Evaluation of the gust-alleviation characteristics and handling qualities of a free-wing aircraft

TABLE O F CONTENTS (Continued) Page APPENDIX C

DESCRIPTIONS O F HYPOTHETICAL AIRCRAFT . . . . . . . . . 117

APPENDIX D

O F COMPLETE AIRCRAFT. . . . 123

AERODYNAMIC CHARACTERISTICS APPENDIX E

METHOD O F COMPUTING TURBULENCE RESPONSES . . . . . . . 135

APPENDIX F

TABULATED NUMERICAL RESULTS . . . . . . . . . . . . . 14 1

SUMMARY A n analytical investigation was performed to evaluate the dynamic characteristics of aircraft employing an unconventional wing, free to pivot freely about a spanwise axis forward of its aerodynamic center and subject only to aerodynamic moments imposed by lift and drag forces and a trailing- edge control tab. The left and right wing panels operate independently, with symmetrical tab displacement being used to control the angleof attack and differential tab deflections causing asymmetric panel deflections for lateral control.

Three hypothetical subsonic aircraft were considered, ranging in gross weight from 3000 to 5 0 , 0 0 0 pounds. The influence of the free-wing concept was determined by comparing the turbulence penetration performance and handling qualities for each free-wing aircraft and the equivalent conventional a i r c r a f t .

It was found that the free-wing concept has natural gust-alleviation char- acteristics which greatly reduce the perturbations in atmospheric turbulence.

The most dramatic reductions are in normal load-factor increments, vertical path displacements, and roll disturbances.

While longitudinal handling qualities appear to be satisfactory, an arti- ficial roll damper was found to be beneficial to the lateral control character- istics because of inherent low roll damping and spiral divergence. With the roll damper augmentation, the lateral-directional control characteristics are excellent. The very powerful roll control provided by differential panel dis- placements, and the reduced gust sensitivity, would be particularly beneficial during low-speed approaches in rough air.

INTRODUCTION Background Operation in atmospheric turbulence is an inescapable fact of life for any aircraft. Although turbulence severe enough to endanger the structure is rare and avoidable, milder regions of rough air are present in even the fine st weather. The unavoidable turbulence is frequently found at lower levels and is caused by thermal drafts or mechanical mixing of the air over rough terrain.

The frequency and extent of exposure to turbulence depends strongly, of course, on the nature of the aircraft's mission. For high performance air- craft engaged in flights over longer distances, the low-level turbulence is en- countered only during climb-out and approach. Furthermore, these aircraft are characterized by high wing loadings which attenuate much of the ride dis- comfort. By contrast, many military and commercial flight operations are conducted at relatively low altitudes and with aircraft having low wing load- ings. Examples of the latter type are aircraft for military observation and liaison missions and commercial pipeline-patrol flights.

Although professional crew members may develop a high tolerance for ride discomfort, prolonged operation in the turbulent environment is physi- cally exhausting and mission performance seems certain to be degraded.

Furthermore, an encounter with gusty air during landing approach forces the pilot to adopt higher approach speeds which compromise the short-field capa- bilities of the aircraft. In particular, the degradation of lateral control power has been a limitation on the maximum performance capability of some STOL aircraft in rough-air approaches.

Attempts have been made to provide artificial gust alleviation by sensing flow changes and actuating either conventional or special control surfaces.

This attack on the problem has merit but has not been completely explored; however, available results indicate that the more effective gust-alleviation systems may require considerable mechanical and electronic complexity.

The unconventional wing concept explored in this study is a more funda- mental approach to the gust-alleviation problem in that no sensing devices or special control actuators are employed. Instead, the free-wing concept makes use of the natural alleviation caused by the fact that a stable lifting surface tends to maintain a prescribed l i f t coefficient by responding to natural pitch- ing moments which accompany changes in flow direction.

The Free-Wine ConceDt As defined in this report, a free-wing aircraft differs from a conven- tional airplane in that the two panels of the fuselage-mounted wing a r e f r e e t o move independently about a spanwise axis and are controlled by means of trailing-edge control tabs. Each wing panel is completely free to rotate about its spanwise axis, subject to aerodynamic moments but otherwise unrestricted by mechanical constraints. To providestaticpitchingstability,theaxis of rotation is located forward of the chordwise center of p r e s s u r e of the wing panel, as shown in Figure 1. Thewing is brought to an equilibrium angle of attack through a balance of moments created by the trailing-edge tab, which is controlled by the pilot, and the torques produced by the lift and drag forces.

Longitudinally, the pitching motion of the wing is mechanically uncoupled f r o m the r e s t of the aircraft, and the vehicle may be considered a "flying wing" with all p a r t s of the aircraft, except the wing itself, hanging freely from the spanwise axis of rotation. For lateral-directional motion, no analogy to any other type of aircraft exists, since the left and right wing panels are free to rotate independently.

The basic concept of the free wing was disclosed in U. S . Patent No. 2 , 347,230, now expired, issued in 1944 to Mr. Daniel R. Zuck, who built a small prototype aircraft in 1945 a s a private venture. This aircraft was never successfully flown, and no analytical work to predict the dynamic be - havior of such an aircraft is known to have been performed.

Several potential benefits of the free-wing design were cited by the inven- tor and these can be supported by intuitive arguments. The most significant claim is that of reduced sensitivity to atmospheric turbulence.

All stable aircraft tend to relieve the normal load-factor response to vertical gusts, for example, by pitching into the relative wind to maintain the equilibrium lift coefficient. The rapidity of the alleviating motion depends upon the pitching moment of inertia. Although reflecting a somewhat oversimplified view, it m a y be argued that significant normal load-factor increments do not occur at frequencies below the natural short-period frequency since these com- ponents are attenuated by the pitching of the aircraft.

The wing alone will certainly have a much lower moment of inertia in pitch than the entire aircraft; consequently, its natural frequency will be much higher, and a greater portion of the gust spectrum will be alleviated. In addi- tion, the energy content of the spectral components of the atmospheric turbu- lence falls off sharply with frequency. This is seen in Figure 2 which r e p r e - sents a typical power spectral density (PSD) function for vertical gust velocity.

The PSD function may be regarded as the relative portion of the total tur- bulence energy which is contained in an infinitesimal bandwidth about a given

wavelength corresponding to the value of the spatial frequency, a. Figure 2

displays the characteristic, common to all such atmospheric turbulence mod- e l s , of sharply reduced energy content at the higher frequencies.

On the basis of the preceding information, it is logical to expect the free- wing aircraft to exhibit reduced vertical turbulence responses.

, Lift - "-,

"-

"- -

" " Tab force FIGURE 1. CROSS-SECTIONALILLUSTRATION O F THE FREE WING loo,ooc 50,OOC L = 1000 f t u g = IO f t / s e c .

I0,OOC I ooc 0.0005 0.001 0005 0.01 0.05

ReducedFrequency , 51 , r a d / f t

FIGURE 2. POWERSPECTRAL DENSITY REPRESENTATION O F ATMOSPHERICTURBULENCE I Additional benefits which might be expected are improved maneuvering capability because of the rapid panel responses and differential deflections, possible beneficial stall behavior because the tab may not be capable of de- veloping a full wing stall, and various configuration improvements made possi- ble by the removal of direct pitch coupling between the fuselage and wing.

It should also be noted at this point that some variations of the basic free-wing concept may have merit, although they are not explored in this study. In particular, it is possible that some beneficial effects may be ob- tained by a p a r t i a l r e s t r a i n t of the wing rotation, in the form of a spring or a damper device. Additional variations might include the use of physical inter- connects between wing and tail surfaces.

On the debit side, some penalties may be inherent in the free-wing con- cept. These might include the weight and drag increments caused by the structural complexity of the pivot supports, and the induced drag penalties caused by imperfect sealing of the wing-fuselage gap.

ScoPe The research effort described in this report is an analysis of the pri- mary effects of the free-wing concept upon turbulence operation and gross handling qualities, designed to provide a first-order evaluation of some of the potential benefits described above and to expose any inherent eccentricities which might offset these advantages.

Attention was confined to linear analyses of turbulence responses and certain handling qualities, for both longitudinal and lateral-directional motion.

No evaluation was made of nonlinear phenomena such as stall c h a r a c t e r i s t i c s , nor was consideration given to performance effects or possible unique design features .

The turbulence responses were evaluated by computing root-mean- s q u a r e (rms) values of pertinent output variables in response to continuous atmospheric turbulence, but no attempt was made to evaluate, directly, the riding qualities as affected by human tolerance factors.

Only those handling qualities phenomena which are inherently affected by the free-wing aircraft were evaluated. These factors include the stability of characteristic modes and the nature of certain open- and closed-loop con- trol behavior, but exclude consideration of control force gradients.

SYMBOLS The following symbols are used in the main body of this report; addi- are defined in each appendix, as required.

tional symbols

-

c = mean aerodynamic chord, feet = roll-damping-stability derivative cQP C Q = slope of roll-momentcoefficientvs.right-wing-panel *p displacement = slope of left-panel pitching-moment coefficient vs. right- cmLgP panel displacement C = slope of right-panel pitching-moment coefficient vs. non- mRp dimensional roll rate Cm = slope of right-panel pitching-moment coefficient vs.

R*p right-panel displacement = slope of right-wing-panel pitching moment vs. slideslip

c m ~ angle

= slope of yawing-moment coefficient vs. nondimensional cnP roll rate

cn = slope of yawing-momentcoefficientvs.right-wing-panel

6P displacement Cp = gain constant, aileron deflection per unit roll rate, seconds Cq = gain constant, aileron deflection per unit roll angle Dy = lateral path displacement, feet g = acceleration of gravity, feet/sec2 h = vertical path displacement, feet

IxxT = total moment of inertia about roll axis, slug-ft2

Ixyp = X-Y product of inertia of right wing panel, slug-ft2 I "

Ixz, = X-Z total product of inertia, slug-ft2

Iy1 = moment of inertia of each wing panel about hinge axis, slug -ft2 Iyzp = Y -Z product of inertia of right wing panel, slug-ft2 I z Z T = total moment of inertia about yaw axis, slug-ft'

j = unitimaginarynumber, fi

KG = gain constant, elevator deflection per unit pitch angle L = scale length of atmospheric turbulence, feet Lp = roll damping coefficient, 1 / second = wing contribution to roll damping coefficient, l/second LPW Lr = coefficient of roll moment due to yaw rate, 1/ second = coefficient of roll moment due to right-wing-panel dis- L 6 P placement, l/second2 Lg = coefficient of rollmoment duetoright-control-tab tR displacement, l/second2

Lp = coefficient of roll moment due to sideslip, l/second2

mp = mass of one wing panel, slugs M = fuselagepitchdampingcoefficient,l/second coefficient of right-wing-panel pitching moment due to MRP = r o l l r a t e , 1 / second = coefficient of right-wing-panel pitching moment due to MRP sideslip, 1/ second2 M = coefficient of right-wing-panel pitching moment due to R6p right-paneldisplacement, 1 / second2 = coefficient of right-wing-panel pitching moment due to MR6L left-panel displacement, 1/ second2 M = coefficient of right-wing-panelpitchingmomentduetoright- R ! t R tabdisplacement, 1 / second2 = coefficient of right-wing-panel pitching moment due to left- MR6 t~ tabdisplacement, 1 / second2 M = coefficient of right-wing-panelpitchingmomentduetoangular Rb rate, 1 /second M, = fuselage pitching-moment coefficient due to angle of attack, 1 / second2 Mk = fuselage pitching-moment coefficient due to angle of attack rate, 1 / second = fuselage pitching moment due to symmetric panel displace- M 6 p ment, 1/ second2 Mb;. = fuselage pitching moment due to panel rotational acceleration = fuselage pitching moment due to vertical gust velocity, Mvg l/feet-second M+ = fuselage pitching moment due to vertical-gust accelera- g tion, l/feet N = coefficient of yawing moment due to roll rate, 1/ second P N = wing contribution to N l/second PW P' Nr = yaw damping coefficient, 1 / second N b = coefficient of yawing moment due to sideslip, l/second2 N = coefficient of yawingmoment due toright-wing-panel 6p displacement, 1 / second2 N6 = coefficient of yawingmoment due toright-control-tab tR displacement, l/secondZ P = a r e a of each free-wing panel, feet2 P = panel-pitching-moment coefficient due to pitch rate, 1 / second PG = panel-pitching-moment coefficient to pitch acceleration Ph = panel-pitching-moment coefficient due to longitudinal acceleration, 1/ second Pv = panel-pitching-moment coefficient due to vertical gust g velocity, l/feet-second Pa = panel-pitching-moment coefficient due to angle of attack, 1 / se cond2 Pir = panel-pitching-moment coefficient due to angle of attack, rate, l/second Pg = panel-pitching-moment coefficient due to symmetrical tab e displacement, 1/ second2 = panel-pitching-moment coefficient due to symmetrical p 6 P panel displacement, 1/ second P = panel-pitching-momentcoefficientduetopaneldisplace- e'p m e n t r a t e , l / s e c o n d Ps = panel-pitching-moment coefficient due to panel acceleration P p = roll rate, radians/second except when indicated r = yaw rate, radians/ second except when indicated U = true airspeed, feet/second u = dimensionless airspeed variable, A U / U Vg = vertical-gust velocity, feet/second Xu = longitudinal force coefficient due to airspeed, l/second Xv = longitudinal force coefficient due to vertical-gust g velocity, l/feet X, = longitudinal force coefficient due to angle of attack, 1 / second X0 = longitudinal force coefficient due to pitch angle, l/second X 6 = longitudinal force coefficient due to wing-panel displace- p ment, l/second - I I - Xkg = position of wing-panel center of gravity forward of hinge axis, in percent of E = coefficient of side force due to roll rate, 1/ second yP Y , = coefficient of side force due t o yaw rate, l/second2 Yp = coefficient of side force due to sideslip, l/second2 Y = coefficient of side force due to asymmetric panel dis- dp placement, 1 / second2 Zq = coefficient of normal force due to pitch rate, l/second Z . = coefficient of normal force due to pitch acceleration Zu = coefficient of n o r m a l f o r c e due to airspeed, l/second2 Zv = coefficient of normal force due to vertical gust velocity, g l/feet-second Za = coefficient of normal force due to angle of attack, 1 / se cond2 Zk = coefficient of normal force due to angle of attack rate, 1/second Z 6 = coefficient of normal force due to symmetrical tab e displacement, 1 / second2 Z = coefficient of normalforcedue to symmetricalwing- 6p panel displacement, l/second2 Z = coefficient of n o r m a lf o r c e due topanelrate,l/second &P Z " = coefficient of normal force due to panel acceleration bP af = inertial angle of attack of fuselage angle between longi- tudinal axis and projection of inertial velocity vector in plane of symmetry, radians p = inertial sideslip angle, angle between longitudinal axis and projection of inertial velocity vector in horizontal plane,radians I 1 I1 I I . I I I

pg = sideslip gust velocity, lateral gust velocity divided by

a i r s p e e d 6, = aileron deflection, or asymmetric tab deflection, numer- ically equal to displacement of right tab, radians de = elevator deflection, or symmetrical tab deflection, radians d P = displacement angle of right wing panel with respect to fuselageaxis,radians 8 = pitch angle of longitudinal fuselage axis with respect to horizon,radians X = Laplace operator, l/second p = atmospheric density, slugs/feet3 0 = rmsgustintensity,feet/second g T R = roll-mode time constant, seconds cp = roll angle, radians = rollinggust,l/second

%

@ = power spectral density of gust velocity, (feet/second)2/ (radians/foot) = power spectral density of normal acceleration, (g units)2/ *nZ (radians/foot) Qn = power spectral density of lateral acceleration, ( g units)2/ Y (radians/foot) @ a = power spectral density of r o l l r a t e ( r a d i a n s / s e c ) 2 / cp (radians/foot) @$ = power spectral density of yaw rate (radians/sec)2/ (radians/foot) Y = yaw angle, radians = reduced,orspatial,frequency,radians/foot.

PROCEDURE F r e e -Wing Static and Dynamic ~. Characteristics An initial decision was made to include unsteady aerodynamic effects in the longitudinal motion of the free wings, but to limit the lateral-directional aerodynamic representation to that provided by conventional stability deriva- tives augmented by new derivatives peculiar to the split free wing.

A s a f i r s t s t e p , a n a s s e s s m e n t was made of the necessary control-tab geometry required to provide trim l i f t coefficients throughout the expected

i linearrange.Thechosengeometrywasthenheldconstantfortheremainder

of the program.

Transfer functions were developed to approximate the unsteady aerody- namic effects on wing panels undergoing symmetrical pitching and plunging, and a brief study was made of the dynamics of the wing free only in pitch.

This cursory examination illustrated the effect of the unsteady aerodynamics on the natural frequency in pitch and provided a rational basis for the quasi- static panel damping derivative (panel pitching moment per unit angular rate) in the lateral-directional equations which follow.

Except for the aforementioned damping derivative, the lateral- directional-stability derivatives of the split free wing were computed from classical lifting line theory, using a finite trigonometric series to describe the spanwise circulation distribution, and an iterative procedure to determine the series coefficients which satisfied the boundary conditions. The desired sta- bility derivatives were then computed from the series coefficients, including new derivatives which occur because of the panel rotational degrees of f r e e - dom. Details of the tasks described above are given in Appendix B.

Hypothetical Aircraft To provide a d i r e c t a s s e s s m e n t of the effects of the free-wing concept, three basic hypothetical aircraft were chosen and each was examined in two versions: as a free-wing aircraft and as a conventional fixed-wing airplane.

The aircraft were chosen to represent light observation and transport vehicles ranging in weight from 3000 pounds to 5 0 , 0 0 0 pounds. Furthermore, two flight conditions were considered for each aircraft: one representative of cruise flight for that class of airplane, and the other an approach condition at roughly 1 . 3 stall speed. These aircraft are described in more detail in

Appendix C , but some of the gross characteristics are given in Table I. A subscript attached to each aircraft designates its wing planform, two aspect ratios and two taper ratios being considered.

Cruise Approach Gross Wing True True Alt., Speed, A l t . , Speed, Desig - Aspect Taper Span, Weight, loading, ft knots nation Description Ratio Ratio ft Ib lb /ft2 ft knots ~ " ~~ " - " . . " 41.4 3,000 5,000 118. 1,000 Light obser - 8. 1.0 14. 74 vation 8. 0.6 41.4 14. 118. 1,000 74 3,000 5,000 1,000 1 4 6. 1 . 0 35.8 14. 118.

3,000 5,000 6. 0.6 35.8 14. 118. 1,000 74 3,000 5,000 197.

Utility trans- 8. 1 . 0 54.6 33.5 10,000 1,000 115 12,500 portation 33.5 197. 1,000 115 8. 0.6 54.6 12,500 10,000 197.

6. 1 . 0 47.3 33.5 1,000 115 12,500 10,000 197. 1,000 6. 0.6 47.3 33.5 115 12,500 10,000 1.0 29.6 219. 1,000 Light 8. 116.3 108 50,000 20,000 freighter 8. 0.6 116.3 29.6 219. 1 , 0 0 0 108 50,000 20,000 29.6 219.

6. 1 . 0 100.7 1 , 0 0 0 108 50,000 20,000 29.6 219. 1 , 0 0 0 6. 0.6 100.7 1 08 50,000 20,000 Equations of Motion The complete nonlinear equations of motion were developed as described in Appendix A . With each wing panel free to move independently, the complete equations described a dynamic system with eight degrees of freedom. These eight degrees of freedom are related to the eight independent variables re- quired to identify the instantaneous state of the system: six conventional variables to define the spatial position and orientation of the fuselage assem- bly, and two additional variables to define the respective left- and right-wing- panel displacements with respect to the fuselage.

The complete set of equations were then linearized about a straight and level equilibrium flight condition. The linearization process permitted the separation of the equations into two uncoupled sets describing the lateral- directional and longitudinal motions separately; made possible the direct com- putation of characteristic roots which greatly simplified the assessment of handling qualities; and permitted the use of conventional power spectral den- sity techniques for turbulence -response calculations.

The linearized set of equations describing the longitudinal motion of the ( 1 ) .

aircraft in response to vertical gust velocities is given in Equation - Z af vg

e

-M. X-M, vg g 6 P V U g V h 0 where the matrix of coefficients is:

-x + xu 0

x , xe 0

x b P 0 Ke 0 0 - 1 -U U 0 0 -x These equations are written with respect to a stability axis system with origin at the center of gravity of the aircraft. The first four equations of the set represent the translational motion normal to the longitudinal axis, pitching

motion of the fuselage, wing-panel pitching about the hinge axis, and longitudi -

nal acceleration. The fifth equation of the set permits feeding back a pitch- angle signal to elevator deflection (symmetrical tab displacement), while the last equation represents the kinematic relationship between inertial flight-path angle and rate of climb.

The set of equations is more complex than Equation ( 2 ) indicates because eight of thecoefficients,namely, Z , , Zq, ZAP, Z h p , Pa, Pq, PAp and P d P , contain a first-order transfer function representing the lag in the circulatory lift buildup following a change in angle of attack. This complication required a special technique for the numerical expansion of the determinant of the co- efficients to obtain the characteristic equation. This is discussed further in Appendix E .

Only the vertical component of turbulence was considered for longitud- inal motion, since the head-on component has little influence except at very low frequencies.

When r e p r e s e n t e d a s a polynomial in the operator X , the determinant of the coefficients of Equation ( 2 ) became a ninth-order expression. One ofthe roots of this characteristic equation was always zero, leaving eight roots to describe the longitudinal modes of the system.

As derived in Appendix A, the lateral-directional motion of the aircraft system was permitted to be perturbed by spanwise gradients of vertical gust velocity, and by lateral gust velocities and gradients. Mathematically the gust disturbances appear as rolling gusts and sideslip gusts as shown in Equation ( 3 ) .

- L Cp

- LP

Pw Y -Np-NrX -NPw

P

- 9

[ BI bg + - 2M "Rp 6P

RP

0 0 'a 0 0 DY

where the matrix of coefficients of the homogeneous equations, 1 B] , is given

by: I x z (-X2 + LpX) ( - X2 + L,X) 0 2L6tR IXXT N6t R [ B l = ZMR (-Xz t MR. X + MR ) (MR6 ) 0 B 6 tR tL ' 1 0 0 0 u 0 0 -x T h e f i r s t t h r e e of the equations of this set are very similar to the ordi- nary rolling, yawing, and lateral translation equations of conventional air- craft. The fourth equation describes the asymmetric motion of thewing panels, the fifth p e r m i t s the use of a closed-loop control of aileron in re- sponse to bank angle and roll rate, while the last is the kinematic relation- ship for lateral path displacement.

The expansion of the determinant of the coefficients of Matrix B yielded an eighth-order characteristic equation, but for stick-fixed motion, two of the roots were zero, leaving six nonzero roots to describe the lateral-directional characteristic modes.

Equations ( 1 ) and ( 3 ) required the estimation of numerous aerodynamic of the complete aircraft.

coefficients and nondimensional stability derivatives The estimation procedure is outlined in Appendix D.

Handling-Qualities Evaluation The primary reference for handling-qualities requirements was the re- visedmilitaryhandling-qualitiesspecification,Reference 1 . Forlongitudinal motion, the phugoid damping and the short-period frequency and damping were compared with the specification requirements. In addition, the ability of the pilot to damp long-period oscillations by monitoring fuselage attitude was ex- amined. This feature is not specifically required by Reference 1,butwasin- corporated in this study because of the unconventional nature of the free-wing a i r c r a f t .

For lateral-directional motion, the characteristic roots were used to check compliance with dutch-roll damping requirements, roll-mode time- constant specifications and permissible rates of divergence in the spiral mode.

In addition, the closed loop roll control characteristics were evaluated using a simple pilot transfer function.

Responses to Atmospheric Turbulence The responses of the aircraft to atmospheric turbulence were computed using the power spectral density techniques outlined in Appendix E. Except in selected instances, all responses were computed for stick-fixed motion to pro- vide a simple basis for comparison.

F o r the purpose of comparing the free-wing and fixed-wing statistical responses, the rms values of selected variables were computed from truncated spectra which eliminated all harmonic components below a temporal frequency of 0 . 3 radians per second. These low-frequency disturbances are easily con- trolled by pilot action, and in some cases, a static instability of the spiral mode would have rendered the output spectrum meaningless at zero frequency.

For longitudinal disturbances, only vertical gust components were con- sidered, and the Dryden PSD function of Figure 2 was used, with a scale 1 foot per second.

length of 1000 feet, and an rms gust intensity of For lateral-directional motion, the combined effects of uncorrelated rolling and side gusts were computed, and these were based upon the same basic spectral density function as was used for the longitudinal motion.

DISCUSSION O F RESULTS Isolated Free-Wing Characteristics Static Characteristics A n e c e s s a r y first step in this study was an investigation of control-tab requirements, and a cursory examination of the dynamic characteristics of the free wing to identify possible difficulties caused by pitching-mode insta- bilities of the wing itself. The details of this work are found in Appendix €3.

Considering the static characteristics, it was found that no difficulty would be encountered in providing sufficient control power for symmetrical pitching. The effects of a 10 percent chord, sealed, plain-flap control were examined in detail, using two-dimensional characteristics from Reference 2, and a symmetrical airfoil section with zero pitching moment at zero l i f t .

Only operation in the linear l i f t curve range was considered.

The control deflection required for trim is shown in Figure 3 f o r a wing of infinite aspect ratio, for several values of hinge margin. The hinge margin used here is the distance, in percent of chord, that the hinge axis is forward of the quarter-chord line.

In Figure 4, the same information is shown for finite wings with aspect ratios of 8 and 6 and taper ratios of 0 . 6 and 1. Although these data are ideal- ized in that the flattening of the lift curve near the stall is ignored and the tab effectiveness is independent of angle of attack, it seems clear that no prob- lems are likely to be encountered in providing sufficient control power for the f r e e -wing panels.

Dynamic Characteristics The dynamic characteristics of the free-wing panels are quite compli- A s mentioned earlier in this report, the evaluation of the stall c h a r - cated.

acteristics was outside the scope of this study, but there is evidence that a possibility exists of single-degree-of-freedom torsion flutter near the stall.

Rainey (Reference 3 ) , for example, examined the characteristics of a two- dimensional wing oscillating about its midchord axis and found large regions of reduced frequency and angle of attack near the stall, where negative damp- ing exi st s.

symmetrical airfoi I sect ion I I I L .< 01 ( 1. L ! 0.3 0.4 0.5 06 0.7 08 0.9 1.0 I I T r i m m e d L i f t C o e f f l c l e n t FIGURE 3 . TWO-DIMENSIONAL TRIMCHARACTERISTICS ~ 1

Taoer ratlo =0.6 - 1

I v) 0.10 C hinge margin W IO percentplainflapcontrol W c - " . .- I O t V W - LC W Aspect ratlos 8 n W V Le L ~ " .. ..

' " 5 - -

t

c c

I

I l l

0 . 2 0.3 0.4 1 0.8 0.9 I O T r i m m e d L i f t C o e f f i c i e n i FIGURE 4. FINITE WING TRIMCHARACTERISTICS 2 0 On the other hand, available data, such as that contained in Reference 4, indicates that single-degree-of-freedom torsion flutter is no threat at m e a n angles of attack in the linear range for the mass parameters and hinge-axis locations envisioned for free-wing aircraft.

Aside from flutter considerations, the pitching dynamics of the free wing are interesting because of the effect of short-period natural frequency upon the response to vertical gusts. The importance of the natural frequency prompted an examination of the influence of unsteady aerodynamic forces upon the pitching mode.

Following Jones (Reference 5), a wing with a n a s p e c t r a t i o of 6 was con- sidered to be free to rotate in pitch about a spanwise axis forward of the aero- dynamic center. The l i f t force on the wing was composed of circulatory con- tributions and virtual mass forces. Using Jones' exponential approximation to the indicia1 l i f t growth function, a transfer function was derived relating circulatory lift coefficient to angle of attack, as outlined in Appendix B.

The characteristic equation of the wing free only in pitch was derived assuming that the pitching moment was caused by both the circulatory and ap- parent mass components of the lifting forces acting through their respective moment arms. This equation is given in Appendix B as Equation B - 2 2 . The dimensionless roots of this characteristic equation, a cubic, were found to be functions of two parameters only, the static hinge margin and a mass pa- r a m e t e r a s shown in Figure 5. The static hinge margin is the distance, in percent of chord, that the hinge axis is located forward of the wing a e r o - dynamic center. Aside from the oscillatory mode shown, a stable real root also exists for all cases studied. It should be mentioned that the hinge margin and mass parameter are not independent for an actual wing because the pitch- ing moment of i n e r t i a i s a function of pivot location.

To assess the importance of the unsteady aerodynamic forces, the fre- quencies of oscillation for a hinge margin of 10 percent were compared with those obtained by ignoring all forces but that caused by multiplying the in- stantaneousangle of attack by the static l i f t curve slope. This comparison is shown in Figure 6 , and the importance of unsteady aerodynamic forces is evident when the mass parameter is small.

T h e m a s s p a r a m e t e r s of the free wings used in this study were typically about 10, so the conclusion was reached that unsteady aerodynamic effects were necessary to describe the longitudinal wing motion. These effects were incorporated into the derivation of the complete longitudinal equations. For lateral-directional motion, unsteady aerodynamic effects, as such, were not used, but the wing-panel damping coefficients were based upon the damping observed in the longitudinal motion.

b Dimensionless root = a k j b 1 . 4 2u

To dimensionalize , multiply roots by -

C 0.25 E hinge margin Aspect ratlo 6 Moss porometer pPE3 I Real Axis a ~ -0 5 - 0.4 -0.3 - 0.2 -01 FIGURE 5. ONE-DEGREE-OF-FREEDOMOSCILLATORY MODE ROOTS

P 5 1 0 1 5 20

Mass Parameter, p PC3 FIGURE 6. E F F E C T O F UNSTEADY AERODYNAMICSON NATURAL FREQUENCY 2 2 Longitudinal Motion Free-Wine Characteristic Modes

Additional Modes. The longitudinal motion of a conventional rigid air -

craft with controls fixed is adequately described by a s e t of equations yielding four characteristic roots. These four roots are typically divided into two complex pairs: one defining the long period phugoid mode, and the other pair representing the longitudinal short period motion. In contrast, as derived in Appendix A, the linearized set of equations describing the longitudinal dy- namics of the free-wing aircraft will generally yield four additional charac- teristic roots. In addition to the phugoid and short-period mode, a rapid os- cillatory mode and two heavily damped aperiodic modes appear for the free wing with unsteady aerodynamic effects included.

The nature of these modes can be illustrated by considering a particular example.Table I1 lists the characteristic roots for the light, observation c l a s s a i r c r a f t with an aspect ratio of 6 , in the cruise condition. The fixed- wing aircraft can be compared with three versions of the free-wing aircraft obtained by varying the hinge axis location and the horizontal tail volume.

TABLE 11. LONGITUDINAL CHARACTERISTIC ROOTS Aircraft A g , Cruise ~ .~ - . ~. ~" . " - ~ . -~ . . . . . .~ .~ . .. . ~~ ~ ~ " - - - - ~ - . . . . , .

1w0 Panel Margin 1w0 Panel Margin 2 w o Panel Margin

Mode Fixed Wing Nominal Tail Volume 1/2 Tail Volume Nominal Tail Volume ~~ . .~ ~. . . ~. , ~~ . ~~ "~ ~ . " -0.0228 f j 0.180 -0.0239 f j 0.226 -0.0239 f j 0.226 -0.0240 f j 0.226 Phugoid -3.11 + j 6.87

-4.41 f j 2.73 -2.89 f j 6.65 -1.49 * j 5.01

Shortperiod Symmetric -8.90 f j 12.2 -8.60 f j 12. 04 -10.5 + j 16.4 Wing-Panel Mode -22.3 -22.9 -23.4 Aperiodic -20.1 -20.5 -20.5 2 3 The phugoid mode is little affected by the free-wing concept, as would be expected in view of the relatively minor effect of short-term pitching dy- namics on this long-period motion.

The short period roots are changed in magnitude when comparing fixed- wing and free-wing versions, and, more importantly, the function of the short- period mode described by these roots is changed considerably. The free-wing short-period roots describe a motion which is largely confined to pitching of the fuselage assembly about the hinge axis, with only a minor normal-load- factor contribution caused by aerodynamic forces associated with pitch-rate- induced aerodynamic forces on the horizontal tail. The short-period mode of a conventional aircraft dominates the normal-load-factor response to turbu- lence and control inputs; in the free-wing aircraft, this function is assumed by the symmetrical wing-panel mode.

SampleLongitudinalResponses.Figure 7 illustrates time histories of the control responses of the fixed-wing aircraft of Table I1 and its free-wing counterpart with 10 percent panel margin and 1/2 the fixed-wing horizontal tail volume. The most striking effect of the free wing in these motions is the greatly reduced time required to reach the peak load factor. As mentioned above, this is a consequence of the fact that the symmetric wing-panel mode dominates the initial response. A s seen in Table II, the natural frequency of the wing-panel mode is more than four times as high as the fixed-wing short period mode. Consequently, peak load factor is reached in approximately one -fourth the time.

The wing-panel-deflection history in Figure 7 appears to contain a residual damped oscillation in the short period mode. This is r a t h e r m i s - leading since the panel deflection is measured with respect to the fuselage, and it is the residual pitching motion of the fuselage which gives this appear- ance to this trace. The wing-panel displacement with respect to a fixed horizontal reference does not contain a significant component in the short- period mode.

The reduction in the damping ratio of the free-wing short period mode is caused primarily by the loss of the Z , damping effect which contributes significantly to the fixed-wing short-period damping, Another interesting observation from Figure 7 is the fact that the angle of attack of the free -wing aircraft's fuselage assembly responds to longitudi- nal control exercised through the wing control tabs even though no mechanical pitch coupling between wing and fuselage exists. This phenomenon is a r e s u l t r-“

.. Fixed wing

w v )

-

m a ) I=!!!

a m

W

, free-wing aircraft

z -7-

E m I I 4 5 6 ..

c Time, seconds

0 I .o-

.- t V

. ~ - 1- ””” 1 ~ ~ ~ I~ I

3 4 5 6 Time, seconds u) u - a l o w Fixed wing , ” ” ”

””” L”-

\Free wing

I I I

2 3 4 5 6 T i m e , seconds FIGURE 7. COMPARISON O F RESPONSES TO S T E P LONGITUDINAL CONTROL INPUT Aircraft A3, Cruise of the increase in the downwash angle at the horizontal tail when the l i f t coeffi- cient of the wing is increased. This is a beneficial effect, from the handling qualities standpoint, and will be discussed later.

Further insight into the longitudinal behavior can be obtained from Fig- u r e 8 which demonstrates the effect of the free-wing on the encounter with an isolated vertical gust. The assumed gust has the commonly used "1-cosine" shape with a period of 1 second, corresponding to a 200-foot wavelength, and a peak velocity of 10 feet per second.

The dramatic reduction in load-factor response is apparent, as i s the r e a s o n - the ability of the wing panel to deflect rapidly into the updraft as op- posed to the relative sluggishness of the fixed-wing aircraft pitch angle re- sponse. The net result is a reduction of over 4 to 1 in the positive load- factor peak and an attenuation of better than 2 . 5 to 1 in the negative transient.

Once again, it should be noted that the wing-panel deflection plotted in Figure 8 is measured with respect to the fuselage whose pitch-angle oscilla- tion is also shown. After the gust has subsided, it should be observed that the fuselage pitch angle and wing-panel deflection traces are virtually equal and opposite, demonstrating that the true wing panel motion has subsided and the predominant motion is in fuselage pitching.

Effect of Parameter Variations. With regard to the fuselage pitching LI1 ~~ " .

motion, it was found that the frequency and damping of the oscillation (the free-wing short period mode) are strongly influenced by the horizontal tail volume. In some cases, reducing the tail volume to 1 /4 the nominal value gave better turbulence response than that obtained with either the nominal or 1/2 nominal values. In other cases, 1 / 2 nominal tail volume seemed best.

R number of possibilities exist for improving the pitch response. Fixed auxil- iary damping surfaces, mechanical interconnects between wing deflection and tail surface displacement, wing pivot restraints by means of springs or dash- pots, or a simple pitch rate SAS operating through the horizontal tail could be investigated in any particular design. Again referring to Table 11, and com- paring the nominal tail volume roots with those for the 1/2-tail-volume case, the evidence is clear that changing the free-wing short period mode charac- teristics has a negligible effect on the other modes. Because of t h i s , a r t i f i - cial improvement in the fuselage pitching motion would not be expected to have any adverse effects on other motions. In fact, since the short-period mode does appear somewhat in the residual normal load-factor response as seen in Figure 8, any artificial improvement in fuselage pitch damping would be ex- pected to improve the overall response characteristics.

0.5- 0.4- "\

" " " _

Fixed wing

:'I

0.3 - I Freewing

I 1

0.2-1 1

I I I 3 4 5 6 Time , seconds

- I /

-0.3

I I

- v

-04 I 3 4 5 6 T i me , seconds -5 Time, seconds FIGURE 8. COMPARISON O F RESPONSESTO A DISCRETE GUST Aircraft A3, Cruise The effect of a forward movement of the hinge axis location is to in- crease the frequency of both the short period and wing panel modes. Time histories are not shown for the 20 percent panel-margin case of Table 11, but further improvement in load-factor turbulence response and control deflection response time could be expected. In fact, the spectral turbulence responses to be discussed later show improved load-factor responses because of the in- creased natural frequency of the wing panel mode with the 20 percent margin.

Offsetting this advantage somewhat is the increased control power required with the greater hinge margin, as was illustrated in Figure 3 . The greater tab deflection requirements reduce the trimmed l i f t curve slope of the wing panels and would have an adverse effect on trim drag.

S e v e r a l o t h e r p a r a m e t e r s w e r e v a r i e d t o a s s e s s t h e i r i m p a c t upon the longitudinal modes. Specifically, the characteristic roots were examined for sensitivity to aspect ratio, fuselage center of gravity with respect to hinge axis, and wing-panel imbalance with respect to hinge axis.

With regard to aspect ratio, values of 6 and 8 were examined, and aside f r o m the expected increase in wing panel mode frequency caused by the larger lift-curve slope with an aspect ratio of 8, no particularly significant differ- ences were noted. No variation of taper ratio was explored because its impact on longitudinal motion could be expected to be smaller than the aspect ratio effects.

Similarly, the effect of locating the center of gravity of the fuselage as- sembly off the hinge axis was insignificant for reasonable locations, either for vertical or longitudinal displacements. Since the assumption of the initial equilibrium state demands that steady mass imbalance effects be t r i m m e d , the primary effect of displacing the center of gravity reduces to a slight al- teration in the pitching moment of inertia of the fuselage assembly.

Displacement of the wing panel center of gravity was found to influence the frequency of the wing panel oscillatory mode almost exclusively. Devia- tions of the wing panel center of gravity ranging from 0 . 1 C forward to 0 . 25 Z aft of the hinge axis were permitted. Intuitively, it had been expected that such imbalances would result in pronounced effects, perhaps undesirable, inthelongitudinalcharacteristics.Consequently,rootlociwerecomputed for all three basic aircraft, with both aspect ratios, for both cruise and ap- proach flight conditions. Rather surprisingly, the effect was mild consider- ing the extent of the permitted imbalance, and, furthermore, all cases were quite similar.

The relatively minor effect of panel imbalance is probably related to the fact that 'the ratio of wing mass to fuselage assembly mass is small. If this t- ratio were large, a panel center-of-gravity location aft of the wing quarter- chord line could be expected to produce a purely divergent motion.

As mentioned previously, for an actual wing the pitching moment of inertia and the location of the pitching axis are not independent without struc- t u r a l mass changes. For simplicity, however, the assumption was made in this part of the study that the pitching moment of inertia is constant about an axis through the center of gravity, regardless of its location.Consequently, the moment of inertia about the hinge axis is a minimum for the nominal c a s e , XLg = 0, and increases parabolically for center-of-gravity offsets in either direction.

A typical root locus illustrating the effect of wing panel imbalance on the wing-panel symmetric mode is shown in Figure 9 for Aircraft C 1 in the cruise condition. Forward center of gravity locations cause an increase in at a relatively mode frequency, while aft locations reduce the mode frequency constant damping ratio. The effect on other characteristic roots is insignifi- cant, although some increase in phugoid frequency was observed for aft panel center of gravity locations. A s discussed in a later portion of this report, the lateral-directional modes are much more strongly affected by wing panel imbalance, and, therefore, no further discussion of longitudinal effects is warranted.

Longitudinal Handling Qualities Evaluation of the longitudinal handling qualities was confined to long- t e r m path control and maneuvering characteristics, and then only to the ex- tent that these features might be modified by the inherent nature of the f r e e - wingconcept.Specifically,attentionwasgivento ( 1 ) the stability of the phugoid oscillation and the pilot's ability to damp this mode, and ( 2 ) the short- term response to longitudinal control inputs.

W i t h regard to phugoid characteristics, Table I11 contains period and damping-ratio data for all cases considered. It may be noted that the free- a slight de- wing version of each aircraft exhibits a reduction in period and terioration in damping ratio except for Aircraft B1 and BQ in the approach condition. In any event, the damping ratio exceeds the standard of Refer- ence 1, which prescribes a minimum damping ratio of 0. 04 for Level 1, the highest level of acceptability.

In a conventional fixed-wing aircraft, oscillations in the phugoid mode are usually damped by the pilot's control of pitch attitude through elevator displacement. It seems highly desirable, therefore, that the free-wing a i r c r a f t phugoid oscillation should be controllable by similar pilot action, -1 2 1 1

-

IO - 9

- a

- 7 m

-

a

X & = -0.25 E

--5

.- E

F

- 4 -

X& is distance from hinge axis - 3 forward to wing-panel center of gravity

I,I= I I + rnpX;g2

yx;co -2

"I

Real Axis

I 1 I

I I I I I I I

-a -7 -6 -5 4 -3 -2 -I

I 2 3 FIGURE 9. E F F E C T O F PANELCENTER-OF-GRAVITYLOCATION ON WING-PANEL SYMMETRIC MODE A i r c r a f t C1, C r u i s e although no mechanical pitch coupling exists between the lifting surfaces and the fuselage assembly, and despite the fact that the longitudinal control is exercised through the trailing-edge control tabs instead of the horizontal tail surface.

111. STICK-FIXEDPHUGOIDCHARACTERISTICS TABLE Fixed Wing F r e e Wing Period, Period, AircraftFlightConditionsecondsDampingRatiosecondsDampingRatio ~- - - . " . - A1 Cruise 34. 7 0 . 1 2 0 2 7 . 8 0 . 1 0 0 A1 Approach 22. 6 0 . 0 8 8 1 7 . 3 0 . 0 7 2 A3 Cruise 34. 8 0 . 126 27. 8 0 . 106 A3 Approach 22. 8 0 . 1 0 2 1 7 . 3 0 . 085 B 1 Cruise 52. 0 0 . 1 0 5 46. 1 0 . 1 0 0 B 1 Approach 3 1 . 8 0 . 070 26. 9 0 . 076 Cruise 51. 9 0 . 1 1 1 4 6 . 1 0 . 106 B3 B3 Approach 3 2 . 0 0 . 0 8 4 2 6 . 9 0 . 0 8 9 C 1 Cruise 6 1 . 5 0 . 1 6 3 5 1 . 5 0 . 1 0 0 c1 Approach 34. 7 0 . 0 9 6 2 5 . 3 0 . 0 7 1 c3 Cruise 62. 1 0 . 1 2 1 5 1 . 5 0 . 1 0 7 c 3 Approach 3 4 . 9 0 . 1 1 0 2 5 . 4 0 . 0 8 4 ~ To determine whether such control was possible, several root loci were computed in which fuselage pitch attitude was fed back to the free-wing con- t r o l t a b s . A typical root locus in Figure 1 0 shows the path of the phugoid mode root as the feedback gain is increased. Notice that the oscillation can be completely damped in this manner, j u s t as in a conventional aircraft.

An explanation of this fortuitous behavior lies in the fact that the fuse- lage tends to align itself with the flight path through the fuselage angle-of- attack stability provided by the horizontal tail surface. As a result, the fuse- is v e r y similar to that of a fixed-wing aircraft lage pitch attitude behavior for long period motions. In addition, for shorter term motions, the pilot is provided some pitch-angle response to his control inputs by the changes in downwash at the horizontal tail caused by changes in wing l i f t coefficient.

3 1 0 4 0 3 a FIGURE10. ROOT LOCUS O F PHUGOID MODE AS AFFECTED BY FUSELAGE PITCH-ANGLE FEEDBACK TO CONTROL TABS With regard to short term maneuvering response to control inputs, the typical response time history in Figure 7 indicates several excellent attri- butes for the free-wing aircraft. The response in normal load factor is much more rapid than with the fixed-wing aircraft, while the fuselage pitch-attitude and angle-of-attack histories are very similar.

Thehandlingqualitiesspecification,Reference 1 , places limits upon both the minimum and maximum short-period frequencies as functions of the ratio of normal load factor to angle of attack in response to rapid longitudinal control displacement. Taken literally, the free-wing responses would fall within the allowable range of frequencies; but because of the unconventional nature of these aircraft, direct application of the specification may not be valid. In the free-wing aircraft, it is the symmetrical panel mode which governs normal load-factor response, and not the short period mode, and the panel mode frequencies are always higher than the maximum acceptable "short period" frequency of thespecification. On theotherhand,fuselage pitching motion is not dominated by the panel mode, but takes place predomi- nantly in the short period mode which is not greatly different in frequency than that of the fixed-wing aircraft. Because of this paradox, it can only be surmised that the rapid load-factor response to control displacement is 3 2 . - I entirely beneficial from the pilot's standpoint; a moving-base piloted sirnula- tion might be required to provide a definitive answer to this question.

With regard to short period damping ratios, the free -wing a i r c r a f t c a n m e e t the requirements of Section 3.2. 2. 1. 2 of Reference 1 if either the wing- panel mode or the short period mode is considered to be appropriate, assum- ing that the horizontal tail volume is sized properly or that wing-fuselage in- t e r c o n n e c t s o r a suitable pitch damper is provided to augment the free-wing short -pe riod m6de.

Longitudinal Turbulence Responses A s discussed previously, the prospect of reduced turbulence responses is, perhaps, the strongest justification for a consideration of the free-wing concept. In particular, intuitive arguments were advanced in a preceding section which would suggest substantial improvements in turbulence flying, particularly with regard to the load-factor response to vertical gust velocities.

To evaluate the promised advantages, the power-spectral-density ap- proach was employed, as described in Appendix E . For longitudinal disturb- ances, only the vertical gust component was considered, and the power spec- t r u m of this component was assumed to be adequately represented by the one-dimensional Dryden model. A plot of this function was presented earlier in Figure 2 , and a scale length, L, of 1000 feet was used for all cases.

To prevent the stick fixed phugoid mode from contributing significantly to the computed responses, all power spectra were truncated at a reduced frequency, S 2 , corresponding to a temporal frequency of 0 . 3 radian per second.

It was reasoned that disturbances in this low-frequency range could be easily controlled and should not be permitted to influence the computed rms perturbations .

Since the rms value of each output variable is computed by evaluating the square root of the area under its spectral density curve, a finite upper limit of integration was needed and was chosen as the reduced frequency correspond- ing to a temporal frequency of 40 radians per second.

Typical power spectral density functions of the load-factor responses to vertical gust disturbances of unit intensity are shown in Figure 11 f o r A i r - c r a f t B1 in cruise for the fixed-wing aircraft and two versions of the free-wing counterpart. The tremendous reductions in the load-factor responses of the f r e e -wing aircraft certainly support the intuitive arguments presented earlier.

3 3 w P radians/sec Note : crq = I f t/sec

\

Free-wing panel (10% margin) Symmetric mode W n = 1 8 . 0 rodians/sec

\

Free-wing panel (20°/. margin)

Free wing \

I Symmetric d e wn =24.6 radians/sec

i /

Hinge axis at 0.1%

t

I margin 1 \ Hinge axis at 0.05 E (20% margin- - . " j v .

" I I ! I I 0.01 0 02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 S2, radianslft FIGURE 11. COMPARISONOFNORMAL-LOAD-FACTOROUTPUTSPECTRA A i r c r a f t B1, C r u i s e The fixed-wing aircraft exhibits the customary response peak'at the short-period frequency (although the peak actually occurs at a slightly lower frequency because of the slope of the input disturbance as shown in Figure 2 ) .

The response of the free-wing aircraft, on the other hand, is governed by the symmetric wing-panel mode which occurs at a much greater frequency where the input power is greatly reduced. At all lower frequencies, the ability of the wing panels to adapt to the random vertical drafts counteracts the in- creased turbulence energy. The slight bump in the 10 p e r c e n t m a r g i n r e - sponse spectrum is located near the free-wing short period frequency and is probably caused by vertical loads on the horizontal tail caused by fuselage pitching in this mode. A similar slight bump occurs in the 20 p e r c e n t m a r - gin spectrum but it is not apparent in the scale of Figure l l .

Integration of the output spectra of Figure 11 yielded an rms normal- load-factor response of 0. 0206 g ' s f o r the fixed-wing aircraft as compared with 0 . 00588 for the 10 percent margin free-wing aircraft and 0 . 00365 g ' s for the 20 percent margin. Expressed another way, the load-factor responses have been attenuated by a factor of 3 . 5 and 5 . 6 5 , respectively.

The reduction of vertical path displacement is even more pronounced, since the rms altitude deviation for the fixed wing case was 0 . 659 f e e t a s compared to 0 . 054 and 0. 041 feet, respectively, for the two free-wing a i r c r a f t .

On the adverse side, the pitch-rate and pitch-acceleration rms re- sponses are larger for the free-wing aircraft, each being over three times as high for the free-wing aircraft as for the fixed-wing version. As discussed previously, however, it is clear that fuselage pitching oscillations can be i m - proved through reductions in horizontal tail size or other passive or active means with no adverse effects.

Figure 12 displays the rms load-factor, pitch-rate, pitch-acceleration, and path-displacement responses to unit turbulence intensity for all three air- craft with rectangular wing planforms and aspect ratio of 8. The fixed-wing responses are shown for comparison with the free-wing results. The f r e e - wing aircraft shown have a 10 percent hinge axis margin and a horizontal tail volume one-fourth that of their fixed-wing counterparts. It can be seen that attenuation of load-factor responses by a factor of about three can easily be ' achieved. A greater reduction in normal load factor could have been displayed if the 20 percent hinge margin cases were used, but the 10 percent value may be more practical because of other penalties associated with the greater con- trol power requirements of the larger hinge margin.

0 F i x e d w i n g

Free wing (with & fixed-wing

horizontal tail volume) AI B l Cruise Cr i se 0.02 0.01 lach Approach Cruise Cruise FIGURE 12. COMPARISON O F LONGITUDINAL TURBULENCE RESPONSES 3 6 " I The reduced tail volume of the free-wing aircraft has an effect which is significant only in the pitch-rate and pitch-acceleration responses, and these could be improved greatly by an artificial fuselage pitch damper as mentioned previously.

The rms responses for the aircraft with aspect ratio of 6 are quite simi- lar to those displayed in Figure 12. Although these are not shown graphically, they are tabulated numerically with the other data contained in Appendix F.

Lateral-Directional Motion Free-Wing Characteristic Modes Comparison W i t h Fixed-Wing Aircraft. As with the longitudinal motion, the lateral-directional characteristics of a conventional fixed-wing aircraft are adequately described by a s e t of differential equations of fourth order.

The four characteristic roots typically are found to include one complex pair, associated with the dutch roll mode, and two real roots defining the aperiodic roll and spiral modes.

F o r a free-wing aircraft, an additional complex pair of roots is obtained which describes an asymmetric mode of wing panel deflection. In addition, the aperiodic roll.and spiral modes are significantly modified; the roll mode becoming far less heavily damped and the spiral mode tending towards insta- bility. The dutch-roll mode roots are not substantially altered by the f r e e - wing concept.

A nominal configuration for each free-wing aircraft was employed in the 10 percent study of lateral-directional motion. This configuration featured a panel hinge margin with the panel center of gravity on the hinge axis, a fuse- lage assembly center of gravity directly below the hinge axis, and a vertical tail volume identical to the fixed-wing equivalent aircraft.

Lateral-directional motion was analyzed for Aircraft Al, B1, C1, A2, B z , C2, A3, B3, and C3 for both the cruise and approach f l i g h t conditions.

For clarity, only the characteristic modes of A1, B 1 , and C1 will be dis - cussed in detail. Some indication of the effects of wing planform variations are discussed later, and all r e s u l t s are tabulated in Appendix F.

Considering first the dutch roll mode, a comparison is presented in Table IV which illustrates the fact that the free-wing concept has virtually no effect upon this oscillation.

3 7 TABLE IV. COMPARISON O F DUTCHROLLMODES Fixed Wing Free Wing Flight Period, Damping Period, Damping Aircraft Condition seconds Ratio seconds Ratio 1.75 0.202 1. 75 0.202 A1 Cruise A1 Approach 2 . 58 0 . 2 1 0 2 . 5 7 0 . 179 Cruise 1. 83 0 . 156 1. 8 2 0 . 154 B1 B1 Approach 2.66 0 . 167 2 . 67 0 . 139 2 . 76 0 . 198 C1 Cruise 2 . 76 0. 198 4. 00 0.212 C1 Approach 4. 07 0 . 258 F o r the spiral mode, on the other hand, the effect of the free-wing is quitepronouncedand is detrimental. As seen in Table V, the fixed-wing air- craft have slightly stable s p i r a l m o d e s i n the cruise condition and mildly in- stable characteristics in the approach. The free-wing aircraft exhibit spiral instability at all flight conditions examined, and although the rates of diver- gence are mild during cruise, they become quite pronounced during approach.

TABLE V. COMPARISON OF SPIRAL MODES Fixed Wing F r e e Wing Stable Unstable Stable Unstable Flight Time to Time to Time to Time to Aircraft Condition 1/2 Amp, sec Double, sec 1/2 Amp, sec Double, sec Cruise 4,780 " " 28. 2 A1 " 20. 7 " 3 . 76 A1 Approach " " 44. 7 B1 Cruise 12,000 " 31.6 - - 5 . 2 5 B1 Approach Cruise 8 , 8 0 0 " " 50. 5 C1 " 30. 1 " 5 . 3 C1 Approach

It should be mentioned that the dihedral effect parameter, Lp, has a

pronounced effect upon the fixed-wing spiral stability. Because of this, some caution was required in selecting the fixed-wing dihedral parameter which would permit a legitimate comparison with the free-wing aircraft. The re- sulting fixed-wing spiral characteristics are believed to be representative.

The free-wing concept also has an important and deleterious effect upon the roll mode because of the reduction in roll damping. A comparison is made in Table VI, where the roll mode root is given along with its reciprocal, the roll-mode time constant.

TABLE VI. COMPARISON O F ROLL MODES Fixed Wing F r e e Wing Time Time Flight Root, Constant, Root, Constant, Aircraft Condition sec-l sec sec-1 sec A1 Cruise -6. 58 0 . 152 -0.639 1. 56 A1 Approach -4.68 0 . 213 -0. 783 1 . 2 8 B1 Cruise -5.35 0 . 129 - 0 . 5 5 4 1 . 8 1 Approach -4. 15 0.241 -0.675 1.48 B1 C1 Cruise -5.73 0 . 175 - 0 . 4 9 7 2 . 0 1 C1 Approach -5. 16 0 . 194 -0.600 1.66 The additional oscillatory mode, peculiar to the free-wing aircraft, describes an asymmetric mode of wing panel displacement, as listed in Table VII. This mode is characterized by a much higher frequency than the dutch-roll oscillation, and is well damped.

TABLE VII. ASYMMETRIC WING PANEL MODE CHARACTERISTICS Aircraft Flight Condition Period, sec Damping Ratio A1 Cruise 0.422 0. 591 A1 C r u i se 0.312 0.614 B1 B1 Approach 0.490 C1 Cruise 0.531 C1 Approach 0.895 0 . 820 SampleLateral-DirectionalResponses.Figure13showsthemotion of the fixed-wing and free-wing versions of Aircraft Ai, with controls fixed, in the approach condition, following release from a steady slip. The initial con- dition was with wings level, but with the nose of the aircraft displaced 10 de- grees to the left of the flight path.

In some respects, the motion of both aircraft is similar: the initial yawing motion is virtually identical, and after an initial roll to the left, both aircraft eventually assume a right turn. The disimilarities which exist are clearly caused by the much more rapid spiral divergence of the free-wing a i r c r a f t . In fact, referring to Table V, the rate of divergence of the free- wing aircraft is m o r e t h a n s e v e n t i m e s a s r a p i d .

These time histories also indicate that the wing panel mode is largely confined to motion of the wing panels themselves, as evidenced by the initial transient in the panel deflection trace, which is not apparent in the other variables.

Effect of Parameter Variations. The sensitivity of thestick-fixed lateral-directional characteristic modes of the free-wing was examined for variations in several of the parameters. Specifically, the influence of wing planform, fuselage center-of-gravity location, wing panel imbalance, hinge axis location, vertical tail volume, and wing pitching moments due to sideslip were examined.

W i t h regard to wing planform variations, the primary effect of reducing the aspect ratio from 8 to 6 was t o cause a reduction in the magnitude of the roll root, and a reduction in the rate of spiral divergence. These trends are similar to those observed for fixed-wing aircraft. A change in taper ratio f r o m 1. 0 to 0 . 6 had a similar beneficial effect on spiral divergence rate, and also improved the roll damping somewhat. Table VI11 i s a listing of t h e s e r e - sults for the light observation class of aircraft. Since the wing panel mode does not couple with the other modes for the nominal configuration, it is not contained in the table but the roots, tabulated in Appendix F, show that the primary effect of planform is on the frequency of this mode.

The variation of the fuselage center of gravity with respect to the hinge axis revealed that neither vertical nor longitudinal displacements had a pro- nounced effect on any of the modes, but the effect of wing panel center-of- gravity displacements can be dramatic.

To investigate the effects of mass imbalance on the wing panels, the panel center of gravity was varied from 20 percent of the chord length forward

mt 15 /

IO Free wing

" " _

Fixed wing

/"" """""""-

- 1 0 L

FIGURE 13. TIME HISTORY O F RELEASE FROM SIDESLIP A i r c r a f t A, Approach

"-

" I " " " bCL". .I 6 7 8 9 1 0 Free wing

""-

Fixed wing

0.5 '1

FIGURE 13. CONCLUDED 4 2 of the hinge axis to 30 percent rearward: These extreme changes had a neg- ligible effect upon the dutch roll mode, and only a minor effect upon the spiral divergence. On the other hand, an interesting coupling between the roll con- vergence and panel mode roots was found to exist at large aft center-of-gravity location is shown in Figure 14 for Aircraft A1 in cruise, and the same phe- nomenon was found to exist for all of the aircraft and flight conditions.

TABLE VIII. EFFECT OF WINGPLANFORMONLATERAL-DIRECTIONAL FREE -WING MODES Spiral Divergence Planform Dutch Roll Time to Roll Mode ~ _ _ _ _ _ ~ _ _ _ _ _ ~ Flight Aspect Taper P T ime eriod, Damping Double Amp, Aircraft Condition Ratio Ratio sec Ratio sec Root Constant, sec A 1 Cruise 8 1.0 1.75 0.202 28.2 -0.639 1.56 A1 Approach 8 1.0 2.51 0.179 3.76 -0.783 1.28 A2 Cruise 8 0.6 1.70 0.215 38.2 -0.117 1.40 A 2 Approach 8 0.6 2.49 0.181 4. 02 -0.860 1.16 A3 Cruise 6 1.0 1.76 0.182 31.2 -0.463 2.16 A3 Approach 6 1.0 4. 05 2.60 0.138 -0.662 1.51 It can be seen that moving the panel center of gravity progressively aft of the hinge axis causes the wing panel mode to diminish in frequency and split into two aperiodic modes. One of these new roots tends to merge with the roll root to form an oscillatory mode which then becomes dynamically unstable. Computed time histories of the divergent oscillations show that the mode is one in which rolling motion is predominant.

Although the coupled mode is technically interesting, its importance should not be overemphasized since the instability can be avoided by r e - stricting the permissible panel center-of-gravity range.

Movement of the hinge axis has no significant effect on any of the modes except the wing panel mode itself, whose frequency increases with increasing hinge margin as would be expected. This relative invariance is more readily understood by examing a simplified mathematical model of the aircraft. Con- sider, for example, the net roll damping and adverse yaw characteristics that may be computed for a quasi-static condition of pure rolling velocity and pitching equilibrium on each panel. Beginning with Equation ( 3 ) , if the total pitching moment on each panel is set to zero and the equilibrium panel dis- of roll rate, these displacements may be placements are found in terms 0.2 E

\ 0. I E

1 5 .c”

x ; ? 0 -

z

X.& is distance from hinge axis towing-panel

center of gravity - positive

forforward center of gravity

a

- 0 . I E ,Dutch-rol I root ( unaffected 1 Unstable rol I oscillation for \x;91-022 E

2 ipira

Real Axis I

- + ( -0.IE

- 2 0 - 1 5 IO -0.2; FIGURE14. E F F E C T O F WING-PANEL CENTER O F GRAVITY ON LATERAL-DIRECTIONAL MODES Aircraft AI, Cruise substituted into the rolling- and yawing-moment equations to arrive at quasi- static effective stability derivatives. The equivalent roll-damping derivative is r o l l r a t e is Similarly, the equivalent yawing-moment derivative due to The significant result of this is that both the numerator and the denom- inator of the additional terms are directly proportional to the distance between the hinge axis and the quarter -chord line. Values of these derivatives are tabulated in Appendix B. It follows that the effective changes in roll damping and yaw due to roll in this prescribed quasi-static condition are independent of wing-panel hinge margin. Similar arguments can be advanced for other stability parameters, supporting the observed fact that hinge margin has little effect on any of the lateral-directional modes except the asymmetric panel mode itself.

The effects of changes in the vertical tail size were mostly confined to a reduction in both the frequency and damping ratio of the dutch roll mode as the tail size was reduced. Some minor improvement was noted in the roll mode root for reduced tail size, but the spiral mode roots were less sensi- tive to the parameter than one might expect from fixed-wing experience. For fixed-wing aircraft, an increase in vertical tail size would invariably be detrimental to spiral stability; but, in the free-wing aircraft, the vertical tail contribution to net dihedral effect is very significant and may tend to counter- act the destabilizing influence of the increased weather-vane effect.

An aerodynamic parameter peculiar to the free-wing aircraft is the wing panel pitching moment, about the hinge axis, caused by sideslip. If the wing has a positive dihedral effect with the wing panels restrained, positive sideslip (to the right) will cause an increase in the lift on the right wing and a d e c r e a s e ontheleft.Intuitively,then,theincrementalpitchingmoments about the hinge axis will be negative on the right wing and positive on the left, resulting in an asymmetric panel deflection in a direction which would reduce the dihedral effect. An accurate determination of these pitching moments would require a theory which could provide chordwise, as well as spanwise, normal force distributions. This capability is beyond the simple lifting line theory used in this study, so an arbitrary value of the pitching-moment de- rivative, CmP, was established for each flight condition, and a sensitivity analysis was conducted to evaluate the influence of this unknown p a r a m e t e r .

The nominal value of Cm was selected as the magnitude required to

P

eliminate the wing contribution to the rolling moment, in the presence of a steady sideslip. In the steady state, then, with the wing panels in equilibrium, the total aircraft dihedral effect is completely dependent upon other compo- nents of the aircraft, particularly the vertical tail.

Figure 15 illustratesthelocus of theaffectedroots as C is varied

mP

through both positive and negative values with absolute magnitudes up to more thanthreetimesthenominalvalue.Thenominalvalue of C is negative

mP

since the sign is governed by the right wing panel, and larger negative values than the nominal can be seen to aggravate the spiral divergence. Some im- provement in the roll mode may also be noted, but the roll convergence root remains quite small by comparison with that for fixed-wing aircraft. Al- though positive values of Cm are not expected, the trend in the positive di-

P

rection is a coupling of the roll and spiral roots into a low-frequency oscil- latory mode. Such coupling would be unacceptable from the handling-qualities standpoint, as discussed later; but, if attention is confined to the expected negative values of C the most significant influence of this derivative is mP , upon the spiral-mode stability.

.- s

-

- 0.5

.r Real Axis FIGURE15. E F F E C T OF PANELPITCHINGMOMENT DUE TOSIDESLIP ON THE ROLL AND SPIRAL MODES Aircraft AI, Cruise . . . . ..

Lateral-Directional Handling Qualities Free-Wing Lateral Dynamics. From the pilot's viewpoint, a p r i m a r y lateral-directional control task is to establish and maintain a prescribed bank angle. This function is required to maintain level flight in the presence of disturbances, and to achieve coordinated turns for heading control.

While not explicitly stated in the handling qualities specifications, evi- dence suggests that the pilot prefers a lateral control system which commands a pure rolling motion at a rate of roll proportional to control deflection.

Figure 16 shows time histories of response to step lateral-control de- flection for both the fixed-wing and free-wing versions of A i r c r a f t A i , f o r both the approach and cruise conditions. It should be noted that the fixed-wing behavior is very near the ideal, in that a relatively steady rate of r o l l i s quickly achieved. The free-wing behavior, on the other hand, is far f r o m ideal; the control deflection appears to command not a roll rate, but a rolling acceleration yielding a monotonic increase in roll rate. This unfortunate be- havior can be attributed to the combination of low roll damping and spiral di- vergence of the free -wing configuration.

The significance of the roll-mode time constant listed in Table VI lies in the fact that if a n a i r c r a f t is assumed to be constrained to pure rolling mo- tion in response to a step control displacement, the roll rate is given by Equaticn ( 7 ) is derived in many texts, for example, Chapter XVIII of Reference 6 . This equation describes a simple first-order exponential rise to the steady-state rolling velocity. The roll-mode time constant is a d i r e c t indication of the time required to achieve the steady rate because when the elapsed time equals this value, the idealized aircraft will reach approximately 63 percent of the steady roll rate regardless of the aileron deflection.

According to Figure 16, the simplified model of Equation ( 7 ) describes the actual time history very well for the fixed-wing aircraft, but the free-wing responses appear quite differently. The roll-mode time constants are appre- ciably longer for the free wing, and it a p p e a r s that the divergent spiral mode begins to dominate the response, particularly in approach, soon after the time exceeds 7 ~ .

Reference 1 specifically disallows any outright coupling of the spiral and roll mode roots, such as the so-called lateral phugoid oscillation seen in

- Free wing , a , = 0.1 1 5 deg

-- - Fixedwing, a , = 2.86 deg

Time ,seconds Time ,seconds 25- V

% 20 - TR

\ Fixed Free 0 1 2 3 4 5 6 Time ,seconds T i me , seconds T i m e , seconds Time, seconds ( b ) Cruise ( a 1 Approach FIGURE 16. RESPONSESTO STEP LATERAL CONTROL INPUTS Aircraft A1 4 8 but no explicit combined effects are covered if Figure 15 for positive C mp., the roots remain real. Evidence suggests, however, that the ratio of absolute values of these real roots should be at l e a s t 30, according to Reference 7.

Intuitively, this would seem to be particularly true if the spiral mode is un- stable, if the synergistic effects in Figure 16 a r e t o be avoided.

The standards of Reference 1 were examined for the roll and spiral modes separately, using the mode data in Tables V and VI. Concerning the spiral mode, all three aircraft exceeded the standards for Level 1;: during cruise, but in approach none were able to satisfy Level 2 requirements and Aircraft A1 was unable to meet even Level 3 specifications. This is the air- c r a f t in Figure 16.

F o r the roll.-mode time constant, the standards are not met for Level 1 operation at all, but are within Level 2 standards during cruise for all three aircraft. For approach, Aircraft A1 meets Level 2 requirements, but B 1 and C 1 fall to Level 3 .

Closed-LoopBank-AngleControl. It is instructive to examine the closed-loop behavior of the pilot aircraft system if the pilot is assumed to a c t a s a pure gain, feeding back a lateral control displacement in response a deviation in bank angle. In practice the pilot is able to adjust his t r a n s - to fer function considerably to compensate for aircraft dynamic deficiencies.

More will be said of this later, but the use of a "pure-gain" pilot illustrates basic differences between the fixed- and free-wing aircraft.

F o r the data in Figure 17, the pilot gain relating aileron deflection to bank-angle error was given by the magnitude of Ccp. With the fixed-wing aircraft, increasing the feedback gain caused the roll and spiral roots to combineinto a stable oscillatory mode. The dutch roll roots were practically unaffected.Bycontrast,thefree-wingcaseshowed a dynamicinstability, if C were sufficiently large, caused by movement of the dutch roll root to the e p positive half plane. Even with lower gains, the coupled roll spiral oscillatory mode would be poorly damped.

*Reference 1 defines three levels of acceptability: Level 1. Flying qualitiesclearlyadequate.

Level 2. Flying qualities adequate to accomplish the mission. . .but some increase in pilot work load or degradation in mission effectiveness exists.

Level 3 . Flying qualities such that the airplane can be controlled safely, but pilot work load is excessive or mission effectiveness is inadequate, or both.

-

-6

-

-5 c&#J= 2 .

"4 .- cn St ck-f ixed c+ = \[,

c+= 2

Stick-f ixed roll mode 1 "I Stick-f ixed spiral mode

1 I I I L I , Real Axis A

- 1 1 r I -6 -5 -4 -3 - 7 -2 - I I 1 1 ---I (a) Fixed- Wing Aircraft $= 0 . 0 5 4 "I Stick-f ixed spiral Stick- fixed roll mode

Amode Real Axis

I I I I I , \ I I

*"

-5 -4 -3 -2 -I -7 -6 I 2 I ( b ) Free- Wing Aircraft 5 0 Since the fundamental problem appears to be the small value of the roll- mode root, artificial stability augmentation in the form of a roll damper was evaluated. It is likely that other possible solutions may exist, such as a spring restraint on wing panel asymmetric displacement, but only the roll damper was evaluated. A system with no actuator lags was conceived which fed back an aileron deflection in response to a rolling rate. In particular, the feedback gain of this damper was selected to yield a roll mode time con- . stant for the augmented free-wing aircraft equal to that of the fixed-wing air- craft. The closed-loop root loci as a function of pilot gain is shown in Figure 18.

The closed-loop behavior of the augmented aircraft is clearly superior to the basic free-wing configuration, even though a dynamic instability is still possible if C y is sufficiently large. A range of values of C e p exists which should provide reasonably tight control with good damping.

As mentioned previously, the actual behavior of a human pilot is vari- able, in that he can adapt his control technique to a wide range of situations.

The matter of defining human transfer functions has been the subject of con- siderable research effort, and a particular representation was chosen to ob- tain a better understanding of the roll control features of the free wing.

In Reference 8 an instability in roll of the aircraft-pilot combination for the X-15 was successfully explained using the transfer function: The evidence cited in Reference 8 suggests that this transfer function provides a good description of the pilot performing a stabilization control task near the limits of pilot controllability. Notice that the roll power character- (8) since it p r e s c r i b e s a r o l l - i s t i c s of the aircraft do not enter into Equation ing moment per unit bank-angle error rather than merely a control deflection.

To apply Equation (8) to the free-wing aircraft, an effective roll power must be derived. From Equation ( 3 ) , if thewingpanel is in static pitching- moment equilibrium under the influence of tab deflection and panel displace- ment only, the resulting panel displacement is: 5 1 -6

Roll mode

r / 9 Real Axis

C+= - 0.05

I , Q = O I C+=-0.05 t J .\ I I I l Y

I c d I I I ” 1- I I I O I -5 - 4 -3 -2 -7 -6 I 2 3 -I - - - I FIGURE 18. E F F E C T O F ROLL ANGLEFEEDBACKTOAILERON A i r c r a f t A I , F r e e Wing With Augmentation Substituting into the rolling-moment equation, the effective roll power derivative is Using appropriate numerical data to compute the effective La and sub- a stituting into Equation (8) produces the desired feedback function Using these procedures for the free-wing versions, the time histories in Figure 19 were computed to illustrate the human pilot's ability t o recover from an initial bank-angle error in both approach and cruise. The c o r r e - sponding behavior with the fixed-wing version of the aircraft is also shown for comparison.

It should be noted that the unaugmented free-wing aircraft is not only controllable, but the pilot is able to remove the bank-angle error in less time than with the fixed-wing aircraft. The smoothness of his recovery with the fixed-wing aircraft is much better, however.

Lateral-Control Responses With Stability Augmentation. Figure 2 0 dis - plays the time histories o f response to step lateral control deflection f o r the free-wing aircraft with roll rate damping augmentation. This figure may br- compared with Figure 16 to demonstrate the tremendous improvement in lateral-control characteristics afforded by the roll-rate damper.

The rate damper not only permits a roll rate response which h-s a nearly ideal shape, but the spiral mode is made stable and the augmented free-wing aircraft displays a roll rate capability, per unit aileron deflection, which is nearly independent of airspeed. This latter feature could be quite important during approach, where available roll rates are reduced for con- ventional aircraft as seen in the fixed-wing traces in Figure 16.

Returning to Equation (i'), and recalling that the dimensional roll damping derivative, Lp, is simply the negative of the reciprocal of T R , the steady-state roll rate response is, ideally: 5 3

-

- -0.05

-

K -0.lOL Approach v - + ” - J I I 4 5 6 7 8 Time, seconds

- -0.05 -

-

B

- 0.10-

Cruise FIGURE 19. PILOTEDRECOVERY FROM ROLL-ANGLEDISPLACEMENT Aircraft A1 U I I ( a 1 Approach ( b ) Cruise FIGURE 20. RESPONSES TO STEPAILERON WITH ROLL-RATE AUGMENTATION A i r c r a f t A1, 6 , = 2. 86 degrees.

If all roll damping is provided by natural aerodynamic means, the ratio of dimensional derivatives is proportional to true airspeed, all other things being equal. It follows that the maximum rate of roll will also then tend to vary directly with speed.

F o r the augmented free-wing aircraft, however, the greatest portion of the effective roll damping is artifically produced and the r o l l rate per unit aileron deflection tends to be constant. Furthermore, the effectiveness of the control tab, in displacing the wing panels for roll control, is very power- ful. It may be surmised, in view of these facts, that any desired roll-rate capability within practical limits could be provided down to very low approach speeds.

Lateral-Directional Turbulence Responses The lateral-directional turbulence responses were computed for the combined effects of uncorrelated side and rolling gusts using the power spec- tral density techniques described in Appendix E. Typical power spectral density functions for selected variables are shown in Figure 21 f o r A i r - craft A1 in cruise. When a comparison is made with the fixed-wing aircraft, the effect of the free-wing configuration in reducing roll rate response is very pronounced, but the effect on yaw rate is v e r y small, with the free-wing response being slightly larger.

As with the longitudinal responses, the output spectra were truncated to include frequency components only within the temporal frequency range f r o m 0. 3 to 40 radians/sec. The rms values are based upon integrating the output spectra in this interval.

A comparison of rms responses is shown graphically in Figure 2 2 for A i r c r a f t A I , B1, and C 1. In addition to decreasing the rolling motion, the free-wing aircraft shows a marked reduction in lateral path displacement and lateral load factor. No really significant differences were observed for the other planforms, although some responses were slightly greater for the re- duced aspect ratio cases. These data are tabulated in Appendix F.

Finally, the performance of the stability augmented free-wing aircraft should be noted. Table IX is a comparison of the responses of the aircraft, with roll rate damping, to the behavior of the unaugmented free-wing and fixed-wing aircraft.

Despite the fact that roll damper gain, Cp, was sized to make the roll mode time constant equal to that of the fixed-wing aircraft, the augmented free-wing aircraft shows great improvement in lateral turbulence responses.

I Roll - Rate Spectrum I Dutch roll w, 3.6 radians/sec

0.0005 - b Free wing -

- 0.0005

n 1

a, mdianlft

/Freewing 0.oa O.00f = I Yaw-RoteSpectrum 0.00; 0.001 r , radion / ft FIGURE 2 1. LATERAL-DIRECTIONALOUTPUTSPECTRA, UNIT TURBULENCE INTENSITY Aircraft Ai, Cruise 5 7

I

-

o.ooo8

I

-

I

0.0007

I

-

o.Oo06 c u-

I I

Roll- Angle Spectrum -

.- E 0.0005

‘CI

I

e

-

” o.oO04 c : Y -e-

-

0 0.0003

-

0.0002

-

0.0001

1 I

0 . 0 1 0.02 0.03 a04 0.05 OO ,radian/ft

I

-

0.0012

-

0.0010 L

-

g 0.0006

h - 0.0004

0.0002 -

t

OO 0.0 I 0.02 0.03 0.04 0.05 5 1 , radionlft FIGURE 21. CONCLUDED 5 8

0 Fixed wing

Free wing 0.6 BI CI roach Amroach 0.2 0. I 0 - 0.5 ,oath

,oath 4

roach

2r

FIGURE 22. COMPARISON O F LATERAL-DLRECTIONAL TURBULENCE RESPONSES 5 9 C I AI Approach

C

Approach

-

v) -0 0.01of- FIGURE 22. CONCLUDED This is because the primary contributor to lateral perturbations is the span- wise gradient of vertical gust velocity, and this "rolling gust" disturbs the airplane in proportion to the aerodynamic roll damping coefficient, Lp. If the natural aerodynamic roll damping is small, the forcing function is r e - duced. The evidence is quite convincing that the combination of low gust sen- sitivity and powerful roll control provides the augmented free-wing aircraft with truly remarkable flying qualities, particularly during low-speed approaches.

TABLE IX. COMPARISON OF RMS LATERAL-DIRECTIONAL RESPONSES TO UNIT TURBULENCE INTENSITY Aircraft AI, Approach Cp = - . 06 sec for augmented aircraft.

Lateral Yaw Lateral Load Roll Roll Yaw Angle , Angle , Rate , Rate, Displacement, F a c t o r , Aircraft de g de g degl sec f t d e g / s e c g units " . __ . -. "" __" ~ ..

" ~~ "" _ _ ~ _ 1. 30 .00748 Fixed wing 0.412 0.382 0.413 0.482

F r e e wing 0.270 0.316 0.335 0.511 0 . 766 . 00470

0. 112 0 . 234 F r e e wing 0.305 0.462 0.252 .00335 with roll damper Conclusions F r o m the results of this investigation, the following conclusions may be drawn: (1) Atmospheric turbulence effects are greatly reduced by the free- wing concept at all flight conditions examined. The most dramatic improvements are in the root-mean-square normal load factor and vertical path displacement responses, but important alleviation effects are also obtained for rolling disturbances. On the other hand, the fuselage pitching mo- tion response can be degraded substantially in comparison with equivalent fixed-wing aircraft.

6 1 All stick fixed modes of motion of free-wing aircraft are stable, except for the spiral mode. The rates of s p i r a l d i v e r g e n c e a r e mild for cruise flight but may be excessively high for the ap- proach configuration. In addition, a dynamic instability in roll is possible if the wing panel center of gravity is permitted to lie well aft of the hinge axis.

The lateral handling qualities are unsatisfactory because of the combination of low roll damping and spiral divergence for the unaugmented free -wing aircraft, although the aircraft appears to be controllable by pilot effort.

Artificial stability augmentation, in the form of a simple roll damper, provides excellent lateral control and turbulence penetration characteristics. The augmented free-wing air- c r a f t is characterized by very powerful roll control by virtue of the differential wing-panel deflections. This unique feature can permit a relatively constant maximum roll rate capa- bility, up to any reasonable value, over the entire speed range.

This feature, coupled with the reduced gust sensitivity, can

provide exceptionally good lateral handling qualities , p a r -

ticularly during low-speed approaches in rough air.

Longitudinal handling qualities appear to be satisfactory.

Pilot control of long t e r m phugoid motion can be exercised exactly as with a conventional aircraft by employing longitudinal control feedback in response to fuselage pitch- attitude cues. In addition, the free-wing aircraft has far more rapid short term normal acceleration response to control inputs; but, because of the unconventional separation between normal load factor and fuselage pitching motion, a moving base piloted simulation may be required to ensure pilot acceptance of the longitudinal maneuvering characteristics.

W i t h regard to fuselage pitching and lateral control improvements, the most obvious approach would be to provide an active stability-augmentation system. The possibility of using purely passive mechanical devices such as pivot springs or dampers or control interconnects should be considered, al- though they were not examined in this study.

6 2 REFERENCES 1. Anon. , "Military Specification - Flying Qualities of Piloted Airplanes", MIL-F-O08785A(USAF)(October31,1968).

2. Abbotand Von Doenhoff,Theory of Wing Sections, McGraw-Hill Book Company ( 1949).

3.Rainey,A. G. , "Measurements of Aerodynamic Forces for Various Mean

Angles of Attack on an Airfoil Oscillating in Pitch and on Two Finite-Span Wings Oscillating in Bending W i t h Emphasis on Damping in the Stall", NACA TN 3643 (1956).

4.Runyan,H. L . , "Single-Degree-of-FreedomFlutterCalculationsfor a Wing in Subsonic Potential Flow and Comparison W i t h an Experiment", NACA Report 1089 (1952).

5. Jones, Robert T. , "TheUnsteadyLift of a Wing of Finite Aspect Ratio", NACA Report 68 1 ( 1940).

6 . Seckel, Edward, Stability and Control of Airplanes and Helicopters, Academic Press, New York (1964).

7. O ' H a r a , F . , "HandlingCriteria",Journal of theRoyalAeronautical Society, 71 (April,1967).

-

8.Taylor, L. W. , J r . , "Analysis of a Pilot-AirplaneLateralInstability

Experienced With the X-15 Airplane", NASA TN D-1059 (1961).

APPENDIX A DEVELOPMENT O F EQUATIONS O F MOTION Introduction In deriving the equations of motion, each wing panel and the fuselage assembly are initially considered as free bodies. After the individual sets of equations with respect to the most convenient axis systems are written, they are combined into a single set, referred to standard aircraft stability axes, The consolidation of equations is accomplished by eliminating the common forces and moments acting between the various components, The equations are then linearized for convenience in the analysis.

Symbols Symbols that are defined explicitly each time they are used have been omitted from this list.

b = wing span, feet

-

c = mean aerodynamic chord length, feet CD = drag coefficient CL = lift coefficient C Q = rolling-moment coefficient, positive for right roll

aCQ/a (g) , per radian

cQP C Q = wing contribution to C Q Pw P

CQ, = a C j / a (*) , per radian

2UO C , = pitching-moment coefficient on fuselage assembly, positive nose up = pitching-moment coefficient on right wing panel, c , R positive L. E. up

- - aCmR/ a(*), per radian

C 2UO mRP

= aCmR/a6p, p e r r a d i a n

bPc‘

C = a C , / a ( -) , per radian

mR6p uO

C = X m R / a 6 t R , a G m R / a d t L , respectively, per radian

G

’ ,R6

mR6 tR tL Cn = a C , / ? ( * ) , perradian P 2u0 C = wing contribution to C “ P W “p

c = acn/a ( z ; ” o ) , - per radian

nr

G = aCn/abp, per radian

C = aCn/aStR, per radian StR = gain constant, aileron deflection per unit roll rate, cP seconds CT = thrust coefficient Cy = sideforce coefficient, positive to right

c = a C y / a (2) , perradian

yP 2UO

c = aCy/ a(">, perradian

Y r 2UO C = aC,/aP, per radian

Y P

C = aC,/aSp, per radian y t i Cy = gain constant, aileron deflection per unit roll angle Dy = lateral path displacement, feet, positive to right E = ratio of wing semiperimeter to span Fx, Fy, F, = force components along X, Y, and Z stability axes, respectively,pounds F = forcescomponentsalonghingeaxessystemassociated FxhR' F yhR'ZhRwithacceleration of rightwingpanel,pounds g = acceleration of gravity, feet/second2 G1 = transfer function relating lift coefficient to angle of attack G2 = transfer function relating lift coefficient to vertical gust velocity

h = altitude increment, feet

-

H = moment of momentum vector, feet-pound-seconds

-

i = unit vector along x axis

$ 1 , Iyl, IZl = moments of inertia of right wing panel measured in

panelaxissystem,slug-feetz

Ixy1, IxZ1,Iyz1 = products of inertia of right wing panel measured in

panel axis system, slug-feetz

& I I = moments of inertia of fuselageassemblymeasuredin

f’ yf’ zf thestabilityaxessystem,slug-feet2

&yf’ LZf 9 1 yz = products of inertia of fuselage assembly measured in

thestability axes system,slug-feet2 IxxT’I Y Y TyI Z Z T = moments of inertia of total aircraft, measured in the stabilityaxessystem,slug-feet2 = component of right-panel pitching moment of inertia I Y P defined by Equation (A-39), slug-feet2

IxzT = product of inertia of total aircraft, measured in the

stabilityaxessystem,slug-feet2 Ixy,, Iyzp = components of right-wing-panel products of inertia defined by Equations (A-40) and (A-41), slug-feet2

-

j = unit vector along Y axis I; = unit vector along z axis Ke = gain constant, elevator deflection per unit pitch-angle e r r o r pUoSb2 pUoSb2 L r = 4kXT

FO2 Sb

L = ‘!XXT

LC = lift due to circulation, pounds

Lm = lift due to apparent mass of air, pounds m = total mass of aircraft, slugs mf = mass of aircraft minus wings, slugs mp = m a s s of one wing panel, slugs = fuselage-assembly pitching-moment coefficients defined M(i) by Equation (A-81) pUoSFb M = Rp 41 C m Y ' RP puo2sz

-

MR

21y' cm

P P M = moment about Xh axis caused by inertial reactions of right, XhR, L orleft, wing panel,foot-pounds M = moment about yh axiscaused by inertial reactions of right, YhR, L orleft,wingpanel,foot-pounds M = moments about Zh axis caused by inertial reactions of right, Zh R, L orleft,wingpanel,foot-pounds MXf, M M = momentsappliedtofuselageassembly,measured in yf' Zf stability axes system, foot-pounds M = aerodynamic moments acting on total aircraft, about roll Xaero' MZaeroandyawstabilityaxes,respectively,foot-pounds = aerodynamic pitching moment acting on fuselage assembly, M Y aero foot-pounds

- pUoSb2

NP - 41 C n

P Z Z T " " " " l . " I . . " I I I " I I I 1 1 . 1 1 1 I1111111 1111111 I 1 I1111

- pUo2Sb

C NP - 212.

rn "P

roll rate about X stability axis, radians/second a r e a of one f r e e wing panel, feet2 coefficients of panel pitching equation, given by Equation (A-83) pitching rate of fuselage, radians/second P O 2

dynamic pressure, - , pounds/foot

yawing rate about Z stability axis, radians/second vector defining spatial position of origin of hinge axes system vector defining spatial position of total aircraft center of gravity total wing area, feet' component of velocity of hinge axis origin lying along Xh axis, feet/ second component of velocity of a i r c r a f t c e n t e r of gravity along X stability axis, feet/ second Component of velocity of hinge axis origin lying along yh axis, feet/second component of velocity of a i r c r a f t c e n t e r of gravity along Y stability axis,feet/second E: vertical gust velocity, positive upward, feet/second vg W h = component of velocity of hinge axis origin lying along zh axis, feet/ second W = component of velocity of a i r c r a f t c e n t e r of gravity along Z stability axis, feet/ second A x = distance from hinge axis to half-chord point, a negative number,feet = coordinate axes in hinge system Xh, yh, "h x', y ' , z' = coordinate axes in wing-panel-fixed system X, Y, Z = primary coordinate axes of stability axes system X' cg, yIcg, zfCg= coordinatesof wing-panel center of gravity measured in panel-fixed axes Xf = longitudinalcoordinate of fuselagecenter of gravity cg measuredinstabilityaxessystem,feet

-

X = longitudinal coordinate of hinge axis measured in stability

axessystem,feet X(i) = coefficients defined by Equation (A-87) zf = coordinate of fuselage center of gravity measured along Z cg stability axis, feet Z(i) = coefficients defined by Equation (A-72)

-

Z = coordinate of hinge axis measured along Z stability axis, feet 7 1 af = inertial angle of attack measured upward from inertial velocity vector to X stability axis, radians /3 = sideslip angle, radians 6, = asymmetric tab displacement defined by Equation (A-47) 6, = symmetrical tab displacement, positive trailing edge down, radians 6p = displacement of right wing panel with respect to fuselage, positive leadingedgeup,radians 6~ = displacement of left wing panel with respect to fuselage, positive leading edge up, radians = displacement of right and left control tabs, respectively, posi- 'tR, L tivetrailingedge down, radians 0 = pitch angle of longitudinal fuselage axis with respect to horizon, radians X = Laplace operator, l/second cp = roll angle, positive right wingdown, radians p = atmospheric density, slugs/ft3 px, p y , p z = components of position vector from origin to wing panel center of gravity measured in hinge axis system, feet

7c/ = yaw angle, positive nose right, radians

-

w = angular velocity vector Subscripts: On unit vectors, h and p denote hinge axes and panel axes, respectively.

o = equilibrium value = measured with respect to earth-fixed reference g = gust w = wing f = fuselage.

Coordinate Svstems Three coordinate systems were employed: Conventional stabilityaxissystem,Followingstandard practice, the basic set of coordinates for describing the aircraft motion has its origin at the center of gravity of the complete aircraft. The X axis isalignedwiththe velocity vector of the aircraft in the reference condition, the Y axis extends to the right of the plane of symmetry, andthe Z axiscompletestheright-handset.Thesecoordi- nates are fixed in the aircraft and rotate with it.

The orientation of the stability axis system with respect to an inertially fixed reference is defined by three standard Euler angles, Thesequence of rotation used to define these angles is ( 1 ) rotation about the Z axis through the yaw angle 7 + b j ( 2 ) rotation about the Y axis through the pitch angle 8, and ( 3 ) rotation about the X axis through the r o l l angle cp.

A sketch of the stability axis system is shown in Figure A- 1.

Hingeaxissystem.Thehingesystem of axes, Xh, Yh, Zh, has its origin in the plane of symmetry of the aircraft.

The positive Y h axis coincides with the axis of rotation of theright wing panel.Forsimplicity,the wing panels are assumed tohave no geometricdihedral.Consequently, Zh lies in the plane of symmetry and the negative Y h axis coincides with the axis of rotation of the left wing panel.

The hinge axis system is parallel to the stability axis system, and is therefore fixed in the fuselage assembly f o r a given flight condition.

A-1 shows the hinge axis system.

Figure Panelaxissystem. Thepanelaxissystem, X I , y', z', i s similar to the hinge axis system but rotates with the wing panel under consideration. When dealingwiththeright wing panel, the panel axis system is rotated about the yh axis through the displacement angle 6p; whereas for the left panel the displacement angle is dL.

The panel axis system is also illustrated in Figure A-1.

7 3 H inse axis

\ Total aircraft

center of gravity FIGURE A-1. ILLUSTRATION O F AXIS SYSTEMS Wing-Panel Force Equations Force equations were developed for each wing panel separately, but only the right-wing-panel equation is discussed. A similar s e t of equations can be written for the left panel, differing only in the use of 6~ to denote panel displacement and the fact the y' has the opposite sign for the left cg panel.

In the hinge axis system of Figure A-1, the position vector of the panel center of gravity is given by where px = x' cos 6 p t z ' sin d P cg

-

Py - Ylcg

- s i n $ t zlcg C O S 6p .

Pz - -xIcg If uh, Vh, and wh a r e the components of the inertial velocity of the origin of the hinge axis system, measured in that system, the inertial velocity of the panel center of gravity is The velocity of the hinge axis origin can be expressed in terms of the velocity of the aircraft center of gravity as 7 5 Since the hinge axis system is parallel to the stability axes in which p, q, and r a r e defined,

- - - -

uh = pih t q j , t rkh .

Differentiating once again, the inertial acceleration of the right-wing- panel center of gravity is obtained:

- -

"p = apx'h . t apjTh t a Pz i;h , where Then, applying the fundamental Newtonian law, the three equations describing the forces existing at the origin of the hinge axis system that are associated with acceleration of the right wing panel are = m a FxhR p px = m a (A-8) F PY yhR = m a F Z ppz hR Wing-Panel Moment Equations The wing-panel moment equations are written most conveniently in the panel axes system (shown in Figure A-1) because in this system, the moments and products of inertia are constants. The moments are then transformed to the hinge axis system for later use.

An unusual feature of the panel axes is that the origin is displaced from the panel center of gravity. Because of this, the more general form of the principle of the conservation of moment of momentum must be used. This is

-

fi = E t ( p x mp R) .

(A-9)

The components of the E vector are the inertial terms found in the con-

ventional Euler equations for the rotation of a rigid body. These are not rederived here because they are developed in many texts.

The second term, caused by the offset center of gravity, requires the development outlined below.

In the panel axis system, the position vector to the panel center of

-

gravity, p , isconstant,andisgiven by (A- 10) The inertial velocity of the origin of the panel axis system may be expressed in that system by noting that the origins of the hinge and panel axes coincide, So (A-1 1 ) This velocity vector m a y be transformed to the panel axes by a simple rotation transformation through the angle tip, for the right panel.

7 7 so

-

s i n 6 ) i t v 3 t (uhsin 6 t w h c o s 6 ) i f (A-13)

R = (uh C O S d P

- Wh

p P h P P P P and (A- 14)

-

t Wh C O S 6 , ) k + (GR X F) , P The rotational rate of the right panel, GR, can be expressed in the hinge axis s y s t e m as

-

(A- 1 5 ) GR = p 7, t (9 t 6 p ) jh t r E h .

Applying the transformation of Equation (A- 12), (A-16) This can be written as (A- 1 7 ) Using these equations, Equation (A-9) m a y be expressed in the panel axis system and then transformed, by means of Equation (A-11), into the hinge axis system. The components of the moment are, for the right panel: F o r the left wing panel, the equations are identical in form. They m a y be written by simplychangingthesign of every term containing y' as a cg factor. It should also be noted that moments of inertia are the same for each panel, but the products of inertias containing the y component change sign.

7 9 (A-2 1) Fuselage Moment Equations ~~ - The fuselage moment equations are written in the stability axis system whose origin lies at the center of gravity of the complete aircraft, Since the center of gravity of the fuselage assembly free-body does not, in general, coincide with that of the entire aircraft, the general form of the equation for the conservation of angular momentum must be used.

..

(A-22) Since the fuselage center of gravity is assumed to lie in the aircraft's plane of symmetry, pf = xfcg i t zf k (A-23 ) cg Since the velocity of the origin is the velocity of the aircraft's center of gravity,

-

Ro = U T t Vy t WE (A-24) and

-

Ro = U : t VT t WG t (w x Eo) , (A-25) where (A-26) So the second term on the right of Equation (A-22) becomes (A-27)

i

The remaining terms on the right side of Equation (A-22) are, as before, the inertial terms found in the conventional Euler equations for the rotation of a rigid body.

The components of the applied fuselage moment defined by Equation (A-22) becomes (A-28)

M = m x (+ - p w + ru) t 1 : + (qr - I;)

Z f fcg f f

- I ( ; I + p r ) t I (qz - p2) t (I - I ~ pq

Y Z f xyf yf f The moments applied to the fuselage assembly, represented by the sides to the left in Equation (A-28), contain contributions from the reversed effec- tive forces and moments of the wing panels. In actuality, they also contain gravity moments due to the weight of the fuselage and wing panels; however, since the origin is a t the total aircraft center of gravity, these weight moments must add to zero.

Total Aircraft Equations The translational equations describing the motion of the m a s s c e n t e r of t h e a i r c r a f t a r e the conventional expression of Newtons law of motion expressed in a rotating axis system, In the stability axes system these are

F = m (V - p w t r u ) (A-29)

Y

F~ = m ( W t p v - q u )

The gravity-force contributions can be expressed as = -mg sin 8 Fx gravity F = m g cos 8 sin cp (A-30) 'gravity F = mg cos 8 cos cp z gravity Finally, the complete set of equations defining the translation and ro- tation of the stability axes system may be written:

= m (U t q w - rv) t mg sin 0

Fx a e r o c (A-3 1 )

F = m ( i r t p v - r u ) - mgcos e ,sin cp

Y a e r o

F = m ( W t p v - q u ) - mgcos e cos

2 : a e r o and

= M t ( M t M x ) - ( F -I" )z

MX a e r o Xf XhR hL yhR yhL

= M t(F, t F x )z-(Fz t F , (A-32)

MYae r o Yf hR hL hR h L Here, M M and M, come from Equation (A-28), and the remain-

Xf' Yf ' f

ing t e r m s a r e the reversed effective forces and moments which m a y be evaluated from Equations (A-8) and (A-18) and equivalent expressions for the contributions of the left wing panel.

Two additional equations are necessary to describe the complete sys- tem.Thesearetheexpressionsrepresentingtherotationaldegrees of free- dom of the two wing panels. One of these was written previously as Equation (A-19) for the right wing panel.

Linearization of Equations The equations are linearized, using conventional techniques, about an equilibrium flight condition of straight and level flight with no angular rates o r accelerations. The equilibrium panel deflections are not assumed to be zero, but they are assumed to be identical. In the following equations, all variables are considered as small perturbations from the reference condition.

Translational equations:

m U = A F , - (mg) 8

m$ = A F - (mu,) r - (mg) CP (A-33)

Y mw = AF, t (mu,) q .

Rotational equations (fuselage assembly): (s, t 5,) + Z(AF, I Y P aero, wings

) 1 (A-34)

aero, wings ..

" + = I

IZZT xzT fi Iyzp (6p - h L ) t AMz

a e r o Wing panel rotational equations : t AM yhR b (A-35) t AM yhL The total moments and products of inertia used in Equations (A-34) and (A-35) are computed from: = I t 4m Z p z t 2 1 ~ ~ cos2 6 , t 2 1 ~ ' s i n 2 6 , 'XX T X P f (A-36)

- 41 s i n 6 c o s 6 + 2 Z 2 m

xz' 0 0 P " " = I t 2m Z (Z t p,) t 2m X (X t p x ) (A-37) I Y Y T Yf P P (A-3 8 ) t 41x,, s i n 6o c o s 6 , t 2K2 m P

= I t 2m ('ii p, t Z p x ) t ( I , , - I ~ , ) s i n 2 d 0

IXZ, XZf P (A-39) " t 2IX,, cos 26, t 2m X Z P (A-40) (A-4 1 ) Ixyp = Ixy, cos 6 , t szl sin 6 , - I cos d o - I I s i n 6 .

(A-42) IYZP - IY.

XY 0 The Lateral-Directional Equations Examination of the wing-panel displacement terms in Equation (A-34) shows that symmetrical wing-panel motion, (6 = h L ) , has no effectupon P therollingandyawingequations.Furthermore,inEquation (A-35), rolling and yawing accelerations are seen to cause only asymmetric panel displace- ments, since the terms containing these variables have the same coefficient, butoppositesign,inthe two equations, In addition,theaerodynamicderiva- tives are such that no coupling exists between lateral-directional variables and symmetricalwing-paneldisplacements.Because of thisseparation,the two uncoupled sets, just as with a con- linearized equations can be split into ventional aircraft.

Since only asymmetric displacement is significant, let (A-43) % = - 6 p .

The lateral-directional equations then become

Ixx, fi = sz E t 21xy gp t MX

T P a e r o 1

'ZZT + - - 1x2, 1 ; t 2 1 y z P b , t MZ a e r o 1

(A-44)

m V = A F y - (mu,) r - (mg) cp

The aerodynamic rolling moment is expressed as follows: (A-45) if The rolling moment coefficient m a y be expanded in a T a y l o r s s e r i e s about the equilibrium zero value. If only the first-order terms are retained, these become the rolling-moment stability derivatives.

Equation (A-45) then becomes For control-tab displacements, only asymmetric control is of i n t e r e s t forlateral-directionalmotion.Because of this,define: (A-48) Using Equations (A-48) and (A-43), and the fact that (A-49)

J

the rolling moment becomes L J (A-50)

f 2c d p + 2 c

P tR ".]

B y similar development, it can beshownthat - - MZ a e r o r (A-5 1) -I- 2 c tip t 2c and (A-52) Similarly, - - ACmR Q S c' , M (A-53) yhR and let The Taylors series expansion of this function, along with Equations (A-43) and (A-48), yields (A-55) Sideslip angle is introduced as the dependent variable in the third equation of Equation (A-43) by the substitution, ‘ V p = - (A-56) UO The set of linear equations describing the lateral-directional motion in still air can now be written as: " pUo2Sb t [cQp p t 2cj 6 p t 2c 6P

IXZ I Y zp pUoSb2

r = - l j t 2 - 8 , t

IZZT 41z z

(A-57)

t= 2m [ c p t 2 c

yP Y 6 dpl

" IX Y pUoSzb

d P = -

I 41y 1 cm Y l RP 1.

puosc pu, 2 sc'

'[ 4IY1 cm R6 1% 21y' [ZCmRdp d p + c , P P

P To these equations, a feedback control expression was added to permit simulation of bank-angle control by a pilot or augmentation system. To per- form this function, aileron deflection is considered as a linear function of r o l l angle and roll rate, with no actuator lags.

(A-58) When flying in turbulence, the air mass is in motion. The relative velocities, both linear and angular, of the aircraft with respect to the local air mass is considered to be made up of two p a r t s : one caused by motion of the aircraft with respect to an earth-fixed reference, and the other caused by air movement.

r = + = i * + G g (A-59) I n Equation (A-59), the subscript (*) denotes displacement with respect to the Earth-fixed frame of reference, and the subscript (g) denotes effective rolling,yawing,andsideslipgusts,respectively.

If the s e t of equations in Equation (A-59) is substituted into the set in Equation (A-57), and proper distinction is made between inertial and aero- dynamic displacements, the set of equations can be written as (A-60) = [GI where [B] is given by Equation (4) in the main body of this report and

-

L 0 0 0 LP Lr P 0 0 0

NP Nr NP

Y 'r 0 0 0

y P

P 0 2MR 0 0 0 [GI = (A-6 1) MRP

P

0 0 0 0 0 0 0 0 0 0 0 0

- -

The rolling gusts of Equation (A-59) result, in reality, from the span- wise gradient of the vertical gust velocity. Similarly, the yawing gust is related to the gradient of side gust velocity along the length of the aircraft.

The yawing gust is therefore related to the sideslip gust, whereas both of these are unrelated to the rolling gust.

It should be mentioned at this point that the use of equivalent rolling and yawing gusts, operating through fixed coefficients to provide the turbu- lence forcing function, is an approximation to the more rigorous technique outlined in Reference A-1. In that work, use was made of power spectra of rolling- and yawing-moment coefficients on wings subjected to continuous isotropic turbulence. These spectra take into account the random distribu- tion of g u s t s a c r o s s the span and along the flight path. Furthermore, the sideslip-dependent coefficients in the third column of Equation (A-61) become frequency-dependent if lateral gust penetration effects are incor- porated as in Reference A-1.

The effective yawing gust of Equation (A-60) includes two independent effects. One is the spanwise gradient of the head-on longitudinal gust velocity which acts predominantly to cause rolling moments through the Lr coefficient, is the gradient of the side gust velocity which acts upon the fuse- and the other lage and vertical tail as an aerodynamic yawing rate.

The results of Reference A-1 show that the spanwise gradient of longi- tudinal gust velocity has a negligible contribution to the total motion; for this r e a s o n the Lr term in the G m a t r i x m a y be ignored. Furthermore, the side force caused by the yawing gust, Y r , is generally a much smaller effect than the yawing moment, and may also be omitted. As an additional and important simplification, the side gust forcing-function coefficients are not treated as 9 1 frequency-dependent stability derivatives, Instead, the lateral gust penetra- tion effects are included only by allowing for the equivalent aerodynamic yawing-rate forcing function in the yawing-moment equation.

With these simplifications, Equation (A-58) becomes

-

-L

-LP

P W -N -Np-Nr: P W

- y P

(A-62)

8, +

" R -2MRp P 0 0 0 0 In Equation (A-62), the subscript w has been added to the coefficients of the rolling and yawing moments caused by the rolling gust. This is in accordance with the rationale of Reference A-1, which recognizes that the spanwise gradient of vertical gust velocity acts almost exclusively on the wing,andnoton other parts of the aircraft, such as the vertical tail, which normally contribute to these derivatives.

The Longitudinal Equations F o r the longitudinal motion, only symmetrical wing-panel displacement need be considered:

6L = 'p . (A-63)

Similarly, only symmetrical control-tab displacement is included, Because of this, let (A-64) The longitudinal equations from Equations (A-33), (A-34), and (A-35) then become

m W = A F , t (mu,) q

IyyT 4 = 21yp g p ( A F , - X (AF, 1

aero, wings aero, wings t AMy a e r o (A-65) m U = A F x t (-mg) 8 The first of these equations can be written in terms of the fuselage angle of attack by noting that W (A-66)

"f = u

The equation becomes (A-67) The increment in normal force, A F , , involves components due to circulatory lift and apparent mass effects, as showninAppendix B. In fact, (A-68) A F , = LC t L , t L 6, ' e LC is the circulatory lift, and from Appendix B, is Where G1 and G2 are complex lift-curve-slope derivatives which for aspect ratios near 6 may be written as the following transfer functions: 0.3611 (A-70) G1 (X) = CL U X + 0.598- aW

[i-

c' 1

0.488 X , 0.272 X 0. 193 X

G2 (1) = CL - - - . (A-7 1 )

a U U U W

I

X t 0 . 4 5 5 7 A t 1.04- X t 4 . 7 1 - C C c' The factor in the brackets of Equation (A-70) describes the lag in circulatory lift following a change in the angle of attack due to wing motion, whereas the bracketed factor in Equation (A-71) represents the transient effects of angle-of-attack changes associated with vertical gusts, F r o m Appendix B, the lift increment due to apparent mass effects is After appropriate substitutions, Equation (A-67) becomes Zhf hf = z a af t z q t zq 4 t z g b p t Z ' 6 , f z - a , 9 P 6 P 6 P (A-73)

t z , u t z vg t z g de

t V e g where psc Z' = 1 t - af m E

-

ps;

z = 1 --(; x "

q 2m psc' -

(F t 6)

zci -a

z ="

pus G1 2m 6 P (A-74) ps: A = - X

z s

mEU, P

z = - - ps G2

2m vg The pitching motion of the fuselage assembly is given by the second equation of the set in Equation (A-65). For simplicity, unsteady aerodynamic effects are not included in the wing-force terms.

Lf the wing-force increments are assumed to be linearly related to wing angle-of-attack and airspeed changes,

*Fx = - QS[CD a w + 2 C u]

aerowings aW D W (A-75) A F = - QS [C " 2 C u] , Z La aw a e r o LW wings W where A U u =- (A-76) U The wing angle of attack is V g

a, = af t 6 p t - (A-77)

UO The aerodynamic pitching moment on the fuselage assembly is AM = A C ~ Q S ~ , (A-78) Yaero of attack of the The vertical gust influences the aerodynamic angle fuselage assembly since a = a f t r vg (A-79) Furthermore, following Reference A-2, the vertical gust imposes an effective pitching rate equal to (A-80) The influence of the variation in downwash at the horizontal tail caused by wing-panel deflections must also be considered in evaluating the increment in the fuselage pitching-moment coefficient. Equation (A-78) becomes (A-8 1 ) (A-82) t M+ > g g where (A-83) - 2

M e g =-e. km; - cmq]

y y T The third equation of the set in Equation (A-65) describes the pitching motion of one wing panel. The aerodynamic moment which appears in that equationinvolvestheunsteadyaerodynamicseffects.Aftersubstitution,the equation may be written as ; i = p a + P a f + P q q + P ; I ; I t P 6 6 p + p i 6 , af f kf P P (A-84) \ where J A Px PUoP? x

-

P a = U m - t

E af P Iyl IY > (A-85) i The last of the longitudinal equations of the set in Equation (A-65) can be written (A-86) Here, F , is one component of the total applied force vector which is composed of the lift force acting normal to the aerodynamic velocity vector and the drag and thrust forces which act parallel to the aerodynamic velocity.

The force term can be written (A-87) Equation (A-86) becomes

u = x , a f t X@ e t Xg 6 t X u u t Xv

? (A-88) f P P g vg where Two additional equations, associated with longitudinal motion, were used in the analysis. The first describes a simple feedback of fuselage pitch attitude to elevator (symmetrical tab) displacement: The second is the kinematic relationship required to compute altitude deviations :

h = Uo (e - q ) . (A-9 1 )

The complete set of linear longitudinal equations, composed of Equations(A-73),(A-82),(A-84), (A-88), (A-90),and(A-91),appearsin as Equation (1) in the main body of this report.

matrix operational form References

A-1. Eggleston,John M. , andPhillips, William H. , "The Lateral Response

of Airplanes to Random Atmospheric Turbulence", NASA Technical Report R-74 (1960).

A-2. Etkin,Bernard,Dynamics of Flight,John Wiley andSons,Inc., New York (1959).

APPENDIX B AERODYNAMIC CHARACTERISTICS O F FREE WINGS Introduction The unique character of the free-wing concept required certain prelimi- nary tasks to (1) define the control-tab geometry; ( 2 ) assess the general nature of the pitching motion, including unsteady aerodynamics effects; and ( 3 ) corn- pute the additional lateral-directional stability derivatives which arise be - cause of the independent movement of the left and right wing panels.

Svrnbols a, = two-dimensional lift-curve slope, l/radian An = coefficients of F o u r i e r s e r i e s b = wing span, feet c = local chord length, feet Ctl = chord length at inboard end of control tab, feet Ct2 = chord length at outboard end of control tab, feet = mean aerodynamic chord length, feet CD = drag coefficient C Q = rolling-moment coefficient C L = l i f t coefficient

acL

c = -

La aa C = l i f t coefficientcausedbytransientapparentmass Lm effects Cm = Free -wing-panel pitching-moment coefficient on each panel about hinge axis = Free -wing-panel pitching-moment coefficient on right cmR panel = Free -wing-panel pitching-moment coefficient on left c m L pane 1 Cn = yawing-moment coefficient dC n

cn =

P a C mR C mLP E = ratio of s e m i p e r i m e t e r of wing to span length G1 = complex lift-curve slope, 1/ r a d i a n

-

G1 = Laplace transform ofG1

-

constants appearing in G1 g129 813 =

-

h = distance, in mean aerodynamic chord length, from quarter-chord point to hinge axis, feet I = pitching moment of inertia of each panel about hinge IY

axis , slug -ft2

-

I = m a s s p a r a m e t e r of wing panel, Equation (B-21) L , = transient l i f t force caused by apparent mass effects, pounds LC = lift force caused by circulation, pounds M = total pitching moment of wing panel about hinge axis, foot-pounds (M)s = pure pitching moment caused by tab deflection, foot-pounds (M)L, D = pitching moment caused by l i f t and drag forces, foot-pounds P = a r e a of one free panel r = yaw rate, radians/second; also number of span seg- ments used for lifting-line calculations s = distance traveled, in half-chord lengths t = time, seconds U = local airspeed, feet/second Uo = trim airspeed, feet/second wc = velocity of free stream normal to half-chord point,

z feet/second

A x = distance from origin of hinge axis system forward to half-chord point, feet yr = distance from center span to inboard end of f r e e - wing panel, feet = distance from center span to inboard end of control Ytl tab, feet = distance from center span to outboard end of control Yt2 tab, feet a = angle of attack, degrees or radians ai = induced angle of attack, degrees

pmk = multipliers for induced-angle -of-attack calculations

6 tR’ 6tL = right and left control tab deflections, respectively, positive trailing edge down, radians 6 p , 6~ = right and left wing panel deflections, respectively, positive leading edge up, radians X = Laplace operator, l/second

9 = nondimensional Laplace operator, Equation ( B - 2 0 )

p = atmospheric density, slugs/ft3 8 = pitch angle, positive leading edge up, radians.

Control-Tab Geometry For simplicity, the tab is considered to run the f u l l span of the free- wing portions, and to be a plain flap design with a sealed gap.

With an arbitrarily chosen tab-chord ratio of 0 . 1, the section tab effectiveness, C L ~ ~ , as given by Figure 96 of ReferenceB-1, is 0 . 3 . The pitching moment effectiveness of the tab is taken from Figure 9 7 of the same reference,where ( C ) = - 0 . 5 5 .

m6t 7

Since the airfoil section is assumed to be without camber, the l i f t co- efficient is related to angle of attack and tab deflection through The ratio of tab deflection to angle of attack within Equation ( B - 2 ) is d e t e r m i n e d f r o m a balance of moments about the hinge axis: Using the last two equations, the two-dimensional trim characteristics in Figure 3 of this report were computed using

c = 6.28/ radian

La

CD = . 0 0 6 . J

A similar approach is used for finite wings. The data in Figure 4 of this report were computed using the lift-curve slope and pitching moment due to tab deflection from the results of the finite wing analysis presented later in this appendix.

" " Pitch Dynamics of Isolated Free Wing - Following Reference B-2, a wing of aspect ratio 6 was considered to be free only to rotate in pitch about a spanwise axis. The physical situation

is depicted in Figure B-1 , where for convenience in the derivation, the hinge

axis is shownin a faraftpositiontomake a positivequantity.Inpractice, the hinge axis must be forward of the quarter-chord point for static stability.

1 0 5 Note : Hingeaxis placed inaft posit ion only for convenience in deriva t ions FIGURE B- 1. PITCHING-MOMENT ARMS Equation ( 2 9 ) of Reference B - 2 provides an approximate expression for the indicia1 response of lift coefficient to a step angle-of-attack change for an elliptic wing of aspect ratio 6 .

cLa ( s ) = C L a [ 1 .- 0 . 3 6 1e - o . 3 8 1 s ] .

This expression is assumed to be a sufficiently accurate approximation for other aspect ratios and planforms, with the only adjustment being to use theappropriatevalues of C .

La The independent variable of Equation ( B - 5 ) is the distance traveled in half-root-chord lengths. This variable may be related to time, in seconds, by using C 4 - c = - c .

-

In the last equation, the mean aerodynamic chord, c , is taken to be identical to the average chord length of the elliptical wing.

Equation ( B - 5 ) then becomes U

-0.598 t 3

C (t) = CL [ 1. - 0 . 3 6 1 e

La a The corresponding transfer function relating l i f t coefficient to angle of attack may be obtained, as outlined in Reference B-3, by taking the Laplace t r a n s f o r m of the time derivative of the indicia1 response of Equation (B-7).

-

c = c

La La X t 0.598 ~ C

[-" -

The desired transfer function is, then,

- -

0. 361X G I = CL ( X ) = C (B-9) a La U

X t 0.598 =

C According to Reference B-2, the circulatory l i f t is determined by G1 acting on the angle of attack as defined by the normal velocity at the half-chord point, plus an incremental angle of attack caused by the effective camber due to pitching: A x * 1 d d a = d - - 6 t - - (B-10) U 2 d s * T h e l a s t t e r m is converted to time dependence as 1 d6 F " X " b . (B-11) 2 d s

flu

The circulatory-lift contribution for one wing panel then becomes (B-12) The lift coefficient arising from the acceleration of the apparent mass of air surrounding the wing is given in Reference B-2 as =" C fl da (B-13) Lm E ds Again converting to time units, 2 " a - " C (B-14) L , E U The angle-of-attack rate is again based upon the local rate of change of normal velocity at the half-chord point.

Since

wc = - 2 s t u a , (B-15)

z

then (B-16) Substituting Equation (B-16) into Equation (B-14), the total lift force caused by apparent mass effects is (B-17) In Chapter 5 of Reference B-4 it is implied that the pitching moments of the wing panel may be computed by considering the circulatory l i f t force to act at the quarter-chord point. In addition, the L, force is divided into two parts for the moment calculation. The first term in Equation (B-17) acts at the three-quarter-chord point, whereas the remaining term acts very near the half-chord position.

Using these moment arms, the equation describing the pitching motion about the hinge axis is pup: * PP: A 2

6 " X . (B-18)

IY I E The characteristic equation used to compute the modes of the pitching motion is obtained by taking the Laplace transform of Equation (B- 18).

Then, GI is written as

I

-

G1 = cLa I

Multiplying the transformed version of Equation (B-18) through by the denominator of Equation (B-19), a characteristic equation is obtained which is the product of a cubic polynomial and the first-order denominator of Equation (B-19). The first-order factor is disregarded because it describes a n uncoupled stable real root. The cubic factor, on the other hand, will generally yield one stable real root and a complex conjugate pair.

The complexity of the cubic equation is reduced by employing a di- mensionless form of the Laplace operator defined by x = " 2 u k .

( B - 2 0 ) C F u r t h e r m o r e , a mass parameter is defined as ( B - 2 1 ) The nondimensional form of the cubic characteristic equation becomes a ( B - 2 2 ) Free-Wing. Aerodvnamic Derivatives Wing Geometry The wing is considered to be composed of a short center section of constant chord, with a free-wing panel on either side. The quarter-chord lines of all sections of the wing are aligned in the spanwise direction with no sweep, and the hinge axis is parallel to the quarter-chord line.

The purpose of the center section is to approximate the effect of the fuselage between the two free panels, and the chosen span of this section, 12. 5 percent, is an arbitrary value. For symmetrical deflections of the free-wing panels, the center-section geometrical angle of attack is taken to be the same as that of the outer panels; but, for a s y m e t r i c conditions, the angle of attack varies linearly between the values at the root sections of the deflected panels.

la " Application of Lifting-Line Theory Reference (B-5) provides a convenient formulation of the application of classical lifting-line theory to the determination of the circulation distribu- tion on finite wings of arbitrary planform and twist. The approach used in Reference (B-5) is followed closely in this study, except that the method is expanded to permit spanwise variation of airspeed caused by yawing r a t e s and spanwise variation of geometrical angle of attack caused by roll rates.

The expanded approach is outlined briefly below.

If y is the spanwise distance measured positive from the plane of symmetry to the right wing tip, a substitution of variables can be made as b - c o s e = y .

A F o u r i e r s e r i e s c a n now be written in terms of 9 to define the span- wise distribution of circulation. At any spanwise location, the strength of the bound vortex is related to, r - 1 c c

C A n s i n n 8 k . (B -23)

n = l Furthermore, the local l i f t coefficient is, by definition, CLk = a, ( a - a i ) k .

(B-24) The induced angle of attack, however, depends uponthe entire circu- lation distribution through r - 1 (B-25) u o m = 1

Here, the p d are multipliers which depend only upon the number of

spanwise segments, r. An expression for these multipliers is contained in Refe rence (B - 5).

In brief, the computational process begins with assuming an initial CL distribution. Combining this with a knowledge of wing geometry, flight

r

speed, and angular rates, the induced angle of attack is computed at each station by means of Equation (B-25). Then, a revised raw estimate of local CL at each station is obtained from Equation (B-24). This raw estimate is refined through a smoothing scheme described in Reference (B-5) and the p r o c e s s is repeated until the change in CL becomes less than 0. 1 percent of the previous value at all wing stations.

Having found the circulation distribution, the left side of Equation (B-23) is known, and the coefficients of the Fourier series can be found as r . - 1 (B -26) k = 1 For this study, 29 Fourier coefficients were obtained in all cases. For e a c h wing planform and angle of attack, the l i f t distribution was computed six times: The first distribution was for zero tab and wing-panel deflec- tions, and no rolling or yawing velocities. This established thewing l i f t coefficient, lift-curve slope, and free-wing- panel pitching moments at the reference angle of attack.

Following this, the control tabs were displaced symmetrically, and by comparison with the first computation, the contribution of symmetrical tab displacement to the wing l i f t coefficient and the panel pitching-moment coefficients was evaluated.

Next,onlytherighttabwasdeflected.Fromthis,thedirect rolling-moment and yawing -moment contributions from single tab displacement were determined, and the direct effect of single tab displacement upon the pitching moment coefficient of each panel was evaluated.

W i t h the control-tab displacements once again set to zero, the right wing panel was displaced and its contribution to the rolling, yawing, and individual panel pitching moments was determined.

Following this, the panel displacements were again set to zero, and a rolling velocity w a s assumed. As before, the rolling-velocity contribution to the rolling, yawing, and individual panel pitching moments was established.

(6) Finally, the rolling velocity was returned to zero, but ayawing velocity was assumed to evaluate the roll-due-to-yaw rate derivative and the effects on each of the wing-panel pitching moments.

Concise expressions were derived for each of the aerodynamic p a r a m e t e r s , i n t e r m s of the series coefficients of Equation (B-26), obviat- ing the need for numerical integration of the forces and moments.

Wing lift coefficient: - 4s A1 - z k ] 2u0 2 .

(B-27) Roll -damping de rivative : (B-28) Roll-due-to-yaw rate derivative: - - A2 + z b (A1 + A , ) ] .

(B-29) r If the input i s a panel deflection or a tab deflection, tii, (B-30) The yaw-due-to-roll derivative is, If a panel o r tab deflection is the input, (B-32) The pitching moment on each free panel is composed of a pure pitching moment caused by tab deflection and the contributions of the lift and drag f o r c e s ( s o m e of which may be caused by tab deflection) acting through their respective moment arms about the hinge axis.

By direct integration, the pure pitching moment caused by tab deflec- tion on the right panel is PUo2 2 2

= c - 6, [ e ( Y t 2 - Y t 1 t ef ( Y t 2 - Y t , ) 1 9 (B-33)

mat 2 where (B-34)

Ct2 - C t l

f = The pitching moment caused by lift a n d d r a g f o r r e s is, for the right pane 1, B-35) r - 1 sin (n t 1) y sin (n - 1) y 2 ( n t 1) 2 (n - 1) n = 2 where y = c o s - l ( b/2 Y r ) .

( B - 3 6 ) F o r the left panel, the equivalent expression is 2 2 C

( M ) L , D ~ = ~ ( I t $ ) Pb Uo (A1 [ - x - 2 sin 2 4 (T - y)

I t

B-37) r - 1

sin (n - 1) ( f l - y) sin (n + 1) (T - y)

2 (n + 1)

n = 2 The total pitching moment on either panel is (B-38) from which the total pitching moment coefficient is 2M

cm =

(B -39) 2 - * PU0 s c Each pitching-moment derivative is then obtained by dividing the pitching-moment coefficient by the appropriate variable.

For each combination of aspect ratio and taper ratio, the preceding computational procedure was performed for three angles of attack and two hinge line positions. The results of the calculations are listed in Table B-I.

References

B-1. Abbott, Ira H. , and Von Doenhoff, Albert E. , Theory of Wing Sec-

tions , Dover Publications , Incorporated, New York ( 1959).

B-2. Jones, Robert T. , "The Unsteady Lift of a Wing of Finite Aspect

Ratio", NACA Report 68 1 (1940).

B-3. Etkin, Bernard, Dynamics of Flight, John Wiley and Sons, Inc. , New York (1959).

Bisplinghoff,Raymond L. , Ashley,Holt,andHalfman,RobertL. , B -4.

Aeroelasticity,Addison-WesleyPublishingCompany,Cambridge, Massachusetts ( 1955).

B-5. Sivells, James C., and Neely, Robert H. , "Method for Calculating

Wing Characteristics by Lifting-Line Theory Using Nonlinear Set- tion Lift Data", NACA TN1269(1947).

TABLE B-I. COMPUTED WING AERODYNAMICCHARACTERISTICS All Dimensions per Radian 10 Percent Sealed Plain Control Tabs, Full Span of Free Panels = 6 Aspect Ratio = 8 Aspect Ratio - Taper Ratio Taper Ratio 0 . 6 1 . 0 0 . 6 1 . 0 Lift Derivative s 4. 75 4 . 5 3 4 . 9 5 4. 84 cLa 1.45 1.428 1. 36 1.485 cL6 e Rolling-Moment Derivatives -0.121 -0. 124 -0.133 -0.139 -0.415 -0.423 -0.454 -0.474 -0.505 -0.523 - 0 . 556 -0.593

ce

P 0.206CL 0 . 216CL 0 . 2 1 8 C ~ 0 . 2 2 9 C ~ Yawing -Moment Derivatives

C 9 - c 0.0007 + 0 . 0008 + 0.0006 + 0 . 0007 +

n 6 t R n%L 0 . 0087CL 0 . 0lOlCL 0 . 0103CL 0.0115CL

+ 0 . 0084 + 0.0082 t 0 . 0081 t

0.0072 0. 0292CL 0 . 0 3 3 6 6 ~ 0 . 0 3 5 1 C ~ 0 . 0 4 0 C L -0.0867CL -0. 0881CL -0.0949CL -0. 0967CL ‘ n P TABLE B -I. (Concluded) AsDect Ratio = 6 AsDect Ratio = 8 TaDer Ratio Taper Ratio 0 . 6 1 . 0 0 . 6 1 . 0 Free-Wing-Panel Pitching- ~ ~ ~~ Moment Derivatives(a) (Hinge axis 10 percent root chord forward of q u a r t e r - chord line) - 0 . 4 1 6 - 0 . 2 9 2 - 0 . 2 9 8 C -0,296 m6e -0.335 -0. 194 - 0 . 2 5 4 - 0 . 2 0 8 cma

crn (b) -0.408 - 0 . 2 8 9 - 0 . 2 9 4 - 0 . 2 9 3

C mL6 ' '"Rg - 0 . 0 0 7 1 1 - 0 . 0 0 3 8 5 - 0 . 0 0 4 2 8 -0. 00315

tR tL - 0 . 2 2 3 -0. 185 C m 9 Cm - 0 . 2 8 6 - 0 . 169 R 6 P L6L -0. 031.5 - 0 . 0 1 6 6 -0. 0 1 9 4 -0. 0143 c m L ? C 6 P mR6L -0. 136 -0.0837 -0. 108 -0.0935 C

mR; - cmL P

(a) All of the pitching moment derivatives, except those dependent on control tab deflection, are directly proportional to hinge margin.

(b) These derivatives are linear with hinge margin and have a value at zero hinge margin equal to the two-dimensional pitching moment due to tab deflection multiplied by the ratio of free panel area to total wing area.

APPENDIX C DESCRIPTIONS O F HYPOTHETICAL AIRCRAFT Introduction Three aircraft, designated A, B, and C, were considered in this study.

These aircraft range in gross weight from 3000 to 5 0 , 0 0 0 pounds. This range of weights was used to uncover any unusual characteristics which might depend upon mass and inertia properties. In addition, four wing planforms were postulated for use with each aircraft. A subscript ranging f r o m 1 to 4 denotes the planform.

These hypothetical aircraft are patterned ina general way after exist- ing aircraft. The design effort has been limited to the selection of the gross arrangement of components to provide a rational basis for the estimation of weights and inertias. Although the outboard hinge axis is externally sup- ported in all three designs, no engineering details regarding support strength, etc. , were considered.

DescriDtions of Aircraft Aircraft A Aircraft A is in the light observation class and is patterned after the Cessna family of aircraft. The high wing configuration seems well suited to a simple type of external support for the outer axis bearing.

In Figure C-1, the A1 version of this aircraft is shown, with aspect ratio of 8 and taper ratio of 1. As in the other aircraft, conventional arrangements have been preserved as much as possible to provide a mean- ingful comparison between the free -wing and fixed-wing counterparts.

Aircraft B Aircraft B i s a twin-engine utility aircraft, patterned loosely after the Short Skyvan utility transport, although a single vertical tail is used and Nominal -1 hinge axis

FIGURE C-1. ALRCRAFT A1 - LIGHTOBSERVATION ALRCRAFT

"_ the engines have been moved from wing t o fuselage mounting. The overall length, general fuselage configuration, wing loading, and gross weight of 12,500 pounds are similar to those of the Skyvan. Aircraft B1 is shown in Figure C-2.

Aircraft C Aircraft C is a transport/freighter aircraft with a gross weight of 50, 000 pounds, patterned roughly after the Bristol Type 170 Mk 32 freighter, although turboprop engines mounted beneath the wings are assumed. (See Figure C-3. ) Weights and Inertia Parameters The estimation of component weights and inertia parameters was re- quired for inclusion in the equations of motion.

Gross estimates of wing weights and structural weights were obtained from Reference C-1. These were then used with the approximate method outlined in Reference C-2 to obtain the inertia parameters.

Table C-I is a listing of the significant parameters describing each of the aircraft.

References C-1. Wood, K. D., Aerospace Vehicle Design,Volume I, AircraftDesign, Johnson Publishing Company, Boulder, Colorado (1 963).

C-2.USAFStabilityandControlDatcom,AirForceFlightDynamics Laboratory, Wright-Patterson Air Force Base, Ohio,Revised November, 1 965.

FIGUREC-2.AIRCRAFT B1 - MEDIUM UTILITYAIRCRAFT

I

FIGUREC-3. ALRCRAFT C2 - TRANSPORT/FREIGHTERAIRCRAFT

TABLE C-I. BASIC CHARACTERISTICS OF THREE SAMPLE FREE WING AIRCRAFT Chords G r o s s Wing Wing Root Weight, Area, Aspect Taper Span, Tip, Y I c g , L ~ J IYf' 1.f' k' 1 Ix', y's 4 ' 3 $ 1 Aircraft lb ft f t slug-ft2 slug-ft2 slug-ft2 slug-ft2 slug-ft2 dug-ft2 dug-ft2 ft2 h t i o Ratio ft 1451 t 2 14 2 , 5 0 2 1,420 31'2 Planform 1 3,000 8 1 . 0 4 1 . 4 2,502 90.6 d C p 5.18 6.27 Span 1271 t 2 14 2 3,000 8 0.6 41.4 46 1 2 , 5 0 2 2 , 5 0 2 1,242 - - 81.7 XIcg 3.76 4 10x1 2 CB A 39.8 t 1123 t 3 3,000 214 6 1 . 0 35.8 2,502 2 , 5 0 2 2 78.5 xtcg 5.97 " 4 3,000 2 14 6 0.6 35.8 2,502 2,502 " " " 4.34 6.83 Span 80.8 t 4721 t Planform 1 12,500 37 3 54.6 8 1.0 261 xtfg 6.83 6.0 8,000 22,700 22,850 4,640 28.7 28.7 xlcg2 85.0 x 4065 t 2 12,500 373 8 0.6 54.6 235 d C g B 7 . 8 8 Span 105.0 t 3590 t 3 12,500 373 6 47.3 1.0 7 , 8 8 6 . 0 8,000 22,700 22,850 226 xIcg 31485 28.7 xsCgz 28.7 d C g 2 9.5 90.5 t 3141 t 4 12,500 37 3 6 0.6 47.3 204 XIc* 5.7 14.52 Span 2055 t 121,355 t Planform 1 50,000 1690 8 116.3 3160 xlCg 1 . 0 14.52 6.0 35, 567 373,512 384,900 119,300 163 163 xtcg2 2120 t 106,720 t 17.7 Span 2 50,000 1690 8 0.6 116.3 6.66 35,567 373,512 384,900 104,600 1 6 3 x , c g 2845 xlCg 163 xlCgz C 2525 t 92,025 t 16.8 Span 3 50,000 1690 6 1 . 0 100.7 35,567 373, 512 384,900 89,500 163 x,cgz 163 x,cg2 2740 xtCg 16.8 6 . 0 2980 t 82,280 t 20.4 Span 4 50,000 1690 6 0.6 100.7 2460 xtCg 12.21 6.66 3 5 9 5 6 7 373,512 3&4,900 79,300 163xtcg2 163 xlcg2

APPENDIX D AERODYNAMIC -. . . - - " " CHARACTERISTICS - O F COMPLETE AIRCRAFT " .. - ~~ Introduction For simplicity, and to delineate the effects of a i r c r a f t s i z e m o r e vividly, the nondimensional stability derivatives are assumed to be the same for all aircraft with a given wing planform. All differences in dynamic characteristics are therefore dependent upon mass and inertia effects as well as the equilibrium flight condition. Furthermore, to reduce the num- ber of p a r a m e t e r s tobe computed, the cruise and approach lift coefficients were held fixed, respectively, for all aircraft.

Those aerodynamic parameters that are dependent only upon the wing, and which are discussed in Appendix B , a r e not treated in this appendix.

Symbols A = aspect ratio at = slope of lift curve for tail surface, l/radian b = wing span, feet

-

c = mean aerodynamic chord length, feet C D = total drag coefficient CD, = profile drag coefficient

acD

-

- , 1 /radian

aa CL = lift coefficient

- , l / r a d i a n

CLCZ - C J = rolling moment coefficient

- aca

C a p - ap , 1/ r a d i a n

aca

-

" , 1/ r a d i a n

'Qgp aa, C , = pitching -moment coefficient

acrn

forfuselageassembly,l/radian = pitching moment of right wing panel about hinge axis cmR aCmR

, l / r a d i a n

CmP = ap

Cn = yawing moment coefficient I1

c = , 1 /radian

"p A / & ) CT = thrust coefficient C = sideforcecoefficient Y dC,, J

c =

, l / r a d i a n

Y , l / r a d i a n

-

= 2 , l / r a d i a n

C 6P dC

C - 2 , l / r a d i a n

Y P - d P e = span efficiency factor

-

I = mass parameter defined in Appendix B 1, = tail moment arm, feet p = roll rate, radians/second q = pitching rate, radian/second r = yaw rate, radians/ second

s = wing area, ft2

Uo = trim airspeed, feet/second S t 4

VH = horizontal tail volume, -

S F S t i t

Vv = vertical tail volume, -

S b Zvt = height of vertical tail center of pressure above roll axis, feet a = angle of attack, radians /3 = sideslip angle, radians 6p = deflection of right wing panel, L.E. up positive, radians dL = deflection of left wing panel, L. E. up positive, radians E = downwash angle at horizontal tail

x = Laplace operator, I/sec

A X = dimensionless operator A XR = real component of complex root.

Subscripts, int = interference of wing and body f = fuselage vt = vertical tail w = wing.

Longitudinal Coefficients A simple parabolic drag polar was assumed: whe r e e = 0 . 8 CL = 0 . 3 4 3 for cruise, 0 . 77 for approach.

This derivative could conceivably be z e r o if the equilibrium attitude of the fuselage is for minimum drag, but an arbitrary small value of .0029 was selected for all cases.

C It is assumed that the profile drag coefficient is independent of angle of attack over a small range about the trim point. Consequently, all drag changes are associated with the induced drag, The neutral point of the wing alone would be a t the quarter-chord point. If theinfluence of thebody is considered, less horizontal tail, the neutral point is shifted forward because of the destabilizing influence of the fore -body and propeller effects.

The forward shift caused by the fuselage is estimated from Figure B. 8. 1 of Reference D-1, and was computed to be approximately 5.8 percent of the mean aerodynamic chord. A further shift of 5 percent was arbitrarily selected to account for propeller effects, placing the aerodynamic center of the wing-body combination at 0. 14 c. By definition, then, any pitching-moment changes with angle of attack about this point a r e c a u s e d by the horizontal tail.

Since 0.14 C, a s a hinge location, gives an 11 percent hinge margin, the Cm, for the fuselage and tail assembly can be computed for this near- nominal location by simply calculating the horizontal tail contribution.

This is Values used throughout were: at = 4.35/radian dc

- = 0 . 4

da The nominal value of horizontal tail volume, VH, was 0 . 68.

From Reference D-1, assuming that horizontal tail is sole contributor, all damping is provided by the From Reference D-1, assuming that horizontal tail, It can be easily shown that the downwash effect of wing-panel deflection is Since all aircraft being considered are propeller driven, the assump- tion is made that the engine delivers constant power over the limited speed rangeneartrim.Thisleadsto Lateral-Directional Coefficients It is assumed that the entire roll-damping derivative is due to the wingcontribution.Accordingly C j i s obtainedfromTableB-I in P Appendix B .

Similarly, the wing contribution to the rolling moment due to yaw rate is assumed to predominate. These values were taken from Table B-I in Appendix B .

The total value of the dihedral-effect derivative is obtained by summing the results of Equations(D-8),(D-9),and(D-12).

The selected nominal value of Vv was .0683.

The wing and vertical tail are both important contributors to the yawing moment due to roll rate. The wing contribution given in Table B - 1 of Appendix B includes only the effect of the tilting of the lift vector. To this must be added the profile drag component which is obtained from Figure B . 1 2 . 2 of Reference D-1. If the wing-profile drag coefficient is taken to be . 0 0 6 , this component becomes (D-13) Cn = .054 , Pwdrag The derivation of the vertical-tail contribution is similar to that of the dihedral effect, but in this case, (D-14) Since the span changes with aspect ratio, for fixed area, two ex- pressions are obtained: = 0.319 V v for A = 8 Cn P vt (D-15) Cnpvt = 0.379 V v for A = 6 The total derivative is obtained by summing the value from Table B - I of Appendix B with the results of Equations (D-13) and (D-15).

It is assumed that all yaw-rate damping comes from the side force on the vertical tail induced by the yawing rate. This is estimated to be Cnr = - 2 . 7 2 VV for A = 8 ( D - 16) Cnr = -2.98 V v for A = 6 1 3 1 The procedure used to compute the directional-stability parameter was to compute the effect of the side force on the vertical tail only, and then to reduce this value by 10 percent to allow for the destabilizing influence of other components of the aircraft. This approach yielded (D-17)

CnP = 2 - 9 6 vv

Only the vertical-tail contribution was considered in computing the sideforcedue to roll rate. This became

cyp = -0.734 vv . (D- 18)

The side-force derivative due to yawing rate is computed by again considering only the vertical-tail contribution: Although this derivative may exist because of pressure differences on the fuselage with asymmetric panel deflections, no convenient means is available to compute its value. It was therefore assumed to be z e r o .

The side force due to sideslip was assumed to be dominated by the vertical-tail force s o this derivative was estimated as

C = -7.57 Vv for A = 8

y P

( D - 2 0 )

Cyp = -6.57 Vv for A = 6

This derivative is peculiar to the free-wing aircraft and represents the pitching moment on the wing panel (right panel) due to sideslip. A s explained in the main body of this report, a nominal value of this derivative was selected. This nominal value was sized to provide steady-state can- cellation of the wing contribution to C a p . F r o m the requirement for equi- librium in both the rolling-moment and pitching-moment equations, the expression for the nominal value is (D-2 1 ) To approximate the damping derivative of the wing panel for asym- metric motion, a technique was developed to make use of the syrnmetrical- oscillatory-mode data in Figure 5 of the main body of this report.

If the stability derivatives used for asymmetric motion were used to describe the symmetric oscillation of the wing, the -characteristic motion of the isolated symmetrical panel mode would be The roots of this equation are of the corresponding dimensionless root is The real component (D-24) Since the primary effect of the damping derivative is upon the r e a l component, the selected value of the derivative is References D - 1 . Etkin,Bernard,Dynamics of Flight,JohnWileyandSons, Inc., New York (1959).

D-2. Campbell, John P., andMcKinney,Marian O . , "Summary of Methods for Calculating Dynamic Lateral Stability and Response and for Estimating Lateral Stability Derivatives I ' , NACA Report 1098 (1952).

APPENDIX E METHOD O F COMPUTING TURBULENCE RESPONSES S ymb ol s b = wing span, feet T = length of mean aerodynamic chord, feet g13 = coefficient defined i n Appendix B g = quantity defined by Equation (E- 1 1) L = scale length of turbulence, feet U = airspeed, feet/second A = numerical value of determinant X = Laplace operator, l/second O x = rms value of variable x @ = power spectral density function Obpg = power spectrum of rolling gust, feet/second power spectrum of sideslip gust, feet

@Ps =

R = spatialfrequency,radius 1 foot.

Longitudinal Responses Equation (1) of the main text of this report describes the deterministic response of the longitudinal system to the vertical gust velocity.

F o r random turbulence responses it is necessary to derive transfer functions for the response of each variable to the gust, and to use these trans- fer functions to compute the spectrum of the response in each variable of

interest. The output spectrum for a variable , x, i s given by

X

where - is the modulus of a frequency response function which defines the

response of the variable to the gust velocity.

The root-mean-square response of the variable is Expressions for the transfer functions of interest can be developed using standard techniques. For example, the transfer function relating pitch angle to vertical gust velocity is

where I [ A1 J I is the determinant of the matrix obtained by substituting the

column matrix on the right side of Equation (1) for the second column of the m a t r i x [ A] .

Expressions for any of the other transfer functions are obtained in a similar fashion, and for the variables related by a differentiation through, for instance, Because of the algebraic complexity of the transfer functions, no analyti- cal derivations of these expressions were performed. Instead, the determin- ants of the respective matrices for the numerators and denominator were expressed in polynomials of the operator X, by purely numerical means.

Assume, for example, that the determinant of the matrix of coefficients i n the denominator of Equation (E-3) can be expressed as an nth-order poly- nomial,with n t l coefficients.Thetechniqueconsists of selecting nS1 a r b i - t r a r y v a l u e s of X. Then, for each one of these, a unique value of the deter- minant of the matrix is found using a standard computer library subroutine for determinant expansion. After a value of the determinant has been found for each of the ntl values of X, a s e t of ntl linear simultaneous algebraic equations can be formed and solved for the coefficients of the characteristic polynomial.

This technique was employed, with some modification as described be- low, for the numerators and the denominator of each of the transfer functions.

1 3 6 Then, setting X = j m , the transfer functions were converted into the complex frequency response functions.

A complication exists in applying the basic technique t o the longitudinal set of equations because some of the elements in the first and third rows of m a t r i x [ A ] include the complex lift-curve slope function G, which contains a denominator that is, itself, a first-order polynomial in X. Because all ele- ments of the determinant are not simple polynomials, the determinant cannot be expressed as a simple polynomial. Instead, the determinant is, Polynomial A = (E-5) / Since the same factor appears in all numerators as well as the denomi- nator of the transfer functions, only the ratio of the polynomials is significant.

For this reason, for each numerical value of X, the corresponding value of A was multiplied by the denominator of Equation (E-5) to obtain The unsteady aerodynamics effects of the buildup of aerodynamic l i f t following penetration of a vertical gust were approximated by multiplying the vertical gust velocity by a smoothing transfer function which approximates the Kus s n e r l i f t growth function.

The approximation was obtained by taking the Laplace transform of the time derivative of the indicial response to a vertical gust. The indicial- response function was obtained from Reference E-1, and although it was given therein for aspect ratio 6 , it was used for aspect ratio 8 as well in this study.

0.448 X 0 . 2 7 2 X 0. 193 X G 2 = 1 - (E - 7 ) U U U

X t 0.455 = X t 1. 04 = X t 4.71 =

C C C Lateral-Directional Responses Equation ( 3 ) of the main body of the report describes the deterministic response of the system to rolling and sideslip gusts. Two forcing functions are present in Equation ( 3 ) , and these functions are uncorrelated in the statis- tical sense.

Since the turbulence is assumed to be homogeneous isotropic, the ver- tical and side gust components measured at the same point on the airplane have the same spectrum, and both components have the same rms value. The sideslip gust is directly related to the side gust velocity, but the rolling gust is based upon the spanwise gradient of the vertical gust velocity.

Because the side and vertical gust components are uncorrelated, the total response of the aircraft, in a variable x, is computed from X X

where I I and I are the moduli of frequency-response functions for the

response of the variable x to the rolling and sideslip gusts, respectively. The s p e c t r u m of the sideslip gust i s simply related to the PSD function plotted in Figure 2 of the main body of this report, and is, for unit gust intensity, 2 2 L 1 t 3 5 2 L @ = - (E-9) 2 2 2 a

pg nu2 [ l t n L ]

The power spectrum of the rolling gust was obtained from Reference E-3, wherein a quantity, CP 2 , is derived which is equivalent to one-half the rolling gust PSD function used in Equation (E-8). This rolling-gust spectrum i s , for unit intensity, 2 2 1 L a 3

@ ( a ) = - ( 2 2 ) g -t

7 T L

%

1 t L a

L where 7~ L / b ( E - 1 1 ) 2 2 2 1 t L t ( n L / B ) The frequency-response functions needed for Equation (E-8) are obtained by the numerical method outlined earlier, and the root-mean-square responses a r e computed from Equation (E -2).

References E-1. Jones, Robert T., "The Unsteady Lift of a Wing of Finite Aspect Ratio", NACA Report 6 8 1 ( 1940).

E-2. Eggleston,John M., andPhillips,William H . , "TheLateralResponse of Airplanes to Random Atmospheric Turbulence", NASA Technical Report R-74 (1960).

APPENDIX F TABULATED NUMERICAL RESULTS Introduction To provide a complete record, the results of each of the computer runs for the longitudinal and lateral-directional turbulence responses are tabulated in this appendix. For longitudinal cases, results are given for the fixed- wing aircraft and four versions of the free-wing counterpart. For lateral- directional responses, resultsare tabulated for the fixed-wing aircraft and the nominal free-wing equivale'nt. This nominal free-wing aircraft has the same vertical-tail volume as the fixed-wing aircraft and a hinge margin o f 10 percent of the root chord.

All rrns values are per unit turbulence intensity.

S ymb ol s

V H = horizontal tail volume

" _ Longitudinal ~ Responses a n = rmsloadfactor,g's Z

aq = rmspitchrate,degrees/second

a* = rmspitchacceleration,degrees/second2

a h p = rms panel deflection, degrees a h = rmsaltitudedeviation,feet Lateral-Directional Responses " 06 = r m s r o l l angle,degrees = rrns yaw angle, degrees

a&, = rmsrollrate,degrees/second

14 1

c$ = rms yawrate,degrees/second

= rms lateral path displacement, feet

“=Y

= r m s lateral load factor, g ’ s .

“3

TABLE F-I. LONGITUDINAL DATA, AIRCRAFT A 1 Nominal Free Wing 1% Hinge Margin 1(pl0 Hinge Margin Fixed Wing l C $ Hinge Margin 2w0 Hinge Margin 1/2 Nominal VH 1/4 Nominal VH Cruise Phugoid Roots -0.0217 f j 0.181 - 0.0226 f j 0.226 -0.0227 f j 0.226 - 0.0225 f j 0.226 - 0.0225 f j 0.226

Short-Period Roots - 4.04 f j 2.96 -0.668 f j 2.72

- 2.59 f j 6.35 - 2.74 f j 6.50 - 1.33 f j 3.72

Symmetric Panel Roots

- 8.96 5 j 13.8 - 10.7 f j 18.6 - 8.75 f j 13.65 - 8.69 f j 13.6

-

Plunging Roots

- 25.2, - 23.8 - 26.4, - 23.8 - 25.8, - 23.3 - 26.0, - 23.2

0.02476 0.00946 0.00634 0.00912 0.009 0.134 0.474 0.486 0.367 0.381 1.46 3.37 3.63 1.15 1.74 O 4

-

0.0823 0.148 0.0899 0.119 1.22 0.145 0.150 0.106 0.150 Oh Approach

-0.026 f j 0.362 -0.0266 f j 0.362 - 0.0257 f j 0.362 -0.0255 f j 0.362

Phugoid Roots -0.0244 f j 0.277

-0.0926 + j 2.44 - 0.462 f j 1.79

Short-Period Roots - 2.4 f j 1.69 - 1.8 f j 4.15

- 1.92 f j 4.26

-

Symmetric Panel Roots

- 6.19 f j 8.85 - 7.38 f j 11.9 - 6.04 f j 8.76 - 5.99 f j 8.74

-

Plunging Roots - 17.4, - 14.5

- 16.5, - 14.5 - 16.6, - 14.5 - 16.6, - 14.5

0.0191 0.00453 0.00673 0.00651 0.00651 0.151 0.476 0.487 0.384 0.372 1.150 2.32 2.49 1.22 * 0.804 0.126 0.139 0.184 0.228 2.37 0.377 0.284 0.400 0.403 c P P TABLE F-11. LONGITUDINAL DATA, AIRCRAFT B 1 Nominal Free Winn 10% Hinge Margin 1 0 % Hinge Margin Fixed Wing 1 0 % Hinge Margin 2070 Hinge Margin 1 / 2 Nominal VH 1/4 Nominal VH Cruise Phugoid Roots -0,0127 f j 0.121 -0.0137 f j 0. 136 -0.0137 f j 0.136 -0.0137 f j 0.136 -0.0137 f j 0.136 Short-Period Roots -2.07 f j 2.77 -1.27 f j 5.33 -1.33 f j 5.38 -0.644 f j 3.06 - 0.321 f j 2.19 Symmetric Panel Roots “ -13.1 f j 18.0 -16.3 f j 24.6 -13.1 f j 18.0 -13.0 f j 18.0

Plunging Roots ” -34.8, - 29.2 -34.7, - 30.3 -34.2, - 29.6 -34.6. - 29.3

0.0206 0.00588 0.00365 0.00571 0.00567

“ “z

0.124 0.413 0.411 0.325 0.318 “ q 0.719 2.33 2.37 1.10 0.749

“ 4

” 0.0829 0.0853 0.116 0.153 “6 P 0.0540 0.0411 0.056 0.0568 “h 0.659 Approach Phugoid Roots -0.0137 f j 0.197 -0.0177 f j 0.233 - 0.0177 f j 0.233 - 0.0176 f j 0.233 -0.0175 f j 0.233 Short-Period Roots

-0.60 f j 1.66 - 0.964 * j 3.51 -1.01 * j 3,56 -0.488 f j 2.02 - 0.241 f j 1.45

” Symmetric Panel Roots

-9.38 f j 10.7 -11.5 * j 14.5 -9.30 f j 10.6 -9.28 f j 10.6

”

Plunging Roots -22.05, - 17.02 -23.05, - 17.04 -22.1. - 17.0 -22.1, - 17.0

0.0154 0.00442 0.00279 0.00429 0.00427 U “Z 0.103 0.406 0.411 0.326 0.321 “q 0.571 1.582 1.62 0.766 0.519

“ci

” 0.126 0.131 0.179 0.235

“ti P

1 . 0 4 0.111 0.0875 0.117 0.119 O h TABLE F-111. LONGITUDINAL DATA, AIRCRAFT C1 Nominal Free Wing 1q0 Hinge Margin l q o Hinge Margin Fixed Wing lW/o Hinge Margin 20% Hinge Margin 1/2 Nominal V H 1/4 Nominal VH Cruise -0.0123 f j 0.122 -0.0124 f j 0.122 -0.0123 f j 0.122 Phugoid Roots -0.0116 f j 0.102 -0.0123 f j 0 . 1 2 2 -1.35 f j 3 . 7 9 -0.332 f j 1 . 5 7 Short- Period Roots -2.04 f j 1 . 8 8 -1.29 f j 3 . 7 3 -0.660 f j 2 . 1 6 " -8.16 f j 9.59 -9.95 f j 1 3 . 0 -8.03 f j 9 . 5 2 Symmetric Panel Roots -8.06 f j 9 . 5 4 "

-19.2, - 15.2 -20.2, - 15.2 -19.3, - 1 5 . 2 -19.3, - 1 5 . 2

Plunging Roots 0.0220 0.00762 0.00487 0.00763 0.00729 U "Z 0.0781 0.2727 0.277 0.212 0.202 O9 1 . 1 7 1.23 0.574 0.312 0.516 0 4 " 0.0804 0.086 0.113 0.142 ug P 0.0177 0.490 0.083 0.085 0.906 'h Approach Phugoid Roots - 0.0174 f j 0.181 -0.0177 f j 0.248 - 0.0180 f j 0.248 - 0.0172 f j 0.248 -0.0170 f j 0.248 -1.15 f j 2.29 -1.21 f j 2.36 -0.589 f j 1.35 -0.297 f j 1.00

Short-period Roots - 1.84 f j 0.938

" -5.54 f j 4.30 -6.7" f j 5.78 -5.44 f j 4.26 -5.40 f j 4.25 Symmetric Panel Roots

- -

-12.9, - 7.53 -13.1, - 7.53 -12.9, - I. 53 - 12.96, - I. 53

Plunging Roots 0.00392 0.00609 0.00603 0.0174 0.00618 "Z 0.261 0.275 0.208 0.197 0.0803 u 9 0.872 0.425 0.266 0.492 0.816

us

" 0.126 0.141 0.181 0.220 u6 P 0.241 0.156 0.265 0.214 'h 1.51 TABLE F-IV. LONGITUDINAL DATA, AIRCRAFT A 3 Nominal Free Wing l w o Hinge Margin 1q0 Hinge Margin Fixed Wing l0Oj'o Hinge Margin 20Oj'o Hinge Margin 1/2 Nominal V H 1/4 Nominal V H Cruise -0.0240 f j 0.226

Phugoid Roots -0.0228 f j 0.180 - 0.0239 f j 0.226 -0.0239 f j 0.226 -0.0239 f j 0.226

Short-Period Roots -4.41 f j 2.73 -1.49 f j 5.01 -2.89 f j 6.65 -3.11 f j 6.87 -0.699 f j 5.19 Symmetric Panel Roots " -8.90 f j 12.2 -10.5 f j 16.4 -8.60 f j 12.04 -8.56 f j 12.05

Plunging Roots " -22.3, - 20.5 -23.4, - 20.5 -22.9, - 20.1

-22.7, - 20.2

"

0.0239 0.00942 0.00627 0.00925 0.00963 "2 CT 0.141 0.465 0.485 0.460 0.641 0 . 1 . 58 3.43 3.77 2.48 3.32 0.0937 0.0735 0.0834 0.0952 0.114 0.159 0.115 0.159 0.155 "h 1.23 Approach Phugoid Roots -0.0282 f j 0.275 -0.0307 f j 0.362 -0.0311 f j 0.362 -0.0306 f j 0.362 -0.0309 f j 0.362 Short-Period Roots -3.11 f j 1.41 -2.01 f j 4.32 -2.18 i j 4.49 -1.03 f j 3.28 -0.483 f j 3.41 " Symmetric Panel Roots -6.13 f j 7.80 -7.20 f j 10.4 -5.92 f j 7.69 -5.88 f j 7.69 "

Plunging Roots - 14.67, - 12,55 -15. 5, - 12.6 -14.7. - 12.5 -14.7, - 12.5

0.0185 0.0067 1 0.00451 0.00662 0,00698 "nz (5 0.157 0.466 0.485 0.461 0.645 1.24 2.36 2.59 1.68 2.20 " 0.113 0.129 0.146 0.172 2.28 0.413 0.314 0.415 0.447 TABLE F-V. LONGITUDINAL DATA, AIRCRAFT B 3 Nominal Free Wine 1% Hinge Margin 10% Hinge Margin Fixed Wing l w o Hinge Margin 20% Hinge Margin 1/2Nominal V H 1/4 Nominal V H Cruise

-

Phugoid Roots -0.0134 f j 0.121 -0.0145 f j 0.136 -0.0145 f j 0.136 -0.0145 f j 0.136 -0.0145 f j 0.136 Short-Period Roots -2.24 f j 2.79 -1.43 f j 5.68 -1.51 f j 5.74 -0.716 f j 4.1 -0.323 f j 4.15 -15.4 j 21.1 Symmetric Panel Roots -12.62 f j 15.5 -12.5 f j 15.4 -12.5 f j 15.4

-32.2, - 25.5

Plunging Roots -30.1, - 26. 0 -30.9, - 25.5 -31.0, - 25.4

0.0196 0.00593 0.00361 0.00590 0.00630 O"Z 0.414 0.414 0.126 0.410 0.605 Os (5.

0.779 2.47 2.53 1.75 2.52 " 0.0765 0.07 99 0.103 0.141 0.0444 0.059 0.660 0.059 0.058 Approach Phugoid Roots - 0.0163 f j 0.196 -0.0207 f j 0.233 - 0.0207 f j 0.233 -0.0207 f j 0.233 -0.0208 f j 0.233 Short-Period Roots -1.15 f j 3.79 -5.41 f j 2.72

- 1.73 f j 1.61 -1.08 f j 3.74 -0.242 f j 2.76

" -8.84 f j 8.97 -10.3 f j 12.2 -8.75 f j 8.92 -8.74 f j 8.92 Symmetric Panel Roots "

Plunging Roots -20.1, - 14.7 -20.9, - 14.8 -20.1, - 14.7 -20.1, - 14.7

0.00448 0.00475 0.0148 0.00283 0.00445 4 1 , 0.106 0.411 0.413 0.408 0.601 Os 0.617 1.67 1.73 1.17 1.67 " 0.116 0.123 0.155 0.209

%

an 1.06 0.128 0.102 0.128 0.124

TABLE F-VI. LONGITUDINAL DATA, AIRCRAFT C3 Nominal Free Wing 10% Hinge Margin 1% Hinge Margin Fixed Wing 1 0 % Hinge Margin 2V% Hinge Margin 1/2 Nominal V H 1/4 Nominal VH Phugoid Roots -0.0122 * j 0.101 -0.0130 f j 0.122 -0.0131 f j 0.122 -0.0130 f j 0.122 -0.0131 f j 0.122 Short-Period Roots -2.22 -t j 1.84 -1.46 f j 3.94 -1.54 f j 4.03 -0.739 + j 2.91 -0.344 f j 2.98 Symmetric Panel Roots -7.99 f j 8.09 -9.68 f j 10.98 -7.87 f j 8.02 -7.85 f j 8.02

Plunging Roots -17.8, - 13.2 -18.6, - 13.2 -17.9, - 13.2 - 17.9, - 13.2

0.00754 0.0048 0.00786 0.0211 0.00746 QZ 0.277 0.0817 0.270 0.265 0.382 Q 1.29 0.845 0.557 1.22 1.15 0.0728 0.0799 0.0948 0.121 0 8 0.903 0.0855 0.0534 0.0857 0.0819 Q1 Approach Phugoid Roots -0.0198 f j 0.180 -0.0206 f j 0.247 -0.0208 f j 0.247 -0.0206 f j 0.247 -0.0209 f j 0.248 Short-Period Roots -2. 01 * j 0.764 -1.30 f j 2.38 -1.37 + j 2.48 -0.664 + j 1.83 - 0.318 f j 1.93 " Symmetric Panel Roots -5.04 f j 3.43 -6.10 f j 4.57 -4.90 f j 3.40 -4.88 f j 3.40 "

Plunging Roots -12.8, - 6.5 -13.0, - 6.52 - 6.52 -12.9, - 6.52

-12.9, 0 0.0168 0.00618 0.00394

q 0.0858 0.256 0.273 0.349

0.

0.528 0.835 0.904 0.561 0.687 " 0.113 0.131 0.143 0.163

9 1 1 . 50 0.272 0.179 0.273 0.253

TABLE F-VII. LATERAL-DIRECTIONAL DATA, AIRCRAFT A Cruise Dutch-Roll Roots .O. 0763 f j 3.58 -0.0759 f j 3.58 -0.808 f j 3.69 -0.812 f j 3.70 -0.671 f j 3.56 -0.664 f j 3.57 -0.000145 -0.00116

Spiral Root 0.0246 0.0181 - 0.00117 0.0224

Roll-Mode Root -6.58 - 0.639 -6.89 -0.717 - 5.38 - 0.463

" -- "

Asymmetric Panel Root -10.9 f j 14.8 -10.7 f j 16.2 -10.7 f j 12.8 0.365 0.297 0.366 0.315 0.375 0.285 0.261 0.262 0.260 0.262 0.266 0.269 0.403 0.303 0.410 0.321 0.425 0.317 0 6 0.470 0.478 0.475 0.481 0.491 0; 502 UJI 1.140 1.000 1.140 1.050 1.170 0.963 ODY 0.00724 0.00578 0.00722 0.00598 0.00556 0.00735 unY Approach Dutch- Roll Roots 0.523 f j 2.44 -0.437 f j 2.44 -0.554 f j 2.52 - 0.462 f j 2.52 - 0.336 f j 2.411 0.446 f j 2.43 Spiral Root 0.0335 0.184 0.0322 0.172 0.0359 0.171

Roll-Mode Root -4.68 - 0.783 -4.90 - 0.860 - 0.662

-3.87 " " " Asymmetric Panel Root -7.55 f j 10.7 -7.56 f j 8.41 -7.65 f j 9.77 0.412 0.270 0.413 0.286 0.271 0.422 0.381 0.382 0.316 0.330 0.391 0.344 0 1 c / 0.413 0.335 0.421 0.363 0.436 0.368 0.482 0.511 0.487 0.520 0.553 0.506 O$ 1.30 0.766 1.30 0.812 1.32 0.729 U ODY 0.00748 0.00470 0.00751 0.00484 0.00759 0.00445 n V TABLE F-VIII. LATERAL-DIRECTIONAL DATA, AIRCRAFT B

- B 1 B2 *3

Fixed Free Fixed Fixed Free Cruise -0.542 f j 3.44 -0. 537 i j 3.45 -0.563 i j 3.51 -0.564 i j 3.52 -0.583 f j 3.56 -0.581 f j 3.57 Dutch-Roll Roots

- 0.000701 0.0112 -0.000709 0.0146

Spiral Root -0.000058 0.0155

- 0.554 - 5.41 - 0.602 - 5.40 -0.473

Roll-Mode Root -5.35 ” -15.6 f j 20.1 ” ” Panel Root -14.9 f j 22.0 -12.6 f j 17.2 Asymmetric 0.228 0.290 0.23 9 0.291 0.230 0.282 O4I 0.185 0.186 0.184 0.186 0.186 0.187 0.346 0.284 0.349 0.279 0.341 0.270 0.406 0.410 0.404 0.411 0.409 0.405

“5/

0.885 0.761 0.884 0.755 0.785 0.889

0 DY 0.00575 0.00435

0.00577 0.00440 0.00449 0.00573 “Y Approach

- 0.412 i j 2.40 f j 2.40 - .O. 322 f j 2.28 -0.261 f j 2.27

Dutch-Roll Roots -0.398 f j 2.36-0.331 i j 2.35 -0.342 0.0210 0.122 0.0234 0.122 Spiral Root 0.0219 0.132

Roll-Mode Root - 0.675 -4.21 - 0.727 -3.18 -0.545

-4.15

” - 11.96 * j 12.81 ” --

-11.5 f j 14.2 -11.8 f j 10.2 Asymmetric Panel Root 0.346 0.261 0.355 0.250 0.345 0.248 OO 0.282 0.27 0 0.294 0.288 0.283 0.267 u$ 0.383 0.363 0.316 0.369 0.343 0.354

%

0.419 0.417 0.451 0.456 0.439 0.488 1. 07 1. 08 1.07 0.701 0.733 0.649 ODY 0 0.00382 0.00629 0.00413 0.00623 0.00424 0.00637 n,, TABLE F-IX. LATERAL-DIRECTIONAL DATA, AIRCRAFT C C1 c2 c3 Fixed Free Fixed Free Fixed Free Cruise Dutch-Roll Roots -0.458 f j 2.28 - 0.458 f j 2.28 - 0.477 f j 2.33 -0.483 f j 2.33 -0.497 f j 2.38 . -0.506 f j 2.38

Spiral Root -0.0000788 0.0137 - 0.000629 0.010 -0.000636 0.0130

Roll-Mode Root - 5.73 - 0.497 -6.00 - 0.567 -6.43 -0.443

" "

Asymmetric Panel Root -11.2 f j 11.8 -11.0 f j 13.05 -- .9.95 f j 10.5

0.239 0.198 0.240 0.210 0.242 0.207 "$ 0.169 0.170 0.168 0.169 0.167 0.169 "J, 0.246 0.208 0.251 0.224 0.256 0.225

"4

0.274 0.27 2 0.275 0.27 5 0.27 5 0.277

"4

0.738 0.621 0.739 0.654 0.742 0.643

" DY

"

0.00404 0.00528 0.00405 0.00530 0.00411 0.00528 nY Approach

-0.350 f j 1.60 -

Dutch-Roll Roots -0.414 f j 1.54 -0.340 f j 1.57 -0.430 f j 1.58 ,0.346 f j 1.52 - 0.273 f j 1.53 0.0230 0.131 0.0221 0.0247 Spiral Root 0.123 0.123

- 5.41

Roll-Mode Root - 5.16 - 0.600 -0.676 -4.36 - 0.506

" " " Asymmetric Panel Root -10.1 f j 7.01 -9.93 f j 8.02 -10.4 f j 5.04 0.295 0.250 0.307 0.250 0.293 0.235

"4

0.272 0.255 0.271 0.258 0.281 0.277 "J, 0.255 0.239 0.260 0.260 0.277 0.267

"4

0.308 0.294 0.284 0.302 0.286 0.326

"9

0.937 0.637 0.939 0.664 0.961 0.627

" Dv

0.00572 0.00424 0.00572 0.00430 0.00591 0.00415

"i