of M.A.C .

. 6

ORIGINAL

OF Poo PA.GE 18

.4 ,~ QUALITy .2 A o 0 -lOo ° ° ° 0° 0 " 20 30 40 5 Figure 5.35 - Lift coefficient, lift curve slope, and aerodynamic center versus skew angle -.3 .3 C Cw.

xO!

-.2 .2 .1 -1_ I I A 0 A (., ° 0° 20° 30 40° 50 10 20 30 .6 -5 .5 C rnO!

-4 .4 -3 .3 -2 .2 -1 .1 A A 0 0° 0° 10° 20° 30° 40° 50° 10 -.15 -12 C CZO!

nO!

-10 -.1 - 8 - 6 -.05 - 4

-

A

A 0

0° 0° 40° Figure 5.36 - O! stability derivatives versus skew angle ORIGINAL PAGE IS

OF P"Y;.R. (U ALITY

.03

-.015 ~

C Ci,fJ .02 ;

xfJ

-.01 .01 A -.01 -.005 -.02 -.03

O~A

0° 10° 20° 30 40 50° .3 -.3

L

I

.2[ ~ -.2 , C C

~

mJ3 yfJ

-.1 .1

~

I I

~~_joA t

0 I

~OA

20 30 40 50 ° 0 10 0° 10° 30° 40 50 .3 -.6

I

-.5 C -.4 .2

nJ3

-.3 .1

l~oA

40 50

Figure 5.37 - J3 stability derivatives

versus skew angle -1. 5 -1.

- .5 A -3 -2 -.04 c np yp -.03 -1 -.0 -.01 o

i A 1l-_L..---1._--1."..-----.L __ °

o A

0° 10° 20° 30° 40° 50° . 03 -.15 Figure 5.38 - P stability derivatives versus skew angle -6.

-5.

Cg,q -4.

-3.

-2.

-l.

A ° ° 0° 10° 30° 40° -.06 -60 C -50 -.05 C yq nq -40 -.04 -30 -.03 -20 -.02 -10 -.01 A o.~A o 1;.0---L-_.l..--.l1-.---..J1-.--J ° ° ° ° ° °

o 10(! 20 30 40 50 o 10° 20° 30° 40° 50°

-.25 -30 -.2 C C nq zq -.15 -20 -.1 -.05 -10 A O"-~~~_-L--..J..---' .05 .~I-A

o 0~0~10~-2~OO 30 40 50°

Figure 5.39 - q stability derivatives versus skew angle .3 1-

~

.2 C~r .1 A 50° 0° 10° .3 .6 I I i .4 .2 C C mr yr .1 .2 0"l~A o 0° 10° 20° 30° 40° 50° -.3 -.2 ORIGINAL PAGE IS OF POOR QUALITY -.1 A o Figure 5.40 - r stability derivatives versus skew angle -1. 2 -1.

c

xu - .8 - .6 - .4 - .2 A -.3 -.2 -2 C x 10 zu -.1 o Figure 5.41 - u stability derivatives versus skew angle 5.5 Dynamic Results 5.5.1 Natural Modes In5.l, Model description, it was noted that a C.G. location of 355 M.A.C. resulted in longitudinal instability. In case 3 of Table 5-111 the instability is not apparent from C which is less than zero, rna however the roots illustrate the unstable short period mode discussed in 5.1: Damping Wn Real Root Imag.Root

+

Phugoid .1265 .063 -.00797 .0652

+

Dutch Roll .1001 1. 058 -.1059 1.053

+

"Rolling-Short-Period" .8007 2.120 -.1713 1. 25 Spiral Mode .0166 Unstable "Short Period" .1542 Table 5-X records similar roots for case 1 and case 5. These configurations are therefore longitudinally unstable as well. The stan- dard short period and rolling convergence modes are recovered for both cases when 40~ SAS is introduced (equivalent to moving the C. G. forward .4 M.A.C.) The added static margin decreased C to -2.85 and -1.35 for rna cases 3 and 5 respectively· Since the numerical integration is sensitive to the number of wing stations, to obtain good results a large number of stations should be considered.

For example, the stability derivatives for cases 3 and 5 (40% SAS) are shown in Tables 5-XI and 5-XII and the corresponding natural modes in Table 5-X for 36 wing stations, compared to 40 for previous cases.

The roots of the dynamic equations were computed at 0 15 0 0

= 0, ,30, and 45 for the configuration having a e.G. at

A -.15 M.A.e., and the accompanying root loci plotted. (Figures 5.42 through 5.46) These reveal the variation of dynamic behavior corresponding to the stability derivative versus skew angle curves plotted earlier.

The effect of skew is to cause minor variation in the natural frequencies of all but the rolling convergence mode, which experiences a large reduction in roll damping. This is traceable to the diminishing of e~p and further to declining e~, ~n~ent arms, and induced velocities.

T~e mode shapes corresponding to the new roots at A = 45

are shown in the form of Argand diagrams. (Figures 5.47 through 5.50) Although the characteristic roots have changed only moderately the dynamic modes no longer resemble the familiar ones of a symmetric airplane.

The diagrams show the phase and magnitude relation of the six state variables describing the dynamic state. When the magnitude of a state variable is insignificant its relative phase angle is still shown by a line. State variables for non-oscillatory modes have only 0 0 0 or 180 phase relations between them. The rolling convergence mode

is one of these. The Argad diagram for A = 45 shows the diminished

roll rate r resulting from the reduction in roll damping.

In summary, the most dramatic change in dynamic character- istics appears to be the reduction in the roll mode damping. No other large changes in root locations due to sweep were encountered for this configuration. The changes in mode shapes due to the long- itudinal/lateral coupling are generally small with the exception of the short period; which takes on a substantial amount of rolling along

with the normal e and a

Figure 5.42 - Root locus versus skew angle for spiral mode.

o -1 Sec. -15 -10 -5 o Figure 5.43 - Root locus versus skew angle for rolling convergence -1 1m (sec. ) .1 .05 .....

-.001 o -.002 -.003 Re -1 (Sec. ) -.05 -.1 I Figure 5.44 - Root locus versus skew angle for phugoid mode -1 ~m(Sec. ) 2.0 o 1.0 o Re -1 (sec. ) -1. 0 -2.0 Figure 5.45 - Root locus versus skew angle for short period mode.

• -1 Iffi(Sel ) o l.0 --I .5

I

-.1 Re -1 -.15 -.05 o (sec. ) -.5 ORIGINAL PAGE IS OF POOR QUALITY Figure 5.46 - Root locus versus skew angle for lateral oscillation or Dutch roll Rolling Convergence Mode Spiral Mode A

I I

-5 u/u- =-4 .. 564xlO -4

--, 0 r--

u/u = 9.075xlO- I 4 o - , fJ = .066

~ a = -1. 063x16 4

I

a = .093 I e = -4.795xlO-

t0r¢- 0 , -3 I

0 I

e = .080

fJ = 5.043xlO te)- I I l' = .815 .039 r =

>

~

i cP =1. 0 cp = 1.0

-

-,=. I I~

:>

'>

t-' CJl -.J -3 -3 = -2.456 x 10 u/u ~ u/u =-1. 260xlO I o I o -4 I 0 ~ I fJ 45 = .052

a =-4.132xlO 4

I I ~

= .059 a

l- e = 1. 299xlO- 3

I

fJ = 4.509xlO- .Oll e

I I :If = .077 I .039 ...

l' = 1")- cp =1.0 CP=l.O I ~

L

Figure 5.47 - Effect of skew on rolling convergence and spiral mode shapes o r = .031@40. 5 o l.P == .821 @ 40

= l.0

u/u R -3 0 o p = 4.36x10 @_178.3 ~oo o l.P == . 013 @ 1. 2

./1". lIIlI<c~~-!!i!!!!!!~~~;S u/u == 1. 0@0

o 4

r = 4.5 x 10- @_8.3°

-3 .306 x 10- CJ. = 9.1 x 10 @ _17.2 @-157 .13 o .011 @ -158 CJ.

~o #'a

gtJ:i

e =1. 476@ _94.3

r-' :1

CIJ

s~

!::'-f t:;} ./jj

fJ

ti1

e = 2.506 @-94. 56

A ~ 0 A ~ 45°

oid Figure 5.48 _ Effect of skew on the phug mode shape

~--===:::::::>Q = 1. 0 @ 0

u/u = .014 @ -40.8

o o == .99 @ -57.87

P = .231 @ 35.5

-3 0 10 @ 4.4

;>

cp - 3.63 @ 2 Figure 5.49 - Effect of skew on short period mode shape.

-4

u/u = 3.09 x 10

o @ 103.27 -3 0

e = 8.38 x 10 @ 33.14

. @ 0

cp = .166 176

o

r = . 99 @ -88. 1

cp = .465 @ 8.3 o ~.-:~~======:;> f3 = 1. 0 @ 0

e = .178 @ 152.3

o

a = .127 @ -144.9

-3

u/u = s.867 x 10 @ -62.30

o

ORIGINAL PAGE IS

o

r = 1.065 @ -85.6

OF POOR QUALITY

Figure 5.50 - Effect of skew on lateral oscillation on Dutch roll 5.5.2 Transient Response The transient responses have been computed for two cases: (1) Rigid wing, 40% SAS added.

(2) Flexible wing, no twist, 40% SAS added

Figure 5.51 compares with a and ~ responses to an aileron im-

pulse for the 40% SAS rigid wing case and shows the cross coupling between the lateral and longitudinal modes.

Figure 5.52 compares the a and ~ responses to an elevator im-

pulse for the elastic wing with 40% SAS. Note that the ~ response is larger than the a response after the initial peak in the first second and will no doubt have some effect on a pilot's evaluation of the handling qualities. This same result was found for the rigid wing with 40% SAS.

ORIGINAL PAGE IS OF POOR QUALITY , I I:) ~

I

,

I

I •

~

I

I

, ,

~

•

\

5i

ffi

...J

I

t-4 a:

I

~ el

I

D ...

(

I

\

J

\

I

~ I

1/ r.n

.

It) ( \ W ,:I:

"

~

,St:

~ O1lIG~

• U'I /-' 0;> 1'ooJl ~jfll ...

c,

'"

"' ......

-

--

/ ,..----

ID'O 20'0 90'0- 01 '0-

111'0 01'0

(10-0 t><J

SNVIOW .

§i

ORIGINAL PAGE IS

OF POOR QUALITY

\A

a: 10' I- a:

>

L&J • ...J

Iff

L&J

t-

~ 8

•

z

• \(1

D- r

"""

u

en

C';j I LLJ LLJ It:I

en

a: It:I ~~

-

l

ra

~

~ , I:z4

t1

•

,

~

en

N C :::tt C!J Z ~ :r; U .....

't-

en

a: ...J IJJ _.

......

- _ ........ -

- - .......

90'0- 01'0- ~O'O 90'0 01'0

(10-0 t xJ

SNVlaw

5.6 Summary The model used and the aerodynamic result obtained with both strip and linear theory were discussed in detail and compared. The strip theory approach, though very simple, showed results not always satisfac- tory; in addition, its limitation to rigid wing case only turned out to be too restrictive, since the elastic wing behavior strongly differs from the rigid one.

The effect of stability derivative changes on the natural modes was tabulated. The derivatives of a longitudinally stable configuration o 0 were graphed for skew angles from 0 to 45 , and discussed in light of the geometric and aerodynamic effects of skew.

Substitution of these in the eighth order dynamic system gave the characteristic root loci as functions of skew angle.

Calculation of the mode shapes revealed significant changes in dynamics, although the natural frequencies had not changed greatly, (apart from the rolling convergence mode).

Transient responses to elevator and aileron deflections were presented.

VI CONCLUS IONS The analysis and computer program for the adaptation of the lifting line aerodynamic theory to the oblique wing have been described.

The existence of a side force component, due both to induced drag and to the tilting of the lift force when the wing is skewed, has been shown. Stability derivatives were obtained for an aircraft used in a Boeing study with the oblique wing placed at 0°, 15°, 30°, and 45°.

The derivatives were generated using the lifting line theory and a

simple strip theory (for sweep = 45). The two results are compared

°

with each other and with the Boeing results and show reasonable agree- ment in most cases The stability derivatives computed using the lifting line theory were used in a linearized dynamic model of the aircraft to determine the effect of sweep on dynamic behavior. No instabilities or large changes occurred in the root locations for sweep angles varying from 0° to 45° with the exception of roll convergence. The damping of the rolling mode was reduced by more than an order of magnitude due in most part to a similar decrease in C~p' A dramatic increase in the characteristic roll angle, in comparison to other state variables, was prominent in the rolling convergence, and

the three oscillatory modes at A = 45°. The rolling motion in the

Dutch roll is exaggerated with increasing skew, and surprisingly both the phugoid and short period modes picked up significant rolling motion.

In the latter mode rolling dominated by a factor of three.

APPENDIX A A.l Relations Between Dimensional and Nondimensional Derivatives (Sta- bility Axes).

The geometric quantities b ' wing span, and c, mean aerodynamic O chord, are referred to the wing in the unswept position.

2~

x [lb ~t sec j

= Cx u U u

o

x = qS CX [lb] ex ex c Cx· [lb • x· = sec] qS 2U ex ex x - mg cose [lb]

e O

= qSCX~ [lb] x~ b .

[lb x· = sec] qS 2U Cx~ ~ O b Cx [lb • x = sec] qS 2U p p c x Cx [lb • sec] qS 2U q q b x Cx [lb • sec] qS 2U r r The same relationship can be obtained for y and z .

[lb • ft] - c c [ lb ft· M. = qSc - em· sec] Lex = qSb 2U C.U~

·

ex 2U ex O = = [lb ft] qSbC 1,~ qsccm~ L~ M~

·

- b b ft • [lb sec] = = L· M· qSb 2U C 1,~ qSc 2U em~

·

~ ~ O 0 - b b ft • [lb sec] L = M = qSc 2U em qSb 2u Cl,

·

p p

o p o p

c - c [ lb ft • sec] = M = L qSc 2U em qSb 2U Cl,

·

q q

o q o q

- b b ft.

[lb sec] L = M = qSc 2U Cm qSb 2U Cl,r

·

r r r O O The relationships for the yawing moment a.re the same as for the ro lling momen t •

APPENDIX B Since only the wing contribution to stability derivatives is a new element in the sta,bility ana.lysis, we shall limit ourselves to deriving only wing derivatives and provide a list of the ones which can be com- puted in a classica.l way and can be found in the literature [Ref. 15, 21 and 22]. The normalizing quantities b ' c are referred to the A= 0 O condition.

B.l "u" Derivatives [Ref. 22].

C x u 0T 1 oF x 1 h oCD )] = --- (B-1) C - 2C + M - -- - - = [ (

x qS ou D a qS oM OM

u a a C z u 1 oF z

C --- (B-2)

=

z qS ou

u B.2 "13" Derivatives.

Figure B-1 shows the case of the aircraft experiencing a sides lip velocity v.

The effect of introducing a 13 corresponds to a, - t:.A for the wing.

The fuselage also is affected, but we neglect its contribution since it is too complicated to evaluate it.

Neglecting the induced drag due to the fin, we consider only the component of the lift due to the fin in the x direction.

s F = - D cos~ + L. s in (~ - 5) x ~1n Since when \3=0 also 6=0 D

C = _ dC + c* (1 _ 00) ~ OCD

X o~ L . 013 013 a I' -:-F1n (B-3) since b cosA O PR= --- (B-4 ) S and assuming (CL)A = (C) cos A (B-S) L A 2 • "0 cosAQ we obtain (B-6) * C~= 0 a.ssuming the trimmed condition does not require any rudder.

0 1) 2

(B-7) (

OJ\. '11 lR /'r;= Au = lR t anAo

and after substituting into (B-3), the final expression becomes (B-8) C x~ or, in terms of the induced drag coefficient c.

J

In a similar way we can compute C The side force coefficient y~ for a sideslip is (B-9) where C derives from fuselage and tail contribution c oC __ c + OCD] C sin~ + CD cos~ + ~

Cy~ = - [ -

c ~ ~=O (B-lO) oC _-.£+ = CD o~ The main constribution fo C usually comes from the body and the c vertica.l tail, for the oblique wing case the side force generated by the wing also should be included in C ,but it turns out that such contri- c bution is of second order and therefore negligible.

The tail contribution is a conventional one and it is given by ,/Ul.3: , ____ , 1 --..

/ : " I I ' ./ (B-ll) derivatives, no attempt is here made to evaluate similarly to the C x~ the fuselage contribution.

C

~

(B-12 ) For simplicity, we shall neglect the influence of fuselage on C~ ~ Wing Contribution (B-13) The magnitude of this term is strongly dependent on the FWLD, there- fore no approximation is possible which would give a meaningful result.

Assuming a zero 40lling moment for the cruise condition, it is possible

to evaluate (C~~)w by computing the wing rolling moment at the per-

turbed sweep angle and dividing it by the increment in sweep.

Tails (Fin) Contribution [Ref. 21J 1., ] ZF (B-14 ) b ~=O

ZF

_C (1 _ 05 SF) ..e

(B-15 ) L~ Qj3 S b

The same approach osed in computing ( c~ L is necessa.ry when

evaluating the wing contribution to (Cm~)w and (Cn~)w

C

..3l

The tail contribution is negligible, therefore (B-16 )

c - (c )

m~ m~ w

c

~ The fin contribution is given by [Ref. 21J (B-1?)

where (B-18 )

v =

v vertical tail volume.

The sidewash factor ~, generally speaking is difficult to estimate

with engineering precision. Suitable wind-tunnel tests are ~equired for this purpose. The contribution from the fuselage arises through its behavior as a lifting body when yawed. Associated with the side force that develops is a vortex wake which induces a lateral-flow-field at the tail. The contribution from the wing is associated with the asymmetric structure of the flow that develops when the airplane is yawed. This phenomenon is especially pronounced with low-aspect-ra.tio swept wings.

When such tests are not available, References 27 and 28 can be used for empirical values.

B.3 !let' Deri va ti ves.

(B-19) where ~- For a rigid wing, dO: - tanA , therefore integra.ting (3.28) over the span and normalizing the result

of b(2

...!.. --Y = ...!.. f (L jI + L tanA - 2k L

CLcJ tanA dy

qS 00: qS 0: 0 0 0 -b(2 (B-20) we obtain (B-2 I) In s ta,bi li ty axes (B-22) The presence of an unsymmetric lift distribution due to the angle of a,tta,ck introduces a new wing contribution to C m a Let us first evaluate this contribution before considering all the other ones which are common to symmetric a,ircraft. We sha,ll refer to Figures 3.5 and B-2 for the symbols used. To be consistent with the conventional notations we shall consider this contribution as a part of

C = C (A) referred to the M.A. C. at zero sweep (Y )'

MAC m m O O The wing moment a,bout is given by Y MAC bj2 (M) - q Z Y dy (B-23) = MAC w A

£/2

where (M) is the wing pitching moment due to the effects of skew.

w A Let us now consider the general problem.

Moments about C.G.

a) Wing contribution (B-24) b) Tail contribution (B-25) c) Total moment L ~, I a • t i ' , t : N. P •..................... "'- I .-.' E: I I .. ' ... - - 1--- I i .~

-

t ,

~ l

he

I I !

.. ~

h c

r

n I Figure B-2 Moment about the C.G. in the Plane of Syrrnnetry.

N.P. = Neutral Point c = M.A.C. at A= 0 a = Angle of attack (wing-body)

o

E = Effective wing downwash at tail it = H-tail trim angle at = a - (E - it) = H-tail angle of attack.

O ':: M + (M) + L (h - h ) c - 1J L (B-26) t

o wA wb t

having assumed

C :;::- M.r (B-27)

1l\r qs~

(B-28) where (B-29) (B-30) C (B-31) m t (B-32)

- C [1 - O€] V

L oa H

at

f O'1

·· -/

j: /

v

Neutral Poiqt (C.G. location about which 0) C m a (B-33) or

- 1 ['0 ( de)]

h = h + - -. (C) - V C 1 - - (B-34)

wb C oa m WAH La da

L a t and by substituting (B-34) into (B-32) (B-35) (B-36) h - h ~ Static Margin (B-3?)

Aerodynamic Center Build-up.

(B-38) where h == .25 Wb C La

'a€) -

x = - ~ c (B-39)

VH~ (1

oa

ACT '1 a~ s La (B-40)

x =

ACpivot and Sweep This is one of the new derivatives and only the wing contributes to it. It is formed of two terms: one due to the lift and one due to the side force.

b/2 (B-4l)

~ (X ~ + Y y) dy

-b/2 The knowledge of the spanwise distribution of X and Y is needed in order to evaluate C nex B.4 lip" Derivatives.

The local angle of attack varies linearly according to ex = ~ (B-42) p U x

o

C z -E.

For the symmetric wing case this derivative is zero since the in- crease in lift on one side of the wing is balanced by an equal decrease on the other side.

This is no longer true for the oblique wing, therefore b/2 C (B-43) dP dy z p

f

-b/2

The evaluation of oZ ca.n be done according to the method outlined

op

in section 3.2.2. However, such an approa.ch will require the evaluation of that will take into a.ccount the behavior of the oblique wing.

To remind us of this detail, we shall add the sUbscript p to the derivatives that must be evaluated in tha.t way.

Since (j ~ (ja ---J (jp - (ja (jp and (B-44 ) Therefore b/2 (B-44)

[(~) + dO ] ~ dy

f

-b/2 P C

.J

Wing Contribution.

Similarly to C) (C) would be zero if the change in lift ( zp w' yp w was antisymmetric.

b/2 b/2 2 (j U

o

E:l.. dy = -

d tanA,) dy

f o~ (1, 'Y

op baS

f

-b/2 -b/2 (B-46) lQ() -LVV

We notice that ~ = 0 for a rigid wing, but it would be different from

op zero for a flexible one.

Fin Contribution [Ref. 21J.

(B-47 )

_ '(0)

= _ SF C ( Z1, FZ op S Lex b O Fin Therefore, for a rigid wing b/2

FZ

SF (z.e 00)

tanA dy - - C -- - -

[(~) YO - (~) ]

S L . b op eL O

f

-bIZ p p Fl.n (B-48) For the flexible wing, the term ~ is no longer zero, therefore

the term .eO ~ should be included in the integration representing the

wing contribution.

Wing Contribution b/2 bIZ 02 - - 2 dy

'Op x dy = T [ (~~) + dO] ~2

f

1 -b/2 bOS

-biZ p (B-49) Fin Contribution (B-S 0) Therefore b/2 (B-Sl)

-2 f

C,e == b 2S p 0 -b/2 C m -P.

No tail contribution to this new derivative b/2 b/2 oZ - 2

01,) - 1--

C

ap Y dy == -=- [(

00: P + dO x Y dy m p

~/2

f

-b/2 bOcS (B-S2) C n -P.

Wing Contribution The side force is the new element in the wing contribution to C n p b/2

OX - OY-]

(B-53)

op x + op Y dy

l

f

-b/2 where (B-54 ) Tail Contribution [Ref. 21J.

(B-55) Therefore, by adding the two contributions, for a rigid wing we obtain C n p (B-56 ) The flexible wing would have the extra term ~- ;"0 dO: y B.5 "q" Derivatives.

The q derivatives are derived in the same way as for the pones.

All the connnents made in the previous .. group of. derivatives can be extended to this one and therefore they will not be repeated.

The local angle of attack now varies linearly according to .c

.3

Wing contribution only b/2 b/2 2U .9.l dy = _0

Oq f (~)q ~c: dy

f

-b/2 cS -b/2 (B-57) b/2 =-

YO - (~)q tanA] Y dy

f [(~)p

cS -b/2 For flexible wing the term t ~y must be included in the integra-

o o:x

tion.

C z -9.

Wing Contribution.

b/2 OZ dy C z

f oq

q w cS -b/2 (B-58) b/2 b/2

2 1 (oz) - -2

- Y dy = -.:::-

~s o:x q cS

-b/2

L2

Horizontal Tail Contribution [Ref. 21] I ) C = (B-59) \ Zq H. Tail and (B-60) Wing contribution only b/2

oz-

dq x dy

f

-b/2 (B-61) -2 =-

, -

b·'cS

o

C m ---S Wing Contribution O

c ) - 2U dZ y dy

( m - -2 oq

q w c S (b-62) -2 = -2 c S Tail Contribution [Ref. 21J (B-63) Therefore b/2

r

-2

C =::::z-

m c S q

~b/2

C n -S Wing Contribution only b/2 2U O

- -

(X x + Y y) dy

=---

L2~

(B-64) = B.6 "r" Derivatives.

derivative, due to the wing aerodynamic Except for the new C m r coupling, and for the wing side force contribution to yaw moments, all the other derivatices are standard.

The method used in evaluating r derivatives in section 3.2.4 will be applied here without any further explanation.

Assuming the steady state flight condition to be straight levelled flight

C = C o

z x r r Fin Contribution only [Ref. 2lJ

_SF [ 21,FH co]

(B-65)

C - S C b + cr

L Yr ~. 0 Fln C.t r Wing Contribution b/2 b/2 -2U -4 -2

o

dy =--

f .to ~ d~;Y) .to x dy

b S

f

-b/2 o -b/2

(B-66) Tail Contribution [Ref. 21J (B-67) Therefore b/2 .tF~ -2 d + C -.!::.

(B-68) .to x y Y b

f r 0

-b/2 C m r Wing Contribution only b/2 b/2

- d" (,,'I -4

C .t y ~ dy = ------

f .to y~ dy

m o dr -

r f

~b/2 b cS O -b/2 (B-69) C n r Wing Contribution (including side force) b/2

x + (.to'Yo - d tan1\) y] dqd~Y) dy

f [do

-b/2 b/2 (B-70)

f [do (~- tanA, y) + .eoyo y) I ~ dy

= b2S

o

-b/2 Tail Contribution [Ref. 21J (B-71) biZ

4 f [-

C = b2 S d

nr 0 -biZ (B-72)

APPENDIX C C.l Similarity Transformation between Inertia Matrices.

The conversion of an Inertia Matrix from body into stability axes occurs according to a similarity transformation [Ref. 26J.

{C-l) [I J s where [I J = Inertia Matrix in stability axes s [IbJ = Inertia Matrix in body axes

[TS/bJ = Rotation (or direction cosine) matrix from body to

stability axes

= Transposed matrix

The stability axes are obtained by rotating the body axes about the Yb axes by an angle a, therefore Yb:= ys • The rotation matrix [Ts/bJ is given by

[ cosa

Si:aJ

°

= 1 (C-2) [Ts/bJ _ sOina cosa

°

and

o - Sina]

[co;a

T = (C-3) [Ts/bJ

1 °

sina ° cosO: In a symmetrical aircraft both the I and I components of xy yz the inertia ma,trix are zero. When the wing is skewed, the I compo- xy nent is no longer zero, therefore for an oblique wing aircraft the inertia matrix, in body axes, looks like I I I xx xy xz = I I 0 0 (C-4 ) [IbJ xy yy I I xz yy • If we know transform [IbJ according to the similarity tra.nsfor- mation (C-l), we shall obtain the inertia. ma.trix expressed in terms of the stability axes, [I J • s After some algebra we obta,in I I I' I' xx xy xz = [I J I' I' I' s xy yy yz I I I' I' xz yz zz where 2 . 2 I' I I sin2a + = cos a + s~n a I xx xx xz zz I' = I yy yy 2 2 I' sin2a + = I cos a sin a - I I zz xx xz zz I' I cosa xy xy I I = I cos2a + (I - I ) sin2a xz xz xx 2 zz I' = sina

- I

yz xy

APPENDIX D D.1 Downwash Matrix [81l..:..

The lift or circulation distribution can be visualized as resulting from a system of horseshoe vortices, each of which is of constant strength (Fig. D-1).

~ r b ~rd ~Ifl

~rN'-''''-r" -~~ -_ .....

...

Actual airload curve Approximation to the actual loading as given by horseshoe ~ vortices ~ ( Figure D-1 [Ref. 14J.

Horseshoe Vortex System The net strength of the trailing vortex at any point on the span of the wing is numerically equal to the rate of change of strength of the bound vortex in the spanwise direction. Little loss in accuracy with respect to the spanwise air load distribution will be entailed if: a) The total strength of the chordwise system of bound vortices is concentrated in one bound vortex located at the local span- wise quarter-chord point.

b) The downwash angle at each vortex station across the span of the !::.

wing, at the local streamwise three-quarter-chord point (= down- wash control point D) is equal to the geometric angle of attack for airfoil having a 2-D lift curve slope equal to 2n. When the section 2-D lift slope is different from 2n, equation D-1 must be used (D-l) The downwash angle at anyone downwash control point is the sum of the incremental downwash angles due to the horseshoes in the system of horseshoes which represent the wing and its lift distribution.

Assuming the geometry of the wing platform, the angle of attack and section 2-D lift curve slope variation are given across the span, the unknowns ar~ the values of the running lift at each point on the span. The strength of each bound vortex represents the average airload over its own portion of the wing span.

The method for determining the downwash matrix for an oblique wing is now illustrated.

In Fig. D-2 a system of horseshoe vortices and the associated down- wash control points are shown.

Il- r--~l

r I

II I

n,~

. I

II

I

Ii

,I

I

I

I I !

i

I

Figure D-2 station the section lift curve slope will be m. and At the ~ the angle of attack of the section zero lift line is a .

fi

Since a linear relationship exists between the strength r of a

j particular horseshoe vortex j and the downwash velocity at the W •• ~J point i (D-2 ) where K is a constant, and the downwash at i due to the entire vortex system is (D-3) W. = ~J' Woo = ~J' K •• r.

~ ~J ~J J From equation (D-l), which expresses the relationship which must W

exist between the downwash angle V at each control point, the wing

angle of attack and the section lift curve slope M for the wing O station at the control point, the following series of equations result and in general W.

-= (D-4 ) V and substituting (D-3) into (D-4) W.

1 1 -= (D-5 ) -V L. K •• r.

V J 1J J The relation between the running load 1 and the circulation at the th i station is (D-6) 1. = pVr.

1 1 Equation (D-5) can therefore be rewritten in terms of the running load 1 Wi -= K •. (D-7) 1.

~ L K r =

L

j ij j j V 2 1J J PV Substituting into (D-4 ) results in (D-7) (D-8) 1 :E. K · lJ.

iJ pV J or qm.

= __ 1 a: (D-9) L ' K .. l.

1J J 'TT i J And in matrix form (D-lO) ' qm ] is a diagonal matrix.

where '~~, [ Defining a new matrix (D-ll) Equation (D-lO) can be rewritten as (D-12 ) The elements of the [8 J matrix are to be influence coefficients l relating the incremental downwash angle at each control point to the intensity of the running lift over each increment of the semispan of the wing.

D.2 Evaluation of K.. Elements.

----------~J

The velocity induced by a vortex of strength r at a point P can

be written as [Ref. 24 J

w = r (cosO: - cos(3)

(D-B) P 47T R where 0: and (3 are the angles between the direction of the vortex segment and lines joining the ends of the segment to the point as shown in Figure D-3.

r

End view R

__ 1 __

o. w

Figure D.3 - Finite Segment of a Straight Vortex Filament [Ref. l4J.

A plan view of the geometry of a typical horseshoe vortex is given in Figure D-4.

In order to eva.lua.te the incremental downwash velocity induced by a single horseshoe vortex it is convenient to consider the following three cases: 1) Control point to the left of the horseshoe vortex.

2) Control point within the horseshoe vortex.

3) Control point to the right of horseshoe vortex.

Fig. D-4 shows the quantity that will be used in the derivation; by the subscript i we shall indicate quantities relative to the control point D. , whereas by j we shall refer to the horseshoe reference ~ point V .

j The origin of the axis system is at root quarter chord point. When- ever the locus of the quarter chord point does not lie on a straight line, it is assumed to be given by an equation f(y) w.r.t. a straight line pasSing through the root qua.rter chord point and aligned with the wing span (unswept case) y

----1-===::::::-::::::::::::J ~ ~y)

X , Y coordinates of horseshoe reference point V.

j j J C.

~ coordinates of control point D Xi - 2 ' Y i i C. chord length at station i ~ X. = - y. tan A + f (y .) tan A ~ ~ ~.

1) Control point to the right of the horseshoe vortex (Fig. D-4a) defining £:,X .. = X. - X.

~J J ~ R .. = b,X .. + Ci/2 ~J ~J b,Y .. = Y. - Y.

~J ~ J Qua.rter-chord Line /

Y. I j

~ I /- Y -_ .. -..j "" Iv.

'- ,J t···_- ----.---.

".

h h ..... _ ..... L ... _ .

.

• Fig. D-4a Control Point to the Right of Horseshoe Vortex.

For the left trailing vortex and equation (D-13) becomes o [1 - cos (ex .. + 90 ) J 1+ sin ex ..

1J

= -:---:-:-_-=1 J,,:- r

(D-I4 ) 4'(((D. Y .. + h) 4'(((D. Y .. +h) j 1J 1J R .• = 1J sin ex .. (D-lS) _I 2 2 i 1J ~ (6 Y .. + h) + R •.

1J 1J For the right trailing vortex ex=900-~ij ~ = 180 [cos(90 - ~ij) - cos l80 J = _ [ ~~r- ~ij + 1 ] r.

(D-16)

(WiJ')R = - r . 4 (6 Y h) 4 (6 Y h)

'(( .. - '(( .. - J J 1J 1J The minus sign derives from having assumed positive downwash veloci- ties.

R •.

. 1J (D-I7) S 1n ~ ij = -_7= '==~===;2;:=="""2=ij "V (D. Y •. - h) + R ..

1J 1J For the bound vortex ex = ex ..

1J ~ = B ..

1J (cos ex .. - cos ~ .. ) 1J 1J (D-I8) W •• = r'J.

4'(( R •.

1J 1J .6Y .. +h _ --;:.==~1:!:!:J ==;;o:::==:;;=; (D-l9) cos a.. = _I 2 2 1 1J -V (.6 Y .. + h) + R ..

1J 1J .6Y .. -h 1J (D-20) cos ~.. = -_-;=: '==~====;2;;;::=='7<2=ii 1J -V (.6 Y .. - h) + R ..

1J 1J The total downwash velocity at the control point D. is therefore given by 1 + sin ~. . cos a .. - cos ~ .. ] r. rl+ sin a ..

_ J 1J 1J + __ __=1~JL....- __ __=1:..LJ (D-2l)

W ij - 471 L .6 Y .. + h .6 Y - h R

ij ij 1J and

__ .6 W __ ..l.[l+sin a _ l+sin ~ij + cos a -cos ~ij]

ij ij ij K (D-22)

ij r j 471 .6 Y + h .6 Y - h R

ij ij ij 2) Control point within the horseshoe vortex (Fig. D-4b).

~1·····- -----·-···-t---·

a\ ->

i~ / / R .• / 11 ... , / .1 U _____ ... '._ .. _ .. _. _ . __ . r i h h Figure D-4b.

Since i == j 6X .. =6Y .. =0 ~J ~~

R . = C./2

i ~ ~ Both trailing vortices will induce the same downwash velocity CX = 0 ~ = CX •• + ~~ (D-23) (D-24) For the bound vortex

r

i (D-25) (w .. )B = 4 Icos CX •• + cos CX ) ~~ ~ R \ ~~ ii ii h (D-26 ) cos cx = -Vr=h92F+~t=c?Rii ii Therefore J (D-27)

l/ii = ~; [~ (1+ sin "ii) + c: cos "ii]

3) Control point to the left of horseshoe vortex (Fig. D-4c), defining Y Y. I i

J 1

" ~

" ....... ! Vi r1,

, ) "

i .......

I

i ""- 1 .......

Xi

C.! ....... "L----+------.r- ~ 2i

Xjl

i .......

IR ..

.......

I ~J i -""" L .......

!:J.Y •.

~J

,-

-

Figure D-4c a' ~ 180 - a ..

~J ij A' ~ 180 - f3 •.

t' ij ~J C.

~ Roo =&··+-2 ~J ~J /::"Yoo = Y. -Yo ~J J ~ For the left trailing vortex a = 0 f3 = 90 + a~ .

~J --202 (D-28) R · i (D-29)

= V J 2 2'

(/::,.Y •. - h) + R •.

~J ~J For the bound vortex ex= ex .. = 180 ex~ .

~J ~J = = 13 ij 13 1.j r.

(D- 30)

(Wij)B = 4~ ~ij (cos ex - cos l3 )

ij ij Therefore /::"Y .. - h

, ~]

(D- 31) cos ex = - cos ex = - -.r= '=:::::=:=:::::;20===""'2" ij ij -V (/::,.Y •. - h) + R ..

~J ~J Simila.rly /::"Y •• + h . ~J cos 13 •• = - cos 13 ~ . = - -;====='~-='"?f""="7f=i (D- 32) ~J ~J

V (/::"Y .. +h)2 + R~.'

~J ~J For the right trailing vortex ex = 90 -131.j p = 180 0 0

r. [cos(90 - ~' •. ) - cos l80 J r. (sin ~ ~ . + 1)

(w) J ' . ~J = J ~ ] (D-33)

ij R = 4fT (!;,Y .. + h) 4fT (!;,Y .. + h)

~J ~J R ..

= ~]

. ~'

nn ij

_I 2 2 '

"" (!;,Y .. + h) + R ..

~J ~J Therefore

r. [ 1 + ~in ex~ .) 1 + sin ~ ~ . cos ex .. - cos 1

w = J _ ~J + ~J + __ "':~:;,.jJ,-- __ ~...:i;:.w..j

(D-34) ij 4fT (!;,Y - h) !;,Y + h R ij ij ij It is now possible to derive the matrix K of equation (D-10) since each element will be given by w ..

- .2:J.

K - r

ij j D.3 Structures Fundamentals.

As done in the previous section, the continuously varying spanwise airload distribution will be replaced by a series of constant intensity running loads.

The assumption of the section aerodynamic center acting at the quarter chord becomes weak when considering the bending and torsion due to the air load distribution whereas it is quite good when computing spanwise lift distribution. This inconvenience,due to the lack of predicting chord-wise lift distribution, can be reduced by introducing a correction factor f which will allow the section aero- dynamic center to be placed in any desired place along the chord. Such a correction factor is here assumed to be known.

Because of reasons which will appear more evident in this section, the horseshoe vortices must be chosen in a way such that the aircraft centerline will coincide with one trailing vortex.

Let us consider the geometry of the structural skeleton of the wing as shown in Figure D-S and define

M , M , ... , M Rolling moment due to total lift of all the

x x.

Xl 2 ~ vortices outboard of this point (positive when raises left wing tip).

,

M M , ... , M Pitching moment due to total lift of all the

Yl Y2 Yi vortices outboard of this point (positive when nose up).

h h X.

1.

T.

1.

/

Figure D-5 - Structural Skeleton of the Wing.

Beam bending moment at elastic axis point about axis perpendicular to local elastic axis (posi- tive when it compresses wing upper surface).

Torsional moment around elastic axis (positive when leading edge up).

Total lift acting on the wing section having span 2 h , numbered from the left wing tip L. = 2hi,.

~ ~ j,. being the intensity of the running lift at ~ station i.

Streamwise distance from horseshoe reference point at a wing station to the corresponding point on the elastic axis (positive when elastic axis point is to rear of horseshoe reference point).

Section aerodynamic center correction factor; (f positive when aircraft is to rear of quarter chord point).

The general form for the bending moment is M. ="' M cos A + M s in A (D-35 ) ~ xi Yi and for the torsional moment T. = M cos A - M s in A (D-36) ~ Y xi i At station 1, on the center line of the horseshoe vortex nearest to the left wing tip (Fig. D-6) the following equations apply:

L1 ( h t;n A)

M = 2 e - f C +

1 1 Y1 and substituting into equations (D-35) and (D-36) (D-37) (D-38)

T = ~1 [<e - f C + ~ tan A) cos A - h Z1 sin A]

1 1 1 At station 2 (D-39) (D-40) and in general, for the left wing (forward), for i > 1 i-1 h (D-41)

M ) = L: 2 h L. (i - k) + L.; -4

( xi L k=l K ...

i-1

(MyJL= E ~(~i + e - fk '1<) + ~i (e - \ C + ~ tanA) (D-42)"

i i i where herefore, for i > 1 , the bending moment and torsion for the left wing are given by: Leading Edge /-- Centroid of the Load L/2 Elastic Axis ~ ..

M i • hv h ~-- -;- -1 I \ I Figure D-6 - Plan View of Left Wing Tip Section.

(D-43) (D-44 )

- [1: 2 h 'l< (i - k) + L ~] s in A

i k=l For i = 1 expressions (D-37) and (D-38) should be used.

For the right wing, assuming n to be the farthest right station L h n n (D-45 )

M = TT

x n L h M = n (e - f C - ~ tan A) (D-46) Y 2 n n n 2 n h L h n ; tan A)sinA (D-47) (Mn)R = - L :: cos A + 2 (en - f C n n n (D-48) and in general, for the right wing, for k <n i+l

M ) = 2: 2 h 1. (i - k) (D-49)

( xi R k.=n tc i+l .) ~ L. h M = £.J ~(~J.. + eJ.. -f C ) + 2J. (e. -f. C. - -2 tanA) ( k k Yi R k=n J. J. J.

(D-50) where - x.

~ Therefore the bending moment and torsion, for the right wing and for k <n , are given by i+l

- L E:] cos A

(Mi)R = L 2 h ~ (i - k)

[ i 4 k=n (D-5l) L.

h sin A e i - f C ) + : ( e i - f i C i k k 2 At the root station of the elastic axis the bending moments of the right and left wing cancel each other in a trinnned flight condition.

The presence of an unsymmetric air load would introduce a non-zero aero- dynamic moment at the root station.

In Figure D-7, such a case is shown.

(Mr)L Figure D-7 where = t.M (Mr)R + (Mr)L Root bending moment due to right wing air load = (Mr)R bending moment due to left wing air load = Root (M \ r r = Root station In such a case the following considerations can be made; a) Instantaneously the wing is assumed clamped at the centerline, allowing for the discontinuity in the bending moment. The structural rotation angle across the span, due to bending, can be computed as shown in the next pages by simply substituting, for the semi-wing considered, the corresponding root bending moment.

b) The t.M produces an angular acceleration which can be decomposed w.r.t. the roll and pitch axes.

This acceleration introduces angular velocities about the roll and pitch axes which results in an appa.rent or aerodynamic twist. This new twist contribution modifies the spanwise air load distribution and, con- sequent ly, the root t. M• A change in t. M changes the angular acce lera- tion and so on.

Because of this discontinuity at the root it is convenient not to consider the root station.

In case such a point were included in order to take into account the two different values of the bending moment it would be necessary to write two separate matrices: one for the left and one for the right wing.

In addition, when computing the streamwise twist due to wing flexibility, this discontinuity in the root moment comes up again.

By using a system of vortices such that the aircraft centerline coincides with one trailing vortex, the root double-valued point will not be computed. In doing so nothing is lost in accuracy but there is a gain in simplicity for the matrix analysis.

In matrix form, equations D-37, D-3S, D-43, D-44, D-45, D-46, D-5l, and D-52 can be opportunely combined and become 1 I

I Ml = [ cos A [r] + sin A [u] J 1 (D-53)

I \ 1 I ITI =

r- sin A [r] + cos A [u] I 1 (D-54 )

, \

I I where M T £'1 l l T £'1 ~ 1 M 2h 2h = =

1= l£,!

1 !=

IT I

T £'R ~ R £, M T n n n ~ , T ' £'1 are the values at the last sta.tion to the left of the air- craft centerline (Fig. D-S).

~ , T ' £'R a.re the values at the first station to the right of the R aircraft centerline (Fig. D-8).

x ----A -A ________ L___________ Yo e- R

h I

h h _ e h

.. -~

~"

-"""""-" "\, -', Figure D-8 - Plan View of Aircraft Centerline.

hI h 2h 4"

o

o

h (i-2) 2h (i - 1) 2h hi

(R - 1) 2h (R - 2) 2h 4' I

I:>:l I-' -1- - - - - - - - - - - - - - - - - - - - - r If - - - - - - - - - - - - - - - - - - -- (Jl (D-55 ) [r] -_ 4" - 2h . • • • • • • • • • • (L - n)2h I I I I h (j - n)2h - 2h - 4" I I

o

I I h I - 2h - 4"

o

I h n -4 and

[A] : [OJ]

(D-56)

[u] =

I [ [oJ I [B] where [A] and [B] are reported in Tables D-I and D-II.

The moment at the aircraft centerline can be computed as follows (D-57) (D-58) h (D-59)

=

M~ - ~ 4"

(MXr)R h

M + ~

(e - f C - '2 ta.n A) (D-60)

=

r R R Y 2 ( MYr)R R where M ) - Rolling moment at the aircra.ft centerline elastic x L - ( r axis due to left wing air load distribution.

M ) = Pitching moment at the aircraft centerline elastic

( Y L r axis due to left wing airloa.d distribution.

Simila.rly (MXr)R and (MyJ indicate the corresponding moment due to the right wing contribution.

The total rolling and pitching moments acting on the airplane are:

for the rolling moment f

(D-61) 1 h l '2(e - flC + T tanA) l l 1 h (l'lX +e - flC ) '2(e - f C +'2 tanA) 2 l 2 2 2

o

. .. .. .. .. .. .. .. .. .... .... .. ..

(")~ I-:ij,.,..

~O

.. ~~

[A] = &) 1 h (l'lX + e - flC ) (l'lX + e - f C ) ••••• '2(e - f C +'2 tan A) li i l 2i i ~~ 2 2 1\:1 i i i

1-'1 >0

,I

Ctrj

~EiJ

.... .. .. .. .. .. .. ..

1 h (l'lX + e - f C ) (l'lX +e - f C ) •.••••••••••••••••••• '2(e - fLC + '2 tanA) 1L L l l n L 2 2 L L TABLE D-I h;:l;:O ....-I 1-00 Od c:z.

~> r ID >-d

d>

>0

Ctsl

~t;j 1 h 2"(e - fRC - 2" tan A) ••.••..••.••••••••••••••••••••••••••• (f,XnR + e - fnC ) R R R n . .. .. . .. .. .. .. .. .. .. ..

= [B] ~, -(e - f C h 2 (n-I) (n-I) (n-l)- 2" tanA) (f,X(n-I)R+ e(n_I)- fnC ) t-" n 00' 1 h -(e - f C - -!! tanA) 2 n n n 2

o

TABLE D-II for the ~itching moment M (D-62) The streamwise angle of a.ttack contribution due to wing flexibility can then be obtained from i i ex -.!. d s cos A (D-63)

= f E~ d s s in A +

s. GJ ~

f

k k where ex = Streamwise angle of attack contribution at station i Si due to values of bending and torsional moments acting inboard and at station i.

k = R Station when i on the right wing

k = L Station when i on the left wing EI Effective beam bending stiffness around elastic axis GJ Effective torsional stiffness around elastic axis.

2h Since dS = cosA for this study, the integrals in equa.tion (D-62) for station 1 can be written as

M(L - 1) 2h ~ 2h]

• •• + -E-~-I--'-- cosA+ ELI cosA L (L-l) (L-1) 2h

= cosA t E~i.]

2 ~= ~ ~

T. ] G)i And similarly it is possible to obtain the expressions for the corresponding values at each station.

In matrix form: M (D-64 )

:= c~~A [sinA [EIJ 1 ! + cosA[GJJ IT!1

where (D-65 ) where [cJ and [DJ are reported in table D-III.

The matrix [GJJ is similar to [EI] and can be ontained from equation (D-65) by simply replacing EI with GJ.

Substituting equations (D-53) and (D-54) into equation (D-64)

I I 2h r

I CXs I := cosA l sinA [EIJ (cosA [r] + sinA [uJ)

(D-66)

+ cosA[GJ] (sinA[r] - cos.t\[uJ) 1) L I

which can be written as

: as i = 4h [ sinA( [EI] + [GJ] )[r] + (s::s1 [EI] , cosA[GJ] ) [U]] i L: (n-67)

It is now possible to define the elasticity matrix [S2 J .

[C] =

o

ORIGINAL OF POOR PAGE IS QUALITY

o

1 1 ERIR 2E(R+l)I(R+l) [D] = . 2E. I.

~ ~ 1 1 2E I n n TABLE D-III

[S2 It- 4h [sinA ([EIJ+ [GJJ) [r] + (s;on:t [EI] - cosA [GJ]) [a] I

J (D-68) Then \ I - [S J \ n I (D-69) I CXs I - 2 IJ'J I The S2 .. element of the [S2 matrix represents the angle of J 1J atta,ck change in radians at station i due to the structural deflection of the wing caused by a unit loa,ding at station J.

D.4 Wing Twist.

The twist in a flexible wing can have several contributions or twists which can be divided into two cla,sses: I) those which would be present even if the wing were rigid, and IP those due to inertia effects, thrusts or drag, and section pitching movements on the flexible wing.

la 1= la I

+ la I (D-70) I gl

I g l I gIl

class I: Aerodynamic Twists jagIl a) Built-in geometric twist due to camber or construction (including eventual dihedral contribution) or both.

b) Interference twist (not considered in this study).

c) Twist due to control surfaces deflection (flap, aileron, spoilers) .

d) Apparent twist due to airplane rolling and/or pitching velocities (the angles of atta,ck due to pitching velocities should be measured at 3C/4).

Class II: Structural Twist due to Wing Deflections caused by Aerodynamic Loading which are independent of Wing Lift Distribution la I I gIll a) Vertical acceleration upon dry wing and internal fuel dead weights.

b) Effects of a,irplane rolling and/or pitching acceleration upon dry wing dead weight, wing internal fuel dead weight.

c) Section pitching moment with control surfa.ces in neutral position (C of the airfoil).

mO d) Incremental section pitching moment due to control surfaces deflection.

The type of twist due to the effects of wing deflections arising from loads which are independent of wing angle of attack, such as those listed under Class II, may be computed with the aid of equation (D-63) (D-71) where M and T are the wing bending moments and torsion along the wing elastic axis due to the loadings of Class II. For cases a) and b) the knowledge of mass for each section of span 2h and the position of the corresponding C.M. w. r. t. the ela,stic axis must be known.

No attempt is made here to compute the section pitching moment with control surfaces in neutra.l position.

D.4.l Twist due to Control Surface Deflection.

The effect of a control surface deflection is of introducing an a.erodynamic a,s well as a structural twist.

The aerodynamic twist can be computed from the following expression for the lift produced by control deflection [Ref. 25J.

(D-72) where

e = cos -1 (2 S - 1)

a

C = control surface chord

f

151 control surface deflection in radians

Equation (D-71) applies for a 2-D lift curve slope of 2n. For a 2-D lift curve slope of rna ' this equation can be written as (D-73) substituting the value for 8 results in (D- 74) 6 :

iCL 16 = h I CU~ fco-. -l-(tS -lH_~_2 -,,~(~ -_s~ 1J 1

And dividing both sides by [ rna j

(D- 75) Structura.1 twist derives from the section pitching movement due to control deflection. Once the spanwise bending moments and torsion distributions are known, equation (D-71) can be used to compute the section structural twist where M and T are replaced by the corre- sponding ones due to controls, JYl and T c c In Figure D- 9 the symbols used are shown.

h h , " "- ......

" ......

M "- co. , 1.

"

I

.th .

1. Stat1.on Figure D-9 - Structural Moments due to Control.

Station pitching moment due to controls M = q 2h C C cO m Oi i Local chord length C Local pitching moment coefficient C m Oi = M M sinA C. ca.

1. 1.

M cosA T c. ca.

1. 1.

The station pitching moments can be computed similarly to what was done in equations (D- 42) for the left wing, and (D-50) for the right wing.

For stations land n = M q h C C (D-76) l cOl mal M q h C C (D-77) cO l man n For stations 2 and n - 1 (D-78) M (D-79)

= q 2h [f C~n_1) C

cO (n-1) m (n-1) O and so forth.

And in general, for the left wing - i-1 2 1 (D-70)

q 2h l L: C C + -2 c: c ]

k=l k m ~ m Ok Oi for the right wing i+1 ] 2 1 (D-81)

M 0) = q 2h L: c C + -2 c: c

[ k ( c i R k=n m ~ m Ok Oi Equations (D-80) and (D-81) are derived for the general case where the control surface can be extended from tip to tip.

In matrix form, (D-80) and (D-81) become (D-82) where I 1/2 I 1/2 I I 1/2 1 I . . .

I I 1 1 1 . . 1/2 (D-83) ~I~----------------------------- I I 1/2 1. 1 I I 1 1/2 I I 1/2 I 227.

If wind tunnel data are not available, theoretical expressions for C in terms of the deflection 0 may be used.

mO From reference [25J (D-84 )

substituting sinS and sin26 = 2sinS cosS results in

O O 0 O (D-85 ) Therefore, the structural bending moment and torsion are (D-86) (D-87) and the structural twist is M

lex (= 2h [sinA [EIJ 1 1+ cosA [GJJ IT !]

c A (gIll cos c

= -

(D-88) The total twist due to control deflection is C08 2 1

= [[\ 1-_- ~1~2S d) + ~ y ?~1-,:- :~~ _ 8 q h [ ·c :s1 [EI]

(D-89)

+ cosA [GJ] ] [D I 1] [-"'tel -_stJ lc~J 1 i 01

Equation (D-89) can be applied to both aileron and/or flap systems with no requirements for symmetry or antisymmetry in their displacement since, once (~}is specified, the deflection vector (6 lcan assume any configuration.

D.4.2 Apparent Twist due to Rolling and Pitching Velocities.

When the wing is at a skew angle, the apparent twist is computed similarly for both the rolling and pitching case, the only difference being in the way the station arms are computed.

A plan view of the geometry of the quantity used in computing the apparent twist is given in Figure D-10.

Consequently, the apparent twist is given by: 1) for rate of'ro11 p 2) for rate of pitch q Quarter-Chord Locus x s d sin k--+------- YO d

i

L~

I

ORIGINAL

i OF Poo PAGE IS I R QUALITY y t i q i th Station x. tan1\. + f(y ) cos1\.

- Y = ~ i i c + d cos1\. + x.

= Y i p ~ C.

~ Y (Y3/4)i = i 2 x d sin1\. + Y.

~ Figure D-IO - Planview of the Geometric Parameters Used in Computing the Apparent Twist due to Angular Velocities.

and in matrix form D.S Unsymmetrica.l Flight Conditions.

Because of the oblique wing, whenever an unsymmetric spanwise lift distribution occurs, the aircraft will experience accelerations about its rolling as well as pitching axis.

A number of unsymmetrical flight conditions are usually investigated in structural design; for the purpose of this study, such investigation will be restricted to those which arise through the use of control surfaces on the wing, such as ailerons.

The load distribution on an elastic wing associated with control deflections may be thought of as the summs.tion of distributions from the following specific loadings.

1) Symmetrical loading with controls in meutral position.

2) Incremental loading due to controls deflection.

3) Incremental loading associated with constant rolling and/or pitching velocity with controls in neutral position.

4) Incremental loading caused by rolling and/or pitching angular acceleration. This loading results from the structural twist described under "Structural Twist" (class II b).

In a steady roll and/or pitch condition, the span load distribution for the elastic wing is given by the first three loadings enumerated.

For this condition the lagl values vary linearly and antisymmetrically b across the span from - (p+q tanN 2V at the left (forward) tip, and b (p + q tanJ\) 2V at the other.

The wing being B.t a skew angle, any unsymmetry in the spanwise lift distribution produces both pitching and rolling moments and, therefore, it introduces angular accelerations about the corresponding axes. Such a condition occurs when deflecting the desired wing control surface, unless some corrective action is taken in compensating the pitching moment by mean of the horizonta.l ta.il.

.

We shall now investigate this case which implies non-zero p and/or .

q and divide it into three parts: initiation, steady state, and termi- nation of the motion. The second part, the steady roll and/or pitch condition, has already been discussed. The first and third ones differ only for the initial condition: no angular velocities for the initiation and a steady angular velocity for the termination. Therefore the analysis is the same for both cases.

Let us therefore consider the initiation of the motion due to an instantaneous deflection of the ailerons.

There will be contributions to the unsymmetric loading from all of the four loadings enumerated above. The first three ones having already been discussed, so only the loading due to the angular accelerations needs to be analyzed. The result of the angular accelerations is a new contribution to the structural twist due to inertia bending and torsional moments. The resulting twist distribution will superimpose an unsymmetric lift distribution on the already existing spanwise lift distribution.

Because of the linear theory assumption this structural twist contribution is independent from the othe~ ones and, therefore, can be analyzed separately.

Figure D-ll shows the geometry of the wing section center of mass.

th The acceleration acting on the i center of mass is (D-90) and the corresponding loading (D-9l) Similarly to what done in "Structures Fundamentals" it is now possible to determine the pitching and rolling moments distributions and therefore the bending moment and torsion distributions.

For the left wing (D-92) (D-93)

and in general (for I < i ~ L where L Ii last sta.tion to the left of air-

cra,ft centerline) i-I

M = 2: 2h ~ (k - i) (~p + Yll\ (1) (D-94 )

Xi k=1 i-I

M L: - mi[~i - (e -~) + ei](~ P+Yll\ q)

k Y K=l i (D-95 ) .

qua.rter-chord locus p .....

......

......

...... ..

y.

..... \ ~ ..... d sil:').1\.

......

... '-\. .

..................

"#-:------.4=------il ... YO 1 , x.

'- .....

~ '- '- ...

~.

xm.

ie.

t~ I ~ !

-... ...

I C!

·li

Elastic Axis I . th. .

(C.M_). = Center of Mass of ~ w~ng sect~on ~

xm. = distance of (C.M.). from elastic axis

~ ~ ym.; = c + d cos1\. + X. - (e. - xm.)

L P ~ ~ ~ x . = - Y. tan1\. + f (y . ) ~ ~ ~ Figure D-ll - Wing Section Center of Mass.

..

For the right wing h m - -

M = - - (x p + ym . q)

.(D-96 ) x 2 2 n n n (D-97)

and in general for (n > i ~ R) where R tt first station to the right of

aircraft centerline H1

M = L 2h Ill. (k - 1) (~ P + vm. q) + 4!! m. (;Z. p + ym. q) (D-98)

xi k=n 1< 1<. 1< 1. 1. 1.

i+l

M = L - ~(6 X - (e - e ) + xmi)(x P + ~q)

ki k i k Y k=n i (D-99) and in matrix form

I M \ = \ A 1 ( P + \ A 2( q

x (D-100)

1 Myl = lB 11 P + ) B 21 it

where IA 1

= h[AJ[-mr..] \;Zj

I \ \-1 IA 2'

= h[AJ [-m:r -J t ym \

I I - I-I

jB 1 \ = [B Jr m J I x I

r

\B 2

= [BJ[-m ..] lyml

I \

r

= mass matrix (diagonal).

I)nlJ (D-10l) where [E J and [F J are reported in Ta,b 1e D- IV • ........

IN· ....

I A '-/ I "'"' p::: I I C'l I ~ '-" C'l I I I I I I I I I C'l I l .... \..;:r I 1- _

- -- - -

I I .... 1"" I I I C'l I I I I I I "'"' H I I C'l I ........

C'l I ........

I H I I C'l ....

........

N I II r-, <C L-J ...

1 ... h ~g - 2" (Xm + 2" tan/\) ......

~S .......

.......

......

·8 rz

o

1...... h ~~ - [&12 - (e -e )+XlD ] - 2" (Xm2 + 2" tan/\) 2 1 ......

.0."

......

...

..............................................

~>

.......

t""'Q ......

...... [E] = 1 . h

~: - [&u - (e -e )+Xm ] - [&2i - (e -e )+XID ]

- 2" (Xm + 2" tanA) 1 i 1 2 i i -......

......

..................................................

. . . . . . . .. ........

-

.......

1 ...... h - [&lL - (e - e ) +Xm ] .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. ..

L 1 •• - 2" (Xm +"2 ~n/\) L ......

1:-:1 CJ.:I "";1 1: (- - h -.2 X~ - 2" tanA) .

. . . . . . . . ." . . .. - [&nR - (en - e ) + Xn\t]

R

--

-

- ~... _ ......... - .. " ...

[F] =

-

-

o

- 1 (Xm _ II tanA) - 2" 'n 2 _ TABLE D - IV Since 1M. I

= I M I cosA + I M I sinA

1-01 I xl I yl

it is now possible to compute la I from equa,tion CD-7l) •

. gulp The solution of the following equation will give the lift distribu-

tion arising from the twist contribution due to p and 4

CD-103) It is now possible to compute the aerodynamic pitching a.nd rolling

moment due to p and q by integrating over the entire span the lift

force times the corresponding moment arm.

!1, I M = 2 h Ll 2 2 lJ L-x. J ...

I Ij x.

J CD-104) \1, I = [--y- .J M h L 1 2 2 ...

lJ

I (j Yj

where j = p 4

By a.ssuming now that p and q are small quantities, it is

possible to linearize even the p a.nd 4 contributions to aerodyna.mic

pitching and rolling moments OM OM =L x .... xl.

M x = op I:' + oq 'i CD-lOS)

oM oM

= M = -.:t.. p' + -.:t.. q' M

Y oJ? 04

where M oM x.

x '" ----l oj - j M

~~~

oj J and

j = p q

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