Integrated Aerodynamic and Control System Design of Oblique Wing Aircraft

Chapter 1

45 __

Introduction

For reduced operating cost, an efficient high speed aircraft design must achieve a high lift to drag ratio at transonic and possibly supersonic speeds. In 1952 Dr. R. T. Jones proved that for any flight Math number, minimum drag at a fixed lift is achieved by an elliptic _iii,_ ¸_/•_ wing planform with an appropriate oblique sweep angle. Since then, wind tunnel tests and i _ _ !._i ii numerical flow models have confirmed that the compressibility drag of oblique wing air- • •L_•_ • craft is lower than similar symmetrical sweep designs (Fig. 1.1). The advantages of oblique wing sweep extend beyond drag reduction. High speed aircraft designs must achieve a compromise between the conflicting demands of performance in the subsonic flight regime (i.e. takeoff, climb to cruise, and landing) and reduced compressibility drag at • _• .i__ transonic and supersonic speeds. Configurations which employ variable wing sweep often show improved performance when the subsonic and supersonic performance requirements are stringent. The oblique wing configuration is ideally suited for variable sweep because only a single wing pivot loaded primarily in tension is required (Fig. 1.2), whereas sym- metric variable sweep configurations require two pivots, each loaded in bending and tor- sion by its respective wing panel. As a result, symmetric variable sweep designs suffer in- creased complexity and structural weight when compared with oblique designs. Kroo [Ref. 1] and Nelms [Ref. 2] describe in greater detail these and other advantages of the oblique wing configuration and give a more complete historical description of the concept's evolution. A summary of the key advantages are listed below.

Advantages of Oblique Wing Configurations 1) Reduced transonic and supersonic drag 2) Reduced structural weight for a variable sweep configuration 3) Reduced aerodynamic center shift as wing is swept 4) Lower sonic boom strength due to reduced •wave drag 5) Low storage space requirements if wing is overswept for storage k Despite the advantages of reduced compressibility drag and variable wing sweep with low ,i _ structural weight, no oblique wing aircraft has achieved operational status. Several un- manned demonstrators have been flown [Ref. 2] to prove the feasibility of the asymmetric i i_ _I_ _ configuration. A subsonic manned demonstrator, the AD-1, was built and flown by NASA i i'_ in 1979 to evaluate the handling qualities with oblique wing sweep. Flight tests of radio controlled models and remotely piloted vehicles (RPV's) (Fig. 1.5) indicated that aircraft with oblique wing sweep as high as 65 degrees could be flown without a stability augmen- tation system. Subsequent flight testing of the AD-1 proved that, although the AD-1 could be flown safely, the handling qualities deteriorated rapidly as oblique sweep and flight dy- namic pressure increased [Ref. 3]. Aerodynamic and inertia couplings inherent to the de- ,! , i_ _ sign's asymmetry cause the handling qualities to degrade. A feasible oblique wing design would therefore require a stability augmentation system (SAS) to restore acceptable han- dling qualities over the flight envelope. From 1984 - 1988 NASA and Rockwell studied the possibility of converting NASA's F-8 fly-by-wire research aircraft into a supersonic oblique wing demonstrator which would investigate handling qualities with an onboard SAS.

The F-8 Oblique Wing Research Aircraft (OWRA) was designed to operate at Mach num- bers as high as 1.6 and oblique sweep angles up to 65 degrees. Although the OWRA has not yet flown, estimates of its closed loop handling qualities were obtained from piloted flight simulations conducted in NASA's Vertical Motion Simulator (VMS) [Ref. 4]. A SAS designed by Honeywell [Ref. 5] using a 'loop shaping' technique was implemented in the simulation trials. Pilots flew the aircraft through various maneuvers including abrupt pitch-ups and wind up turns and issued Cooper Harper ratings ranging from 3 (acceptable) to 7 (unacceptable) for the tasks performed. Pilots were most critical of the severe lateral accelerations during pitch maneuvers and the asymmetry in roll tendencies during turns. In general, handling quality ratings decreased with increasing oblique sweep and flight dy- namic pressure. The studies showed that even with a SAS onboard the OWRA lacked suf- t.

flcient controllability to achieve acceptable handling qualities. The OWRA configuration must be changed to improve its controllability, but the impact of aerodynamic and inertia coupling on handling qualities is non-intuitive and therefore complicates the redesign proce- i • _ •i dure.

40 ,- I1 A=O" L _ I0:! ELLIPTICAL PLANFORM

! iiii • \ _45"

L/D 2O IO • i_ ,i _ :k ¸¸ .G .8 LO 1.2 1.4 MACH NUMBER Figure 1.1 Maximum lift to drag ratio, oblique wing vs. symmetric sweep AEROOYNAMIC ADVANTAGE STRUCTURAL ADVANTAGE OIK_E SVmBP SYliE_IC SWE_ ...... LSvu --__ • VOtUME 06TI_OUTIO_ (AREA mJI.L r) i LCIK ....... WAVE 04IAG COiM_DNENTS 0- Y_: o- _ ._-j_ t_ L2 • SWEI9 EFFECT (AT ROOT SEC11OWt • mOT TOAOLq Ak_ IIIENO_WG tOA_5 AV(N_(9 • mmokqo WING TO_XIE tO_ AVOIDED • _ PIVOT • ACI1JATOR LOAi_i AVOIDED

7/

• AC IUF't' AVOIDED Figure 1.2 Summary of oblique wing advantages The VMS studies of the F-80WRA did not account for the influences of elastic wing def- ormation on the aircraft's dynamic response. Flight tests of the AD- 1 subsonic manned demonstrator showed that wing flexibility can significantly influence the aerodynamic coupling and in some cases reverse the coupling present in the rigid aircraft [Ref. 6]. In addition to influencing handling qualities, the combination of wing elasticity and planform asymmetries leads to unusual modes of aeroelastic instability. Jones and Nisbet [Ref. 7] studied a simple mathematical model for an elastic oblique wing with infinite roll degree-of- freedom. Previous to their study, designers felt that oblique wing aeroelastic stability would be limited by the divergence speed of the forward swept wing panel. Jones and Nisbet showed that the aeroelastic instability was not a divergence, but a flutter mode con- _ •i ¸ !

sisting of combined wing bending and fuselage rolling motion. Most significantly, this in- stability occurs at a dynamic pressure considerably higher than the clamped divergence dy- namic pressure of the forward swept wing panel.

The success of the oblique wing concept depends upon improving the aircraft's closed loop handling qualities without increasing its drag or structural weight significantly. This thesis will show how this can be achieved and will also introduce a new approach to the integrat- ed design of aircraft and their control system. Improving the handling qualities of oblique wing aircraft is a highly integrated design problem because the disciplines of aerodynamics, structures, and controls must be considered simultaneously to evaluate performance. A de- sign synthesis which accounts for these effects and systematically alters the configuration of a proposed oblique wing design to improve its handling qualities is presented in this work.

The integrated design of aircraft and their control systems has been previously investigated by Zeiler and Weisshaar [Ref. 8] and Sakawi et. al. [Ref. 9]. In both of these works han- dling qualities are measured in terms of a dynamic cost function identical to that used in op- timal control regulator theory (LQR design). The design variables which define the air 7 craft's configuration are treated as additional degrees of freedom with which to further improve the aircraft's handling qualities. Sakawi et al. optimized the wing and tail geome- • •(ji_" (' try of an actively-controlled aircraft excited by gust disturbances. The synthesis assumed a ii _i _7 full-state feedback control architecture and used a random search procedure to solve for the i _i_ i i _ , "i,I,: : _ • _ .

f .to_.nsuomop _u_ onb.qqo o.mos.t_ns Va_O 8-:I 17"I _n_!d .to_nsuomop _u.t,_, onb!Iqo p_;}ds ,_oI I-(IV £'I _n_!:I L • "i _ , S JS ssSS , i _ 7.6m

, ]

'._ 6.8m " Figure 1.5 Oblique wing RPV 3-view global minimum. Zeiler and Weisshaar describe the integrated design of a four degree-of- freedom aeroservoelastic system. The design variables consisted of the elastic axis location i!!i: and the full state feedback control gains. A multi-level linear decomposition scheme was used to solve for the optimal system variables.

In the work presented here, an integrated synthesis procedure is extended to handle any lin- ear control system architecture including reduced-order controllers and passive (no control- ler) designs. The procedure is used to optimize the system commanded response as well as its unforced response. Multidisciplinary performance indices are considered, illustrating the trade-offs between handling qualities and other measures of performance. Chapter 4 in- troduces the integrated design procedure and demonstrates its application in two example syntheses: the design of a wing-tail system for minimum trimmed drag with handling qual- ity constraints and the design of a tailless aircraft for minimum weight with aeroelastic c_on- straints. The results show how that this method can solve a variety of highly integrated i ¸ • _ • _. problems and achieve performance that sequential design procedures would have over- looked.

/iiI • Before an integrated design procedure can bc implemented, all of the analyses required to calculate performance (giventhedesignvariables) must bc available. Aerodynamic, con- i' ' _i, ,i _ trois, and acroclastic analysesused tomodel the F-80WRA arc describedin the initial chaptersof thisthe, sis.The aerodynamic characteristics of obliquewing aircraft arc dis- cussed in chapter 2 and a vortexlattice model ispresentedwhich estimates theforces and moments acting on the aircraft with a minimum computation time. Chapter 3 describesthe fuU-statc feedback and reduced-order controller architectures consideredfor the SAS. The sixdegree-of-freedom equations of motion fora frccflyingaircraft with an elastic wing arc derived to model wing flexibility effects in chapter 6. All of these analysesarc implement- ed in the integrated synthesisdesign.

The integrated synthesis method is then applied to the design of the F-80WRA in chapter 5. Five configuration variables which define the placement of the wing relative to the fuse- • _,j lage arc included along with the controller gains to further decouplc the aircraft's dynamic response. The changes in the configuration arc limited to ensure that the aircraft's drag and weight is not affected and that aim in 1-g level flight may bc achieved with small control deflections. Initially, OWRA designs are synthesized assuming a rigid aircraft with a full- state feedback explicit model following SAS. Multiple flight conditions, reduced-order- controllers, and wing elasticity arc included in the integrated synthesis procedure and the OWRA configuration is re-synthesized. In all cases, the resulting optimal integrated de- signs show improved handling qualities when compared with the original OWRA configu- ration.

.ii !

Chapter 2

!i.....

Oblique Wing Aerodynamics

• "L,_I ¸¸7_¸ : : The aerodynamic coupling experienced by oblique wing aircraft at high sweep angles iliil, _i_ _i¢i (greater than 30 degrees) is a key factor in the degradation of handling qualities. A method of accurately estimating the aerodynamic loads with low computation cost is required by the '_.L, integrated design synthesis so that the influence of configuration geometry changes on han- ding qualifies can be calculated. In this chapter the origin of the aerodynamic coupling are studied using wind tunnel data and numerical modelling results. A vortex lattice method which calculates the forces and moments for the complete aircraft with sufficient accuracy and efficiency are discussed. The results from this code are compared with other more elaborate numerical methods and wind tunnel results.

When R. T. Jones initially proposed the concept of oblique wing sweep [Ref. 10], The re- duction of supersonic wave drag was the perceived major advantage of the configuration.

A simplified expression for the supersonic drag of an aircraft is given by Jones: L 2 ( M 2- 1) L 2 128 q Vol 2 (2.1) Drag = q S Cop + t- + 2 4 q x b 2 2 x q X 1 x X 2 where (Vol) is the aircraft volume and Xl, X 2 are averaged lengths computed from super- sonic area rules. If the wing lies near the Mach cone, Xland X 2 are essentially equal to the vehicle length, which has a powerful effect on the lift and volume dependent wave drag (the third and fourth terms in equation 2.1, respectively). For a fixed aerodynamic span (b) the total drag is minimized by a configuration with the greatest length. From figure 1.2 it is obvious that for equal spans and sweeps, the oblique sweep configuration has nearly twice the length of the symmetric sweep planform, and therefore lower wave drag. When an oblique wing with a fuselage is considered, the distribution of wing volume along the fuse- i? _, i lage length provides a natural area ruling effect, which creates a favorable volume distribu- tion for low wave drag without altering the fuselage cross sectional area near the wing root /i _ ii: (i.e. 'coke bottle fuselage').

2.1 Wind Tunnel Test Results

The advantages of reduced wave drag are to some extent countered by the disadvantages of aerodynamic coupling inherent to the asymmetric planform. Extensive wind tunnel tests of the F-80WRA configuration (Fig. 2.1-2.2) have been conducted in the NASA Ames tran- sonic wind tunnel [Ref. 11-14] to help analyze the magnitude and nature of the coupling terms. These tests investigated several wing designs in combination with the F-8 fuselage ,!

and empennage. Table 2.1 summarizes the configurations tested and figures 2.1-2.2 dis- play isometric drawings of the different versions.

Airfoil Skew Pivot Planform Configuration Wing Area L.E. Sweep no 1 250 (sq. ft.) SC(2)-0714 sym.

2 250 OWT0 14/12 asym. no 3 300 OW70 14/12 4 ° sym. yes 4 300 SC(2)-0714 0 ° sym. yes Table 2.1 F-80WRA wind tunnel test configurations ;.Y :: Initially, a design with small wing area (250 sq. ft.) and supercritical airfoil was consid- ered (configuration 1). Studies by Kroo and Kennelly showed that an asymmetric plan- form, and improved airfoil sections resulted in reduced drag when swept obliquely (config- uration 2). Wing planform asymmetry, however, complicates the wing construction and a version with a symmetric wing planform was designed as a simpler alternative. Configura- tion 3 is the updated version of Kroo and KenneUy's design which incorporates the sym- metric planform and has 300 square feet of wing area for an expanded flight envelope.

Configuration 4 has the same wing area as configuration 3, but the planform has reduced leading edge sweep and different supercritical airfoil sections are used. Configurations 3 and 4 employ a skewed wing pivot which allows the wing to bank relative to the fuselage e..

f, .

as the wing is swept obliquely. Wing bank reduces the sideforce produced by leading edge suction at large oblique sweep angles.

24 i73 J 0.40 Figure 2.1 F-80WRA wind tunnel model (configuration 3 shown)

!

Configuration # 1&4 2 Figure 2.2 F-80WRA wing planforms tested • i ¸ Results from the wind tunnel tests of OWRA configuration 3 at Mach 0.8 and 45 degrees oblique sweep are shown in figures 2.3-2.8. This data clearly shows the nonzero values of the coupling terms Cy, C 1, and C N and their nonlinear variation with changing angle of at- tack.

Lift Coefficient vs Angle of Attack 45 deg. sweep, M=0.8 1.2 _'_ i _ 1.0 .8 .6 C L .4 .2 .0 -.2 -5. 0. 5. 10. 15.

Alpha (deg) 2.3 lift coefficient vs. angle of attack Figure Drag Coefficient vs Lift Coefficient 45 deg. sweep, M=O.8 .25 .20 .15 C L

i!!!!!!!!!!!!!!!!!

.10 .05 i ¸ , _.... i....... _.... :i....... :....... '............ _ ....... ....... ........

t | | t I ! | | | .00 1.0 .0 •1 .2 .3 .4 .5 .6 .7 .8 .9 C D Figure 2.4 lift coefficient vs. drag coefficient Sideforce Coefficient vs Lift Coefficient 45 deg. sweep, M=0.8 .040 .035 ....................... "........ :........ : ........ : ...................................

i : : oooJolol,_ollleee .030 ........................................

.025 Cy .020 .015 .010 .005 .000 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

cL

Figure 2.5 sideforce coefficient vs. lift coefficient Rolling Moment Coefficient vs Lift Coefficient 45 deg. sweep, M=0.8 .03 ............... _ ........ 7 ........ • ........ r ........ _ ........ _ ........ _ ................

.02 iiolj*o_*It*_*_I_ojtIii_ijoooo_o_°_*lIer_*_*_i-_°_*e*_*Ii_°_it_ .01 Cl .00 -.01 -.02 _.o.,oo.....o.oo..oo.oo.o..oooo,o._o**oo.o.ooo.oooo.ooo.oooo.oooo.,. .......... , ...... .

| i i i i i I a l 1.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Figure 2.6 roll moment coefficient vs. lift coefficient Pitching Moment Coefficient vs Lift Coefficient 45 deg. sweep, M--0.8 .10 .05 0.00 L_ -.05 C M -.10 -.15 -.20 • °,o,.,° °°o.,,., .,.°..° °o°°°°°° °°°,.°o° o°°°°°°° ,.,,°°°°.°°°°°°°°°°°°°*°°°°°° ..... ° -.25 -.30 .0 .1 ,2 .3 .4 .5 .6 .7 .8 .9 1.0 C L Figure 2.7 pitch moment coefficient vs. lift coefficient Yawing Moment Coefficient vs Lift Coefficient i r.

45 deg sweep, M=0.8 .015 ...°.° * °**..*.°°. o°° 1.°****,.......***** °...o*,o°°oo°o°'t. ° °°°. °.*° ° ° °°°° ° .'o • ° °°°*°'. ° °° ° ° ° .010 ! ! i i i i ! i ! ,

i i i i i i i i

%

°°°o,.**_ °,°°oooo°,.,,,°°°r°°***.o,_,.,oo° °°,°,°°°o*°'to**°° °°*',, °° .... .'.°.°°° °° .... ,° .005 • ° .000 -.005 1.0 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 C L Figure 2.8 yaw moment coefficient vs. lift coefficient , , = ? ,, The influence of wing sweep on coupling is more clearly shown in figures 2.9 - 2.14 where Cy, C1, and CN are plotted for sweep angles between 0 and 65 degrees. These re- L • sults show that significant coupling occurs for oblique wing sweep greater than 30 degrees.

i ¸ The F-80WRA is designed to operate at Mach numbers up to 1.6, where the optimal wing sweep for low drag is 65 degrees. At this high speed flight condition the sweep is large enough to insure subsonic leading edges and a subsonic Mach number (0.7) for the flow perpendicular to the quarter chord (i.e. normal Mach number). Because the flow normal to the wing is subsonic and the wing lies inside the Mach cone, the effect of Mach number on the aerodynamic loads is small (except of course for its effect on drag). Wind tunnel data shown in figures 2.15 - 2.20 verifies this claim (at least at low angles of attack), where CL, Cy, C M, C 1, and CN versus at are plotted for Mach numbers up to 1.6 at 65 degrees • ,_ i_!_ sweep. These results justify the use of a subsonic aerodynamics model for numerical esti- mation of the airloads.

Lift Coefficient vs Angle of Attack M--0.6 1.4 ooo,,,,,°oooo°,,°o°o, ,, 1.2 1.0 .8 .6

%

.4 iiiiiiii .............. '.........................................

.2 i ..... • i ___ ...... [] 30 _g. sweep 0.0 _" __eg. sweep -.2 ................... i ..................... i.................. v 65 deg. sweep -.4 -5. , 5. 10. 15.

:f L': Alpha (deg) " 'ii Figure 2.9 lift coefficient vs. angle of attack '!

/ "// _ ii '_ Drag Coefficient vs Lift Coefficient M=0.6 .30 • • i . ° . _.

.25 [A 0deg. sweep J i i :: :: '-_ ! j J a 30 deg. sweep 1:: i i ..... i..._...! ....... i...../.

.20 ........ 10 45 deg. sweep J i ....... i........ i'" i/" " _...... i_ .15

cL

.10 .05 .00 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 .0 C D Figure 2.10 lift coefficient vs. drag coefficient Sideforce Coefficient vs Lift Coefficient M=0.6 .08 ...... 7. ................ -"........ _" ........ "_ .......

.06 : • . .

.04 .02 .- .- .- • .

0.00 °°o°°**°°°°°°o°o°°.°°,°°......°.,°°., ,°°°°°°° °°.° ....

Cy : • i • -.02 -.04 /X 0 deg. sweep -.06 a 30 deg. sweep • 45 deg. sweep -.08

v 65 deg. sweep Ii iiiiii!!i!

-.10 -.12 , f , .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 i !i • • , !

C L i¸ , Figure 2.11 sideforce coefficient vs. lift coefficient Rolling Moment Coefficient vs Lift Coefficient M--O.6 .02 . . . : .01 .00

i!i!!!!!!!!!!!! i i ........ ........ ..... ....... iiii!!!i if!!!

q

-.01 i [] 30 deg. sweep i :: i i \'_ -.02 ........ ";....... • 45 deg. sweep ........ • ........ :........ !........ : ......

v 65 deg. sweep I° -" -.03 .0 .1 .2 .3 .4 .5 .6 .7 .8 ,9 1.0 C L Figure 2.12 roll moment coefficient vs. lift coefficient Pitching Moment Coefficient vs Lift Coefficient M=0.6 .2 I i i i i ::A 0deg. sweep I:: i i i i i i [] 30 deg. sweep _ .... :..

.1 "'- ..... i ........ _....... _......... :: ........ ::........ :: * 45deg. sweep[" _i ........ i........ i........ v 6:degsw_ep! ........

0.0 -.1 C M

: i

-.2 °o..°...ao.o.....,.o ....o.***.....o...°o.°o.._.....***j........_.. o.......... ..... • ........

-.3 -.4 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 .0

cL

Figure 2.13 pitch moment coefficient vs. lift coefficient Yawing Moment Coefficient vs Lift Coefficient M=0.6

o15 I ,, Od g. sweep I

[ ta 30 deg. sweepl:: I o 45 deg. sweep[i . ./.. , .010 ....... :i ........ v 65 deg . sweep ]!? ....... :i........ i" ........ : ....... i"...... ; CN .005 .000 -.005 •0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

eL

Figure 2.14 yaw moment coefficient vs. lift coefficient Lift Coefficient vs Angle of Attack 65 deg. sweep 1.0 A M=.6 [] M=.8 _.,,_.

.8 ................. • M-1.1 .......... ].............. ._,_r ........

v M=l.2 i : .6

CL .4

.2 .0 -5. 0. 5. 10. 15. 20.

Alpha (deg) Figure 2.15 lift coefficient vs. angle of attack Drag Coefficient vs Lift Coefficient 65 deg. sweep .35 i ¸ .30 .............. • M=-6 I _ I : ..... ............... i........... i...........

.25 ............... et M=.8 [ ..... " ............. :........... _ .........

# M=I.1 I i ! /'/_" .15 ...............................

: iiiiiiii i!i!!!i

.10 .00 : : -.2 .0 .2 .4 .6 .8 1.0

%

Figure 2.16 drag coefficient vs. lift coefficient Sideforee Coefficient vs Lift Coefficient 65 deg sweep .02 0.00

..... i........ "' "'"' ........ i.............. i.............. i.......... "

-.02 .............. : ............... -. ............. .: ..... _.............. :, ..............

-.04 Cy A M=.6 :: i ........ .....

-.06 I * M=I.1 I i i_."_'X_'_._ -.08 -.10 -.12 1.0 -.2 .0 .2 .4 .6 .8 sideforce coefficient vs. lift coefficient Figure 2.17 Rolling Moment Coefficient vs Lift Coefficient 65 deg sweep .005 .000 • "f 'f f: -.005 |

q

A M=.6 -.010 rl M=.8 Q M=I.1 v M=l.2 -.015 • , ( -.020 -.2 .0 .2 .4 .6 .8 1.0 Figure 2.18 roll moment coefficient vs. lift coefficient Pitching Moment Coefficient vs Lift Coefficient 65 deg. sweep .1"2 _...

i i [ _ M--.6

I _ i i I u M=.8

0.0 l" .......... _'_ .............. ":..... I * M=I.1 .............

C M -.1 -.2 -.3 i,,,,.

-.4 -.2 .0 .2 .4 .6 .8 1.0

c,_

Figure 2.19 pitch moment coefficient vs. lift coefficient Coefficient vs Lift Coefficient Yawing Moment 65 deg. sweep .O20 _ _ _i _ [] M=.8 .015 O M=I.1 .................................

A M=.6 ] v M=l.2 *°,.°.

.010 .005

%

.000 -.005 -.010 -.2 .0 .2 .4 .6 .8 .0

cL

Figure 2.20 yaw moment coefficient vs. lift coefficient Over most of the operational flight regime the angle of attack is small enough that the flow remains attached to the wing. At higher angles of attack (and oblique sweep greater than 30 degrees) a complicated vortical flow structure occurs as the flow separates from the wing. Figure 2.21 is a photo of the F-80WRA in a water tunnel flow visualization test and shows the separated flowfield at 65 degrees sweep and 10 degrees angle of attack. Ini- tially, flow separation occurs on the aft swept wing tip due to increased boundary layer thickness and higher local lift coefficient (both caused by the oblique sweep). At a slightly greater or, a separated vortical flow forms on the leading edge of the forward swept wing panel until it hits the fuselage which acts as a flow fence and forces the vortex into the wake. Flow visualization studies of the wing alone (Fig. 2.22) show that the vortical flow exits near the region of aft tip flow separation when the fuselage is not present.

e..

_ ! ill:: J 2O 'i OR_G_._L PAO_ IS OF rOOR QUAbn'Y ,, i:I Figure 2.21 water tunnel flow visualization, F-8 OWRA, 65 ° sweep ........ _ _ , _:I i '_i,ii ¸' Figure 2.22 water tunnel flow visualization, F-80WRA, 65 ° sweep, wing only The leading edge vortex produces nonlinear lift and reduces leading edge suction by the :i same mechanisms that govern the flow over delta wings at high angles of attack. As a re- suit, the lift coefficient at high angles of attack increases at a rate greater than linear theory / Jf predicts, because the nonlinear lift increment from the leading edge vortex exceeds the lift lost from the aft tip flow separation. Figure 2.23 shows this effect for the OWRA with 65 degree wing sweep. Flow separation occurs at 7 degrees angle of attack and the lift coeffi- cient continues to increase (nonlinearly) in the separated flow region.

The influence of flow separation on the aerodynamic coupling terms is to contribute addi- tional nonlinear behavior which is governed by mechanisms different from those in the at- tached flow regime. This study does not require the ability to predict aerodynamic loads in the separated flow regime because we seek to redesign the aircraft for improved handling qualities in the maneuver and cruise flight conditions where the flow is attached and most of the flying time is spent. Of course, a feasible oblique wing aircraft must remain stable and controllable when flow separation occurs and the flight tests of the AD-1 have shown that this safety issue can easily be achieved in an oblique wing design, although handling qualities may degrade rapidly in this regime. Air combat missions may require extensive operation in the separated flow regime and oblique wing aircraft are less suited for this role due to degraded handling qualities in this flight condition, but for other oblique wing mis- sions the key issue is improved handling in the lower angle of attack flight regimes.

Lift Coefficient vs Angle of Attack 65 deg. sweep, M=0.6 1.0 .8 A Wind Tunnel Data '1 ]_'_

.6o Jiiiii iiiiiiii

.4.............. .... ....

_:::i_ iiii -.2 -5. O. 5. 10. 15. 20.

Alpha (deg) Figure 2.23 nonlinear lift due to leading edge vortex The linear lift shown in figure 2.23 is given by: _:,, ')! { (2.2)

CL = CLo + CLa tx

and AR cos (A) (2.3) CLa = AR + 2 cos (A) 2 An estimate of the angle of attack at which flow separation first occurs can be obtained from simple sweep theory. If the average wing lift coefficient is CL then the average lift coefficient for airfoil sections taken normal to the quarter chord (CL_I_): >!il _ '/: ?

1 2 1 2 (2.4) 2 PVo CLSref = 2 PVN CLL Sref :i i: (2.5) V N = V o cos ( A ) therefore:

C

L (2.6) C = _ - LL-- 2 cos ( A ) !i_ii! ii

C/4

VN # V o A Top Figure 2.24 normal and streamwise flow directions ,,ii _i:_, An estimation of the angle of attack at which local flow separation occurs can be made by comparing the section normal C 1with Clmax for the two dimensional airfoil section normal to the flow. The effect of wing sweep on the C L where separation first occurs is shown in figure 2.25 (for a 2-D section Clmax of 1.0). This plot shows that at high sweep angles the range of C L over which the flow is attached is small. For example, at 65 degrees of sweep, flow separation occurs at C L of 0.18. This simplified study also shows that the de- sign of airfoil sections for oblique wing aircraft requires high lift airfoil sections that have low transonic drag [Ref. 1].

Maximum Lift Coefficient vs Wing Sweep 1.0 .9 .8 .7 .6 _wn CLma x .5 .4 .3

c -10

.2 .1 .0 0. 10. 20. 30. 40. 50. 60. 70.

Sweep (deg) Figure 2.25 maximum lift coefficient for attached flow vs. wing sweep In the attached flow regime the aerodynamic coupling terms display nonlinear behavior (see Fig. 2.5 - 2.6) which might lead one to conclude that a "linearized" aerodynamic theory will not capture these effects. This conclusion is incorrect because linearized theory can ac- count for induced velocities and nonplanar geometry influences which can produce nonline- ar variations in the aerodynamic loads. This section will explain the physical phenomenon contributing to each of the coupling terms and show how the vortex lattice method (present- ed in the next section) predicts the coupling terms and their nonlinear trends.

,i _ The sideforce (Cy), as angle of attack is varied, is produced by the low pressure region acting along the leading edge of the wing. It is possible to derive a simple expression for Cy based on the induced drag of the wing and the suction force CS which acts normal to the leading edge.

V o / . • "> !

I

' A

I V o A

C/4

Top Side (section A-A) Figure 2.26 sideforce due to leading edge suction For small o_ : ili!> (2.7) CL_- C Z (2.8)

%---Cx+q

Assuming the drag consists only of induced drag:

c L

(2.9) CD= CDi- _ ARe Solving for C X •

%

(2.10) - +%(X CX _ ARe The suction and sideforce are related to C X by: CX (2.11) Cs - cos( A ) (2.12) Cy =- CX tan( A ) !_!i!i Therefore, the suction force is equal to:

% 1

(2.13) Cs=(- +CLa)

ARe cos( A ) and the sideforce becomes: (2.14) Cy= - tan( A ) Substituting % = C ( o_ + {x ) La o 2 2 C L C L (2.15) Cy=tan(A)[ _ARe C + CLao] l_xt Equation 2.15 shows that the sideforce is quadratic in C L, based solely upon leading edge suction and induced drag considerations. For a symmetrically swept wing the sideforce produced by the leading edge suction of each panel cancels leaving only the drag force act- ing in the (-x) direction, whereas oblique wings experience a nonlinear sideforce due to their wing planform asymmetry. It is possible to orient an oblique wing, such that for h given sweep angle and specified lift coefficient, the sideforce at a single flight condition is zero. This is accomplished by requiring that the relative wind axis and the (x) body axis be parallel at the trim condition and that the wing lie in a plane parallel to the x-y plane. If the wing is then rotated about its quarter chord (or long axis) the desired lift can be achieved with zero sideforce. As angle of attack is changed from this trim setting (by gust distur- bances or maneuvering) the sideforce will not necessarily remain zero and large lateral ac- celerations may occur. At higher angles of attack, the suction force decreases as flow sep- aration occurs, and the magnitude of Cy will also decrease. A comparison between the simplified theory (Eqn. 2.15) and the experimental value for the sideforce as angle of attack i_ :i_i_i_,:/_i varies is shown in figure 2.27.

Sideforce vs Lift Coefficient 45 ° sweep .05 i-_ [Aft wing separation !

=----__ / Iand leading edge vortex 0.00 -.05 -.15 -.20 13 Wiffd Tunnel Results I A Simplified Theory Results I -.25 0. 2. 4. 6. 8. 10. 12.

Alpha (deg) Figure 2.27 sideforce comparison, theory and wind tunnel data i _ , , The coupling between rolling moment and angle of attack is caused by induced velocity and leading edge suction effects. When a planar wing operates with oblique sweep greater than 30 degrees the velocities induced on the wing by the wake cause the aft swept panel to in- crease in loading more quickly than the forward one for increased angle of attack. For de- signs with counter clockwise oblique sweep (right wing forward), this induced loading produces a positive (right) roll moment as angle of attack increases. Figure 2.28 shows the _ variation in span loading predicted by a vortex lattice model of a 45 degree obliquely swept wing at two angles of attack. The difference between the load distributions clearly shows the increased loading of the aft tip which produces a linear coupling between angle of attack .... '_H I i i ....

and rolling moment.

:i ' (/ i ; iI_.

The wing's leading edge suction produces a rolling moment when the aircraft's center of _ .'ILl. _ gravity does not lie in the plane of the wing. For the case shown in figure 2.29 the side- i? : :, force component acting above the center of gravity results in a nonlinear variation in rolling ii li ¸:/ moment which opposes moments produced by induced effects. At higher angles of attack the nonlinear effects dominate and the rolling moment reverses in direction as shown in fig ......

/ ure 2.7.

Span Loading at 65 Degree Wing Sweep 1.0 • ° .9 .8 .7 Span .6 oacLill .5 Loading .5 (C l c) .4 / .3 .2 .1 .0 -15. -10. -5. 0. 5. 10. 15.

Aerodynamic Span (ft) Figure 2.28 rolling moment due to induced asymmetric span loading Rolling moment due to ,r_ Cy acting _ above C.G.

opposes roll moment due to span loading.

Figure 2.29 rolling moment due to sideforce At high angle of attack, flow separation influences the roll moment coupling by altering the wing's span loading and reducing Cy. Leading edge vortex formation and flow separation i,i _:::ili over the aft swept wing tip distort the span loading to produce a negative (left) roll moment • : i- _, for oblique wings with "right-side-forward" sweep. This opposes the roll moment incre- ii :: ::i _ ! i ment due to the loss of leading edge suction (and therefore sideforce, Cy) which acts above the center of gravity. Wind tunnel results show that the latter effect is stronger and the rolling moment slope becomes more negative at flow separation for the F-80WRA configuration (Fig. 2.6).

Yawing moments are produced by the aerodynamic influence of the wing alone and the wing-fuselage interference. The wing alone influence is two-fold. As the angle of attack is increased, The wake induces an asymmetric local angle of attack distribution on the wing which produces a pure yaw couple by altering the distribution of leading edge suction.

This couple may be countered by the wing's sideforce component acting ahead of the cen- ter of gravity. Interference of the wing and fuselage contributes to yawing moment through lift and thickness-dependent induced effects. The circulation created by the wing to produce lift induces an asymmetric velocity distribution along the fuselage which causes a negative (nose left ) yaw moment (Fig. 2.30).

Bound vorticity induces velocities which cause suction on the fuselage in these regions CN Figure 2.30 yawing moment due to wing-induced fuselage loading _ i; J At zero lift the contribution of induced effects to C N becomes negligible, but the combined influence of the wing and fuselage thickness does not. Wind tunnel results show that at a i _ C L of zero the value of C N is nonzero and the slope of C N vs C L is distinctly negative in this region (Fig. 2.31). These trends cannot be caused by induced affects (which diminish 'i at low C L ) but instead must be related to an interference of the wing-fuselage thickness distributions. A panel code which models thickness effects (PANAIR [Ref. 15]) was used to study the yawing moment contributions and the results showed that these unusual yaw- ing moment trends eould be predicted using this type of panel code. The results also showed that the yawing moment in the low C L attached flow regime is dominated by wing- fuselage thickness-interference. The displacement of fluid caused by the wing and fuselage thickness (only) produces a low pressure region on the front right side of the F-8 OWRA fuselage at zero CL(Fig. 2.32) which decreases in strength as angle of attack increases, producing a strong negative C N slope. If either the wing or fuselage is modeled as a thin surface this effect disappears, indicating the dependence of this phenomenon on thickness.

The vortex lattice method used in the design syntheses presented in this work cannot mod- el this thickness interference effect. Therefore, the value of C N and CN( x are estimated from the wind tunnel data (F-80WRA) at various sweeps and angles of attack. This ne- cessary approximation for C N will not significantly affect the design synthesis results be- cause the wind tunnel data shows that these C N trends are insensitive to wing placement and planform (Fig. 2.34).

Yawing Moment Coefficient vs Lift Coefficient 65 o sweep, M=O.6 .010 .008 Results .006 & WindTunnel .004 C N .002 .000 -.002 -.004 -.006 -.2 -.1 0.0 .1 .2 .3 .4 .5 .6

cL

Figure 2.31 yawing moment coefficient vs. lift coefficient ?

Pariah" solution for the fuselage pressure distribution at zero lift.

CN :ii:_i:" • Figure 2.32 yawing moment due to wing-fuselage thickness-interference Yawing Moment vs Lift Various OWRA Configurations, M--0.6, Sweep=65 ° .015 Configuration# 4 [__.._ Configuration# 2 Confi_naration# 3 .010 .005 C N __l_m -_ Aft tip flow separation .000 -.005 | ! i I I -.010 -.1 0.0 .1 .2 .3 .4 .5 -.3 -.2 CL yawing moment vs. lift for various OWRA configurations Figure 2.33 At higher angles of attack, flow separation produces nonlinear trends in C N. The leading edge vortex which leaves the wing and trails downstream along the fuselage can strike the vertical tail (Fig. 2.21) producing large variations in C N that are difficult to predict.

2.2 Vortex Lattice Aerodynamic Model

The aerodynamic coupling loads are predominantly due to the forces and moments generat- ed by the wing. Consequently, the controllability and handling qualities of the aircraft can be influenced by the relative positions of the wing and fuselage. An integrated design syn- thesis method will be used to investigate tiffs possibility (chapter 4), but a method for effi- ciently calculating the aerodynamic loads as the wing is repositioned is needed first. Refer- ence [ 16] describes a vortex lattice method 'LINAIR', by Kroo, which is ideally suited for this task.

For a steady, irrotational flow Laplace's equation for the perturbation velocity potential must be satisfied. Assuming small perturbation velocities (relative to the freestream magni- tude) we have: (2.16) V 2 • = 0 A (2.17) boundary conditions: V ° n = 0 (2.18) where V = ( U o + u, v, w ) Vortex singularities placed along the quarter chord of the lifting panel and its wake satisfy Laplace's equation and produce the velocity perturbations required to meet the boundary conditions. Figure 2.34 shows how the vortex filaments are distributed in a 'horseshoe' shape to model a segment of the tiffing surface.

A c Leading Edge n Vortex, strength = F Trailing Vortex Control V o Trailing Edge ) Figure 2.34 typical panel, vortex lattice model • , : .... , " • . i ¸ ,,',_.- .?_ ,• i , _:_'_!_ ,_ The lifting panels may be any trapezoidal geometry and are not restricted to be in a single plane. The vortex strengths, F, are computed by enforcing the boundary conditions. This results in a linear system of equations for F.

; , , 7 i (2.19) [AIC] {r'} = {B} n = total number of panels nxn nxl nxl where: (2.20) {B} = {V-n+cox r } (2.21) tO= { p ,q,r } The [AIC] matrix is composed of terms representing the influence of the jth vortex at the ith control point, where i and j are indices that span the set of n panels. The vector {B } repre- sents the velocity that must be induced at each control point to achieve flow tangency. The first component of {B } accounts for the freestream flow and the second term represents the velocities induced by rotational motion of the aircraft.

The vortex strengths (1") are obtained by solving the linear system of equations (2.19 - 2.21 ) above. The forces acting on each panel are: (2.22) F i=p(V cXF )A Bi where: (2.23) V c = { [VIC x] {F}, [gICy] {1"}, [VIC z] {F} } F = Bound vorticity on panel (i) Bi A = Bound vortex length Where the matrices [VICx,y,z ] represent the velocity acting at the center of each panel's quarter chord as a function of the vortex strengths. The VIC influence coefficients are cal- culated with the bound vortices arranged perpendicular to the longitudinal (x) axis (Fig.

2.35). This prevents the bound vorticity from over-influencing the flow in the vicinity of the quarter chord, which would otherwise occur if the VIC terms were calculated with 'swept' bound vorticity.

i :_.

R I ! m I n_ i m _!i¸ :!

m m ,i Bound vorticity for [VIC] Bound vorticity for [AIC] calculation calculation Figure 2.35 bound vorticity placement for near-field velocity estimation A complete aircraft configuration is modeled by placing trapezoidal lifting elements on the mean surface in an appropriate manner. Several factors determine whether or not a lifting surface model is appropriate. Thin lifting surfaces (such as wings and tails) are best mod- eled by placing lifting elements along the mean camber line. If control surfaces are present, these are approximated by dividing the lifting surface chordwise at the hingeline into a sep- arate control surface element.

The number of panels required to compute accurate airloads is best determined by a trial and error procedure where the number of panels are increased until the desired accuracy is /i achieved. Typically, the estimation of drag loads requires significantly more panels than the prediction of other forces and moments. The best results are obtained when the panel spacing does not vary abruptly and the trailing wake vorticity of one element does not pass i_ :i _, too close to the control points of another element. Fuselages and other bodies that do not i have distinct and sharp trailing edges may be modeled by placing a set of vertical and hod-.

zontal lifting panels along the bodies' centerline, where the panels have the same projected shape as the body (Fig. 2.36). Figure 2.37 shows the results of an analysis where this fu- selage model is compared with experimental data [Ref. 18]. This approximation accurately predicts the wing-fuselage interference that reduces CLo _and static pitch stability.

© +

Front Front - r Fuselage Vortex Lattice Model Actual Fuselage Geometry Fuselage is modeled as two low aspect ratio lifting surfaces.

Figure 2.36 fuselage vortex lattice model Pitching Moment vs Lift Coefficient Fuselage Vortex Lattice Model .10 ,)i I:! Wind Tunnel Results [18] & inair Results .05

% 0.00

-.05

I1

Top -.10 .;i _ , ;: .7 .8 .9 .2 .3 .4 .5 .6 -.1 .0 .1

cL

fuselage-wing model verification Figure 2.37 The vortex lattice model for the F-80WRA with 45 degree oblique sweep is shown in fig- ure 2.38. There are 30 spanwise panels along the wing and 48 more panels which com- prise the fuselage and tail. One aspect of the wing panelling, which improves the modell- ing accuracy for oblique wing aircraft, is that the wing's trailing vorticity is parallel to the streamwise direction. This type of panelling is preferred because it places the vorticity i/i _ closer to its true physical location and the spacing of the wake vorticity remains uniform.

Figure 2.39 shows how a wing with a fixed planform is panelled at several oblique sweep angles using streamwise trailing vorticity.

/ i¸ ' 3/4 view Front Top Side Figure 2.38 F-80WRA vortex lattice model, 45 ° sweep ;_ i!, !/!,i I i_ _,i! /; _ii_i !! i_ !i'_(!ii:i_ ¸ Streamwise Direction v

300 400 500 600

Increasing Sweep streamwise trailing vorticity used for wing vortex lattice model Figure 2.39 The accuracy of LINAIR in modelling the F-80WRA configuration is shown in figures 2.42 - 2.50 where the results from LINAIR are compared to both PANAIR and wind tun- nel results, from references [17] and [13] respectively. The wind tunnel results compari- son (Fig. 2.46 - 2.50) are presented for OWRA configuration 3 (Fig. 2.2) and the PA- NAIR results (Fig. 2.41 - 2.45) are for configuration 2. Figure 2.40 shows the surface panel model used in the PANAIR calculations. Notice that the wind tunnel comparisons in- clude a limiting mark which indicates the largest value of C L at which attached flow is ex- pected (based upon equation 2.6). The LINAIR prediction for the airloads shows accepta- ble agreement with both the PANAIR predictions and the wind tunnel results (in the attached flow region) for all forces and moments except yawing moment. As was men- tioned previously, the yawing moment is dominated by a fuselage-wing interference effect that depends upon the thickness of both bodies. LINAIR is not capable of modelling this ,!. ,;, i,_ phenomenon because it is a thin lifting surface method. Therefore, the variation of C N with changing (x will be estimated from wind tunnel data in the integrated design synthesis ••_•_ i• _ procedure. As was shown in figure 2.33, C N is relatively insensitive to configuration _' •i •'_II ,i ¸ changes if the sweep angle remains constant, therefore little loss in accuracy is expected from using wind tunnel results to predict C N trends.

• Z Figure 2.40 PANAIR surface panel model, F-80WRA, 45 ° sweep , } Lift Coefficient vs Angle of Attack _,_ :! i_ "_ _, 65 deg. sweep, M=0.6 .5, _:' _' .7 i:'_I 'i •_:_ / i" _ .6 ''**°°°°°°°**'*°_°°**°°°°°°°*'I°°°S'°'°'°°°°°°°°°*°*'°°°°°*°°°°°°°°°°°:°°°° .5

"" I A L_a_ 1 ........... _ ...... _

.4 ................. I [] Panair I : -- : ................. t I ........... "............. i................

.3

cL

.2 .1 0.0

ill

-.1 -.2 i ° • • i -.3 -10. -5. 0. 5. 10. 15.

_ (deg) Figure 2.41 lift coefficient vs. angle of attack Sideforce Coefficient vs Lift Coefficient 65 deg. sweep, M=0.6 .02 0.00 -.02 -.04 Cy -.06 -.08 •ill -.10 -.12 'i_ _ _ ;_ ,_ -.4 -.2 0.0 .2 .4 .6 .8 i ¸ •_i:: , CL Figure 2.42 sideforce coefficient vs. lift coefficient • _ L'¸¸ : , i, ¸ "_ , _, • Rolling Moment Coefficient vs Lift Coefficient 65 deg. sweep, M=0.6 .OO8 ,°°°°°_.°,,°°°°°,°°° °p°°, ° °°, °° .... ° .006 J°°°°°°°o°°°°°_°°°°°°''**°°°°_°'*'°°°°*°°°° _.'°°°°°°* .

.004 .002 .0_ -.002 i I° =l i ......... ............

-.004 iiiiiiiii ............

-.006 -.008 .8 -.2 0.0 .2 .4 .6 -.4 Figure 2.43 rolling moment coefficient vs. lift coefficient Pitching Moment Coefficient vs Lift Coefficient 65 deg. sweep, M=O.6 .1 0.0 -.1 -.2

iiiiiiiiiiiiiiiiiii! !!!!iiiiiiiiii iiiiii

-.3 C M ............. I A ........... i.......... i........... !.............

-.4 -.5

iiiiiiiii iiiiiiiiiiiiiiiiiill

-.6 -.7 -.4 -.2 0.0 .2 .4 .6 .8

eL

Figure 2.44 pitching moment coefficient vs. lift coefficient Yawing Moment Coefficient vs Lift Coefficient 65 deg. sweep, M--0.6 (i!:i, LI_'" I .015 .010 .......... % .......... :: .... I, _m_.... :......... '-: ............

.005 _'ii:_ :

%

.000 , i• ¸ : _°_°_°°_**_°°__)_°_°°_°_-_°_°_- -.005 -.010 -.4 -.2 0.0 .2 .4 .6 .8

cL

Figure 2.45 yawing moment coefficient vs. lift coefficient Lift Coefficient vs Angle of Attack 65 deg. sweep, M=0.6 1.0 .8

..................... .... =: ........ :.................... .............. I

.6

cL .4

.2 .0 il'e on w° L ........

-.2 -5. 0. 5. 10. 15.

(x (deg) Figure 2.46 lift coefficient vs. angle of attack Sideforce Coefficient vs Lift Coefficient 65 deg. sweep, M=0.6 .04 i_ High Angle of Attack .02 "1 I .......................................... Region (flow separation) .......

.00 -.02 .........

Cy -.04

::::::::::::::::::::::: IIIIIIIIFIIII I

-.06 -.08 -.10 -.1 0.0 .1 .2 .3 .4 .5 .6 -.2

c,_

sideforce coefficient vs. lift coefficient Figure 2.47 Pitching Moment Coefficient vs Lift Coefficient 65 deg. sweep, M=0.6 .1 0.0 '_. ''__"i .............. I High Angle of Attack I ........

ii ii _ [ Regi°n (fl°w separati°n) [

_ _ i -_ i i i

-.1

%

.......... .......... .......... ...........

-.2 a Linair _.

_1 Wind Tunnel -.3 ......... . ........ i" I -.4 .1 .2 .3 .4 .5 .6

ct

Figure 2.48 pitching moment coefficient vs. lift coefficient Rolling Moment Coefficient vs Lift Coefficient 65 deg. sweep, M=0.6 .0O4 High Angle of Attack L

1 Region (flow separation)

, i. _' ,,, °,, ° ,, ,., ,,, ,, ,.,,,,,, ,, ,°,, t , %°,,°,**o,,, , °°'*''' .002 .i i!i .... ....... i.... ..... i.........

-.004 -.006 ii!i _ -.008 .6 -.1 0.0 .1 .2 .3 .4 .5 -.2

cL

Figure 2.49 rolling moment coefficient vs. lift coefficient Yawing Moment Coefficient vs Lift Coefficient 65 deg. sweep, M=0.6 .020 • ° ° .......... • .......... _.............. , ..... _ .......... _ .......... _ ........... . ..........

.015 I& Linair I I Hidl Angle of Attack [ _-.

[] Wind Tunnel ] [Region (flow separation)[./_ .i ..........

.010 ''° ° " °*'' °'''e _" "°" .005 C N .000 -.005 -.010 .6 -.1 0.0 .1 .2 .3 .4 .5 -.2

ct

Figure 2.50 yawing moment coefficient vs. lift coefficient , i'i_ _ / Figures 2.51 - 2.55 display the comparison between Linair and wind tunnel results for the aircraft at (+ & -) 5 degrees sideslip. These plots show that Linair predictions are in good i_ • _ ,, agreement with the wind tunnel results for cases with sideslip. The results for C N show that the change in yawing moment with sideslip is accurately predicted although the C N va- riation with angle of attack must be estimated from wind tunnel results.

Lift Coefficient vs Angle of Attack 65 deg. Sweep, M=0.8 1.0 T Wind Tunnel, Beta=-5 deg.

o..° Wind Tunnel, Betaffi5 deg.

Linair, Beta=-5 deg.

Linair Beta=5 deg.

.6

cL

.4 .2 _.1High Angle of Attack l i..............

i "l Region <now _a"on)li

-5. 0. 5. 10. 15. 20.

Ct (deg) Figure 2.51 lift coefficient vs. angle of attack Sideforce vs Lift Coefficient 65 deg. Sweep, M--0.8 .15 .10

........ ....... ": ............ i...... i

.05

".......... i............ !_

0.00 RHJ;g_iion_oe°fAt_ on, :: :: "_ ::

%

-.05 : : & Wind Tunnel, Beta=-5 deg.

--_ffi._._._. _ _ 13 Wind Tunnel, Beta=5 deg.

-.10 ...... _.i_-v.._ .-..:_.r: _ .... "" • Linair, Beta=--5 deg.

__ -'.._

-.15 -.20 i ..,: ............

: : _ _ -.25 1.0 -.2 .0 .2 .4 .6 .8

c_

Figure 2.52 sideforce coefficient vs. lift coefficient Rolling Moment Coefficient vs Lift Coefficient 65 deg. Sweep, M=0.8 .015 .010 ol,..oooooo.,.i* .005 : High Angle of Attack .000 Region (flow separation)

q

.:. :...

-.005 %° -.010 Wind Tunnel, Beta=-5 deg.

Wind Tunnel, Beta=5 deg.

-.015 Linair, Beta=-5 deg.

Linair, Beta=5 deg.

-.020 1.0 .2 .4 -.2 .0 .6 .8 Figure 2.53 rolling moment coefficient vs. lift coefficient Pitching Moment Coefficient vs Lift Coefficient 65 deg. Sweep, M=0.8 .2 ": :: [A Wind Tunnel, Beta=-5 deg.

I_ i i Ira Wind Tunnel, Beta=5 deg.

.1 -' _ ..... _ .............. i" _" .......... :"" ] O Linair, Beta=-5 deg.

0.0 _ _._. ....

iiii ......... i!,,!i:i!i!iii il ................... .....i.............. i.............

CM -.1 , o, .oo,,, °oo. ,'o,, *.°o.,o... °''°°" -.2 High Angle of Attack Region (flow separation) -.3 -.4 .0 .2 .4 .6 .8 1.0 -.2 C L Figure 2.54 pitching moment coefficient vs. lift coefficient Yawing Moment Coefficient vs Lift Coefficient 65 deg. Sweep, M=0.8 .04 .- ." • . ..

k, .............. !.............. :ql,, .......... i .............. ";............. _..............

.03 .02 .01 .00 C N -.01 L T t ,°°°oo...,,°,._oo, -.02 & Wind Tunnel, Beta=-5 deg.

High Angle of Attack rn Wind Tunnel, Beta=5 deg.

Region (flow separation) .............. :....

-.03 o Linair, Beta=-5 deg.

V Linair, Beta=5 deg.

. m_ .

-.04 .6 .8 .0 -.2 .0 .2 .4

CL

Figure 2.55 Yawing moment coefficient vs. lift coefficient For each calculation of the airloads, the [AIC] and [VIC] matrices must be evaluated and a linear system with n unknowns (equation 2.19) must be solved. In the integrated design synthesis the airloads must be calculated each time the configuration changes and this may require excessive CPU time for a design with a large number of panels. One method to re- duce the amount of numerical calculation is to assume that the [AIC] and [VIC] matrices do not change significantly as the wing is repositioned on the fuselage. Referring to equation 2.19 this implies that only the vector {B }, which represents the flow boundary conditions, changes as the configuration is altered. This assumption is only justifiable if the planform of the wing remains unchanged and the relative position of the wing and fuselage does not change by a qarge' amount. Figures 2.58 - 2.62 compare the airloads calculated using the approximated [AIC] and [VIC] matrices, to those calculated with exact matrices. Thea'e- suits are shown for configurations that have large variations in wing position relative to the fuselage. For the approximated cases, the original [AIC] and [VIC] matrices used corre- spond to the nominal aircraft shown in figure 2.56. The results clearly show that assuming the [AIC] and [VIC] matrices are constant has only a small effect on the accuracy of the es- : _ '1 • i timated airloads, but significantly reduces the required CPU time for repeated aerodynamic calculations.

?

i_!! _ i Figure 2.56 nominal configuration (zero wing bank) ?• modified configuration (15 deg. wing bank) Figure 2.57 Lift Coefficient vs Angle of Attack 45 deg. sweep, M=0.6 1.0 .8 j A ExactLinair ' I i i"/ .6

......... l a...... _,o__._ I i .............i ........ i........ i........ !

CL .4 L .2 .0 L | i | i -.2 -4. -2. O. 2. 4. 6. 8. 10. 12. 14.

(deg) Figure 2.58 lift coefficient vs. angle of attack Sideforce Coefficient vs Lift Coefficient 45 deg. sweep, M=0.6 .12 • ° .,,,, ,,,,..,.¢-..,,,,,,,,,,o.L',,,,,,*,,,,,,,',,o.**•,,,,,,* ¢'oo-oo,,oooooo, Iooo ooo o o'ooo o o .10 .08 .06 .04

%

.02 .00 -.02

......................... : ............... ..... iii))

-.04 -.06 1.0 .0 .2 .4 .6 .8 -.2

cL

Figure 2.59 sideforce coefficient vs. lift coefficient Pitching Moment Coefficient vs Lift Coefficient 45 deg., sweep M--0.6 .1 . . .. • 0.0 -.1 ......................... _ .............. I . i l ..........

%

-.2 i iiiii, -.3 -.4 1.0 .0 .2 .4 .6 .8 -.2

%

Figure 2.60 pitching moment coefficient vs. lift Coefficient .020 .015 .010 .005

q

.000 -.005 -.010 .2 .4 .6 .8

cL

rolling moment coefficient vs. lift coefficient Figure 2.61 Yawing Moment Coefficient vs Lift Coefficient .... / 45 deg. sweep, M=0.6 .O2O .015 .010 C N .0O5 .._! ......

.000 i • ° -.005 1.0 .2 .4 .6 .8 -.2 .0

%

yawing moment coefficient vs. lift coefficient Figure 2.62 i • 5O

Chapter 3

,i ¸¸ ,?_

Dynamic Response

of Oblique Wing Aircraft

The dynamic response of an oblique wing aircraft to control inputs and gust disturbances differs from a conventional aircraft because of aerodynamic and inertial couplings inherent to the asymmetric configuration. In the fast section of this chapter the impact of configura- tion asymmetry on dynamic response and controllability will be investigated using linear- ized dynamics models of the six degree of freedom aircraft. Aerodynamic stability and control derivatives for the equations of motion are calculated using the vortex lattice method presented in chapter 2. The results obtained by analyzing the equations of motion demon- strate the significance of both aerodynamic and inertia coupling on the degradation of oblique wing aircraft handling qualities. A simple analysis provides explanations for the lack of controllability in some proposed oblique wing configurations. The problem of in- sufficient dynamic controllability is further investigated in section 3.3 where the OWRA's closed loop response is studied, using several types of controllers. A new model following scheme (reduced order model following) is presented and its advantages for oblique wing control are investigated.

3.1 Linearized 6 D.O.F. Equations of Motion for the

Rigid Oblique Wing Aircraft

Figure 3.1 shows the body axis coordinate system 03) in which the equations of motion are derived. Wind axes are also shown to clarify the definition of angles of attack and sideslip.

The body axes are aligned with the x axis along the fuselage centerline and the z axis in the vertical plane of symmetry of the fuselage. The origin of the axes lies at the aircraft's mass center.

C L It Wind _l_ Cr_ = Cns ( i e n 1 ed.

/ 2. Origins Of wlnd and stability axes have been displaced from center of gravity Stability _xes for ulal'lty.

Relative _1 nd Figure 3.1 axes systems for the rigid aircraft Newton's laws of motion applied to a rigid body with three translational and three rotation- al degrees of freedom are: The right hand side of equations 3.1 and 3.2 can be linearized by defining the following _.13-1

(3.1)

_=m[v+O_ ×_]

-- - B -_-; = _B-I

(3.2) M = I • _+ o_ x I • oo ._.B-I O_ where: = Angular velocity with respect to inertial space

V

= Velocity of the mass center with respect to inertial space B l = Time derivative of V in the (B) axis system

V

= ,• • ,• • : . . c/_¸ B = Time derivative of w in the (B) axis system R

I

= Inertia tensor expressed in (B) axes Ixx Ixy Ixz = Iyy Iy z

(SYM) Izz

m.,=@ F, M = Net external force and moment acting on the aircraft perturbation expressions for V & to B-I (which correspond to 1-g level flight).

Where (u, v, w, p, q, r) are perturbation quantities. Substituting for V & tt_ B-I into equa, B

P

q

(3.3, 3.4) V =

r tions 3.1 and 3.2 and dropping higher order terms: The net forces and moments consist of aerodynamic, propulsive, and gravity terms: w o 0 u m u -=o

m 0

0 v

F ,_WoO uo

m

_o -Uo o w

+ P q

0 0

Y

i :I _, : _ r i m m Equation (3.5) Because the moment reference is at the aircraft mass center, the moments due to gravity (3.6) F = Faero + Fgra v + Fprop •_i _::ii (3.7) M= Macro + Mgrav + Mpro p forces are always zero. The external loads are now linearized using a Taylor Series expan- sion and dropping all terms with order greater than one: Where it is assumed that the propulsive forces are appropriate for 1-g level flight aim and ... _.o ..,o ....o [_SFi] [_SFi]

(3.9) .=M

where: x=(u, v, w, p, q, r, _g, O, ¢) = ( droll, dpitch, dyaw) are not affected by changes in the aircraft's motion. Aircraft trim therefore implies: The state vector,x, contains three Euler angles which define the aircraft's orientation rela- -..40 ---40 Macro + Mprop =0 ...¢0 ..¢0 ...oO Faero + Fgra v + Fprop = 0 five to inertial space as a set of successive rotations about non-orthogonal axes. The linear- ized expression for the gravity forces in terms of the perturbation Euler angle quantities is: The linear expansion for the aerodynamic forces and moments is explicitly written as: t_ "-(mg)sin(0o)" = 0 'i', A (3.11) F .,2' gray

+/OOmg/

i:__I }i: :_ i

l ooo j

(mg)cos(0 o) •_ _i¸ _ :; 54.

The control vector (d) consists of generic controls that correspond to deflections of the ap- n m n m Fxv Fxr Fxu U _ O I V (3.12) Faero F I + Fy u v I d • W O I Fzr Fzu

Fz I

- p q m m r drol Fx droll Fx dpitch Fx dyaw \ °-_itc + Fy droll •\ •\ dya F F z droll z yaw M M M XU XV xr (3.13) V M x, Maero- yu "_ W Mzu Mzr P q r roll I Mx droll Mx dpitch Mx dyaw + pitch I My droll \ "_.

lY aw I Mz droll ......... Mz dyaw propriate control surfaces to produce roll, pitch, or yaw moments. For the F-80WRA roll is caused by anfisymmetric elevator deflection, pitch by symmetric elevator deflection, and yaw by rudder deflection. Equations 3.12 - 3.13 are substituted into equation 3.5 to obtain the f'mal form of the linearized equations of motion. Notice that the trim condition has been enforced and that the moments M x, My, M z, have been re-named L, M, N.

Equation 3.14 has the form: g i m

ul

m 0 0 V m I ,,**,,,,,p,,,.,,,,.,,o,ol,..t W P

o o

at q °,°°o°o°°°°°.....°°°°°°**_.,° r ¥ 0 0 1 n m m m n U F 0 -mg F F F (F+ w o) F 0 xu xv xw xp xr V mg Fy u ................. (Fyp- Wo) Fyq

(F+ u o)0

W p F .................. Fzp (Fzq-U o) F ZU Zr q L ,,.**,,,..*,o,,,,,o,oo..,*,.,o,,...,°.''°'''" L U T r M .,,,,o.,,,.,, .... ..,,,,,,,,.,oo,,,,,,,oo,,,*o M u r N ,,.'",o'o'"" .... • .... o'o'""*'°°'""'"" N u r m_ °'°°°°°°'°°°°'°°''°''°°°°° 0 0

o

0 0 .......................... 0 1 0 0

0 0 0 1 see(0)

m m Fx_r Fx_p Fxsy pitch[ LSyawl L6, + Nsr ............. Nsy 0 0 0 i m Equation (3.14) Equation 3.15 shows the standard linearized form in which the equations of motion will be

[M] x = [F] x + [G] d

• ) or x = [A]x+ [B]d (3.15) !_ii _ : where: -1

(3.16) [A] = [M] [F]

-1 (3.17) [B] = [M] [(3] _:L ¸ ' •• ; _/:!iiii :, !

• ,ii _ • • _ written when studying dynamic response and control. Notice that the matrix [M] is always invertible because it is positive definite and symmetric, by def'mition of the mass properties.

To evaluate the terms in the matrices [M], [F], and [G] the aircraft's mass properties, flight velocity and altitude, and aerodynamic stability and control derivatives must be known.

The aerodynamic model presented in chapter 2 (i.e. LINAIR) can efficiently estimate all of the quasi-steady aerodynamic properties for arbitrary oblique wing configurations. The " ?

non-dimensional force and moment coefficient derivatives are calculated by using LINAIR r results in a numerical central difference approximation for the derivative.

Where _ is the ith dimensionless force coefficient, xj is the jth element of the state vector, (3.19) 5x. - 2 A.

J J Central difference approximation about the x ° state.

and Aj is the difference interval for xj. This approximation is accurate to order A2 [Ref.

19]. First order differencing schemes (such as forward or backward differencing) do not provide sufficient accuracy when calculating stability derivatives. For example, first order t.

schemes cannot accurately predict the slope of symmetric functions when the derivative is evaluated at the point of symmetry. This implies that terms such as Fxl 3 ,which are nearly symmetric for small sweep angles, are erroneously calculated to have nonzero value when first order schemes are used.

3.2 Open Loop Stability and Control

The aerodynamic coefficients from LINAIR are combined with the mass property and flight condition information to form the linearized equations of motion in a FORTRAN sub- routine. The dynamic response of the uncompensated (open loop) aircraft is analyzed us- ing numerical simulation and eigensystem analysis of the linear equations of motion. The dynamic coupling of the aircraft's unforced response is shown in figures 3.2 - 3.3 which contain numerical simulation results of the aircraft's motion after being excited by a 50 ft/ see vertical step gust. These results are for the F-80WRA with 40 degrees oblique wing sweep and a cruise Mach number of 0.75. For a symmetric aircraft only the longitudinal motion is excited by vertical gusts, but the simulation results show that for large oblique sweep angles, the aerodynamic and inertia coupling of the asymmetric configuration results in excitation of the lateral states [3, _, 0, _ also.

The effect of wing sweep on aerodynamic and inertia coupling is shown in figures 3.4 - Dynamic Response to a 50 (f/s) Vertical Step Gust 40 (deg) Sweep OWRA .

. ............... e........-..................,......., e........s.. • • •. • .'... • ... • .'. •. .....

.

° ....... ........ } ........ } ....... i........ + ........ } ........ } ....... i........

° Angle (deg) .

-1.

| • m m m m am m m -2.

10.

0. 1. 2. 3. 4. 5. 6. 7. 8. 9.

Time (sec) Figure 3.20WRA open loop unforced response i+;i Dynamic Response to a 50 (f/s) Vertical Step Gust 40 (deg) sweep OWRA

5. i i : i i

'i i _ 4. ... .: ........ _........ :......... :........ _........ : ........ _........ -" ....... " ........

3. •......... _......... _....... : ........ : ........ : .........................

1 • .-illloo, g gl.i ,oil, 0 jl.l**eo jgg bll o o g g i io I olg iI, oj ol, llel,_ o ol01 t. t... g .. • •.

1.

Angle 0.

0.

(deg) -1. '"" ................. "................

-2. ._........ : ......... : ........ _........ : ........ , ........ ." ....... "- ........

-3.

-4.

-5.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

o Time (sec) Figure 3.30WRA open loop unforced response (continued) 3.6. The variation of the three of the aerodynamic coupling derivatives (Cyo_, Clo _, and CNo_) with wing sweep is given in figure 3.4. At zero sweep no aerodynamic coupling ex- ists and as the sweep increases the amount of coupling varies considerably. Figures 3.5 - 3.6 show the effect of wing sweep on the terms in the inertia tensor. As sweep increases, Ixx decreases and approaches the value of the fuselage only, where as Iyy increases and approaches the value of Izz. The most significant inertia coupling term, Ixy, has a maxi- mum value at 45 degrees wing sweep.

The magnitude of the inertia coupling's impact on the dynamic response can be seen from the following example. Consider the aircraft trimmed in 1-g level flight at time = 0, when a pure pitching moment is applied to the aircraft using the control surfaces. For zero inertia coupling this moment should cause only a pitch acceleration, but inertia coupling causes an additional roll acceleration. The magnitude of roll acceleration relative to pitch acceleration is described by equation 3.19.

For the F-80WRA Ixy / Ixx ---0.48, which implies that a pure pitch input causes a roll ac- d_-[tq] Ixy (attin_ O) (3.19) Ixx celeration of approximately one half the induced pitch acceleration based solely upon con- sideration of inertia coupling. These results further emphasize the importance of consider- ing both aerodynamic and inertial influences when calculating dynamic response.

• i i _ The influence of aerodynamic and inertia coupling is not limited to the coupling of the dy- namic response, but it also affects the overall dynamic stability of the aircraft. Figure 3.7 is a root locus plot of the open loop aircraft's natural modes for wing sweep ranging from zero to 65 degrees. In each case the aircraft is trimmed in 1-g level flight. The root traces are labelled with the classical names for each dynamic mode as they occur at zero wing sweep. In reality, all of the modes become coupled as wing sweep increases and the label ......

ling of the modes based on the aircraft's motion with zero wing sweep is purely a matter of convenience. These results show that as wing sweep increases the stability of all of the modes (except for the low frequency phugoid and spiral modes) is significantly affected by Effect of Sweep on Coupling Stability Derivatives 300 sq. ft. wing OWRA, LINAIR Results .2 .1 0.0 -.1 Coefficient -.2 -.3 -.4 -.5

I °

-.6 10. 20. 30. 40. 50. 60. 70.

* Sweep (deg) Figure 3.4 effect of oblique sweep on aerodynamic coupling Effect of Wing Sweep on Inertia ? '_r ! ii xl0 5 300 sq. ft. OWRA 1.1 A A i '¸ ::i 1.0 v v .9 .8 ,_ i_, _ .7 Inertia A Ixx (slug-f t^2) .6 [] lyy 0 Izz .5 .4 .3 .2 A .1 0. 10. 20. 30. 40. 50. 60. 70.

Sweep (deg) Effect of Wing Sweep On Inertia • _ _i_ _, _ 300 sq. ft. OWRA -1000.

0. _ -2000.

-3000.

Inertia -4000.

(slug-ft^2)-5000. ii!

-6000.

-7000.

-8000.

-9000 a t.

0. 10. 20. 30. 40. 50. 60. 70.

Sweep (deg)

Figure 3.5 effect of oblique sweep on inertia c aircraft asymmetry. A more severe example is the dutch roll mode which becomes unstable at wing sweep greater than 50 degrees.

/j "//, Root Locus On Wing Sweep 300 ft^20WRA, 1-g flight ° ° Increasing Sweep

g

0< A <65(deg) odl_fo . _l_tch Roll I'Short Peri .

i_ii, Imaginary (rad/sec) ° '_iral and Phugoid' d| ° -1.

-7° -6° _5. -4. -3° -2. -1 ° 0. 1.

Real (rad/sec) Figure 3.7 effect of oblique sweep on dynamic stability For the integrated design synthesis it is useful to estimate the key undesirable aspects of the aircraft's dynamic response using the minimum number of state variables. Because of the large frequency separation between the spiral, phugoid and the dutch roll, short period, roll modes it is reasonable to assume that these lower frequency modes, which do not signifi- cantly contribute to handling qualities degradation, may be omitted without affecting the re- sponse of the higher frequency dynamics. Studies of aircraft handling qualities [Ref. 20] have shown that it is the higher frequency dynamics that most influence handling qualities.

Furthermore, the low frequency modes can easily be controlled by the pilot or by a simple controller loop that is designed after the high frequency dynamics have been compensated.

Figure 3.8 shows simulation results for the F-80WRA response to a gust, with and with- out the low frequency modes. As is expected, there is negligible difference between the time histories of the two cases for the first five seconds of motion. The results also show that the higher frequency dynamics are unchanged and that the differences that occur be- yond five seconds are solely due to the contribution of the low frequency modes. For the :/_,i; _i _ i!_'I iii /:: cases studied in the remainder of this thesis the low frequency spiral and phugoid dynamics will be omitted by eliminating the u, _, 0 states from the equations of motion. These dy- namics will be assumed to be controlled by the pilot or by a low frequency controller which is synthesized in the final design.

Response to a 50 (f/s) Vertical Step Gust 40 deg Sweep OWRA o -1.

-2.

Theta (deg) -3. _ - . . ,, -4. I V "_ V, 0, u, States Omitted [ -5.

0. 5. 10. 15. 20.

Time (sec) Figure 3.8 open loop response with low frequency modes omitted Figure 3.9 shows the layout of the proposed aerodynamic control surfaces for the F-8 OWRA. The rudder is used to produce yawing moment, while the full-flying elevator can be deflected symmetrically for pitching moment and asymmetrically for rolling moment.

Ailerons produce both rolling and pitching moment when the wing is obliquely swept and loose their effectiveness at high sweep angles due to reduced dynamic pressure in the nor- mal flow directionl Using the tail surfaces to produce rolling moment has the advantage that elevator control effectiveness does not change with wing sweep. However, the elevator _:, _ i_ deflection may be more likely to saturate during combined pitch and roll commands and ap- plying the rolling moment to the fuselage instead of the wing creates increased structural loads in the fuselage and wing pivot. In previous studies of the F-80WRA [Ref. 4] it was assumed that only the elevators are used for roll control, because of the control authority

advantages athigh sweep angles and the simplified wing structure that results when aile-

rons are omitted. In order to be consistent with previous studies, the same assumption will be made in this thesis.

Figure 3.9 also shows the layout of proposed wing tip control surfaces which are effective in producing pitching and rolling moments at high sweep angles. The relative sweep of the hingeline of these control surfaces is small when the wing itself is highly skewed, thus maintaining control effectiveness at high oblique sweep. These control surfaces were test- ed on wind tunnel models [Ref. 11] but the additional structure and actuators required by wing tip control surfaces have limited the interest shown in these devices.

Aileron g Tip Control Surfaces Rudder Elevator Figure 3.9 oblique wing control surfaces The dynamically-coupled response of the oblique wing aircraft to control inputs and gusts could easily be solved using a SAS if the aircraft were adequately controllable. Controlla- bility requires that all of the dynamic modes may be influenced using the available controls and that the control authority is adequate to significantly affect these modes without control saturation. It is precisely the lack of controllability inherent to the original OWRA configu- ?

ration that degrades the handling qualities of the closed loop aircraft. The nature of this i lack of controllability can be seen when one considers minimizing lateral accelerations and rolling motion during a pitch-up maneuver. For the OWRA with an unbanked wing, the sideforce generated during a pitch maneuver is large enough that excessive sideslip and rudder deflection would be required to trim. Therefore, the aircraft must roll to allow a • i i portion of the aircraft's weight to cancel the sideforce from the wing. In the next section, simulations of the closed loop aircraft (original OWRA configuration) further substantiate these claims. Designing additional direct sideforce control surfaces would be one solution to this problem, but there is a drag and weight penalty associated with these surfaces. A better solution is to re-configure the aircraft so that the SAS can more easily decouple the aircraft's motion. The configuration is changed in ways specifically chosen to not signifi- • i_ii _ cantly alter the aircraft's weight and drag. The integrated design synthesis method present- ed in chapter 4 accomplishes this task and is applied to the F-80WRA problem in chapters 5 and 6.

L

3.3 Closed Loop Stability and Control

The undesirable influences of aircraft asymmetry on dynamic response (as demonstrated in the previous section) require that some type of stability augmentation system (SAS) be used to improve aircraft handling qualities. Previous attempts to improve handing quali- fies through SAS implementation have produced less than satisfactory results because of an inherent lack of controllability in the proposed OWRA design. In this section, criteria for acceptable handling qualities are established and several control law synthesis methods are studied to see how they improve the dynamic response of the nominal OWRA configura- tion. Handling qualities using the various SAS schemes are compared with each other and with previous studies performed using aircraft motion simulators. The results show how the lack of controllability in the nominal OWRA configuration degrades handling qualities for all SAS designs. Two SAS synthesis schemes are chosen for use in the integrated de- sign synthesis study because of their superior performance and ease of implementation in a numerical optimization procedure.

3.3.1 Oblique Wing Handling Qualities Specifications Military Specification F-8785-C [Ref. 20] lists criteria which an aircraft must meet in order to have acceptable handling qualities. Specifications are given for different types of aircraft (class I-IV, fighter, bomber, etc.) in various phases of flight (category A-C, takeoff, land- ing, air-to-air combat, etc.) and for three levels of handling qualities G-Ill, acceptable to unacceptable). The specifications are given in terms of the aircraft's natural response and gust response, response to pilot commands for various maneuvers, and stick force charac- teristics. A successful oblique must meet the Mil-Spec F-8785-C requirements for han- dling qualities. The F-80WRA is a fighter-interceptor aircraft (class IV) which must have acceptable handling qualities (level I) in its most critical flight regimes ( category A, air-to- air combat, formation flying, etc.). The handling requirements for a level (I), class (IV), category (A) aircraft are given in table 3.1. Specifications for the stick force gradients are not considered in this study because they are a function of the control surface actuation mechanism which is assumed to be designed separately from the stability augmentation system.

MIL - F - 8785C Handling Quality Requirements Class IV, Category A, Level 1 (Fighter Aircraft) Longitudinal Dynamics Short Period 3.5 <= ¢Osp <=14 (rad/sec) .35 <= _sp <= 1.3 i Phugoid _ph >= 0.04 Commanded Response - Must be able to achieve trim without control saturation between -1 'g' and limit load factor in positive 'g'.

Lateral Dynamics Dutch Roll 1.0 <= ¢Odr <= (rad/sec) ;dr >= 0.4 Roll x r <= 1.0 (sec) Spiral t2 >= 12 (sec) Commanded Response - Roll rate oscillations must remain sufficiently small after a step command in roll rate.

- Sideslip excursions for a roll rate step command must be less than 6 degrees when the bank angle changes 60 degrees in 2 seconds.

- Roll authority must be sufficient to achieve 90 degree bank in 1.3 seconds.

Table 3.1 Mil-F-8785C handling quality requirements

One obvious problem in using the Mil.-Spec. criteria to evaluate the handling qualities of

oblique wing aircraft is that acceptable limits for the coupling of lateral and longitudinal motions is not specified. This coupling is a significant factor in the degradation of the OWRA's handling qualities and therefore provides a key performance measure for a candi- date SAS design. A quantitative measure of how coupling affects handling qualities is giv- en in reference [4] which describes pilot ratings (on a Cooper Harper scale [24]) for the F- 80WRA (with SAS) as obtained from experiments in NASA's vertical motion simulator (VMS). In this study the F-80WRA with a 200 square foot wing (configuration 1) was modelled in the simulator's software using wind tunnel data as the primary aerodynamic database. The simulator has sufficient degrees of freedom to permit rotational motion and longitudinal and lateral accelerations of the cockpit. Acceleration degrees of freedom are a key factor in the results of the VMS study because previous simulation studies (without ac- celeration capability) yielded much more favorable pilot ratings for the OWRA's handling qualities [Ref. 4]. These erroneous results were due to a lack of motion cues that the pilots would otherwise experience in free flight. The VMS results showed that pilots were most critical of the coupling between pitch motion and lateral acceleration as normal load factor was increased. Criticism was also made of the coupling of pitch and roll motion in high 'g' turns. In general, the handling qualities of the closed loop aircraft were found to de- grade with increasing wing sweep and flight dynamic pressure, the worst case being Mach = 1.6 and sweep = 65 degrees. Figure 3.10 [Ref. 4] shows the deterioration of handling quality (Cooper Harper rating) as the ratio of maximum lateral acceleration to achieved nor- mal acceleration increases for a pitch maneuver. The parameter A(y/n) is approximately equal to the ratio of maximum lateral acceleration to achieved normal acceleration.

In order to achieve acceptable handling qualities (i.e. a Cooper-Harper rating of (1-3)) the parameter A(y/n) must be less than 0.05, which implies that the peak lateral acceleration must be less than 1/20th the normal acceleration during a pitch maneuver. The VMS study did not specify acceptable levels of pitch to roll coupling for level flight or in aLmS, but did indicate that this coupling should be minimized also. Similarly, coupling from the lateral commands to the longitudinal motion was found to be small and had little impact on pilot handling quality ratings. Based upon these results, the criteria for acceptable handling quality is given by the Mil.-Spec. F-8785C requirements with the additional stipulation that L lateral and longitudinal commanded response be decoupled; particularly the pitch-to-lateral acceleration and the pitch-to-roll motions.

./ Average pilot rating +- RMS B Average Pilot 5 n Ratings (Cooper-Harper) ..:_i_i'_:".'_"-'!_

I I I I

0 0.1 0.2 0.3 0.4 Side Acceleration Parameter, A (y/n) Figure 3.10 Pilot rating vs. lateral acceleration during pitch-up maneuver The VMS study did not consider the aircraft's response to atmospheric turbulence as part of the handling quality assessment. Mil.-Spec. F- 8785C addresses this issue in terms of fre- quency and damping requirements for the natural modes, but does specify any require- ments for acceptable decoupling of the gust response. For example, an oblique wing air- craft experiences lateral accelerations when excited by a vertical gust. The coupled transients that result may degrade ride quality when compared to an aircraft with decoupled gust response. No data is presently available to verify this claim.

3.3.2 Stability Augmentation System Design : . ; i, ¸' r- Preliminary design studies of the F-80WRA considered two SAS schemes for restoring : )!i:_¸ :}!!

acceptable handling qualities to the asymmetric aircraft. The first is an Eigenstructure Syn- thesis technique [Ref. 21] which uses state feedback to drive the OWRA's closed loop ei- gensystem as close as possible to that of a model case which has decoupled lateral and lon- gitudinal dynamics and meets the Mil.-Spec. F-8785C criteria. The second is a 'Loop • / Shaping' technique [Ref. 22] which uses state feedback and a precompensator to meet fre- :f quency response characteristics obtained from the Mil.-Spec. F-8785C criteria.

Early simulation studies, which used fixed based simulators to evaluate handling qualities, failed to recognize the severe performance degradation caused by pitch to lateral accelera- tion coupling. As a result, both the loop shaping and eigenstructure synthesis SAS were designed to minimize roll excursions more so than lateral acceleration. The inherent lack of controllability in the original F-80WRA configuration makes it impossible to minimize ./,{ . • both of these coupling motions simultaneously. When the performance of the two candi- date SAS was initially evaluated (fixed base simulation) pilot ratings favored the loop shap- ing controller, which was then implemented in the VMS simulator trials. A unique SAS was synthesized at five different flight conditions corresponding to 1-g level flight with dif- ferent Mach numbers, altitudes, and oblique wing sweeps (Table 3.2). Gain scheduling was used during transition between flight flight conditions to define the complete control al- gorithm.

Trimmed 1-g flight conditions with zero wing bank NO.

Mach Altitude (ft) Sweep (deg) q(lb/ft^2) Vo(f/s) 0.8 20,000 45 436 830 1.2 29,000 65 665 1199 1.4 29,000 55 905 1399 1.6 29,000 65 1181 1599 0.9 500 65 1179 1003 Table 3.2 flight conditions investigated in the VMS study Because the VMS simulator allows the pilot to experience acceleration as well as rotational motion, the degradation in handling qualities during pitch maneuvers was discovered and quantified (Fig. 3.10). Figure 3.11 shows time histories from the VMS study where the closed loop F-80WRA is performing pitch maneuvers at flight condition 5. The large coupling between pitch and lateral acceleration is clearly shown.

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V VV

- -- -vvv V v _j _ l"elll 6ep _, 'o _!i: i__ to el_UV _, _, ,_i I ,r_, ¸'¸ control Linear Quadratic Regulator (LQR) theory which should also be considered. In the next section three types of control systems based on LQR theory are synthesized for the nominal F-80WRA configuration. The closed loop dynamic response of the OWRA with !_i •• each SAS is compared and two of these schemes are chosen for implementation in the inte- grated design procedure. In these studies controllers are synthesized for one flight condi- tion, although it is assumed that in a final design the SAS gains will be determined at many flight conditions (to account for the aerodynamic nonlinearities) and gain scheduling will be implemented.

3.3.2.1 Explicit Model Following SAS An Explicit Model Following (EMF) LQR controller causes the outputs of the plant to track the outputs of a model system, when the model system is excited by pilot commands. The controller structure consists of a pre-compensator, which includes the model dynamics, and a feedback compensator which stabilizes the open loop plant and insures adequate band- width for the feedback compensated plant to track the model's outputs. A block diagram of the system is shown in figure 3.12.

Gust Disturbances r Pilot Pre-Compensator Plant Open Loop Y Commands Compensator __ Feedback Figure 3.12 Explicit Model Following (EMF) controller block diagram The EMF controller architecture assumes full-state feedback which implies that the entire state vector is either sensed or estimated from a reduced number of sensed variables (x_sing a Kalman filter). The pre-compensator and feedback gains are determined by solving a sin- gle Algebraic Riccatti Equation (ARE) which results in a control law that minimizes the LQR cost function (Ref. [22]). The synthesis of the EMF controller can be stated more ex- plicitly as the following LQR problem: Given: n=number of outputs (3.20) y= [A]y+ [B]u yisnxl m=number of controls uismx 1

(3.21) Ym = [Am] Ym + IBm] Um

_i_i _ i Find [K] where: (3.22)

u=[K] y

such that Jd is minimized OO (3.23) Jd- [ (Y-Ym) Q (Y'Ym) + u R u ] dt 'i: !

Where [Q] & [R] are positive definite weighting matrices multiplying the model following state error and control effort, respectively. The solution for the controller gains [K] is found by solving the following ARE: The matrix P is obtained from the following ARE _T _ _- 1 _T (3.24)

A P+PA-PBR B P+Q-0

Then K is obtained from: _-I_T

(3.25) K=-R BP

where: "A 0 0 (2n+m x 1) 0 A m B m (2n+m x 2n+m)

A=

(3.26) 0 0 D (2n+m x 2n+m) ii •

(3.27) .. Q

(m x m)

[!0i]

[ 0!lE0

"_, _ _ ,_i_ i Notice that [K] is (m x 2n) in size and contains both the pre-compensator and feedback gains. The pre-compensator outputs are a function of Ym which implies that the model dy- namics must be calculated as part of the control algorithm. The model following controller design requires that the pilot commands be modeled within the augmented system so that these commands may be excited by an appropriately placed initial condition. In this formu- lation, the pilot commands are modeled as washed-out step commands applied to the mod- el's control inputs. The augmented system [A] contains the additional terms, [D], which are washout filter's pole locations. This approach permits calculation of a feedforward •i_ • control (from the pilot inputs directly to (u)) while avoiding singularities that would occur in the solution of the ARE if there were rows of zeros in the matrix [A]. The matrix [D] is diagonal with elements that represent the speed of decay for each washed-out step com- mand issued by the pilot. The elements of [D] are chosen to be very small (.001) to model a non-decaying step input. When [D] is large, the pilot commands decay to zero quickly and the influence of steady state errors on the dynamic response performance is reduced.

The significance of the [D] matrix will be greatest during the integrated design synthesis where [D] is chosen to appropriately stress the importance of higher frequency dynamics in the overall measure of handling qualities.

The synthesis of an EMF controller for the F-80WRA requires an appropriate dynamics model which meets the Mil.-Spec. F-8785C Level 1, Class IV, Category A, handling qual- ifies criteria. The linearized equations of motion given in figure 3.13 meet the Mil.-Spec.

requirements for gust and commanded response and are used as the model dynamics in the controller synthesis. Figures 3.14 - 3.18 show the dynamic response of the model to pilot step commands in roll, pitch, yaw and excitation by 50 (f/s) vertical and lateral gusts. The transient and commanded response display the decoupled lateral and longitudinal motions characteristic of a conventional symmetric aircraft.

Mode Complex Frequency (rad/sex short period -2.6 + 4.1j phugoid -.007 5: .106j roll -.95 + 0.0j dutch roll -2.03 + 4.1j spiral -.027 + 0.0j Table 3.3 Mil-Spec F-8785-C model eigenvalues m_ .000 "6.000 .000 .284 0.000 -69.317 u -789.516 v 0.000 -.272 0.000 69.433 0.000 0.000 w -.230 0.000 -1.535 0.000 786.091 d .151 P 0.000 -.005 0.000 -.905 0.000 m q = 0.000 .002 0.000 -.021 0.000 -3.606 & r -3.863 0.000 .025 0.000 -.097 0.000 1.000 ¥ .000 .000 .000 .000 .000 i:i ¸ .000 e .000 .000 .000 .000 1.000 .000 .000 .000 .000 1.000 .000 .t.

mm u .000 .000 -32.052 V 32.052 .000 .000 W .000 .000 -2.804 P .000 .000 .000 q (dynamic matrix continued) .000 .000 .000 r .000 .000 .000 ¥ .000 .000 .000 e .000 .000 .000 m .000 .000 .000 .000 .000 -83.000 .000 30.000 .000 roll'] 15.000 .000 -5.000 dpitcl1 + .000 20.000 .000 .000 .000 37.000 d_yawJ .000 .000 .000 .000 .000 .000 .000 .000 .000 model equations of motion (ft-sec-rad) Figure 3.13 Mil.-Spec. F-8785-C Pitch Input Dynamic Response, 10 (deg) Elevator Deflection Mil Spec F-8785-C Model 30.

25.

Pitch Rate, q (deg/sec) 20.

15.

10.

gle of Attack (deg) 0. - I 0. 1. 2. 3. 4. 5.

Time (sec) Figure 3.14 model pitch input response Yaw Input Dynamic Response, 5 (deg) Rudder Deflection Mil Spec F-8785-C Model 50.

Roll Angle, _b (deg) Heading Angle, _(deg) 20.

10.

O.

Sideslip Angle, _ (deg) -10.

-20. I I I I 0. 1. 2. 3. 4. 5.

Time (sec) Figure 3.15 model yaw input response Roll Input Dynamic Response, 10 (deg) Aileron Deflection Mil Spec F-8785-C Model . 16a 7.

Angle, 6.

5.

Sideslil-_ p (deg/sec) 4.

3.

2.

/, 1.

! ! ! I 0.

"0. 1. 2. 3. 4. 5.

0. 1. 2. '3. '4.

.

Time (sec) Time (sec) Figure 3.16 model roll input response i.

Dynamic Response, $0 (f/s) Lateral Gust _•/iii', ¸ Mil Spec F-8785-C Model iii:i I i_, , 4.

Heading Angle, W(deg) 3.

2.

13 (deg) 1.

0.

-1.

Roll Angle, _b (deg) iI • -2.

I I I I -3.

0. 1. 2. 3. 4. .

Time (sec) Figure 3.17 model response, lateral gust Dynamic Response, 50 (f/s) Vertical Gust Mil Spec F-8785-C Model , k, (z (deg) , ° -2.

,i_ i! _ .4, -6.

_/ _ Pitch Rate, q (deg/sec) -8. I I I I ° l. 2. 3. 4. ° Time (see) Figure 3.18 model response, vertical gust The state vector for the model and the open loop OWRA is given by: (3.28) x-- [v, w, p, q, r, _]T where the states u, V, 0 have been omitted to remove the low frequency spiral and phugoid 'i modes from the dynamic response. The regulated (and sensed) variables are: (3.29) _ = [gy' gz' P' q' r, o]'r which includes the outputs to be minimized during the pitch-up maneuver, gy and 0. The state and output vectors (x, y) are augmented to include three additional states correspond- ing to a fh'st order lag model of the pitch, roll, and yaw actuators. Figure 3.19 shows the actuator dynamics model in block diagram form.

I; a _c +t, S I

5e 5 )%

where: _c = actuator commanded position = actuator achieved position "Ca = actuator pole location Figure 3.19 f'wst order actuator model For the EMF design synthesis the actuator poles are placed at s=-23.0 (rad/sec) which sets the actuator bandwidth to 4Hz.

The augmented equations of motion required for an EMF SAS design for the OWRA are e.

obtained as follows: Let: n = number of plant states = (6) m = number of actuators = (3) ' i¸;! : i,i The state vector (x) is augmented to include the additional actuator states, resulting in the following plant EOM's: (3.30)

[i][: :][:]+[°]

/

(n+m) × (n+m) (n+ (m) where: x=(v,w,p,q, r, _) = (_roll' _pitch' _iyaw ) arid "-'ga 0 0 'I;a 0 0" 0 -'1; a 0 .-- 0 "_a 0 G_ 0 0 "Xa 0 0 X a These equations may be written in abbreviated notation as: (3.31)

x-[X]x+[_]_

Similarly the model EOM's may be written: , !

(3.32)

[ ml [Am Bm][Xm] [O]E l

8mJ = 0 Fm _m + Gm

or in abreviated form: (3.33)

Xm=[_]_+[_]_m_

where[F m] and [Gin] are identical to [F] and [G] where

x a is replaced by 'gL-n- In this study x a is assumed equal to zm.

is now transformed to the output vector of sensed states _:

(3.34) - , )T

Y- ( gy' gz' P' q' r, _ 8roll, 8pitch, 5yaw The transform between _ and_ (i.e.[F]) is given by: (n+m) x (n+m) where: -W U o 0 0 o 0 0

Loo0oo

g g g 1 -U o 0 0 0 __ 0 0 0 -- 0 0 0 0 g g E _- Dm.

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 Transforming the plant and model EOM's into the _ coordinates: plant (3.36) model (3.37) y"_ = [l"m _m 1-"1] _m + [1-'m _m] _m c Transforming the plant and model EOM's into the _ coordinates: plant (3.36) model (3.37) _m= [rm_m r'l] _m+ [rm _m] _rnc The augmented (2n+m x 2n+m) system used for the EMF SAS design is obtained by combining the plant and model equations:

0 0

m

0 F

(3.38) F"I F m B m _+ Am m m

0 0 D

or: (3.39)

= + [B]

where [D] is a diagonal matrix which defines the pole location (washout) for the input excitation: D - -e [e I = pole location for the input excitation

[O

The EMF LQR problem may now be written in terms of the augmented system state vector _. The control law is given by the values of _ie which minimize the dynamic cost integral: Oo T (3.40)

Jd=j" {_T 6_ + 6c RSc} dt

, ...... , .: , ,, , • Where the output error and control effort weighting matrices ([Q] and[R] ) are given by: JR] = (m x m) matrix of control effort weights (2n+m x 2n+m) matrix of output error weights [Q] = -Q 0" (3.41)

[Q] = 0

0 0 0 and m Q = weighting on the output error ( y - Ym ) The solution for 8e which minimizesJ d is given by : [Ref. 23] (3.42) _Scopt = [K]_ Where the full-state feedback gain matrix is determined from: _I~T (3.43) [K] =-R B P and the matrix [P] is from the solution of the following A.R.E.: • ,,T 1,3' (3.44) AP+PA-PBR- BP+Q=0 Note that [K] is (m x 2n+m) and contains all of the gains for the pre-compensator and the feedback compensator : (3.45) [K]_ = [Kpc] ym+ [Kfc] y + [Kff] _rn c The SAS synthesized for the F-80WRA in this way is shown in a more explicit form in the block diagram of figure 3.20. Notice that the SAS gains are shown as three distinct com- ponents; [Kff] - feedforward gains, [Kpc] - pre-compensator gains, [Kfc] - feedback com- pensator gains.

Noise and gust disturbances :ii_, m i ÷, I [ Mil-Spec Ym Y open-_p I F-8785-C F-80WRA - I Model m 8me IDynamics (pilot commands) Kfc I I Figure 3.20 F-80WRA EMF SAS block diagram Implementation of the EMF SAS requires that all elements of the {y } vector be sensed (or estimated) including the actuator states. The dynamics of the model must also be simulated as part of the SAS control algorithm in order to generate the required pre-compensator out- puts. Figure 3.20 illustrates that the EMF SAS attempts to drive the output error (y - Ym) to zero. A less obvious point is that this statement is only true when the model dynamics are excited by commands from the pilot (Smc) and not when the aircraft is excited by gusts.

Gust disturbances only excite the (y) outputs of the system and the feedback compensator attempts to suppress gust disturbances without any regard to how the model would respond if excited by the same gust. The EMF SAS is therefore only capable of decoupling the commanded response of the aircraft and attenuates gust response instead of decoupling it.

If the plant and model experience simultaneous excitation from gust disturbances in the SAS design procedure, then the transient response of the plant will attempt to follow the model's. This possibility, however, is not feasible in the EMF SAS synthesis. The re- duced-order model following SAS synthesis will show how commanded and transient re- sponse decoupling can be achieved.

An EMF SAS was designed for the F-80WRA at the M=0.8, A = 45 ° flight condition.

Figures 3.21 - 3.22 show the plant and model matrices used in the synthesis. The weight- ing matrices [Q] and JR] were chosen to maximize dynamic response decoupling without excessive control effort. This is difficult to do because the EMF SAS has the tendency to require excessive feedback bandwidth such that the feedback compensated plant can easily follow the pre-compensator commands (and therefore track the model's response). As a result, very fast closed loop natural dynamics will occur unless the control effort weighting is large. For the results presented in this work the control effort weighting was chosen to i'__ '/jl _ constrain the actuator bandwidth to be no greater than 5Hz. The resulting SAS gains for the F-80WRA are shown in figure 3.23. The response of the closed loop aircraft to pilot commands and gusts is shown in figures 3.24 - 3.35.

i _ :=_i The model, open, and closed loop OWRA eigenvalues are plotted in figure 3.36. The EMF SAS has increased the amount of damping in all the modes beyond that present in the mod- el. This is typical for an EMF SAS because fast natural response is required in order to track the model's commanded response transients. The fastest eigenvalues (not plotted in figure 3.36 ) correspond to the actuator states (s = - 33.0 + 0j rad/sec) whose bandwidth has been "artificially" increased by full-state feedback. Because the actuator position can be ) ::: fed-back to the servo motor torque, the bandwidth characteristics of the servos can be arbi- trarily changed. The control weighting matrix [R] was chosen specifically to limit the bandwidth of these modes to less than 5Hz.

The gust response time histories (Fig. 3.30 - 3.35 ) show that the transient motion is still coupled (laterally and longitudinally) but highly damped. Therefore, the compensated OWRA has no distinct short period, dutch roll, etc. modes but instead has coupled highly damped transients. The roots plotted in figure 3.36 for the closed loop OWRA are not identified with the classical names because of their coupled nature.

The commanded response time histories (Fig. 3.24 - 3.29 ) show the model following per- formance of the closed loop OWRA. For the 4-g pitch-up maneuver, the normal accelera- tion and pitch rate track the model's almost exactly. The lateral motions are reduced but not completely attenuated by the SAS. The SAS attempts to limit the lateral motions as much as possible for the given control effort weighting but still the excursions in gy and _ are un- acceptably large (A(y/n) = .075 yields a Cooper-Harper rating of 4 based on Fig. 3.10).

From the simulation results it can be seen that the aircraft initially accelerates to the left as the leading edge suction (Cy) increases with angle of attack and then rolls fight in an at- iii ,_" tempt to minimize gy without excessive sideslip. This behavior reflects the basic lack of

controllabilitythatplagues thenominalOWRA configuration.Thereareno surfaces that

canindependently producesignificantsideforce andthereforetheaircraftmustroll (or side-

slip) to reducegy. The weightingmatrices[Q] and[R] (Fig. 3.23)usedin the SAS syn-

thesisrepresent only onereasonable choicefor theseterms. It is possibleto increase the

weightingon ¢ (to minimizerolling motionat theexpense of _ andgy excursions) andob-

tainresultsmorelike thosepredictedby theLoop ShapingSAS(Fig.3.11). Experimenta-

tion with various [Q] and[R] valueshaveshownthatfor all cases theOWRA's controlla-

bility deficiencyprevents theclosedloop handlingqualitiesfrom beingacceptable.

In a final versionof the SASdesignthelow frequency dynamicsmustalsobecompensated

sothattheaircraft aims atthecommanded flight condition. The resultsshownin figures

3.24- 3.29clearly showthattheaim requirement is not achieved by theproposed EMF

controller. Integralfeedback of the trackingerror(Y-Ym) will minimizethesesteadystate

errors. This portionof the SAScanbedesigned aftertheEMF gainsaredetermined.The

dynamicresponse mostcrucialto handlingqualitiesoccursin the first few seconds of mo-

tion (aftera command is issued by thepilot). In this thesis,theinitial dynamicresponse of

the aircraftwill bestudiedmostcarefullybecause of its stronginfluenceon handlingquali-

ties. The designof the low frequencyportion of the SASis not considered in this work,

althoughtheintegrated designsyntheses will ensure adequate controlauthorityto aim at the

flight conditionstudied.

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II II Lateral Acceleration vs Time EMF SAS, 45 ° Sweep OWRA .4 :j i _ / ,3 ,,****,,,oo***** J, ooo,o,o,,s***,o,o-,*****so,,,,,,,,,-,,,o,,,,***,,,o,,Foo,,ooo, ooo,,,o,, ii_ : : J o°°,,,°°,,,,,,,,_°,°, ,°°,,°,,,,,,,o**D,°,,o,,,,. i [_ Model { "°'"°'°'°'°°°°" j k ¸

I : : I-"_ F-SOWRA {

.1 ................ ; ................. : ..............................

_/i ii_' -.1 -.2 -.3 i • -.4 O. 1. 2. 3. 4. 5.

Time (sec) Figure 3.24 lateral acceleration vs time Normal Acceleration vs Time EMF SAS, 45 ° Sweep OWRA ° ° ............ _: ................. ,_i................. _o_o, ,................. ::. .................

"_ : • I---13-- F-S OW_A I : -1.

g_

-2. ............... .: ................. : ................. : ................. • .................

-3. .............. ;... I_lllMllllllll_l i 4, .

. 2. 3. 4.

° Time (sec) Figure 3.25 normal acceleration vs time • • z Pitch Rate vs Time EMF SAS, 45 ° Sweep OWRA 25.

i' I o,ooo°o°ooo.*,oo° °°°°o ...... °.,o, (deg/sec) q °°b°,*,Ot,,*,lB,,,W*iIDtIOIJ,eOllall°b,°°Ol°**lIOll***,o°°°*°°°**lO°°°°° 10.

5.

0.

.

0. 1. 2. 3. 4.

Time (sec) Figure 3.28 pitch rate vs time Yaw Rate vs Time EMF SAS, 45 ° Sweep OWRA ° . .- .- .- 4. °o,oo°,°oo,, .... -'o°°° .... °,ooo,°°°_°°°°°°°°°°°°°°°°_°°°°o°°o°,°°°oo,o_°°°°°°°°°,,°°°°o_ 3.

2.

............................ I...e7 F-sow

° • 1.

I I Illlll_lll[lll _0 I Illllllllllllll_l b III I I I l IIIII _ll_ll I IIIIIlll II III I_ IIll I IIII I 0 0 I 0 I I r 0.

.-..-..-..-..-..-..-..-..-./t -__-._-_ _-_-_:._-_ _-_-_:._-. _-_-_ _-_ _-. :.:_:_-. _:..-_ _-__-..-..-_ _-__-. ,:,G .:.:. ,:_&&,_.G,:, (deg/sec) i ,_ __._._..... : :: : ..':, ,. --':..-m._i-,_ ._,_.... _ _ _ _- ] -1.

• ,,,,,,,.,,.o,**_,, • ,.,o,°*,,o.a,, ,, ,,, ,,,,,, ,,,,.,, ,.o,*,,,,,,,,o,., ,,,,, ,.,°,. ,, ,,,.

-2.

-3. • , ,,o,,,,,D. ,, p,,,,, ,,,,,,,,,,,..',o**,o°_,,,,.,,,*_',, ,,o,,,,,,,,,.,,_ ...... ,.,,,o,.o, ...... ................. i................. i.............

-4.

-5.

.

0. 1. 2. 3. 4.

Time (sec) Figure 3.29 yaw rate vs time Roll Angle vs Time EMF SAS, 45 ° Sweep OWRA 25.

• • _ _..._L_ - .

20.

................i......... i.................i................. i.................

15.

//_/i%1 10. ,**°°°oo°o °°°°_o°oo****°°,°°°°,4.**°°°*l*°°°ooo*°°°°°o.°° oo*.°°o,°oo'_ ooo0oooooooooo°°° (deg) i I--6- F_ow.AI " ....

o • /•:i_• ¸' ° -5.

1. 2. 3. 4. 5.

° Time (sec) Figure 3.28 roll angle vs time Control Surface Deflection vs Time ElVlF SAS, 45 ° Sweep OWRA 15.

10.

Deflection " " _ right elevator [ (deg) 5. i i --*-_ I.................

-,) f, oo,,°°°o_oo°°***°ooo o,°°.

O.

-5.

J ° ° 1. 2. 3. 4.

Time (sec) Figure 3.29 control surface deflection vs time Lateral Acceleration vs Time EMF SAS, 1-g Vertical Gust, 45 ° Sweep OWRA .020 /i , .015 _ I .010 ...__ Model I .005 gy .000 -.005 -.010 -.015 I I I I -.020 .

0 1 2 3 4 Time (see) Figure 3.30 lateral acceleration vs time Normal Acceleration vs Time EMF SAS, 1-g Vertical Gust, 45 ° Sweep OWRA 1.2 1.0 .8 F-80WRA _ Model ] .6 gz .4 .2 .0 I I I I ' -.2 0. 1. 2. 3. 4. 5.

Time (sec) Figure 3.31 normal acceleration vs time Pitch Rate vs Time EMF SAS, 1-g Vertical Gust, 45 ° Sweep OWRA i i_ ° 5.

4.

.

F-80WRA Model [ q 2.

(deg/sec) 1.

° -1.

I I I I -2.

0. 1. 2. 3. 4. 5.

T_e(sec) Figure 3.32 pitch rate vs time Yaw Rate vs Time EMF SAS, 1-g Vertical Gust, 45 ° Sweep OWRA 2.0

[

1.5 1.0 F-80WRA

Model I

r .5 (deg/sec) .0 -.5 I I I I -1.0 4. 5.

2. 3.

0. 1.

Time (sec) Figure 3.33 yaw rate vs time i.

Roll Angle vs Time EMF SAS, 1-g Vertical Gust, 45 ° Sweep OWRA / iv, 1.0 , _i • F-80WRA

5 t I

.0 Phi (deg) -.5 -1.0 -1.5 I -2.0 4.

0. 1. 2. 3.

Time (sec) Figure 3.34 roll angle vs time Control Surface Deflection vs Time EMF SAS, 1-g Vertical Gust, 45 ° Sweep OWRA .

.

.

right elevator I___ left elevator redder Deflection (deg) 1.

° -1.

I I I I -2.

° 1. 2. 3. 4, Time (sec) Figure 3.35 control surface deflection vs time EMF SAS Closed Loop Natural Modes 45 (deg) Sweep OWRA .

i ." ." " " dl" . ...........i...........i............i............i. ......._..:.........................

........ ....... i....

.

Dli0 ****al;,.** ,l**,.*.l* *,,******_, Imaginary 2.

• Model (rad/sec) 0 o_ Loop F-S OWRA • ladF ClosedLoop ,,o°°.°,**'-,°°.°,°°°°°q.°°°,0°..o** .

o°°*°*,°°,°°-°°°* .

i: _ :: _......roll spiral | -1.

° -8. -6. -4. -2. 0.

-12. -10.

Real (rad/sec) Figure 3.36 natural modes, OWRA with EMF SAS 3.3.2.2 Implicit Model Following SAS Neglecting the inherent lack of controllability of the F-8 OWRA, the EMF SAS properly at- tempts to decouple the commanded response of the aircraft. The principle disadvantages of the EMF controller are: 1) The closed loop aircraft has natural dynamic modes significantly faster than the model's (for good tracking accuracy). The EMF SAS may therefore require high-speed, expensive actuators. Excessive bandwidth in the feedback path may make the system overly sensitive \ to unmodeled high frequency dynamic modes (such as structural flexibility modes) and noise disturbances.

2) The natural (unforced) response does not display the decoupling present in the model's response.

3) Full-state feedback (or an estimator ) is required.

The disadvantage of high bandwidth actuators and coupled transient response may be alle -_ viated by designing the feedback path of the EMF SAS so that it attempts to match the tran- sient response of the closed loop aircraft to that of the model. Reference [23] explains that the feedback path of the EMF SAS may be designed by techniques other than minimizing <

the quadratic dynamiccostintegral(equation 3.23)if specificconstraints on the transient

response areto bemet. Implicit ModelFollowing (IMF) [Ref.22] is a type of modelfol-

lowing controllerthatattempts to matchthe transient response of theclosedloop plantto

the model'sthroughuse of full-state feedback. The IMF method minimizes a quadratic

cost integral that contains an "implicit" representation of the model tracking error. Because the model following error is minimized through the feedback path, the resulting SAS will have unforced dynamics similar to the model's. The pre-compensator and feedforward gains of the SAS must still be designed using the EMF technique and the solution for the IMF feedback gains is obtained by solving a single ARE. Figure 3.37 shows the feedback portion of the IMF SAS in block diagram form.

Gust and Noise Disturbances .... =o.o... ............... o ........ =.. o o o o = o.. = ° = - ,.IDD._ I I i I I Open-Loop Plant _=Ay+Bu Compensator -- Feedback l Kfb IMF SAS minimizes (J): J= (E:T Q e + uTR u) dt Figure 3.37 IMF SAS block diagram The following mathematical description of the IMF SAS synthesis explains "implicit" mod- el tracking and shows how the feedback gains are determined.

Given: Plant Model o _= [A]y + [B]u Ym = [Am] Ym + IBm] Um Find (u) that minimizes (J) where the dynamic cost (J) is given by: (3.46)

u) j = (_- [Am] y) [Q] (_- [Am ] y) + u [R] dt

Note that ( y- tam] y )is approximately equal to( y- ym )

( i.e. if y --- Ym then [Am] y -- [Am] Ym )"

_i_i_: _i

Minimization of{( y- [A m] y )1"[Q] ( y_ [Am] y )}is implicitly related to )T )}which explicitly defines the model minimization of{( y - Ym [Q] ( Y" Ym following performance.

The solution for {u} is given in reference [23] as: (3.47) Uop t = [Kfb] y where: - 1 ^ ^ T (3.48) [Kfb] = -R ( S + B P) [P] is obtained from the solution of the following A.R.E.: A-l^ T A-l^ A -1 T A AT^-IA (3.49) (A-BR S) P+P(A-BR S)-PBR B P+Q-S R S=0 where the matrices _, 1_, _ are given by: A T (3.50) Q = (A- Am) Q (A- Am) A T (3.51) R=R+B QB ^ T (3.52) S = B Q (A-Am) It is essential to note that the IMF synthesis assumes that ( y = Ym ) and that the minimiza- tion of the implicit dynamic cost integral: (3.53) Jd = S(_m- _ )TQ (_m- _ ) dt i ¸ • r- will produce the same effect as the minimization of the explicit dynamic cost integral: o,o L (3.54) Jd -- _ (Ym" Y )TQ (Ym" Y ) dt i" 'i r .

Forlinearsystems with reduced controllability it may not be possible to force { y } to be

sufficiently close to {Ym] such that the previous assumption is true. As a result, the IMF designed SAS may produce inferior closed loop performance.

A simple linear system which demonstrates how an IMF controller produces model follow- ing transient response is given by the two degree of freedom torsion disc system shown in figure 3.38.

:i:i N../' I = Disc Iinertia GJ = Torsional stiffness T = External torque acting on disc O = Rotational angle of disc Figure 3.38 2-disc torsional vibration dynamic system The linearized equations of motion for the 2-disc torsion system are: m OO , ®

z 0]

0 -2k k 1 I

"'] .o

OO ® 0 -'_ k (3.58)

O 1 0 0 11[ + o Ti T2

00l

k 00

1 0 ® 00l The linearized equations of motion may be written in the form: x= [A]x+ [B]u r • / where:

x = (61, 62 , O1, 02)T

Torsional Stiffness GJ k--- _ Disc Inertia I C '17 = Damping Coefficient = -- I and the assumed values of the parameters are: I = 1.0 (slug-ft 2) i iiI ( lbf.

k=2.0 _)

_ iii! ii_i i

lbf-sec ) '_ = 0.5 (slug_ft2 For this example a model system, whose dynamic response is decoupled in 01 and 02, will be used to specify the desired transient response.

Model System Equations of Motion • 00 o . • °

O

O

"-1 0 -1 0

oo

O

O

0 -1 0 -1

2 2 (3.56) • =

O 1 0 0 0 O

0 1 0 0

O

O

• 2. • 2.

An IMF SAS of the form {u } = [K] {x } (where [K] is a full-state feedback gain matrix) is designed assuming [Q] = [1] and [R] = o[1]. Control systems are designed for several val- ues of the control effort weighting parameter o and a root locus of each closed loop sys- tem's eigenvalues is shown in figure 3.39.

t.

2-Disc Root Locus, IMF SAS Root Trajectories Indicate Increasing Control Effort Weighting

3.0

Loop Pole I 2.5 2.0 Imaginary 1.5 (rad/sec) 1.0 .5 Open Loop Pole

Model Poles I

.0 .0 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 0.0 Real (rad/sec) Figure 3.39 2-disc closed loop natural modes, IMF SAS For o = 0 (no control effort weight) the closed loop eigenvalues match the model's exactly.

Furthermore, the closed loop system matrix is identical to the model's indicating that the transient response of the closed loop plant will also match the model's exactly. As control effort weight is increased the response reverts to the coupled transients of the open loop system. In this example the IMF SAS is able to match the model's eigensystem exactly be- cause the system is sufficiently controllable.

Now consider the synthesis of an IMF SAS for the feedback compensation of the F-8 OWRA. The goal of the feedback controller is to force the transient response of the closed loop aircraft to be most like that of the Mil.-Spec. F-8785C model. A pre-compensator SAS which decouples the commanded response can then be designed after the feedback compensator is known. The system matrices for the open loop OWRA at 45 degree oblique sweep, M=0.8 and the model were previously given in figures 3.21 and 3.22. Ini- • ',_i _ tially the weighting matrices are chosen to be: [Q] = I [R] = .001 [I] This choice of [Q] and [R] ensures a SAS design which attempts to decouple the response as much as possible, with little regard to the control effort required. Applying the IMF , i SAS procedure (equations 3.46 - 3.52) the following full-state feedback controller gains are obtained: F-80WRA, 45 ° sweep, M=0.8, Q = I, R = 0 q r gy gz P 12.071 -40.027 -38.937 -.057 -.497 -.444 454 -.152 -.184 Kfb = 39.780 -12.136 -15.618 7[072 2.163 2.795 -2.178 7.060 6.925 ro_ pith yaw -21.981 69.547 171.461-] (matrix continued) -.284 1.380 2.002 I 4.090 -12.405 -30.913__] i i _i__ Figure 3.40 IMF SAS feedback gain solution for the F-80WRA The transient response of the closed loop system to an initial condition of a 1 "g" vertical acceleration is plotted in figures 3.41 - 3.45 along with the response of the model to the same initial condition. This excitation is similar to a vertical gust disturbance which pro- duces a phugoid and short period motion in a decoupled aircraft.

Lateral Acceleration vs Time 45 (deg) sweep OWRA, 1- g Vertical Gust .5 .4 .3 .2 .1 _//EMF SAS gy 0.0 -.1 -.2 -.3 -._ IMF SAS I I I I -.5 O. 1. 2. 3. 4. 5.

Time (sec) • ,_i(i I'_/ lateral acceleration vs time Figure 3.41 _:'i ¸ _ : Normal Acceleration vs Time jji" ", i 45 (dog) Sweep OWRA, 1-g Ve_ical Gust 1.5 , )ii_ Ii!_ _i i i,i 1.0 IMFSAS g .5 z .0 I I I I -.5 0. 1. 2. 3. 4. 5.

Time (sec) Figure 3.42 normal acceleration vs time Pitch Rate vs Time 45 (deg) Sweep OWRA, 1-g Vertical Gust .

5.

EMF SAS .

.

q (deg/sec) 2.

I y MOdel .

0.

MF SAS I I I I -1.

4.

. 1. 2. 3.

Time (sec) Figure 3.43 pitch rate vs time , _i_ , _ Yaw Rate vs Time 45 (deg) Sweep OWRA, 1-g Vertical Gust .

°

/

_MFSAS / Model r 0.

r (deg/sec)

\

-1.

F • I I I I -2.

° 1. 2. 3. 4. 5.

Time (sec) Figure 3.44 yaw rate vs time Roll Angle vs Time 45 (deg) Sweep OWRA, 1-g Vertical Gust .

O° I _ Model -5.

(deg)-10.

N_ EMF SAS -15.

-20.

I I I I -25.

1. 2. 3. 4. 5.

O° , i :i i '_ !i Time (sec) Figure 3.45 roll angle vs time _i'!:_i ¸•¸ i ::

The time historiesof theclosedloop transient response show that substantial coupling of

the lateral and longitudinal motions still exists. The longitudinal response (gz, q) appears to be following the model, but not very closely. The response of the same aircraft with the previously designed EMF SAS is also plotted in figures 3.41 - 3.45 and although the longi- tudinal response is not following the model any better than the IMF SAS, the coupling to the lateral motions (gy, ¢) is substantially less. This is not to say that the EMF SAS is ac- tually decoupling the transient response (it is not) but that the IMF SAS, which should have shown improved decoupling over the EMF SAS, has failed to do so. The reason for the poor performance of the IMF compensated aircraft is that for dynamic systems with re- duced controllability (such as the OWRA) the assumption that: (3.57, J = S( (y-[Am] y)T[Q] (Y - [Am] Y) +uT[R] u}dt is similar to: o_ f uT[R] ym)T[Q] (3.58) J=I{ (y- (y-ym)+ ul dt is no longer valid! As a result the IMF SAS solution for systems with reduced controllabil- ity may be unacceptable regardless of the choice of [Q] and [R]. In the next section a new SAS design method will be presented which solves the problem of achieving model follow- ing transient and commanded response.

3.3.2.3 Reduced Order Model Following SAS The previous sections have shown that both the IMF and EMF SAS have performance defi- ciencies when attempting to decouple the natural and forced response of an oblique wing aircraft. In this section a new type of SAS synthesis will be presented which decouples the commanded and unforced response, does not require excessive actuator bandwidth, and does not assume full-state feedback. The synthesis of this SAS is based on robust, re- duced order controller design techniques presented by Ly in reference 23. The SAS gains are found by minimizing a quadratic dynamic cost function using a nonlinear numeridal op- • i timization procedure. This requires significantly greater computation time than solving a single ARE (i.e. EMF or IMF SAS design) but the advantages of improved closed loop performance and increased flexibility in the controller architecture justify this cost.

Ly [Ref. 23] presents a controller synthesis method that provides robustness to plant uncer- tainties with a controller that does not require full-state feedback. Because the gains are found using a numerical optimizer, the controller architecture can have any linear form. Ly shows how to analytically evaluate the gradient of the dynamic cost function (J) with re- L spect to the controller gains and does this for inf'mite and finite time integrals of the cost function. In this thesis, some of the basic ideas in Ly's method will be incorporated into a L SAS synthesis scheme where the objective function and the SAS architecture are carefully ? ; ,L chosen to produce the desired model following controller. Controllers synthesized using

_4i_i;i _i

this new method will be called Reduced order Model Following SAS or RMF SAS. The synthesis of an RMF SAS is as follows: Given the plant and model dynamic systems: Model Plant :_m = [Am] Xm+ [Bm] Um

]= [A]x+ [B]u

Ym = [Zm] Xm

y = [Z]x

n = number of outputs (y) m= number of inputs (u) Consider the augmented dynamic systems: -1

0 0

ZAZ

U

0 -1 Bm

(3.57)

y=

Z m A m Z m Zm

0 e

(2n+m x 2n+m) (2n+m x m) b -1 Z m A m Z m 0 Z m B m

(3.58)

0 e 0 _m

0 0 e

(2n+m x 2n+m) I • : • i_ . ? ,, _i_ ,i_ , Where:

o[Y, Ym, Um]

' , ' :3 ,, _:/_, _;i "_.

! ,_ _ ii__ _m- [ Ym, arbitrarY, Um] T The dynamic cost function, which includes terms related to the model following error and required control effort, is defined as follows: (3.59) j= (y- ym) (_ (y- _m ) + u R l dt •: ::_' ii _'i where: , %', _= 0 0 (3.60)

[Q0i]

0 0 [Q] is the weighting on the model following error { y - Ym } and (J) represents a quadratic cost proportional to the weighted model following error where only the differences between { Y - Ym } contribute. It is important to note that this choice of { y - Ym } guarantees that initial conditions that excite the plant (Yo) will also excite the model transients if it is as- sumed that Yo = Ymo" This produces a true model following error when the plant is excited by any initial condition, i.e. those representing pilot inputs and those representing gusts.

The {Ym} outputs are included in the {y} vector so that the model outputs are available for the pre-compensator portion of the SAS. This method distinguishes between the model states implemented in the SAS pre-compensator and the additional model states which are excited by the same disturbances (gusts and pilot commands) as the closed loop aircraft.

This distinction is of key importance if the total response is to be decoupled. The resulting RMF SAS architecture can be expressed as: (3.61) u= [Kly Where any term in [K] may be arbitrarily held at zero to reduce the order of the contr611er i!i!•¸ i from that of full-state feedback. For example, in the OWRA synthesis, the actuator states have zero feedback gain associated with them so that these states are not required by the feedback and pre-compensator SAS segments. By carefully choosing {y} and [K] it is i possible to model any desired linear feedback control architecture in the RMF SAS synthe- i_ _: ii , • sis.

The solution for the nonzero terms in [K] is obtained by minimizing (J) using a numerical optimization scheme. The successive evaluation of (J) necessary for this type of solution is accomplished by solving the following system of Lyapunov matrix equations [Ref. 25]: ~ T T (3.62) (A+I3K) PI+PI(A+I3K)+(Q+K RK)=0 i (3.63) (A + g K)TP2 + V2(_m) + ((_) = 0 T (3.64) (_m) P3 + P3 (/_m) + ((_1 = 0 (3.65) p- P1-P2-P2 +P 3

[ ]

The dynamic cost (J) is given by: T (3.66) J = _¢o P -Yo Notice that the value of the initial condition {Yo} is required to evaluate (J). Full state feed- back LQR controllers have the same solution for the optimal controller gains regardless of the initial conditions considered. The design of reduced order LQR controllers depends on the initial conditions which excite the plant because full state feedback is not assumed.

Typically, more than one initial condition is critical to performance, therefore, a sum of the dynamic cost for various initial conditions is used to calculate the total dynamic cost func- tion: No

(3.67) X

Nic = Number of ic' s considered _o P Po i=1 " " The solution of each Lyapunov equation (equations 3.62 - 3.64) represents an integral of a portion of the dynamic cost from time equal to zero to infinity. If the solution to the Lyapu- nov equations is to be bounded, the closed loop system must be dynamically stable.

Dynamicstabilityrequiresthat: for i = 1 to 2n+m, _ is assumed positive R e (oi) <-_ where oi is the ith eigenvalue of the closed loop system and _ is a minimum damping re- quirement based on the feasibility of numerically solving the Lyapunov equations. This stability constraint is enforced by appending a penalty function to (Jd) that becomes active when the closed loop damping drops below the specified limit: N.

(3.68) J._ = z..a _'_ _ + ,e T PYo [0 if Re (c_ <-_ i i _'ki ( IRe03"_l - _)2 -I I if Re(_i) > -_ where k i is a penalty function weight.

Equation 3.68 is the complete form of the objective function (Jd) to be minimized. In refer- ence [23] Ly avoids the problem of enforcing a stability constraint by evaluating the dy- namic cost over a finite time interval.

Finite time dynamic cost function may be implemented: Tf , Jd = (y Qy+u Ru)dt instead of: Jd = (y Qy+u Ru)dt The optimization procedure to minimize Jd is then repeated for increasing values Tf until the value of J) converges to a minimum. This approach permits initial guesses for [K] which may produce an unstable plant, where as the inf'mite time integral approach requires a stable initial guess. Ly's finite time method is more robust to stability constraints but re- quires significantly more computation effort due to repeated minimization of Jd at various Tf. In the case of a SAS design for an oblique wing aircraft, it is easy to produce initial guesses for [K] that yield a stable closed loop system. Therefore, the infinite time formula- tion of the dynamic cost integral will be used in this work.

An RMF SAS is synthesized by using a Quasi-Newton (or Variable Metric) numerical opti- mizer [Ref. 19] to find the gains [K] which minimize Jd- This optimization procedure searches for a minimum along descent directions which improve in accuracy (based upon a second order model of the objective function) as the optimization progresses. The Quasi- Newton method is used in this work because it represents a satisfactory compromise be- tween the high computation costs of second order methods and the poor convergence speed i: i__ j: of fast order schemes. Gradient calculations are obtained from a numerical derivative ap- proximation (central difference) for the derivative of Jd with respect to the control gains.

Analytical expressions for the gradient given by Ly [23] could be implemented to minimize computation costs and this option is recommended for future work.

A block diagram of the RMF controller is shown in figure 3.46.

Model Dynamics Ym ym = [Am]Ym + [Bm]Um '.

Comk_ensator Feed-Forward (Noise & Gust Disturbance) (_ e

t

Open Loop Plant [ Ym= [Am]Ym + [Bin]urn Piio'U'- Pre-Compensator y = [A]y + [B]u & %f

P

Commands[ _ Feedback Compensator Keo Figure 3.46 RMF controller block diagram The model dynamics appear twice, fast in the pre-compensator to facilitate commanded re- sponse decoupling, and then in the formulation ofe = { y - Ym} so that the plant and model experience the same noise excitation for transient response decoupling.

t- .'i To better understand how the RMF SAS performs and is synthesized, a controller for the 2-disc torsion system presented in section 3.3.2.2 is synthesized and the results compared with the IMF controller performance for the same system. Two cases are considered, the i • first having a torque actuator on each disk, and the second having an actuator only on the first disc for reduced controllability.

_!ii_i _i , _: : r._ ,,, T 1 T 1 T2 Case 2 Case 1 (1) Torque Actuator (2) Torque Actuators Figure 3.47 2-disc torsion dynamic system The plant and model dynamic matrices are: Plant _= [A]y+ [B]u Y = (O1' O2' O1' o2)T Case 2 _Case 1 ]'1" "-.5 0 -4 2 1 0 0 1 0 -.5 2 -2

[B]=

[B]=

[A] = 0 0 1 0 0 0 0 0 0 1 0 0 _Model Ym = [Am] Ym -1 0 1 0 0 -1 0 -1 [Am]= 1 0 0 0 0 1 0 0 In order to maintain simplicity, only feedback compensation will be considered for the 2- disc controller. The goal of the control system is identical to that of the IMF example, to force the transient response of the closed loop plant to approach that of the model's.

i • The RMF controller dynamic performance is calculated as follows: i,__ :,_ i Plant y - [A] y + [B] u Model Ym = [Am] Ym Control Law u =[K]y ' ,c _ Dynamic Cost OO T R u) dt + Penalty Function for Stability Constraint Jd=f (:Q I_ + u o where: = (y- Ym ) Jd is computed by solving three Lyapunov equations T T 1) (A+BK) PI+PI(A+BK)+(Q+K RK)=0 T 2) (A + B K) P2 + P2(Am ) + (Q) = 0 T 3) (A m) P3 + P3 (Am) + (Q) =0 p- P1 - P2- P2 +P3

[ T ]

N.

lC T Jd= E Yo. P Yo. + P.F. (stability) i=l l 1 Full state feedback is assumed in this example, so that the results may be compared directly with those for the IMF SAS. The initial conditions used in the RMF synthesis are: 2 Disc RMF SAS initial conditions Y01=l 0 0 0 Y02 = 0 1 0 0 Y03 = 0 0 1 0 y04=0 0 0 1 Using these initial conditions, the total dynamic cost is simply:

Jd - Trace [P] + penalty functions on dynamic stability

_ J Case 1 Results Figure 3.48 is a root locus plot for the closed loop plant as the control effort weighting is increased. As in the case of the IMF SAS, the roots move from the open loop plant values at high control effort weight to those of the model at low control effort weight. Because the system is controllable, the closed loop dynamics can exactly match the model's when the control effort weight is small. It is important to note that although both methods yield the same dynamic response at the extreme range of the root locus, the root traces themselves are different indicating that the dynamic performance is not identical for all values of control effort weight.

2-Disc System Natural Modes Root Traces Indicate Direction of Increasing Control Effort Weighting ) _i _ 2.5 i i .

2.0 .............. __oo_ I ....

1.5 ............. !_ ......... i._i .............. i.............

Imaginary (rad/sec) 1.0 ............. _ .............. ..... _._.::._......: .............

: / _---_.

.5

......................... .............. iMPel _oo_ _iI O_n _o_ _oo_ i

.0 0.0 -1.0 -.8 -.6 -.4 -.2 -1.2 Real (rad/sec) Figure 3.48 2-disc system closed loop natural modes Case 2 Results In case 2 there is only one torque actuator available and it is therefore impossible to com- pletely match the dynamic response the model because of the closed loop system's inherent lack of controllability. Feedback controllers synthesized using IMF and RMF techniques show different performance for cases with reduced controllability. For the case where the control effort weighting is zero, the solution for the full-state feedback gain matrices and the closed loop eigenvalues are given in figure 3.49. Simulated time histories of the closed loop system response to the initial condition {y} -- { 1 0 0 0 }T are given in figures 3.50 - 3.53 for both the RMF and IMF SAS solutions.

R=0 B = [1, 0, 0, 0] T Q- [I]

RMF

IMF

K= [-1.38, .34, 1.731, -1.15] K = [ -3.96, -.20, -4.20, 1.48 ] Closed Loop Eigenvalues Closed Loop Eigenvalues i • S1,2 = -0.5 + 0.87j $1, 2 = -1.86+ 1.69j $3,4 --0.69+ 1.53j $3, 4 =-0.62+ 1.05j Figure 3.49 controller gains for case #2, 2-disc controller yl vs Time 2-Disc System yo=[1 0 0 0]T 1.0 .8 .6 .4 --='O-- RMF SAS Model I + IMF SAS yl .2 0.0 -.2 I I I I -.4 4. 5. 6. 7.

Time (sec) Figure 3.50 yl vs time y2 vs Time 2-Disc System yo=[1 0 0 0]T .4 .3 ?

RMF SAS .2 Model [ IMF SAS .1 0.0 y2 • i_ .

-.1 -.2 -.3 I I I I -.4 O° 3. 4. 5. 6.

Time (sec) Figure 3.51 y2 vs time y3 vs Time 2- Disc System, yo=[1 0 0 0]T .6 .5 .4 .3 RMF SAS _ Model IMF SAS y3 .2 .1 0.0 -.]

-.2 ° ° 4. 5. 6. 7.

Time (sec) Figure 3.52 y3 vs time • / y4 vs Time 2- Disc System, yo=[ 1 0 0 0]T (iii:i:!ii .5 ' i !:! "i "_¸:_:'; .4 .3 RMF SAS I_ Model I IMF SAS .2 .1 y4 0.0 -.1 -.2 I I I I I I I I I 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Time (sec) Figure 3.53 y4 vs time Inspection of figures 3.50 - 3.53 shows that the RMF SAS displays improved decoupling of the Y2 and Y4 states where as the IMF SAS shows slightly better model following per- formance of the Yl and Y3 states. A quantitative measure of the closed loop performance for each case can be obtained by evaluating the dynamic cost integral: Jd = I (ET[Q] E) dt (for zero control effort weight) Equations 3.62 - 3.64 describe how this is done and the results are given in figure 3.54 for the IMF and RMF controllers.

Jd-f'((Y-ym)TQ (Y-Ym) + UTR u} dt o.

Q= [I] R = [0] Yo = [1 0 0 0] T, [0 1 0 0] T, [0 0 1 0] T, [0 0 0 1] T :!i!i IMF: Jd = 5.30_ RMF SAS has 25% better performance than IMF SAS RMF: Jd = 3.97/ Figure 3.54 SAS performance - RMF and IMF The results show that the RMF SAS performance exceeds that of the IMF SAS by 25%.

This is because the RMF controller minimizes the cost function :

ii'_ ' • Jd=SJtQ]edt

where as the IMF controller minimizes the implicit form of this cost function: J d = f (Y-[Am] y)T[Q] (y_[Am ] y)dt As was noted in the IMF SAS results, the implicit formulation of the cost function can yield inferior SAS performance for systems with reduced controllability. The RMF SAS design uses the explicit formulation of the dynamic cost and therefore achieves superior perfor- mance in dynamic response, even in this extreme example where the control authority weighting is zero.

The 2-disc elastic system example shows how an RMF SAS can achieve improved model following transient response (when compared with other SAS designs) in problems with reduced controllability. In the next section an RMF SAS will be synthesized for the F-8 OWRA. The goal of the control system is to provide model following response in both the commanded and unforced response of the aircraft. The controller architecture will be re- duced order (i.e. not full-state feedback) so that the actuator bandwidth limitations will be preserved in the final closed loop design. The resulting closed loop OWRA's response to pilot commands and gusts will be compared to the OWRA's response with the previously designed EMF SAS.

The linear dynamics models used in the RMF SAS synthesis (F-8 OWRA with 45 ° oblique sweep and the Mil.-Spec. F-8785C model) are the same as those given in figure 3.22. The state and output vectors are assumed to be: (3.69) x = [v, w, p, q, r, t_, _iroll, _pitch' _yaw ]T T _ (3.70) x m = Iv m, w m, Pm' qm' rm' _ m' _mroll' _impitch' myaw ]

, ]T

_ i _ (3.71) Y = [gy' gz' P' q' r, _b _roll' _ipitch' _yaw

, IT

(3.71) Y = [gy' gz' P' q' r, t_ 8roll, _pitch' _yaw

5 T

(3.72) Ym = [gym' gzm' Pm' qm' rm' t_ m' 5mroll' 8mpitch' myaw ] • %:' : _ The control inputs are: •, , : (3.73) u c = [Scrol 1, _icpitch, 8cyaw ]T The Pilot Inputs Are: 5 T (3.74) Urnc = [Smcroll' 5tin.pitch, rocyaw ] All of the outputs are assumed to be sensed and available to the controller excepting the ac- tuator states: i:i _Sroll' _pitch' _yaw' 8mroll' _mpitch' _myaw This insures that the SAS will not artificially increase the actuator's bandwidth by feeding back actuator position to the actuator input command. The sensed quantities are thus: Ysensed = { gy, gz, P, q, r, _, 8mcroll, 8mcyaw, 8mcpitc h }T. Figure 3.55 shows the RMF SAS for the F-80WRA in block diagram form.

The controller uses 9 pieces of sensed information (Ysensed and Umc ), has 6 internal states (Ym), and affects 3 controls (Scrol 1, 8cpitch, 8cyaw). This implies that there are 45 control gains to compute in the RMF SAS.

The augmented system equations (Eqn. 3.57 - 3.58) are formed from the plant and model dynamics matrices. Also required is the value of the pilot command washout parameter, (e), which is used to stress the importance of high bandwidth dynamics relative to low fre- quency and steady state response, e is also used to define the eigenvalues of the arbitrary states included in the augmented model dynamics matrix (Eqn. 3.58). These arbitrary states are an artifact of the requirement that y and Ym have the same dimension and in no way do they affect the SAS design or the dynamic performance. For the RMF SAS syn- thesis, the actuator bandwidth is chosen to be 1.6 Hz. This fixes the actuator pole locations at s = -10.0 (1/sec). A bandwidth of 2 Hz or less reflects a more realistic model of the actu- atorsandis substantially lower thanthe5Hz bandwidthassumed in the EMF SASdesign.

Noise and Gust Disturbances) Open Loop F-80WRA v

--+4

x _ Ysensed x= [Alx+ [Blu + Feedback Compensator

lKfb]

Precompensator

Ym

A [Kpfl Feed-Forward Compensator [Kff] Model Dynamics

I

U mc Xm = [Am] Xm+ IBm] Um Pilot Commands Figure 3.55 RMF SAS for the F-80WRA The weighting matrices [Q] and [R] are chosen to produce the best possible closed loop re- sponse without excessive control effort. For the results presented in this section the follow- ing values were used: Control effort weighting Output error weighting _roll _pitch _yaw gy gz P q r [R] = diag [ 100, 100, 100 ] [Q] = diag [ 10, 5, 1, 1, 1, 10 ] Pilot command washout filter pole location e= 2.0 Actuator pole location s -- 10.0 (1.6 Hz B.W.)

Figure 3.56 RMF SAS synthesis input parameters

Unlike theEMF andIMF SAS's,theRMF SAS design is sensitive to the initial conditions

used to evaluate Jd" Ideally, the initial conditions should reflect the type of excitation that the aircraft will experience in flight (i.e. pilot commands and gusts). Figure 3.57 lists the initial conditions assumed for the RMF SAS design.

T vertical gust yo = [ 0, 0.5, 0, 0, 0, 0, 0, 0, 0, ] T lateral gust Yo = [ 0.5, 0, 0, 0, 0, 0, 0, 0, 0 ] T roll input 9o - [ 0, 0, 0, 0, 0, 0, 0.5, 0, 0, ] pitch input Yo = [ 0, 0, 0, 0, 0, 0, 0, 1.0, 0 ]T T yaw input -Yo = [ 0, 0, 0, 0, 0, 0, 0, 0, 0.5 ] Figure 3.57 RMF SAS initial conditions If the dynamic response to initial disturbances of each state variable were instead used to calculate Jd (i.e. Jd =Trace[P]), then the closed loop performance with respect to the subset of physical initial conditions would be reduced.

The RMF SAS synthesis is carried out using the Quasi-Newton numerical optimization procedure to solve for the 45 control gains which minimize Jd" In this example, 2000 ob- jective function evaluations and 30 line searches were required to converge to a solution, when the initial guess for the gains was zero. This computation cost includes all of the ob- jective function evaluations necessary to numerically estimate the gradient of Jd with re- spect to the 45 SAS gains.

The closed loop performance of the F-80WRA with the RMF SAS is shown in figures 3.58 - 3.68. Response to gust disturbances and pilot command inputs are presented along with similar results for the OWRA with an EMF SAS.

t.

The simulation results show that the RMF and EMF SAS have nearly identical performance in decoupling the commanded response of the OWRA. For the 4-g pitch up maneuver, both SAS designs produce a 20 degree bank and a 0.3 g lateral acceleration. These large excursions in lateral motion are due to a lack of independent sideforce control, as was pre- , ii ¸¸" , ._

viously described in theEMF SASsection.Theresponse to verticalgustinitial conditions

showsthattheRMF SASattempts to decouple thegustresponse, but it is not obviousthat

itsperformance is anybetterthantheEMF SASwhich merelyattenuates thegustresponse.

In reality, thereis a significantdifferencein thenaturalresponse of thesetwo cases.The

differenceis mostclearlyseen in figure3.69wherethe eigenvalues corresponding to the

four rigid bodydynamicmodes of theclosedloop aircraftareplotted. Figure3.69clearly

showsthatthe RMF SAShaseigenvalues thatvery closelyapproximate thoseof the model

in bothfrequencyanddamping,whereastheEMF SASeigenvalues areall substantially

fasterthanthe models.The "closeness" of theRMF eigenvalues to themodel'sindicates

thattheRMF SASproduces anunforcedresponse thatattempts to follow thatof themodel

andis not merelya very fast,attenuated response asin thecaseof theEMF SAS.

Not shownin figure3.69aretheeigenvalues corresponding to theactuatorstates.The

EMF SASactuators havemorethantwice thebandwidthof the RMF SAS(i.e.5 Hz vs

1.6Hz). Nonetheless, theRMF SASis still capable of decouplingthe commanded re-

sponse aswell astheEMF SAS. This is because theeigenvalues of the RMF closedloop

systemarevery closeto thoseof themodelandthecommanded response transients are

generated naturallyin thefeedback pathandneednot begenerated in thepre-compensator (

EMF SAS) ascommands to betracked.Overall,theRMF SASappears to beanideal

controllerfor obliquewing aircraftbecause it decouples boththecommanded andunforced

response of theaircraftwithoutexcessive requirements in actuator performance.

_i_ _ i _ Lateral Acceleration vs Time i_i _ •i !

45 (deg) Sweep OWRA, 4-g Pitch-Up .2 Model I .1 I ................................... _ RMFSAS I .............................

: I

0.0 !

-.2 I., /i_ !ii_ _ _i _ii/_ _i/¸ • -,4 z 0. 1. 2. 3. 4. 5.

Time (sec) Figure 3.58 lateral acceleration vs time Normal Acceleration vs Time 45 (deg) Sweep OWRA, 4-g Pitch-Up .

o I _ Model

k I_ _FSAS

-1.

gz -2.

i ..... " ............ { _ EMFSAS ................ , .................

oo -3.

i • -4.

•_ i "• L _ - i ° o 1. 2. 3. 4.

Time (see) Figure 3.59 normal acceleration vs time _ } ..... .......

Pitch Rate vs Time 45 (deg) Sweep OWRA, 4-g Pitch-Up i;?

,,,, °°°,,°._°.,,,,,lIl..,,a°b,,**°°°.,,.,,,.*,_,°°., .o,,,,ooo.oo °,.,.°°..,o°..., : _ Model

I _/ I. : I--¢-- _VSAS I : [

15. . ..... "-,.. ....... , ............. ' ......................

i*;/_: q (deg/sec) <i 10 ......................................................... :.................

_i _• " • ii_ : 5............ 2._i_ ............ ,.......................

<F 0.

.

.

2. 3.

0. 1.

Time (sex:) Figure 3.60 pitch rate vs time Yaw Rate vs Time 45 (deg) Sweep OWRA, 4-g Pitch-Up .

1.

................. i.............. i ................. i................. i ...............

0.

-1.

r -2.

(deg/sec)

i m Modol i

-3.

_..ooooooo,,o.o.oo, -4.

ii_iiiiiiiiiiiiiiiiiiiiiiiill ............... i...... iiiiiiiiiiiiiiiiiii

-5, -6.

.

1. 2. 3. 4.

° Time (sex) Figure 3.61 yaw rate vs time Roll Angle vs Time 45 (deg) Sweep OWRA, 4-g Pitch-Up 25.

20.

15.

Model RMF SAS EMF SAS ° i¸¸ -5.

° 1. 2. 3. 4. .

Time (see) Figure 3.62 roll angle vs time Control Deflection vs Time 45 (deg) Sweep OWRA, 4-g Pitch-Up 12.

10.

p°,°°°°o°o°°o°°°°o'* ° ° °°*°°*o°°.'°°°, J*° o °o°. °..° 8.

i ................. Left el vator _ ........

6.

.o°.°°,.o,°°°°..°°°°.,°.°°°°_°

i -. ...............i........... i

4.

.:. Right el ator .....

Deflection (deg) 2.

......... ; ................. ? ................ ; ................

0.

-2.

°.. ,°.* 0 ,,,,,.*...,_.*,*******,,,°° ° -4.

,,,,,,,°,,.,°°,,°_***,,,,,°l********r***,,*******.,,,,o° ,,,°,°°,***°°...,%°°,,.,..... ,, ,, -6.

° ° 2. 3. 4.

Time (sec) • i .¸ Figure 3.63 control surface deflection vs time Lateral Acceleration vs Time 45 (deg) Sweep OWRA, 1-g Vertical Gust ,, •_i ¸ .06 .05 .04 .03 gy .02 .01 .00 -.01 -.02 ° 1. 2. 3. 4. .

Time (sec) Figure 3.64 lateral acceleration vs time Normal Acceleration vs Time 45 (deg) Sweep OWRA, 1-g Vertical Gust 1.2 1.0 .8 .6 gz .4 .2 .0 -.2 o 1. 2. 3. 4. ° Time (sec) Figure 3.65 normal acceleration vs time Pitch Rate vs Time 45 (deg) Sweep OWRA, 1-g Vertical Gust i _II,: i ° 5.

4.

, q (deg/sec) 2.

.

°,,°°q°o°oo,°°,°o°l*,,_ o° °°°°°°°°oo°°°°°°'°°°°°°°°°°°°°°°°°°ooo .... °°°.°°°°° O° -1.

° Oo 1. 2. 3. 4.

Time (sec) Figure 3.66 pitch rate vs time Yaw Rate vs Time 45 (deg) Sweep OWRA, 1-g Vertical Gust ° ° 0°°°°ooooo°o.o.°,°°o°o°°o°_°,°,o°°°°°oo,.°°°°,°°°°.°oo° ...... o,°°o°0°o ...... °o° | | RMF SAS r l ._ Model ............. EMF SAS .................................

o (deg/sec) o ° 2. 3. 4. 5.

Time (sec) Figure 3.67 yaw rate vs time : i. • i Roll Angle vs Time L" 'ii _'! i _i _ 45 (deg) Sweep, 1-g Vertical Gust .5 .0 -.5 -1.0 / " I_ (deg) .................. i...... ..... i............

-1.5 -2.0 i_ , ,,,.,,,,,,,,,,,,**,,_,,,,,, i _,,,,._ .... ,,,., .......

-2.5 I I I I I IlllllllII IIlIlllIIlllllll IlIll I I I I_lll I III I I I IlIII I Illl IIII I I IIllll Ill -3.0 ° 0. 1. 2. 3. 4.

Time (sec) Figure 3.68 roll angle vs time RMF SAS Natural Modes 45 (deg) Sweep OWRA .

. - 2, :. ?, .

............ 0"..:- i........... ..... ...... i............ ..... ".-"i ............ ..... ii. ........ IN_ ,_.m .......................

.

............ "........... i.... .*..:.._.-_._._ ...... !..._ ..... .._',. ............

Imaginary 0 Open Loop °°°.°°°° .... _°°°, _°°°°°°°°o°°°_.°°°°°°.°°°°_°°° ......... • .... ° .......

(rad/sec) 2.

• Model IF RMFSAS 0 EMFSAS . _°°°°°°...°°_,.,°. ....... i ..... °°°°°°°i,°°,°° ..... ° °°°°°°°°°.,,_°,,.

: rr_ rz_rq

.

I_ -10. -8. -6. -4. -2. ° .

Real (rad/sec)

Figure 3.69 RMF SAS natural modes

3.4 SAS Choices for Integrated Design Synthesis

The integrated design synthesis method, which will be presented in the next chapter, at- tempts to improve the OWRA's handling qualities by simultaneously synthesizing the air- craft's configuration and its SAS. A numerical optimization procedure, similar to that used in the RMF SAS synthesis, is used to solve for the optimal configuration and SAS gains.

The numerical solution technique requires that feasible SAS architectures be automatable in the numerical solution for the dynamic performance. Both the EMF and RMF controllers L,'•. • • fulfill this requirement and have been shown to produce closed loop performance represen- tative of what is achievable given the reduced controllability of the nominal F-80WRA configuration. In the initial integrated OWRA design syntheses, the EMF controller will be implemented, because it simplifies the optimization procedure substantially by solving for all of the control gains in a single A.R.E. solution. After the capability of the integrated de- sign method has been demonstrated with EMF controllers, RMF controllers will then be synthesized to show how more realistic, practical, and efficient control may be accom- plished in the integrated synthesis.

: ",_ :_i _

Chapter 4

Integrated Design Synthesis Method

(MIDSM)

The closed loop handling qualities of the F-80WRA do not meet the desired Mil.-Spec.

level I requirements due to an inherent lack of controllability in the proposed design. A possible solution to this problem involves adding control surfaces that will provide the air- craft with independent sideforce control. This course of action is undesirable because the additional weight, drag, and complexity of these surfaces will d_crease the performance of the oblique wing configuration considerably. A more desirable solution is to somehow re- configure the existing aircraft such that the weight and drag are essentially unchanged, but the controllability is substantially improved. In this asymmetric design more than others, configuration design changes that might improve controllability are not obvious because of the strong aerodynamic and inertial coupling that affects the dynamic response. An inte- grated design technique which can simultaneously account for the influence of configura- tion and SAS changes on the closed loop handling qualities is required to synthesize an in improved design. In this chapter a multidisciplinary, integrated, design synthesis method (MIDSM) will be presented which is capable of re-configuring the F-80WRA for im- proved flying qualities. Although the original motivation for creating MIDSM came from the need to improve oblique wing handling qualities, the method is extended to include per- formance measures from other disciplines (structures aerodynamics, aeroelastics, etc.) so that the trade-off between handling qualities and more conventional measures of perfor- • :( mance (drag, weight, etc.) may be studied. In this chapter the integrated design method MIDSM is presented and is used to solve two example problems: tail sizing for minimum trimmed drag with longitudinal handling qualities constraints and tailless aircraft wing weight minimization with aeroelastic constraints. Chapters 5 and 6 will present the results for the integrated OWRA synthesis.

i _i. _ ' • i ¸ ': "

4.1 Introduction to MIDSM

The dynamic response of an aircraft is often an important aspect of its performance; yet the analysis of handling qualities and control system design are often performed after the major aerodynamic and structural properties have been established. In many cases, this sequen- tial approach to multidisciplinary design leads to suboptimal results. The method presented , i¸.

here (MIDSM) integrates the design of the aircraft and its control system in order to obtain better mission performance than could be achieved in a sequential design procedure. By minimizing a cost t_nction consisting of both conventional performance criteria and a meas- ure of aircraft handling qualities, a design with maximum performance for a specified level of handling can be achieved. Handling qualities are measured using a quadratic cost func- tion similar to that used in the design of optimal feedback control systems. This function is proportional to the difference between the dynamic response of the aircraft and a "model" case with dynamics that are considered acceptable. The variables to be optimized may in- clude both aircraft configuration parameters (e.g. span, tail area, skin thickness) and con- trol system feedback gains. The design variables are determined by an unconstrained nu- merical optimization procedure, using penalty functions to enforce both explicit and implicit constraints. The method is most useful in the simultaneous synthesis of airframe and flight control systems to achieve improved handling or improve performance with a specified lev- el of handling quality. In certain cases results obtained by this integrated synthesis proce- dure are substantially better than those obtained by the usual sequential design methods.

Previous research on the integrated synthesis of dynamic systems and full state feedback controllers is described in references 8 and 9. Sakawi et. al [Ref. 9] optimized the wing •i and tail geometry of an actively-controlled aircraft excited by gust disturbances. The syn- thesis assumed a full state feedback control architecture and used a random search proce- dure to solve for the global minimum. In reference 8, Zeiler and Weisshaar describe the in- tegrated design of a four degree-of-freedom aeroservoelastic system. The design variables • i"( :- .i,'_ consisted of the elastic axis location and the full state feedback control gains. A multi-level- linear decomposition scheme was used to solve for the optimal system variables.

In the work presented here, an integrated synthesis procedure is extended to handle any lin-

earcontrolsystem architecture includingreduced-order controllersandpassive (no control-

ler) designs.The procedure is usedto optimizethesystem's commanded response aswell

asits unforcedresponse. Multidisciplinaryperformance indicesareconsidered, illustrating

thetrade-offsbetween handlingqualitiesandothermeasures of performance.

Two examples areusedto demonstrate themethod. Reduced-order controllersandmulti-

disciplinaryperformance indicesarefirst introduced in the synthesis of anaft-tail aircraft.

The proposed methodis usedtofind thetail sizeandsmilemarginthatproduceacceptable

handlingqualitieswith minimumdrag. Whenthecontrolsystem gainsareincludedasde-

signvariables,theprocedure automatically synthesizes staticallyunstable configurations

andan appropriate reducedordercontrolsystem.

The minimumweightaeroservoelasilc designof afree flying aircraftrepresents a highly in-

tegrated structuralandcontroldesign.The second exampleconsiders this problemby syn-

thesizinga taillessaircraftwhichexperiences a coupledshort-period/wing-bending flutter

mode. The synthesis procedure determines thesparcapthickness(asa functionof span)

for minimum wing weightwith acceptable handlingqualities. In somecases a reducedor-

dercontrolsystemis alsodesigned.This synthesis differs from conventionalflutter sup-

pressionstudiesbecause not only musttheflutter modebestable,but thephugoidand

shortperioddynamics mustmeetspecifiedhandlingquality criterion.

In somecases, handlingqualitiesmaybethe onlyperformance measure thatthe designer

wishesto improve. This typeof problemis solvedin chapters5 and6, whereanoblique

wing aircraftis designed for improvedhandlingqualitiesby simultaneously optimizingthe

wing geometryandcontrolsystemgains.The integrated synthesis procedure usesconfigu-

rationvariablesto enhance thecontrollabilityof theclosedloop system.This produces an

aircraftwith handlingqualitiessuperiorto thoseachievable in a sequential designproce-

dure.

The basicapproach is outlinedin figure4.1. The designmethodis anunconstrained op-

timizationprocedure which minimizesa composite objectivefunction,J, consistingof three

terms:the non-dynamic performance measure, Jnd,theweighteddynamicperformance, Jd, and the constraint violation penalty function, Jpen: (4.1) J = Kd Jd + Jnd + Jpen _i ":.:i:i) ¢ i _ Quasi-Newton Optimizer 5, ",Z ,

I

( Objective Function Calculation) Non-Dynamic Penalty Dynamic Performance (_: /i: i' i Function Performance Constraints ! Jd) (Jnd) 0pen) Aerodynamics • Mass Properties / _i _i ¸ i:i,: • Flight Condition / • Linear EOM s / •L yapunov or A.R.E.: / Jd 71 (ET [I_ E+ uT [R] u ) dt / _.__............__/, Stability Cons.

[ Total Objective Function [ A ] J = KdJd+ Jnd+ Jp en ] Figure 4.1 MIDSM flowchart The non-dynamic performance measure describes the mission performance that is not di- rectly related to handling qualities. Typically, this function is the structural weight, drag, or direct operating cost, etc., of the aircraft. The content of this term is problem-specific and will be discussed further within each design example, but for cases in which only the handling qualities are to be improved, this term is excluded.

The dynamic performance, Jd, provides a measure of the aircraft's handling qualities and t- its computation was described in chapter 3 for both EMF and RMF SAS's. It is calculated from a quadratic cost function identical to that used in the design of optimal control sys- tems; but the control system architecture is not restricted to full state feedback when a RMF SAS is assumed. Dynamic performance is related to the difference between the aircraft's • _i "¸

response anda modelcasechosen for its desirable handlingqualities. In general,the con-

trol gainsareincludedasdesignvariablesin thesynthesis.This permitsthedesignerto

choose anycontrollinearsystemstructure: reduced order,full statefeedback, or a passive

designwith no controlleratall. Controlsystems canbe synthesized which tailor theforced

and/orthe unforcedresponse of theaircraft (aswasshownin chapter3). In thecomposite

objectivefunction,Jdis weightedby a constant, Kd, which is usedto specifytherelative

importanceof handlingqualitiesin thedesignsynthesis.As Kd is variedfrom zeroto a

very largenumber,theoptimalsolutionmovesfromonein which dynamics arenot consid-

ered,to onewhich is requiredto achieveacceptable handlingqualitiesatthe expense of all

othertypesof performance. Configurationdesignvariablesaffectthe handlingqualities

(andthereforeJd)by alteringtheaerodynamic, flexibility, andmass propertiesof thede-

signbeingconsidered.Particulardesigns mayrequirea completeaerodynamic analysisof

anunusualaircraftconfiguration whosegeometry is changed duringthesynthesis.An aer-

odynamicanalysismethodthatcanaccurately predictthe aircraft'sforcesandmoments

with little CPUtime is desired.Thevortexlatticemethod(LINAIR) described in chapter2

fulfills theserequirements for designs in which a thin lifting surface modelis appropriate.

Thethird termof theobjectivefunction,Jpen,consistsof penaltyfunctionswhich areused

to enforceanyexplicit andimplicit constraints in thesynthesis.

A numericaloptimizerbased onthe Quasi-Newton or VariableMetric GradientMethodis

usedto solvefor thedesignvariablesthatminimizeJ. This optimizersearches for a mini-

mumalongdescent directionswhich improvein accuracy (based on a second ordermodel

of the objectivefunction)astheoptimizationprogresses [Ref. 19]. This optimizationpro-

cedure waschosen because it represents a satisfactory compromise between the high com-

putationcostsof second ordermethodsandthepoorconvergence speed of first order

schemes.Otheroptimizersmay bebettersuitedfor particularproblems.For example, if

the non-dynamic performance index is knownto havelocalminima,a morerobustoptimiz-

er may berequired.Theexamples presented heredemonstrate thata varietyof intere_sting

problemsmay beformulated suchthatlocalminimaareavoided,thuseliminatingtheneed

for suchrobust,but inefficient,optimizers.

4.2 Calculation of the Dynamic Performance Index

ii! ¸ Calculation of the dynamic performance index is identical to the method shown in chapter 3 for the RMF SAS. Jd is a scalar equal to the weighted integral over time of the difference between the state vectors of the aircraft being synthesized and a model system whose dy- namic response is considered ideal. An additional term representing the control surface ac- tivity is also included.

Interpretation of J d Jd is a scalar measure of the aircraft's handling qualities. Because there is no way to relate this number directly to a Cooper-Harper rating or a Mil. Spec. F-8785-C classification, ad- ditional analysis of the synthesized design (simulations, frequency response, etc.) must be performed to determine the adequacy of a design's handling qualities. The formulation of Jd does, however, guarantee that for fixed [Q] and [R] matrices the handling qualities will improve as Jd decreases. It is this fact that enables this synthesis method to improve the handling qualities in each optimization iteration and allows the designer a means of trading dynamic performance for non-dyn',,Lrnic performance.

The dynamic performance index calculation requires that the linearized equations of motion for the aircraft be created as a function of the design variables at each objective function evaluation. This portion of the synthesis can be the most costly in terms of CPU time, par- ticularly if the aerodynamic stability derivatives must be re-evaluated. The overall utility of this method relies on the careful choice of the analysis routines which evaluate the [A] and [B] matrices. Methods which capture the essential physical phenomena and minimize com- putation time are desired.

4.3 Example Applications

t_ 4.3.1 Aft-Tail Design for Minimum Trimmed Drag Two examples are presented to illustrate the use of this method. The first of these, and the simplest, is the design of a wing and tail system. The configuration is required to trim at a selected lift coefficient while minimizing drag and retaining adequate longitudinal handling [ qualities and control authority. The design variables include horizontal tail area and wing location; in some of the designs a reduced order controller (consisting of angle of attack / feedback to the elevator) is also synthesized.

_i i ,,i _ _i :i_ _ 8=k

O3

Tail Area (S t ) Wing Location (X) W Goal: Minimize Trimmed Drag With a Given Level ij" ; of Handling Quality Design Variables: Tail Area, Wing Location, Angle of Attack Feedback (optional) Figure 4.2 aft tall design for minimum trimmed drag with handling quality constraints Dynamic Performance The dynamic performance Jd is computed based on a model case with wing location and tail area sized to provide Mil. Spec. 8785C level 1 response in the short period and phu- goid dynamics. The longitudinal dynamics are modeled using axial velocity, u, plunge ve- locity, w, pitch rate, q, and pitch angle, 0, as states. The design variables are horizontal tail area, S t, wing position (relative to the c.g.), X w, and a feedback gain, Ko_, from sensed angle of attack to elevator deflection for the cases with a reduced-order control sys- tem. Jd is calculated from the difference between the model and subject aircraft's dynamic response to the same initial disturbances. This guarantees that as Jd is minimized the short period and phugoid dynamics of the aircraft will approach those of the model case.

Non-Dynamic Performance The size of the horizontal tail and static margin required for acceptable handling qualities may result in a design with excessive trimmed drag. The drag coefficient is used as the non-dynamic performance measure so that trade-offs between handling qualities and drag may be studied. The drag coefficient is calculated analytically assuming elliptic loading on the wing and tail: 1 C 2 Jnd = %- (_+ CD2w ) Lw + CDow W i _ _ii_i ii:i ¸¸: + ( X-'_ + CD2T ) _T + CDoT I + 2( g ARw ) Cl.w CLT This formulation accounts for parasite drag, lift dependent viscous drag, and the vortex drag associated with the interfering lifting surfaces (under the assumption that they are cop- lanar).

Constraints Three constraints are enforced during the design synthesis: 1) Pitch trim at a specified lift coefficient 2) Trim at maximum lift without tail stall 3) Dynamic stability Dynamic stability implies that the largest real part of any eigenvalue must be less than zero.

This constraint must be enforced explicitly because the Lyapunov equation solution for Jd is only valid if the system is dynamically stable.

Aft -Tail Design Results Figure 4.3 shows the trade-off between trimmed drag and the handling quality parameter for the aft tail design synthesis. Curves are shown for designs with and without a feedback control system. Each point on the curves represents a unique design that is optimal for a fixed weighting of handling qualities. As the weighting on handling is increased, Jd de- creases and the dynamic response of the aircraft approaches that of the model case. Note also that the trimmed drag increases with improved handling quality. This occurs because the optimal tail size and static margin increase as Jd decreases, with a subsequent increase in parasite and trim drag. Designs with feedback control show reduced trimmed drag for a fixed level of handling compared to designs without control systems. The synthesis meth- od has recognized that relaxed static stability and smaller tail size can reduce trimmed drag, while feedback control can ensure adequate handling qualities by providing artificial stabili- •, " ; i 7-?,; ". " :' ' " " " ty. As a result, designs that are statically unstable and have optimally designed reduced or- der controllers to provide stability are automatically synthesized. The values of Jd for which the longitudinal dynamics me_t the Mil. Spec. 8785C level 1 handling quality re- quirements are marked on each curve.

Aft Tail Design Results Trim Drag vs Handling Quality 1.00 .99 .98 .97 C D .96 •!i • .95 .94 Level (1) Handhng .93 .92 10 1° 10 -s 10 -6 10 -4 10-2 10 o Handling Quality Error ( Jd ) Figure 4.3 aft tail drag vs handling quality Because the selected control system does not provide rate feedback, adequate damping re- quires some tail area; thus it is not possible to eliminate the tall completely. Even if large values of Jd (tx)°r handling) are accepted, trim constraints still yield a non-zero tail area when the wing pitching moment at zero lift is not zero. This leads to the fiat part of the curve with feedback at higher values of Jd" The smile margin and tail volume of the optimal aft tail designs are plotted in figures 4.4 and 4.5 as a function of dynamic cost weighting. Designs with negative static margin are inherently unstable and rely on the active control system to provide stability. The design with the smallest tail (and consequently the lowest drag) which meets the level 1 handling quality criteria is shown in figure 4.4 to be an inherently unstable design.

i _ , !iil i, Figure 4.6 shows the eigenvalues of each optimal design as the handling qualities weight- _/i_ ¸, ing factor K d is increased. When the weighting is large, the eigenvalues associated with both the short period and phugoid modes are driven to those of the model case. This ei- genvalue analysis is one means of assigning a physical interpretation to the handling quali- fies parameter, Jd. Another way of determining the handling qualities of a given design is t ¸¸' • / to study the time history of its dynamic motion. Figure 4.7 shows the dynamic response of two aft tail designs, with and without feedback, to a vertical gust disturbance. Both de- signs meet the Mil. Spec. level 1 handling quality criteria because their short period re- sponse is adequately similar to that of the model's.

Static Margin vs Dynamic Cost Weighting Optimal Aft Tail Designs ; ; : : ] 1_ with feedback ----O--- no feedback [ Level 1 Handling Qualit_

L

Static i!iiiiiiiiiiiiiill .............. ............. .............

Margin 5.

(% stable) ° ::Unstable -5.

-5. -4. -3. -2. -1. 0. 1.

Log (dynamic cost weight) Figure 4.4 static margin vs handling quality • :i /i _ _ • , i¸ ': _ _ Te.il Volume vs Dynamic Cost weighting Optimal Aft Tail Designs .25 I _ with feedback I IT °vel 1 Handling, t_,alitv I I-..o-- no feedback I I_ . _' "_" _( !, .20 ,,,,,,o,,,,,,.,,,,,,,,°°,,,,_,,,.,,,°,,,,.,*,,,,,,,, ,. •,,,,,,.,,°,,,* Tail .15 Volume .10 .05 ,,,,,,,,,,,,,, .00 ° -3. -2. -1. 0.

-5. -4.

Log (dynamic cost weight) Figure 4.5 tail volume vs handling quality Optimal Design Root Locus Aft Tail Synthesis ° o without feedback Short Period with feedback °

<> Model root locations

° Increasing Kwd Imaginary " " O (l/sec) O.

V oid -1.

-2.

-3.

.5 -1.5 -1.0 -.5 .0 -3.0 -2.5 -2.0 Real (1/sec) aft tail handling quality root locus Figure 4.6 Pitch Angle vs Time Optimal Aft Tail Designs, Vertical Gust Step Response .

• • ° ° • ° ° •

:: no f ga k !

.

....... :................ I t • :...._¢_.. : ...... ,_ : ..... - ! i i ! i " ! : ..__ , Theta (deg) 0.

-2.

_iii!iiiiiliiiii_iiiiiii!iiiiiiii_ ........i........i........i .. " u;_,llwi_ _'c I.."........

( • _ i _ -6.

0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Time (sec) Figure 4.7 optimal aft tail gust response 4.3.2 Tailless Aircraft Flutter Suppression The second example deals with a tailless aircraft designed for minimum wing weight and an acceptable level of handling quality. Tailless aircraft with swept wings may exhibit a unique flutter mode, characterized by a coupling of the short period dynamics with the wing bending modes. The frequency of this flutter mode is slow (typically 1-2Hz) and corresponds to the wing bending and short period frequencies. Reference 26 describes a high performance tailless sailplane (the SB-13, figure 4.8) for which this flutter mode was the critical factor in the structural design.

l

Figure 4.8 SB-13 sailplane 3-view

Usingthe designprocedure presented in this paper,a swept-wingtaillessaircraft,operating

in a flight regimewhere flutter is critical, is designed for minimum structural weight with specified longitudinal handling quality. The spar cap thickness at various stations along the span are used as design variables. Some designs include a feedback control system with elevons deflected in proportion to the wing tip deflection to help control the flutter. The de- sign synthesis involves solving for the skin thickness distribution (and elevon feedback gain for cases with active control) that yield a stable aircraft with distinct short period and phugoid modes while using the least amount of structural material in the wings. The inte- grated design procedure improves the handling by simultaneously suppressing the flutter and driving the unrestrained dynamic modes to be most like that of a rigid aircraft. This

•i!i_!i:_

differs from conventional flutter suppression techniques which only guarantee flutter stabil- ity and do not attempt to restore acceptable handling qualities to the unrestrained modes of the free flying aircraft.

Active control of elevons (optional) Bending structure model is _e = Ks_bt upper and lower spar bt Vortex lattice aerodynamics model Goal: Minimize structural weight with acceptable longitudinal handling qualities Design variables: Spar cap thickness (at 20 spanwise stations), optional active control of elevons 7 I ' • •: • Figure 4.9 tailless aeroelastic design synthesis _', _ • _I_ Figure 4.10 shows the layout of the swept wing tailless aircraft whose structure (and possi- bly control system ) arc to be synthesized. The design flight speed is chosen so that a wing sized to support the static structural loads will not be stiff enough to avoid flutter. This choice of trimmed flight condition ensures that the dynamic response will be dominated by aeroelastic phenomenon. The root locus shown in figure 4.11 indicates that the short period mode and the lowest frequency wing bending mode become coupled as airspeed increases, ultimately resulting in flutter at 200 if/s) airspeed.

I _y (0,0) Draying !s to .scale,

(3.5,0)

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(5.31,0)

Fuselage (1001bs) 5: ¸ " ' located at CG (22,11.39) Stiffness Distribution (22,14.02)

_ (lin_)

EI [ ] I [ [__xe4 (lb-ft^2) , Mass Distribution / m 3 0b-f t) Figure 4.10 tailless aircraft planform and structure Tailless Flutter Root Locus (Minimum Wing Weight Aircraft) 50.

I

40.

Wing Bend Flutter 30.

V = 200 (f/s) Imaginary 20.

(rad/see) Short Period 10.

.

-20. -15. -10. -5. 0.

Real (rad/sec) Figure 4.11 tailless aircraft flutter root locus • _i_' I¸ iI_ :i Dynamic Performance A flutter problem of this type is unusual because acceptable handling qualities requires not i_'_• • only that the flutter be suppressed, but also that the short period and phugoid dynamics be acceptable. To achieve these results, a rigid aircraft with longitudinal dynamics meeting Mil. Spec. F-8785-C level 1 requirements is used as the model for the design synthesis.

As Jd is minimized, the tailless aircraft's response will approach that of the rigid model case, ensuring a suppressed flutter mode and acceptable short period and phugoid re- sponse.

In reference 8 it is suggested that the present definition of Jd is not a sufficient measure of handling qualities for the flutter suppression problem. If the design synthesis considers only one flight condition, it is possible that the optimized design will be unstable at veloci- ties below the design value. This situation can be avoided, however, by including in the dynamic performance index, the sum of Jd'S evaluated at several flight velocities, up to and including the design flight condition. Designs synthesized using this composite Jd will be dynamically stable and have acceptable handling qualities over the entire flight envelope. In this example problem only one flight condition will be considered.

The linearized equations of motion (E.O.M.) for an unrestrained elastic flight vehicle must be formed to evaluate Jd" Lagrange's method is used to derive the E.O.M. (see chapter 6) assuming two wing bending modes (rl 1,r12) and all longitudinal degrees of freedom (u,w,q,0). Linearization with a finite state model is possible because the wing bending is modeled using two assumed modes and the aerodynamics model is quasi-steady. This for- mulation of the E.O.M. has the advantage that the rigid aircraft dynamics are separable from aeroelastic effects, permitting the designer to see the influence of structural flexibility explicitly in the E.O.M.. Figure 4.12 shows the linearized EOM for the tailless aircraft with wing flexibility (assuming a typical wing mass and stiffness distribution) and the rigid wing model. In some cases a reduced order controller is also designed with feedback of the sensed wing tip deflection to elevon deflection. Aerodynamic stability and control de- rivatives are predicted by a quasi-steady vortex lattice method, appropriate because of the low reduced frequency of this flutter mode. Wing mass and stiffness properties are recom- puted as the skin thickness is varied during the optimization.

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Constraints Penalty function constraints are imposed to limit the maximum stress in the spar caps at a 3- g load factor, a minimum skin gauge is also imposed. Maximum skin stresses are calculat- ed using a static aeroelastic analysis that accounts for inertia relief and the effect of wing deformation on the spanwise loading. A third penalty function is used to ensure dynamic stability of the system. This is identical to the constraint implemented in the aft tail design synthesis. The combination of these three constraints ensures that all designs will be strong enough to meet the static load requirements and be aeroelastically stable.

Tailless Flutter Design Results Results for the tailless aircraft design example are shown in figure 4.13. As in the previous case, each point on the curves represents an optimal design with a specific value of han- dling quality weighting. Smaller values of Jd indicate improved handling, and increased values of weight ratio correspond to increased wing weight. The regions in which han- dling quality becomes acceptable lie to the left of the "level 1" limiting marks. Designs with feedback control show reduced wing weight since elevon deflection can provide artificial stiffness without additional material in the skins.

In a sequential design procedure the wing structure is first sized for minimum weight based on staile aeroelastic loading and minimum gauge requirements. The reduced order control system is then designed for the best handling quality with a fixed wing design. The result- ing sequentially-designed aircraft has a stable flutter mode but its short period and phugoid dynamics are still highly coupled to the wing bending mode giving poor handling qualities.

This design is represented by the point marked with an asterisk in figure 4.13.

Tailless Flutter Design Results Wing Weight vs Handling Quality

.45

.40

.35

Without Feedback

.30

• i I Weight .25 Ratio

.20

With Feedback .15 .10

.05

162 ld _ 1.0

Handling Quality Error (Jd) tailless flutter results, wing weight vs. handling quality Figure 4.13 By contrast, the integrated design procedure achieves a stable flutter mode with acceptable rigid body dynamics and does so with the least penalty in wing weight. The eigenvalues and eigenvectors approach those of the model case as the handling quality weighting is in- creased. This is important to note, because the handling qualities for this example only be- come acceptable when the short period, phugoid, and wing bending modes are distinct and properly damped. Figures 4.14 - 4.16 show the response of the Level 1 design with feed- back and the best sequential design to a step vertical gust. The phugoid mode of the se- quentially designed aircraft has lower damping and shows distinct coupling to the wing bend and short period dynamics. The level 1 integrated design, however, displays dynam L ic response more like that of the model's and does so with substantially less control effort from the elevons.

'i' (u) Velocity vs Time Tailless Aircraft, Vo=200 (f/s), Step Vertical Gust 30.

20.

10.

level 1 integrated design, w/feedback u if/s) O.

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. ,i _' best sequential design, w/feedback : -20.

\ 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Time (sec) Figure 4.14 (u) velocity vs time Angle of Attack vs Time Tailless Aircraft, Vo=200 (f/s), Step Vertical Gust ° .

ll_SQQ,°°*_Q,ll,° 4. ....... _ ......

3.

..._....°°._°,...o offdeg) 2.

best sequeniial design, w/feedback 1.

: : : : : : 0.

level 1 w/feedback : -1.

O. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Time (sec) Figure 4.15 angle of attack vs time H ;' Wing Tip Deflection vs Time Tailless Aircraft, Vo=200 (f/s), Step Vertical Gust 1.0 : ; "_ • . : : : : .8 _levei 1 in_ated "_ desi_, w/f_edbac_k I ..... i ....... i ......

.6 .4 ,i_ i.

'"i ....... i ....... _'best _'quen.tiai delsign' w/feed!ack [ Tip Deflection .2

fit)

0.0 -.2 -.4 • _ : : : -.6 10.

1. 2. 3. 4. 5. 6. 7. 8. 9.

.

Time (sec) Figure 4.16 wing tip deflection vs time Optimal Spar Cap Thickness Distribution Spar Cap Thickness vs Semi-Span Fraction .14 .12 •_- wing designed for static loads only .10 • level 1 handling, with feedback Skin .08 0 level 1 handlin[, without feedback Thickness

fit) .06

.04 .02 .00 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 Semi-Span Fraction optimal spar cap thickness vs semi-span Figure 4.17 The optimal solutions for the skin thickness as a function of semi-span are shown in figur_ 4.17. Aeroelastic stability requires that the skin thickness be increased at the wing root, with greater thickness required for cases without a feedback controller. Interestingly, the results also show increased skin thickness at the wing tip. The presence of this additional mass further separates the frequencies of wing bending and short period dynamic modes and is a significant factor in 'he achieving flutter stability.

• i i

4.4 Summary

The method described here for aircraft design optimization with dynamic response consid- erations provides an inexpensive means of integrating dynamics into aircraft preliminary design. By defining a dynamic performance index that can be added to a conventional ob- jective function, a designer can investigate the trade-off between performance and handling.

The procedure is formulated to permit the use of control system gains as design variables, but does not require full-state feedback. The examples discussed here show how such an approach can lead to significant improvements in the design when compared with the more common sequential design of system and control law.

This integrated design procedure is most useful for studying problems in which handling qualities and other types of mission performance are highly coupled. Many problems of current interest fall into this category: 1) Aeroservoelastic design of flexible aircraft 2) Aeroelastic tailoring of aircraft with composite structures 3) Design of unstable aircraft for minimum trimmed drag Problems in which only the handling qualities are to be improved are also readily solved by this method. Examples of this type of problem include the design of unstable aircraft for super-maneuverability and the integrated synthesis of oblique wing aircraft for improved controllability.

Chapters 5 and 6 will show how the MIDSM method is used to re-design the F,8 OWRA and its SAS for improved handling qualifies. Initially, aircraft with an EMF SAS will be considered at single and then multiple flight conditions. The RMF SAS will then be imple- mented and the influence of wing flexibility will be included in the synthesis.

Chapter 5

Integrated Design of Oblique Wing Aircraft

In this chapter the integrated design synthesis method, MIDSM, is used to simultaneously re-configure the F-80WRA and re-design its controller to obtain handling qualities which exceed those achieved by the closed loop nominal configuration. Five configuration design variables, that determine the location and orientation of the wing relative to the fuselage, are considered (along with the SAS gains) in the MIDSM synthesis to further decouple the OWRA's dynamic response. Because most of the aerodynamic coupling (which signifi- cantly affects handling qualities) is caused by the airloads acting on the wing, optimization of the configuration variables which determine the wing placement provides a rational ap- proach for improving the dynamic response without significantly affecting drag and weight. Constraints are placed on the range of the wing displacement to ensure that the configuration is feasible from a structural standpoint and so that drag is not significantly ef- fected. Trim in 1-g level flight is guaranteed by placing constraints on the maximum con- trol surface deflections and sideslip angle required to trim. The dynamic performance is measured using the same quadratic dynamic cost integrals described in chapter 3 for the EMF and RMF SAS designs. The MIDSM OWRA synthesis is very much like the OWRA SAS design problem considered in chapter 3, except that now there are configuration varia- bles which provide additional degrees of freedom with which to further drive the OWRA's dynamic response toward that of the Mil.-Spec. F-8785C model.

Several types of MIDSM syntheses are performed in this chapter. First, the OWRA with an EMF SAS is synthesized for improved handling qualities at a single flight condition.

This example demonstrates that MIDSM is capable of significantly improving the OWRA's • •r .•i ¸ _ i• • handling qualities and shows the details of how the synthesis is carded out. In the second example a single OWRA design with an EMF SAS is synthesized for improved handling qualities at several flight conditions. Although the control gains in the SAS may be changed as flight condition varies, the configuration of the aircraft remains fixed after it is built and must therefore be optimal for performance over a range of flight conditions.

These multiple flight condition syntheses demonstrate that MIDSM is capable of 'robust' configuration synthesis. The last example in this chapter synthesizes the OWRA with an • _ 7 • _i!_/_i_ RMF SAS instead of an EMF SAS. This requires that the optimizer search through a de- sign space that includes all of the controller gains (45) and the configuration design varia- ::!:4,¸•¸i? :_!

bles (4). The results of this example demonstrate that the configuration changes which im- • 7!:. :i::_: !_ prove the OWRA's controllability are independent of the SAS architectures considered.

This analysis is continued in chapter 6, where the influence of wing flexibility on dynamic performance and the optimal MIDSM OWRA configuration is studied.

-. :%

5.1 Aerodynamics and Dynamics Models

Chapters 2 and 3 described how the aerodynamic •loads and the dynamic response of an ii__ _ !i::' oblique wing aircraft are accurately predicted with a minimum computational cost by using a vortex lattice aerodynamic analysis and linearized equations of motion. Figure 5.1 is a

i!:i :•:II

block diagram which illustrates how the components of the LINAIR vortex lattice analysis are incorporated into the MIDSM synthesis. In chapter 2 it was shown that for relatively large changes in the wing's position relative to the fuselage, the assumption that the [AIC] and [VIC] matrices remain constant during these configuration changes is quite good. This assumption is therefore implemented in all of the F-80WRA MIDSM synthesis studies presented in this work. The [AIC] and [VIC] matrices are computed once for the initial guess configuration and as the optimizer re-positions the aircraft's wing, only the boundary conditions are updated in the solution for the airloads. Omitting the re-calculation of these matrices substantially reduces the computation cost of the aerodynamics calculations. It is important to note that the validity of this approximation is strongly dependent on a choice of design variables which only alter the wing's position and orientation. If the wing planform or sweep were to be changed, the assumption that the[AIC] and [VIC] matrices are not sig- nificantly affected would be in gross error. In all of the cases presented in this work, the final MIDSM configuration performance results are predicted using [AIC] and [VIC] ma- trices calculated for the actual configuration (and not those of the initial guess).

The stability and control derivatives are estimated by perturbing the states {_ 13,p, q, r } and the control surface deflections to numerically calculate the required derivatives. As was mentioned in chapter 2, LINAIR provides accurate predictions of all of the aerodynamic of initial guess _nfiguration from I Create vortex lattice geometry 1 inputs I Com ute and store AIC and VIC_ m_nPce_ _ for the initial guess ) C Objective Function Calculation )

I_a_ newvor_ laui_ geometry

yen the configuration design variable_

V

ISolve for the trim state variables (_, _, etc t __ Compute vortex s_ength from:

[A_C]_'= {gas } l

I /" I whereRHS is a function of the new[

f ' -x / [ geometry and the state vector l '[Calculate stability and control 1./ "_ I / _l derivatives using central difference [,. ]_ ,_M derivatives J_ •

i - omputofor esan, mo

Figure 5.1 aerodynamic loads calculation in MIDSM loads except for yawing moment (CN). The unusual interference which occurs between the wing and fuselage (due to thickness) cannot be predicted by a thin surface aerodynam- ics model. Fortunately, the influence of the wing's position relative to the fuselage has little effect on the pitch-yaw coupling (for a fixed wing sweep - see figure 2.33). There- fore, the wind tunnel data for C N as a function of wing sweep and angle of attack may be used to predict the yawing moment for all of the configurations considered in this work.

As the wing is re-positioned on the fuselage the aircraft's inertia properties are changed.

The importance of inertial coupling on handling qualities dictates that this effect be included in the design synthesis. Given the wing and fuselage mass properties (relative to their own mass centers) the following relations describe the total aircraft inertia for any wing position and orientation: Inertia Tensor Calculation -d 1 4 3

(5.1) itota 1= if& t + A [iw] AT +

-02 43 mw

SYM..._ 2 2 d 1 + d 2 where: coordinates Iw = wing inertia tensor about the wing C.G., expressed in wing m If& t = fuselage and tail inertia tensor about the aircraft C.G. expressed in body ax Itota 1 = total aircraft inertia about the aircraft C.G., expressed in body axes d = vector from the aircraft C.G. to the wing C.G. in body axes m w = wing mass A = transform from wing local axes to body axes The linearized equations of motion for a proposed OWRA configuration are obtained by substituting the aerodynamic stability and control derivatives, flight condition, and mass property data into equations 3.14 - 3.17. The CPU time required to form the EOM is very small in comparison to that required to evaluate the airloads.

In all of the MIDSM OWRA syntheses the following constraints are enforced by appending the appropriate penalty functions to the dynamic cost: 1) Trim at 1-g level flight with small control surface deflections and sideslip angle.

2) Dynamic stability 3) Explicit limits on the feasible range of each configuration design variable.

The trim constraint is enforced by computing the control deflections and sideslip angle re- quired to yield no sideforce or moment about the C.G. and a specified lift coefficient. If the deflections exceed specified limits then a penalty is added to the objective function. The following system of linear equations is used to find the aircraft's trim deflections: Linearized Trim State Solution m u B _ Cya Cyl3 CySroll CySpitch CySyaw A(I " m 5 :

C z

roll (5.2) pitch] yaw] ................................... Cngyaw Because the angle of attack (o0 is computed as part of the trim solution it cannot be known in advance for appropriately linearizing the aerodynamic derivatives. Therefore, an iterative solution is required, stopping when the angle of attack value converges. Typically, this takes less than three iterations. The final computed value for the trimmed angle of attack is used as the condition for linearizing the stability and control derivatives for the dynamics analysis. Table 5.1 list the maximum acceptable deflections for the trim variables.

Trim state Maximum value (absolute) 2 (deg) 5 5 (deg) roll 5 5 (deg) pitch 5 5 (deg) yaw Table 5.1 1-g level flight trim constraints

5.2 Application of MIDSM to the F-80WRA

Because the loads acting on the wing contribute most to the aerodynamic coupling, it is rea- sonable to try to improve the handling qualities by re-positioning the wing relative to the fu- selage so that the aerodynamic coupling is favorably altered (without significantly affecting drag and structural weight). Five configuration design variables are chosen which deter- mine the relative position and orientation of the wing and fuselage, along with the wing's symmetric dihedral. Figure 5.2 defines these variables. The location of the wing pivot along the fuselage (Xpf) determines the wing's placement relative to the C.G. and therefore significantly affects static pitch stability. The chordwise location of the wing pivot (Xpw) influences the amount of wing area projected on either side of the fuselage as the wing sweeps obliquely. Small changes in this variable can strongly influence the roll-pitch and pitch-roll coupling. If the wing pivot axis is not parallel to the z body axis, then the wing's orientation relative to the fuselage will change as sweep varies (Fig. 5.13). In all cases studied, it is assumed that the plane of the wing is parallel to the x-y body axis plane when the wing is unswept. This assumption forces the configuration to remain symmetric at zero wing sweep. Pivot axis tilt and wing banking most significantly influence the pitch to sideforce and the pitch to roll coupling of the aircraft by tilting the resultant aerodynamic force vector as the wing banks.

The presence of a skewed pivot axis introduces some ambiguity in the definition of wing sweep because the wing does not rotate about the z body axis. In this study, sweep is de- fined as a rotation about the wing pivot axis. This produces differences in sweep (between the wing pivot and the z body axis) of less than 1 degree at 65 degrees sweep for a wing bank angle of 15 degrees. Wing incidence (at zero wing sweep) is the only variable that needs to be defined to completely describe the wing's orientation relative to the fuselage.

This parameter is not considered as a design variable because it merely defines the decalage between the wing and tail, which can be arbitrarily changed by a symmetric elevator deflec- tion of the full-flying elevators. The final design variable is the symmetric dihedral built into the wing (assumed parabolic). Dihedral affects the pitching and rolling moments_re - quired to trim and also influences dynamic stability. The dimensionless wing tip deflection (rl) fixes the amount of dihedral in a given design.

Pivot Axis Orientation P_abolic Dihedral ii _ z z' 01 _Y _\ ] / _ _ Wing Tip __ _ Deflection _ rl= wing tip deflection / reference Pivot Axis sp_m _ii¸ _ _¸ Pivot Location Relative to the Fuselage Pivot Location Relative to the Wing Root Chord Xpf g Pivot Xpw • :: i: _I / Figure 5.2 confi_ration design variables i _• !i ¸•¸'¸• i Increasing Wing P • • i ¸, i Tilted Wing Pivot \ [ __ Increasing ---.,==g • !==,--- wing sweep 1' causes the _._ wing to bank Top Front Figure 5.3 effect of pivot axis tilt on wing bank angle Each design variable is constrained during the synthesis to not exceed a range over which structural weight or total drag would be adversely affected. Table 5.2 lists the limits of the design variables assumed for all design synthesis considered.

/ • Cef abs(aXpw) _ abs(AXpf) < Cre f abs( dihedral ) < .05 abs(01) < 8.0 (deg) / Implies that the wing bank angle can not exceed 16 (deg) at any

/

abs(82) < 20.0 (deg) sweep angle.

Table 5.2 configuration design variable constraints The analysis methods required to evaluate the dynamic cost and constraint violations for an arbitrary oblique wing design have been presented in chapters (1-4) and may now be incor- porated into the MIDSM objective function calculation. This will be done in the next sec- tions for the cases of several types of OWRA syntheses. Initially, the SAS architecture is assumed to be EMF, so that the optimal SAS gains are computed as part of an optimal con- (,',i_ : ,_ trol sub-problem at each objective function calculation. This removes the burden from the 7_/i ¸:, optimizer of having to search for the optimal gains explicitly. Single flight condition syn- theses are performed fast to show how the MIDSM method is capable of improving oblique wing handling qualities with the configuration variables considered. Because the configuration variables cannot be altered in flight, they must be chosen for improved dy- namic response over a range of flight conditions which is determined by the aircraft's par- ticular mission. The second MIDSM example addresses this issue by synthesizing a a sin- gle OWRA for improved handling at both M=0.8, 45 degrees sweep and M=I.6, 65 degrees sweep. In chapter 3 it was noted that the RMF SAS offered advantages in terms of robustness, transient response decoupling, and actuator bandwidth requirements when compared to the EMF SAS. The remaining MIDSM syntheses assume an RMF SAS archi- tecture for the F-80WRA which requires that the optimizer explicitly solve for the control gains. This dramatically increases the computation costs of a synthesis when compared to the EMF SAS designs. The influence of SAS architecture on the optimal OWRA configu- ration is revealed by these results.

5.2.1 MIDSM OWRA Synthesis, EMF SAS, 45 ° Sweep, M = 0.8 The simplest MIDSM oblique wing synthesis is to simultaneously re-configure the F-8 OWRA and design its SAS for improved handling qualities at a single flight condition (SFC). The flight condition considered in this example is 45 ° wing sweep, M--0.8, and dynamic pressure = 440 (lb/ft2). The optimizer solves for four configuration design varia- bles explicitly and 63 control gains implicitly (in the EMF SAS synthesis) to minimize the dynamic cost and satisfy the constraints. The configuration variables are the location of the wing pivot along the fuselage (Xpf), chordwise location of the pivot along the wing (Xpw), wing bank angle (_D), and parabolic wing dihedral (rl). The wing bank angle (_b) defines the wing pivot axis angles 01 and 02 such that when the wing is swept 45 ° it ap- pears to banked (¢_b) (left wing low when viewed from front) with zero incidence change . i' ._ • i_ (Fig. 5.4 ).

Tilted Wing i," :i, " :.

/¢ _'_ Specified wing bank I _ _ angle (¢b) at a given i¸¸ • Y

\ I / sweep determines the

y value of 01 & 02.

Z 1 Front Figure 5.4 specified wing bank angle Because the EMF SAS architecture assumes full-state feedback, the optimal control gains are computed implicitly by solving the associated ARE at each objective function calcula- tion. This offers the advantage of reduced computational costs for the MIDSM synthesis, but restricts the SAS architecture to the less practical EMF scheme.

The dynamic cost is evaluated as follows: Dynamic Cost ~ T T ) Jd=_ (y-ym) Q(y-_m)+U Ru dt Jd is calculated from the [P] matrix (obtained from the solution of the EMF SAS Riccatti equation) and initial condition information.

N.

IC T Jd = E Yo. P Yo. Nic = Number of I.C.'s considered i=l 1 1 This calculation for Jd is similar to that presented in the EMF SAS design procedure in chapter 3. One important difference is that when only the SAS is synthesized, the resulting values of the optimal control gains are independent of the initial conditions (Yoi) thafexcite the plant. The dynamic cost is dependent upon the plant excitation, but the values of the SAS gains which minimize Jd do not depend on these initial conditions if the system is lin- ear, time invariant, and controllable. In the MIDSM synthesis the type of the initial condi- tions that excite the plant do influence the value of the optimal configuration variables and SAS gains. This is because the integrated design problem no longer has the same mathe- matical form as the linear optimal control problem. The additional degrees of freedom pro- vided by the configuration variables produce a dependence between Jd and the plant exci- tation initial conditions. Therefore, it is important in the MIDSM procedure to choose these initial conditions so that they represent anticipated physical disturbances. Examples of these disturbances include pilot commands inputs and atmospheric turbulence. The initial conditions assumed in the F-80WRA MIDSM synthesis are: Plant Excitation for MIDSM Design, F-80WRA • i, I i ' - , _i T Y = [gy' gz' P' q' r, _ _mcroll' 8mcpitch' mcyaw ] Initial Conditions: T step lateral gust _01 = [ 1 0 0 0 0 0 0 0 0 ] T step vertical gust Y02 = [ 0 1 0 0 0 0 0 0 0 ] T step roll input Y03 = [ 0 0 0 0 0 0.5 0 0 ] T step pitch input _'04 = [ 0 0 0 0 0 0 0 1 0] T step yaw input _05 = [ 0 0 0 0 0 0 0 0.5 ] The magnitude of each initial condition reflects the relative importance of the disturbance on the total dynamic response.

The total objective function is given by the the dynamic cost plus penalty functions for the trim and design variable constraints.

Jtot = Jd + P'F'trim + P.F. design varibles If the plant can be stabilized, the EMF SAS is guaranteed to produce a dynamically stable closed loop system. Therefore, dynamic stability constraints need not be enforced explicit- ly in the design synthesis. A constraint that the plant can be stabilized may be enforced [Ref. 9] but for the range of configuration variables considered in the MIDSM OWRA syn- , ' i thesis this constraint was found to be unnecessary because it is always possible to stabilize the aircraft.

The Quasi-Newton optimizer solves for the 4 configuration variables and the 63 SAS gains that minimize Jtot" Figure 5.5 is a flowchart showing how the integrated design synthesis is accomplished for the F-80WRA at a single flight condition. The design synthesis as- sumes the same model dynamics (Mil.-Spec. F-8785C), [Q], and [R] weighting matrices used in the EMF SAS design described in chapter 3. The initial guess for the configuration variables corresponds to the nominal F-80WRA configuration: Xpf = 37.292 (ft) X =3.1392 (ft) pw b = 0.0 (deg) Dihedral = 0.0 Table 5.3 configuration variable initial guess values Figures 5.6 - 5.10 show how the configuration design variables were changed by the opti- mizer to minimize the total cost. A converged answer was achieved in 20 line searches which required 331 objective function evaluations. The final solution for the optimal con- trol gains and configuration variables is listed in figures 5.11 - 5.13 along with the control surface deflections required for trim.

Figure 5.14 shows a 3-view of the nominal and the MIDSM optimized OWRA configura- tions. The optimal design has its wing banked (forward wing low) and the entire wing is shifted towards the forward swept tip (planform view). These small changes in the air- craft's configuration have a significant impact on its closed loop dynamic response as is shown in figures 5.15 - 5.20. A 4-g pull up maneuver is simulated for the model, nomi- nal, and MIDSM OWRA. These figures clearly show that the MIDSM OWRA experiences reduced peak lateral acceleration and reduced maximum roll angle. Table 5.4 summarizes the improvements in dynamic decoupling for the F-8 performing the pitch-up maneuver.

rpeak (deg/sec) peak (deg) gy peak Nominal OWRA -0.38 23.2 -5.0 MIDSM OWRA -0.034 5.7 1.0 80% 75% 91% % Reduction Table 5.4 MIDSM OWRA performance, 45 ° sweep, EMF SAS ?

i iii i_/_i!_ iii ;

! i_¸_,_iLi !!ill/!_¸¸ i ¸

ective Function Calculafio_ "_ I --'_ _ Read inputs &

I

(Vortex lattice geometry "_/ itialize variables_ _(current configuration variables)Jl I

i /

• (Vortex lattice geometry_ _of initial guess J (Trim state solution)

I

QNMDIF

I

I

Quasi-Newton

Optimizer

( [AIC] & [VIC] ) matrices (store) CStability & control derivative_ _enalty functions)

I

I Ox

I

[AIC] L-U decomposition = Jd + penalty functio I Total cost ns_ Call optimizer I I (Output results)

I

Figure 5.5 MIDSM OWRA synthesis, single flight condition, EMF SAS Total Cost MIDSM Results, EMF SAS, SFC, 45 (deg) Sweep i_ ,_/i .8 .7 .6 Jtot .5 .4 .3 .2 I I I ° 20.

5. 10. 15.

Iteration Number Figure 5.6 total cost vs. optimizer iteration Wing Bank Angle MIDSM Results, EMF SAS, SFC, 45 (deg) Sweep .20 .15 .10 .05 _b 0.00 (rad) -.05 -.10 -.15 I I I -.20 •.i ¸ ._ •• ° 5. 10. 15. 20, Iteration Number Figure 5.7 wing bank angle vs. optimizer iteration Chordwise Location of Wing Pivot MIDSM Results, EMF SAS, SFC, 45 (deg) Sweep 3.5 3.0 i¸ 2.5 Xpw (ft) 2.0 J ' •?i:: !

1.5 ) • " ii i: 1.0 ° 5. 10. 15. 0.

Iteration Number Figure 5.8 chordwise wing pivot location vs. optimizer iteration /• Wing Pivot Location Along Fuselage MIDSM Results, EMF SAS, SFC, 45(deg) Sweep 37.5 37.0 36.5 36.0 / Xpf fit) 35.5 35.0 34.5 .._ _ ._ - _-_ .-_ ..._ 34.0 ! !

:i • , ° 5. 10. 15. 20.

Iteration Number Figure 5.9 fuselage wing pivot location vs. optimizer iteration Parabolic Dihedral MIDSM Results, EMF SAS, SFC, 45 (deg) Sweep .008 .007 .006 .005 Dihedral .004 .004 1] .003 .002 .001 .000 I 0. 5. 10. 15. 20.

Iteration Number Figure 5.10 parabolic dihedral vs. optimizer iteration MIDSM Results OWRA Synthesis 45 ° Sweep Qdiag = [1 1 1 0.1 1 1] Rdiag = [ 30 30 30 ]

Kff = 0214 i. 7277 0425 1

5165 -.1792 4527 I 0202 -.0095 .1530 .0344 .0390 .1692 .0864 .0300 .0979-] Kpf= 0025 0647 0041 7438 .0033 .0092 .0006 .6606 -.0002 1 0153 0058 0993 0391 .1155 .1441 .0633 -.0346 .2735 I .0027 -.0878 -.0345 -.2903 -.0249 -.1268 -.0395 -.4197"] 1251 -.0254 -.8638 -.1436 .0860 -.0658 -.8056 -.17401 Kfb= 0869 1271 -.0279 -.0170 0759 -.5648 -.0183 -.1149 0958 -_70_ Figure 5.11 optimal EMF SAS gains / n -.6692 .0168 .7290 -.1451 6.3665 .4201 9.9194 -.3257 -.9614 .1888 -24.2262 -2.4482 .4202 -.2616 8.2778 -.0_38 -3.1383 2.4444 .3048 -8.2664 15.1480 •1050 .2866 -1.1148 -.0722 -.7123 .6816

[A] = -i 3627

•1275 -.0929 .2010 -.4856 1.3608 1.3831 •0000 1.0000 .0000 .0524 .0000 .0000 •0000 .0000 .0000 .0000 .0000 -25.0000 •0000 .0000 .0000 .0000 .0000 .0000 •0000 .0000 .0000 .0000 .0000 .0000 m I B -.3399 57.7463 -10.0276 .0862 -56.5160 -67.8286 .0332 -.4146 65.3697 .0147 7.0174 -.2327 .0000 .0000 .0000 ([A] 12.6879 -.3822 .0000 .0000 .0000 continued) -.1019 7.2081 .0000 .0000 .0000

IB]=

.0000 .0000 .0000 .0000 .0000 .0000 .0000 25.0000 .0000 .0000 -25.0000 .0000 .0000 25.0000 .0000 .0000 -25.0000 .0000 .0000 25.0000 Figure 5.12 MIDSM OWRA linearized equations of motion Nominal Configuration MIDSM Configuration Trim State Solution 3.20 (deg) a = 2.88 (deg) -.283 = .690

=

8 = 3.89 Exceeds Trim /selev right = elev right -6.38 Constraint 8 - -5.00 /5 = 11.89j elev left elev left 1.33 8 ffi -1.83 /5 = rudder rudder Configuration Variables Xpw 1.425 (it) Xpw= 3.14 (It) Xpf= 34.45 (It) Xpf= 37.29 (It) t_ = -4.04 (deg) t_13 -- 0 (deg) dih = .0037 dih = 0 Figure 5.13 optimal trim state and configuration variables Nominal OWRA

I I

MIDSM OWRA • i Top Front Side Figure 5.14 nominal and MIDSM OWRA 3-view Lateral Acceleration vs Time 45 (deg) sweep OWRA, EMF SAS = .4 .3 .2 .1 0.0

gy

-.1 /, -.2 -.3 ?i _ -.4 ° Figure 5.15 lateral acceleration vs. time Normal Acceleration vs Time 45 (deg) sweep OWRA, EMF SAS .

o i I_ N°minal I i -1.

gz -2.

........... i:: ............. 1---_ i'::[" __iii'-" i'!'i'i i" _S_l ................. !i ::' ................

-3.

4.

Oo . 2. 3. 4.

Time (sec) % Figure 5.16 normal acceleration vs. time L Pitch Rate vs Time L 45 ((leg) sweep OWRA, EMF SAS 25.

20 ............. ! ................ ................. i ............... i..........

!el ........... i.................

_e_' lo"h_ I--'- _l i i

_. If.. .......... _.C=._I.._._:--.!::__.:.::__._.:::__._ .....

o. ][ !"" :. '; :: :. .

O. 1. 2. 3. 4. 5.

Time (sec) Figure 5.17 pitch rate vs. time Yaw rate vs Time 45 (deg) sweep OWRA, EMF SAS 0.

r -2. I'T" ............. _. ............. :: ......... I-- Model l....i.................

(deg/sec) 11 ¢i i 1_ Nominal[ !

II _/i : I_ _sMI !

-3 ............. ; ................. :....................................................

-5.

0. 1. 2. 3. 4. 5.

Time (see) Figure 5.18 yaw rate vs. time Roll Angle vs Time 45 (deg) sweep OWRA, EMF SAS 25.

. . .. .

20.

15.

............... J.-" ....... _1.---- Mo_l [........... !.................

d"i ! I--B-- Nominal I : ........... _.._-.............. _l_.Z_ MIDSM t........... i.................

10.

(deg) .

ili _ •

.

I -5.

. 1. 2. 3. 4. , r Time (sec) Figure 5.19 roll angle vs. time Control Surface Deflection vs. Time 45 (deg) Sweep OWRA, EMF SAS 10.

8.

left elevator (nominal) right elevator (nominal) rudder (nominal) 6.

• o. ,°°o.oo** left elevator (MIDSM) right elevator (MIDSM) 4.

rudder (MIDSM) Deflection (deg) 2.

0.

-2.

-4. | | Oo . 2. 3. 4. .

Time (sec) Figure 5.20 control surface deflection vs. time _i i_._ I Figure 5.20 shows that the control authority requirements for the MIDSM OWRA are less than for the nominal aircraft.

The reason for the improvement in handling qualities for the MIDSM configured OWRA ; • ' 3.: '¸ lies in the values of the aerodynamic coupling terms Clo _and Cy_. Table 5.5 shows the : _/'_, : values of these stability derivatives for the two designs: Nominal MIDSM %Reduction Cyo_ -.326 -.026 92 % Clo _ .0715 .004 94 % Table 5.5 aerodynamic coupling 45 ° sweep OWRA These results show that the MIDSM design has nearly zero values for the sideforce and roll coupling terms. The wing bank (-4.4 ° ) has tilted the resultant of the wing's additional load distribution to be parallel to the x-z body axis plane, thus cancelling the sideforce change due to angle of attack. The chordwise displacement of the wing pivot shifts the wing such that more area projects over the aircraft's right side, placing the centroid of additional load- ing over the x-z plane and cancelling the pitch-roll aerodynamic coupling. The nominal configuration experiences large lateral accelerations due to the leading edge suction compo- nent acting on the unbanked wing. The lack of independent sideforce control dictates that the nominal aircraft must either roll or sideslip excessively to reduce lateral accelerations.

The MIDSM configured OWRA avoids this problem by having a minimum coupling of pitch to roll and sideforce. This permits the aircraft's motion to be more easily decoupled by active control of the existing control surfaces.

The configuration changes proposed for the optimized OWRA effect all of the aerodynamic and inertia properties of the aircraft, but the main influence is on the roll and sideforce coupling terms. To further prove this idea, the time histories of lateral acceleration (gy) and bank angle (d_) are plotted for the MIDSM configuration and for the nominal configuration ?i : ' with an EMF SAS designed for the case where C!¢z and Cyo_ have been deliberately set to zero (Fig. 5.21 - 5.23). These results show that forcing C10 _and Cy0_ to zero accounts for most of the handling qualities improvement while the small nonzero values of these ' ; 5. ¸' •_ : • c' 170 terms in the MIDSM results compensate for the affects of inertial coupling and other aero- dynamic effects. (Notice that the MIDSM response is slightly more decoupled than the contrived nominal case).

Lateral Acceleration vs Time Effect of Aero Coupling %1'i/, _ .1 ::: , ' _,i _, ,

o

-.1 .................... [_--.O--- MIDsMN°minalcya, Cla =0 1 .................................

-.2 / .

-.3 -.4 . 1. 2. 3. 4. ° Time (sec) Figure 5.21 lateral acceleration vs. time Roll Angle vs Time Effect of Aero Coupling 25.

20.

°°'°.°*°°°*°.o°°.,°°_o1,°°°.*°°..°°°°._°°°°*°°°°°°°°°°° 15.

°°°°°°°°°°o°°°Qio°°°°°°°.°°°.°°°2°°°°.°.°°°°°o°°°°,°°°°°°°°°°°°°°.° "-/k'-- Nominal MIDSM Cl_, Cy. = 0 10.

°°°.,°°°o.*°,°o°°°°° °°°°.°°°od.°°...,.°.l*°°° ° ° O° ° 2. 3. ° .

Time (sec) ;_! i:,ii!!: ¸, i Figure 5.22 roll angle vs. time Control Surface Deflection vs Time Effect of Aero Coupling 10.

.

.

o .-m-- right elevator (C'ya--Cla---0) .... "................. i..........

rudder (Cya=Cla=0) T Deflection _____ left elevator (Cya=Cla=0) .......... right elevator (MIDSM) i left elevator (MIDSM) (deg) 4.

rudder (MIDSM) %:.!

i, •i(_i%/, !

2.

0.

-2.

0. 1. 2. 3. 4. 5.

Time (sec) z Figure 5.24 control surface deflection vs. time Figure 5.24 shows also that the MIDSM configured OWRA is able to achieve its perfor- mance with lower control effort than the contrived nominal case.

5.2.2 MIDSM OWRA Synthesis, EMF SAS, 65 ° Sweep, M=l.6 The MIDSM procedure is now repeated for a flight condition corresponding to 65 ° wing sweep and M=1.6 (q=1180 lb/ft 2) to see if similar performance gains may be obtained in other flight regimes. The formulation of the problem is identical to the 45 ° sweep synthe- sis, excepting that the vortex lattice geometry, input weighting matrices, and initial condi- tions are different. Table 5.6 lists the values of these inputs.

weighting matrices Qdiag =[1 1 1 0.1 0.1 1 ] Rdiag = [ 50 50 50 ] Table 5.6 MIDSM OWRA synthesis 65 ° Sweep, M=l.6, EMF SAS ,_ _i _ i,, ¸¸¸_ i 172 !, _ i_ .¸ Plant Excitation for MIDSM Design

_5 T

_r = [gy, gz' P' q' r, _, _rmroll, _mcpitch' mcyaw ] Initial Conditions: step lateral gust _rO1 = [ 100000000] T T

step vertical gust Y02 " [ 0 1 0 0 0 0 0 0 0 ]

T step roll input Y03 = [ 0 0 0 0 0 0.5 0 0 ] T

step pitch input _04 - [ 0 0 0 0 0 0 0 1 0]

T step yaw input _05 = [ 0 0 0 0 0 0 0 0.5 ] Table 5.6 (cont.) MIDSM OWRA synthesis 65 ° Sweep, M=l.6, EMF SAS The initial guess configuration is once again the nominal F-80WRA with no wing bank and 65 ° wing sweep. The MIDSM synthesis was executed and converged to a solution in 50 line searches (700 objective function evaluations). Figure 5.25 shows the resulting MIDSM optimized OWRA and the nominal configuration. The displacement and banking of the wing are similar to the results obtained at 45 ° sweep. Table 5.7 lists the values of the optimal configuration variables.

Nominal Confi_mtion MIDSM Configuration Trim State Solution = 1.98 (deg) ot = 2.49 (deg) 13 -- 1.84 13 = .633 _i - .8511 8 = -.925 elev right elev right _i - -5.00 8 -- 4.02 elev left elev left 8 = -3.87 8 = -1.82 rudder rudder Configuration Variables MIDSM Confima'ation Nominal Confimaration Xpw-- 2.565 (ft) Xpw-- 3.14 (ft) Xpf= 33.54 (ft) Xpf= 37.29 (ft) ¢PI_= "14.2 (deg) _PI_ - 0 (deg) dih = .0105 dih = 0 Table 5.7 MIDSM results 65 ° sweep OWRA synthesis The dynamic performance of the nominal and MIDSM configurations is compared in fig- ures 5.26 - 5.30 for the 4-g pull-up maneuver. Once again the MIDSM design achieves significantly better dynamic decoupling than the nominal OWRA and with reduced control effort. The maximum lateral acceleration and roll angles achieved by the two cases are: i:i:• ¸'_ :::i!ii:_ i rpeak (deg/sec) _) peak (deg) ! :•i:: :•_ _:• i¸_ i:' :) -0.34 8.0 1.5 Nominal OWRA -0.10 4.3 -0.7 MIDSM OWRA 67% 46% 53% % Reduction Table 5.8 MIDSM OWRA performance, 65 ° sweep EMF SAS )) i' 2 In the 45 ° sweep synthesis, improved decoupling occurred because the aerodynamic coupling terms C10 t and Cyot were minimized. This appears to not be the case for the MIDSM OWRA at 65 ° sweep because the value of these coupling terms is distinctly non- zero (table 5.9).

Nominal MIDSM %Reduction Cyot -.172 -.116 32 % Clot .016 .025 -56 % Table 5.9 aerodynamic coupling 65 ° sweep OWRA To better understand why the 65 ° MIDSM OWRA shows large aerodynamic coupling and the 45 ° case does not, a test case was studied where Clo t and Cyot were deliberately forced to zero for the nominal 65 ° sweep configuration with an EMF SAS. Figures 5.31 and 5.32 show the simulation results comparing the MIDSM design to this contrived nominal config- uration.

Both aircraft have similar dynamic response, but the contrived nominal configuratiori / shows slightly better dynamic coupling. The control effort, however, for the MIDSM con- figured design is significantly less. These results indicate that forcing Clo t and Cyot to zero will significantly decouple the OWRA's response, but the MIDSM method has found a de- : ! _i_r::_: _: _f _i? I:¸:i!

Top Nominal

?-\

Nominal Front Side Figure 5.25 MIDSM optimal OWRA configuration, 65 ° sweep, EMF SAS i, Lateral Acceleration vs Time 65 ° sweep OWRA, EMF SAS .1 , i _ , 0.0 -.l _, 1'----- Model ' I

gy

....... ............... ........ .............

_i i " i

.o | i i i ° O. 1. 2. 3. 4.

Time (see) Figure 5.26 lateral acceleration vs. time Normal Acceleration vs Time 65 ° Sweep OWRA, EMF SAS ° .

._*_._..._*_*_*_._*_**_.._°_._°_°_._*_ -1.

g_

-2.

-3.

1. 2. 3. 4. ° Time (see) Figure 5.27 normal acceleration vs. time Yaw Rate vs Time 65 ° sweep OWRA, EMF SAS 2.0 1.5 .,... .... -. .... . ... ,. ,,.,,,r, ....o...........'_ ........ ........¢.,,.. ............

i'% H :¸¸ 1.0 _. , ................................

....... • "............. I_ Model ' I

! I-..m- Nominal i

(deg/sec) r .5

.o - i i i

/ ii ¸ IIL ¸¸ _ o 5 ....... i................. i.......................

-1.0 O° 1. 2. 3. 4. 5.

Time (sec) Figure 5.28 yaw rate vs. time Roll Angle vs Time 65 ° sweep OWRA, EMF SAS .

7.

6.

5.

.............. :. .............. " - - - t,,_.----, ....... "_....- .. _..- - _...-....-"'X'. ....

4.

'i (deg) 3.

......

u_ i _ I Model I 2.

• .._. • .-_.- _ina_ r ...........

1.

i "

O.

-1.

O° ° 1. 2. 3. 4.

_ r Time (sec) Figure 5.29 roll angle vs. time Control Deflection vs Time 65 ° Sweep OWRA, EMF SAS 5.

i i i !

4 ........

1_- I_ : --,I-- fl'ghtdevator (nominal)

I_=Z¢ i i I _ rudder (nominal)

Deflection 2.

I f _"__ _ left elevator (MIDSM) (deg) I _ _r" / : : _ght elevator (MIDSM) ........ 1.

• v'y_ v v t_?. _ .- .... _ ...... .:.,,..,.; ......... _ ...........

: ............. _._:_::__:=:;__:.:x;; :.::.: _'_;:__, I

o I

-1.

0. 1. 2. 3. 4. 5.

Time (sec) Figure 5.30 control surface deflection vs. time sign with nearly as great decoupling and significantly lower control effort where Clo t and Cyot are nonzero. The explanation as to why the MIDSM design strives to minimize con- ,) trol effort at the expense of improved dynamic decoupling can be found in the large values of the weighting matrix [R]. As was noted in chapter 3, the EMF SAS favors higher band- width control in the feedback path, and therefore large control effort weighting is required to enforce an actuator bandwidth restriction (5 Hz in this example). For the 65 ° sweep flight condition, [R] must be substantially greater than at 45 ° sweep to meet the actuator bandwidth limit of 5 Hz. As a result, the total dynamic cost consists largely of control ef- fort cost and MIDSM minimizes this more than the dynamic coupling. The MIDSM design actually has lower dynamic cost than the nominal design with C10 t and Cyot set to zero be- cause of its lower control activity (see table 5.10).

MIDSM Clot, Cya = 0 : ii • Jd _" 25.47 29.29 13% Lower Table 5.10 effect of aero-coupling on dynamic cost, 65 ° sweep OWRA Lateral Acceleration vs Time 65 ° sweep OWRA, EMF SAS .04 • • .

.02 * °.. °* °* °***.°..°. °** °.*** o°.*..*,° ° °.* •* ° ° ° ° ° ° °.....*** ***.... ° °***o J-° ° ° ° °... ° ° ° ° • ° ° °.

0.00 .............. ! .......... . i ,. c_ b' .............. _ .................

, k : I--d,-- _..ya, e = I :

_W :: I---o- _sM I !

-.02 gy -.04 _AAA AAAA AA_A AAAA .AAA IIAA& AAAA A, L -.06

-.08 ....... .'3. ......."........ - ----- _,,__ __,__" __-,._,e_ __-,,_-____.,m

-.10 -.12 | | | | ° 1. 2. 3. 4. , Time (sec) Figure 5.31 effect of aerodynamic coupling on handling qualities lateral acceleration vs. time Roll Angle vs Time 65 ° sweep OWRA, EMF SAS , .

.

°°_°°°°*_°.°°°°°.° ........

(deg) 2.

.**.bIi *oiIe J°*.°*°lii .

CI_ Cyct = 0 ] MIDSM ..........................

° i | 1. 2. 3. 4. .

Time (sec) Figure 5.32 effect of aerodynamic coupling on handling qualities roll angle vs. time MIDSM has accurately identified the extra weighting placed on control activity and has found a configuration with highly decoupled response that uses very little control effort.

Because it is impossible to distinguish between control effort due to large deflections and that due to large frequency of motion in the LQR cost function, the designer is forced to place excessive weight on control activity to enforce bandwidth constraints. When the SAS architecture is switched to the RMF SAS, this problem is avoided because the actuator states are not fed-back, and therefore their bandwidth remains unchanged for any value of [Q] and [R]. It is reasonable to suspect that the 65 ° sweep MIDSM results are more a func- tion of the assumed SAS architecture (than is the case for the 45 ° sweep results) because of their strong dependence on control effort weighting.

5.2.3 MIDSM OWRA Synthesis at Multiple Flight Conditions with EMF SAS In the two previous examples the F-80WRA was re-configured for improved handling qualities at 45 and 65 ° sweep. Unfortunately, the configuration changes required at these flight conditions are not identical, which implies that a configuration optimized for one flight condition will not be optimal at others. It is possible, however, to synthesize a single aircraft with improved performance over a wide range of flight conditions by incorporating the dynamic cost at several flight conditions into the total cost. This idea is similar to the method proposed by Ly [Ref. 24] for the synthesis of robust controllers for linear systems with plant model uncertainties. Ly's method synthesizes a single controller which must im- prove the response of not only the nominal plant, but also plant models perturbed to reflect plant uncertainties. This idea is extended to the oblique wing synthesis, where a single air- craft must perform well over a range of flight conditions. The example presented here con- figures the OWRA for improved handling at two flight conditions. New EMF SAS gains are calculated at each flight condition and gain scheduling is assumed to be used for transi- tioning between flight regimes. This basic technique can be extended to account for the in- fluence of changing wing sweep, nonlinearity of the aerodynamic loads, influence of changing dynamic pressure, etc. on the optimal OWRA configuration.

,i• ' • // The two flight conditions considered are described below.

Multiple Flight Condition OWRA Synthesis Flight Conditions Considered Flight condition _ Math Dynamic Pressure (lb/ft^2) 1 45 0.8 440 2 65 1.6 1180 An optimal EMF SAS is synthesized at each flight condition using [Q] and [R] matrices ap- propriate for each wing sweep. Penalty function constraints are implemented to enforce : i aircraft trim in 1-g level flight at •both flight conditions. The total cost function for the mul- tiple flight condition synthesis (MFC) is: +Jd65 +P'F" trim 45 Jtot = Jd 45 + P.F. trim 65 + P.F. design var's This formulation of the total cost weights each flight condition equally, but if the mission requirements invalidate this assumption, weighting factors may be included to reflect the relative importance of each flight mode. Figure 5.33 is a flowchart of the MIDSM MFC design procedure. The synthesis is nearly identical to the SFC cases previously described, except that the dynamic cost must now be evaluated at more than one flight condition.

In the single flight condition synthesis the wing pivot axis orientation was defined in terms of a single variable, Cb, the wing bank angle at a fixed wing sweep. The variable, Cb, has no meaning for the multiple flight condition synthesis because it is impossible to find a sin- gle pivot axis orientation which will produce arbitrary wing bank at a specified sweep angle with zero wing incidence. Therefore, the design variables 01 and 02 are included explicit- i_!! , / ly in the MFC synthesis.

The MFC synthesis assumes the nominal OWRA configuration for the initial configuration guess. Table 5.12 lists the input parameters used in the synthesis. The optimizer con- verged to a final solution after 47 line searches (756 objective function evaluations). The resulting optimized configuration variables are: 0 = - 1.826 (deg) X --2.01(ft) pw 02 = -6. 138 (deg) Xpf = 34.80 (ft) Dihedral --. 0058 Table 5.11 MIDSM OWRA optimal design variables, multiple flight condition case 65 (des) Sweet_ 45 (deg) Sweep Qdiag=[1 1 1 0.1 1 1] Qdiag=[1 1 1 0.1 0.1 1] Rdiag = [ 30 30 30 ] Rdiag = [ 50 50 50 ] Plant Excitation for MIDSM Design (assumed for all flight conditions) , _ T Jr = [gy, gz' P' q' r, ¢ 8tin.roll, _mcpitch' rncyaw ] Initial Conditions: T step lateral gust _1=[100000000] step vertical gust _2=[010000000] T T step roll input _3=[ 000000.500 l T step pitch input y_=[000000010] T step yaw input

5=[oooooooo.51

Configuration Design Variables Initial Guess (Nominal OWRA Config.)

Xpf = 37.292 (ft) X = 3.1392 (ft) pw 01 = 0.0 (deg) 02 = 0.0 (deg) Dihedral = 0.0 Table 5.12 multiple flight condition MIDSM inputs H •

• _• >•i: i_•Ji i_i •• _i

! >>_+! _> = i _ ; i<

_initialize variables_.]

I

Dynamic Cost Flight Condition (1_ _VL Geometry , "_ Vfortex lattice geometry "_ Trim State & P.F. s , initial guess at 45 & 65J 45 deg sweep J_ Stability & Control Deriv s

I

Mass Properties QNMDIF

I m,v

Linear EOM _5 C] & [VIC] matrices_ (Dynamic Cost Flight Condition (h EMF SAS oo & 65 (store) J

,DynamicCost ,)

_, 65 (deg) sweep J I AIC]45 &

I

-U decomposition

(Pon t

I

Total Cost D + Jd2 +P.F.

Figure 5.33 multiple flight condition synthesis flowchart, EMF SAS Figures 5.34 and 5.35 show the nominal and optimized OWRA configurations in 3-view.

For both flight conditions the MIDSM design has its wing banked right wing low and the wing is shifted to the aircraft's right side. This is the same trend displayed in the single flight condition synthesis results. The aircraft satisfies the trim constraints at both flight conditions even though the nominal configuration does not. Figures 5.36 - 5.40 are time histories of the various OWRA configurations performing the 4-g pitch-up maneuver. Re- suits from the single flight condition syntheses are also shown for comparison. These re- suits show that the dynamic performance of the MIDSM OWRA is markedly improved over that of the nominal configuration at both flight conditions. The peak lateral acceleration and roll angle are listed in table 5.13 for the various cases.

MIDSM-MFC Nominal % Reduction gy peak -.07 -.38 82 % 45 (deg) Sweep qt Opeak 12 (deg) 23 (deg) 48 % gy peak . -.04 . -.33 . 88 % [ I I I I I I I J [ I [ [ [ I J [ i [llllllllllll[[[[[[[[[[[I I _ll[ [ Illllll I Illl Ill I I I [II I I Ill [ [ I[I Illl II IIlll 65 (deg),r Sweep _peak 1.5 (deg) 7.8 (deg) 81% Table 5.13 OWRA performance, MIDSM multiple flight condition results /

Nominal

Top

MIDSM-MFC

_ _ Nominal

Front MID_ i•ii• Nominal Side Figure 5.34 MIDSM-MFC results, OWRA configuration with 45 ° sweep i_:_ _i' i:•'_ii_i!i _I _ _ ii_:i •_' !

i _ _ii__ ' ?

.: ' i ¸ : i!_!,i_i:: ' MIDSM-MFC !,i,_!i ¸i: Top / Nominal MIDSM-MFC Front MIDSM-MFC Side Table 5.35 MIDSM-MFC results, OWRA configuration with 65 ° sweep i¸ i:,:i i:ii _ ,.i_ ¸_•_: Lateral Acceleration vs Time 45 ° Sweep OWRA, EMF SAS .08 ***...°°°°°**, ,,,°°°°°.***.°.**'.°°**.°°°°°*°.°..*_..°° °**.°°°°°''°'_°°°°'°°°°'°°°° °°" i¸ ,• .06 • J_._ Mo_l ! .............. i.................

.04

.............. i......... I_ MIDSM-SFC I i

, i ¸ , :_ r ¸ • , : ---0--- MIDSM-MFC : .02 .............. - ......... 1 . . ! .............. !.................

.00

gy _- . : ______

-.02 .., .......

-.04 -.06 :i: iiiiiiiiii iiiii : i .... .......... , ..... 2 .............

i _, , • _ .

-.08 0. 1. 2. 3. 4. 5.

Time (sec) Figure 5.36 lateral acceleration vs. time Roll Angle vs Time 45 ° Sweep OWRA, EMF SAS 14.

12.

10.

8.

6.

¢ (deg) 4.

2.

• c O.

-2.

: i_.i_!_ii i:_ _ ° ) :!i ii_:II: : Figure 5.37 roll angle vs. time :: iLi_ , ::i _ _ _:'i ; _:! i I" Lateral Acceleration vs Time 65 ° Sweep OWRA, EMF SAS .04 o oO ........ ................. ............... i I ............ !i ........

<o

•}:i i% Y•}

, "i_" _ _ _ gy ..04 :: Model -.06 [_ .................

i} !_ :i _

08,0

-.12 0. 1. 2. 3. 4. 5.

Time (sec) Figure 5.38 lateral acceleration vs. time Roll Angle vs Time 65 ° Sweep OWRA, EMF SAS , -! mm = .. m m _ _ ....... 1 ' I ................. • ................ "_... .......... ..'-77. ...............

° ............ _' Model I.................

.

.A n : : I + MIDSM-SFC I ¢ .

...... •./. .... _ ................. i......... _--'-_-_C__ ....

(deg) ° ..... .............. ..........

O° I I I !

-1.

° 1. 2. 3. 4.

° i_ Time (sec) Figure 5.39 roll angle vs. time •4{' •'/: i i{ '_ Control Surface Deflection vs Time 65 ° Sweep OWRA, EMF SAS .

:/ i_ i ¸ , . t • • i_ ii: ii : : ---b-- left elevator (SFC) .

right elevator (SFC) Deflection rudder (SFC) (deg) + left elevator (MFC) + right elevator (MFC) -- rudder (MFC) ,/i: i_:•; -!'I O.

1. 2. 3. 4. 5.

Time (see) Figure 5.40 control surface deflection vs. time Figures 5.36 - 5.37 show that at the 45 ° sweep flight condition the lateral acceleration and roll angle excursions for the OWRA optimized at multiple flight conditions are greater than the SFC optimized OWRA. This result is to be expected, because the MFC design repre- sents a compromise design at any single flight condition. The dynamic response of the MFC OWRA at the 65 ° sweep flight condition shows lower peak lateral acceleration and roll angle than the SFC result. At first these results seem incorrect, because the SFC de- sign should have superior dynamic response. A more careful examination of the results shows that the dynamic cost is indeed lower for the SFC OWRA, but most of the cost re- duction comes from reduced control effort and not reduced peak lateral acceleration and roll / angle. This idea is confirmed by figure 5.40 which shows that the MFC OWRA requires more control deflection to achieve its performance than the SFC design. The significant in- fluence of control effort on dynamic cost at the 65 ° sweep flight condition is once again due to the EMF SAS architecture which desires excessive bandwidth in the feedback path.

Changing the SAS to RMF will permit the synthesis of designs with acceptable actuator .i/: _ .,!

bandwidth without resorting to excessive control effort weight.

• , k , i _¸ The results of this synthesis show that it is possible to improve the F-80WRA's handling qualities significantly over a wide range of flight conditions by re-configuring the aircraft.

The required configuration changes are small enough that aircraft drag and structural weight will not be seriously affected. The resulting optimized OWRA is more easily trimmed over the flight envelope and has approximately 1/10 the lateral acceleration response and 1/2 to : ; :_i _ 1/3 the roll angle excursion during abrupt pitch maneuvers, as the originally proposed F-8 OWRA configuration.

5.2.4 MIDSM OWRA Synthesis, RMF SAS, 45 ° Sweep, M=0.8 In chapter 3 the reduced order model following (RMF) SAS was shown to have several ad- vantages over the EMF SAS architecture. Because the RMF controller does not require full-state feedback, it is possible to enforce actuator bandwidth constraints by not including the actuator states in the feedback path. Furthermore, the RMF SAS attempts to drive the closed loop plant's unforced response to that of the model's without excessive bandwidth requirements. These features make the RMF SAS a logical choice for a controller that is more practical and well suited for oblique wing aircraft than the EMF SAS. The previous MIDSM synthesis examples have shown how significant improvement in handling qualities may be obtained by simultaneously re-configuring the F-80WRA and its SAS. In this ex- ample the MIDSM method will be used to perform the same task except that an RMF SAS will be implemented instead of an EMF SAS. This example will also show if the MIDSM OWRA configuration is sensitive to the SAS architecture for the case studied.

The integrated synthesis with an RMF SAS requires a slightly different formulation of the objective function than was used in the previous examples. The RMF synthesis requires :,+ that the SAS gains be computed explicitly by the optimizer. The biggest disadvantage of i!_ i ?

this approach is the increased computational burden when compared to full-state feedback schemes which solve for all of the gains in a single ARE. Dynamic stability constraints must be enforced explicitly in the RMF synthesis, otherwise the solution for the dynamic cost may become unbounded. The total cost for SFC RMF synthesis is given by: Jtot = Jd + P'F'trim + P.F. design var's + P.F. stability The penalty functions for trim and limits on the design variables are identical to those pre- sented in the EMF MIDSM examples. The penalty function for dynamic stability is given in chapter 3 by equation 3.68. Figure 5.41 is a flowchart of the RMF synthesis method which shows how the control gains are solved for explicitly by the optimizer.

The integrated design synthesis is carried out for the nominal OWRA configuration at 45 ° sweep, M=0.8 flight condition. The values of the initial guess variables are identical to those used in the EMF SAS example, except for the following parameters: Qdiag=[10 5 1 1 1 10] • i Rdiag = [ 100 100 100 ] Actuator pole: s = -10.0 + 0.0j (rad/sec) (1.6 Hz Bandwidth Actuator) MIDSM inputs single flight condition, RMF SAS, 45 ° sweep Notice that the actuators are modelled as 1.6 Hz bandwidth devices and that their band- width can not be altered by the SAS because only the quantities {gy, gz, P, q, r, @} are sensed and fed-back. The initial guess for the control gains is all zero except for a small amount of roll and yaw damping which is required to stabilize the dutch roll mode (for the initial guess). Because MIDSM must now minimize the total cost with respect to 49 varia- bles (i.e. 45 SAS gains and 4 configuration variables) the number of line searches and the computational cost of each gradient evaluation is much greater than in the EMF MIDSM ex- amples. A converged solution for the RMF MIDSM synthesis was achieved after 56 line ,i i _ .; ".. • searches and 3697 objective function evaluations. Table 5.14 lists the optimal configura- tion variable values and the optimized RMF SAS gains. A 3-view of the nominal OWRA and the optimal MIDSM configuration is given in figure 5.42.

OptimalRMF SASGains

-.2206 r. 3549 .6895--] i. 9225 - I-. 0525 2346 I Kff [_. 2564 -.2725 1.4245_] i_ / / -.0016 .0449 2.1049 -.2960 .2059--] .0742 -.3710 4.2548 -1.0057 - 1918 1 Kpf = .2380 " 0772 .1466 -.0410 -.0640 2.0628 -.3334 1213_] F2129 .0131 -.0866 -1.4096 .0067 -.3269q Kfb = b3004 .0026 .2710 -3.4882 2.0186 .1056| [.2412 .0465 .0446 -1.4056 .1104 .2070..]

Nominal Configuration MIDSM Configuration Trim State Solution tx = 3.20 (deg) ot = 3.25 (deg) 13 = 2.0 [3 = -.283 8 = 4.24 _elev fight-- Exceeds elev fight - -6.38 } = 11.89 Trim 8 = -5.00 _elev left Constraint elev left 8 = -1.82 8 = 1.33 rudder rudder Configuration Variables Xpw= 1.734 (ft) Xpw-- 3.14 (ft) Xpf= 35.05 (ft) Xpr= 37.29 (ft) (_i; -- -5.42 (deg) _1; = 0 (deg) dih = .0037 dih = 0 Table 5.14 MIDSM results OWRA synthesis, 45 ° sweep, RMF SAS

•• • •i••_ •_•• • •_•i! •/_ii•: !_i_¸ !/••(_i_!_i/!_•• _k_

.... _i i/_/I • ve Function Cal_ ,/'Configuration Variables ", ' and Control Gains _ Read inputs & "_

(Vortex lattice Igeometry I

itialize variablesJ k,, (current configuration variables)J

Dy_c co_'_

I I Vfortex lattice geometry'_ d m = [P]Y (Trim state solution) initial guess J QNMDIF I

I

I

Quasi-Newton AIC] & [VIC] Optimizer store) CStability & control derivatives_ _enalty functions)

I

_D I

I

[AIC] ttOt= Total cost Jd + Penalty Fmcliom 3 Call I I Augmented Dynarnic'_ (Output results) System _,} I Solve Lyapunov'_ Equations fl t.

Figure 5.41 MIDSM synthesis, single flight condition, RMF SAS flowchart _i:¸ i,_!:• • i_,_!I /! !

Nominal OWRA MIDSM OWRA Top Front L Side '•y Figure 5.42 MIDSM results, F-80WRA configuration i ••, • f f The time histories of the optimized RMF design and the nominal design with RMF SAS

i_ii: '!_i_ , i:i/

are compared to the model's response for the 4-g pull-up maneuver in figures 5.43 - 5.46.

Lateral Acceleration vs Time '_'_ :i /i I 45 ° Sweep, RMF SAS .1 _ • _i_ , 0.0 -.1 oo****o,,oo,,*oll_*o**oo.o* 0 i :, ii!_ _

gy

_ii _ • i Nominal .................................

-.2 /,i, _ :,_ :_ Model t MIDSM -.3 | | | -.4 o 2. 3. 4.

o Time (sec) Figure 5.43 lateral acceleration vs. time : i Yaw Rate vs Time 45" Sweep, RMF SAS .

o __: __---.-,-,,,,,o, ...... , .....

I ml:: f :. _ ........ _.._.-, -1.

r (deg/sec) -2.

.............. i................ ..........."M i............. i................

-3.

... .... ._ ............. i..l'e-- MIDSM I 4.

1. 2. 3. 4. 5.

° Time (sec) Figure 5.44 yaw rate vs. time Roll Angle vs Time 45 ° Sweep OWRA, RMF SAS 25.

20.

................. ____._:-.:.!:-.-..-.:.:-._

15.

.............•"¢ .................?.... I---- Mo_l I .........i................

d i I_ N°minall : ........... ( ....................i .... I-o- MIDSMI ......... ::................

10.

(deg) ° ........ i................ i .............. i .........

O° -5.

.

1. 2. 3. 4.

Oo Time (sec) Figure 5.45 roll angle vs. time Control Surface Deflection vs Time 45 ° Sweep OWRA, RMF SAS 12.

10.

----b--- left elevator (nominal) ---S--- right elevator (nominal) 8.

rudder (nominal) --.-.V-- left elevator (MIDSM) right elevator (MIDSM) 6.

rudder (MIDSM) Deflection (dcg) 4.

2.

0.

-2.

° 2. 3. 4.

o Time (sec) Figure 5.46 control surface deflection vs. time The optimized configuration with RMF SAS is nearly identical to the EMF OWRA, as can be seen from figures 5.14 and 5.42. In addition, the time histories for the 4-g pull-up ma- neuver show similar dynamic decoupling and transient response. This indicates that for the 45 ° sweep flight condition, the optimized OWRA configuration is not very sensitive to the SAS architecture. The RMF OWRA has appropriate wing bank and pivot displacement to force the coupling terms Clo t and Cyot to be nearly zero, just as is the case for the EMF OWRA. Table 5.15 lists the performance of the aircraft with RMF SAS.

rpeak(deg/sec)

peak(deg)

gyr

Nominal OWRA -0.31 20.16 '3.7 MIDSM OWRA -0.10 6.7 -1.1 % Reduction 67% 67% 70% Table 5.15 MIDSM OWRA performance, 45 ° sweep RMF SAS A key aspect of the optimized RMF OWRA is that it achieves dynamic performance similar to the EMF case, but it does so with actuators whose bandwidth is only 1.6 Hz instead of 5 Hz. In figure 5.47 the closed loop eigenvalues of both the RMF and EMF designs are shown. It is clear from this figure that the RMF design has eigenvalues much closer to those of the model (in frequency) than the EMF case. As was discussed in chapter 3, one key advantage of the RMF SAS is that it attempts to match the poles of the closed loop plant to those of the model, which produces the desired model following performance with- out excessive bandwidth requirements in the feedback path. Figure 5.47 clearly shows that most of the EMF OWRA closed loop poles are at a frequency of 1.6 Hz or greater where most of the RMF OWRA poles are less than 1.6 Hz. All of the remaining synthesis in this work will assume an RMF SAS architecture.

L Closed Loop Roots 45 ° Sweep OWRA 40.

+ I

i Ir RMF poles are lower _ .....

30.

"'"_"M_lei ................ i"'i ................. ._in frequency than EMF3 I 0 MIDSM-RMFSAS I i i- / : 20.

[] mrDSM-E_SAS l...i .................i....../.. .....i................

Imaginary (rad/sec) II° 10. ,.°...,..,...... o ..,,...,..,...p........,.....°. ,......,....oo.,..t..°......°.° .... °.

0 q" q.

I° ,_ii!

I:I [] _' ,: ° ........m ................ [].................,',:-<.<, ........_ ...............

-I0.

-40. -30. -20. -10. 0. 10.

Re_l (rad/sec) Figure 5.47 closed loop roots, EMF and RMF SAS

5.3 Global Versus Local Minimum in the

Optimization Procedure

All of the results presented here used the nominal OWRA configuration as the initial guess for the optimal configuration and the synthesis procedure changed the design until a mini- mum in the total cost was found. Because all possible combinations of the design variables are not considered, it is possible that there are other configurations that produce a smaller total cost than those designs presented here. To test this theory syntheses were repeated for several different values of the initial guess for the configuration design variables to see if the optimizer would converge to a different answer. In all cases, the converged solution was identical to the one obtained when the nominal OWRA configuration was used as the initial guess. This indicates a reduced likelihood of local minima in the design space of the total cost. This in no way guarantees that an arbitrary synthesis will produce a single mini- mum and it is the designers responsibility to formulate the total cost function so as to mini- mize the chances of local minimum. Optimizers which are more robust to local minimum may be used [Ref. 9] but they can have much higher computation costs. The examples pre- ' : • • • u,_! i :_ sented here show that a wide class of problems may be solved with a straight-forward op- timization technique (i.e. Quasi-Newton method) and produce vastly improved designs ,_i_, .

without evidence of local minima.

•r

5.4 Nonlinear Aerodynamics and

Multiple Flight Condition Analyses

In chapter 2 it was shown that many important aerodynamic coupling forces acting on the oblique wing aircraft have significant nonlinear behavior. One shortcoming of a linear dy- namics analysis is that it is not possible to accurately account for these nonlinear trends un- less the plant model is re-linearized as the state vector changes. For the oblique wing de- if ¸¸ •2•,, sign problem this is a significant issue. For example, the simulation results presented for i_i i/i, i_ the nominal OWRA at 65 ° sweep, M=l.6 flight condition do not display nearly as dramatic sideforce and roll coupling as the VMS simulator results do (figure 3.11). This is because the point of linearization for the airloads (Cz= 0.1) happens to occur where the slope of the Clct and Cytx terms is very small, giving the illusion of reduced coupling. In reality, as the angle of attack changes during the 4-g pitch-up, the values of Cla and Cytx grow nonline- arly and the performance degrades rapidly. To properly account for this in an integrated synthesis (which relies on a linear dynamics analysis) the dynamic cost should be the sum of the cost at several angles of attack that span the range of o_ experienced during the ma- neuver. This is identical to idea already presented in the multiple flight condition synthesis where two wing sweeps and Mach numbers were considered. (Note: The results presented for the 45 ° sweep flight conditions do not experience this 'favorable linearization error' be- cause the point of linearization for their dynamics has nearly maximum slope in all of the coupling variables). All of the analysis tools and techniques required to perform such a synthesis have been presented here, but this work is left for future study so that the present work can focus on the important effects of wing flexibility on oblique wing handling quali- ties.

Chapter 6

Influence of Wing Flexibility on Oblique

Wing Stability and Handling Qualities

In the analyses presented in chapters 1-5, the dynamics model for the oblique wing designs studied assumed a rigid aircraft. Flight test results from the AD-1 oblique wing demonstra- tor program have shown that aeroelastic deformation of the wing can strongly influence air- craft aim and handling qualities [Ref. 6]. Furthermore, studies of oblique wing flutter [Ref. 27] have shown that the critical aeroelastic instability for wing sweep greater than 25 ° is a low frequency motion involving wing-bending and aircraft rolling. This flutter insta- bility is unique to oblique wing configurations and may be a critical factor in the structural design of the wing. In this chapter the influence of wing flexibility is included in the linear- ized equations of motion and in the MIDSM synthesis. First, the 6 DOF EOM are re- derived to include explicitly the influence of wing deformation using an assumed modes model for the wing bending. This formulation retains the rigid aircraft terms in the equa- tions of motion, so that the influence of the flexibility appears explicitly, in addition to the familiar rigid aircraft dynamics. These EOM are then used as a tool to study the aeroelastic instability of the AD-1 configuration and compare the results to previous flutter analyses.

Once the validity of the elastic aircraft dynamics model is established, the EOM are incorpo- rated into the MIDSM synthesis so that the F-80WRA can be re-designed for improved handling qualities with a deformable wing. The results show that wing flexibility signifi- cantly alters the handling qualities of the nominal OWRA configuration and that the opti- mized OWRA configuration is different from the previous optimized rigid aircraft results.

In all cases, the MIDSM OWRA shows dramatic reductions in dynamic coupling and im- proved handling qualities when compared to both the flexible and rigid wing nominal OWRA.

i',_i i )

6.1 Linearized Equations of Motion for a

Free-Flying Aircraft with a Flexible Wing

/ When the oblique wing configuration was first proposed, designers believed that some of its potential advantages would be countered by a structural weight penalty required to keep the forward swept wing from diverging aeroelastically. A study by Jones and Nisbet [Ref.

7] showed, however, that the critical mode of instability is a coupled roll-bend flutter which • L occurs at speeds significantly higher than the clamped divergence speed of the forward swept wing. Because this instability significantly affects the structural design of the air- , < , craft, it is important to accurately predict this dynamic behavior. The problem is complicat- ed by the fact that the wing flexibility is highly coupled to the rigid aircraft dynamics during flutter. To properly account for this interaction, the linearized equations of motion for a free flying aircraft with a flexible wing are derived. Wing deformation is modeled by N as- sumed bending modes for the wing. Torsional deformation is not considered initially be- cause its effect on handling qualities and aeroelastic instability (at sweep > 25 o) has been shown to be small [Ref. 6 and 27]. The derivation presented here may be easily extended to include torsional effects if desired. Similarly, a quasi-steady aerodynamics model (i.e.

LINAIR) is used to calculate the airloads because the reduced frequency of the flutter mode is typically small (< 0.05). For other problems where unsteady aerodynamic effects are significant, the method may be extended to include apparent mass terms and circulatory aer- odynamic terms (using Pade' Approximants).

The equations of motion are derived using Lagrange's energy method presented in a form convenient for analyzing a deformable body. The generalized coordinates (qi) and speeds (ui) are carefully chosen to retain the rigid aircraft equations of motion with the effect of flexibility appearing as additional forces and inertia acting on the rigid aircraft.

Lagrange's equations of motion for a single continuous body with possible elastic proper- ties and holonomic constraints may be written as: La_m'ange's Equation of Motion --°_ dVol + E_W.. =% i=1,2, ...r

lii

(6.1) ,, P g _)tl i j=l _)tli jl where: L p = Material density I -- = Inertial acceleration V V = Inertial velocity • _•_ii i:!:i;'_ _ U = Potential energy !ili_i :! i/, tli = ith generalized coordinate tl. = ith generalized speed f W.. = Element of the transform matrix between tx & i:l jl (_ = ith generalized non-conservative force r = Number of degrees of freedom _••_ ,:,'i ¸ The generalized coordinates and speeds are chosen to be: q = x_, Yz, 7-a, _, 0, 4, rl_, rl2, ..., rlN tt=u, v, w,p, q, r, rll, _2' ""' _N The assumed mode shape amplitudes are rl 1 .... tin and X I, YI, ZI are the inertial position of the undeformed aircraft's mass center expressed in an inertial reference frame. The trans- form between u and q is given in equation 6.2.

,io,°ooooo,oooooooo-'oo o

TBIi o i

t

(6.2) W = 0 iTx_)(_ - pqr )) °|,)*l,i,°,g,g***i***,,,a'go°,°**o

0 0',II

where: = Transform from body to inertial coordinates TB_I = Transform from body axis angular rates m T_0(_ - pqr f Euler angles = Identity matrix _: i _, / Because only the wing has modelled flexibility, it is convenient m re-write Lagrange's equations of motion for the rigid fuselage-tail and the elastic wing separately (Eqn. 6.3).

il iiii::_i) iI_ il P £ 3Uwing (6.3) Wing I _V ] V°_ui dVol+ Wji / ,i _, ? , wing j=l _qi Fuselage & Tail + P V" dVol +E _Ufuse&tail Wji = I I _Ui] r fuse&tail j=l _qi % wing + % fuse&tail i = 1, 2, ... r : _-_i:_iiiii I_I: • _: i, i _ The wing potential energy term accounts for both the vertical displacement of each wing mass element and the elastic strain energy stored in the deformed structure.

• _, _'__i_ -% In order to evaluate the kinetic energy terms (volume integrals) in the equations of motion the inertial velocity, VP -°, of each material point (P) must be known (Fig. 6.1).

Y' N.A.

Material Point (p)) Y Wing) yt!

Win Aircraft C.M._ !

X Figure 6.1 definition of axes for the deformable aircraft !

• _: ;,'ii" i' N ,!ii ¸ :•_ !

(Sb = _ rli fi (Yi) i=l where: 8 b = local bending deflection " • i_ _' fi = ith mode shape rli = ith mode shape amplitude Figure 6.2 wing assumed bending modes Rem -p - Distance from the undeformed aircraft C.M. to the point (p) B

[ AI _,_ rx

(6.6) +

[0zXc

B =

L- Bj L°J

L egj / • Where [TB,_B] & [TB,,_B] are coordinate transform matrices from the (') and (") coordinate system to body axes.

The inertial velocity of material point (p) is _B -° (6.7) _B -o B B-I -¢m-P V_Bm'° = + CO XR B + RB-P where: T (6.8) oB-I= p q r T (6.9) V_B "° - u v w Substituting into the volume integral the wing contribution to the i.

kinetic energy term (linearized about 1-g level flight) is given by equations 6.10 - 6.15.

/•. ,17!•'_ '• ¸1,1¸ _ii i_ i_ ,., Elastic Wing Contribution to Kinetic Energy Generalized spe_ Linearized wing KE term (6.10) u : {1 M w + Sm(-r YNA + (_ ZNA ) ds' + qwoM w (6.11) v: M w + Sm (-_) ZNA + r XNA )ds'+ r UoM ff PWoM w w" • )i _ i•• (6.12) 6' M + S m(-cl XNA + t) YNA )ds' N -S(m j_=l fjt[j)ds'-qUoM w N f_I +/tlxyw + i'I + S(-Q2Ixx ., _ 0fj ,_. m Ef._.)ds' (6.13) p .

xxw xzw ___IO_S ' "'j--YNA j=lJ J + Sm ( _ ZNA- r UoZNA - qUoYNA + W YNA + P WoZNA)dS' afj.. , (6.14) q: P Ixyw + _t Iyw + r Izyw + S(m XNA _j=l fJ _J- Ixx" t_l j-_l- =_=s'rlj)ds + Sm(fi ZNK ¢e XNA + q U ° XNA + q w ° ZNA)dS' r • (6.15) IS Ixz w + Cl lyzw + r Izz w + Sm( -fi YNA + V XNA+ r UoXNA=qWoYNA-PWoXNA)dS '

, of, ., of i

_/ (6.16) 1 1_J(-Ixx,,Q2_-'_-)ds' + q J(-Ixx,,Ql_-'_')ds' i=l,... N . Ofi N Of. N

+ J(',x' 'J + m fi j_lfJ_J ) ds'.=

+ S -m f, ( -Cl XNA + [) YNA + 6"- qUo) ds' * all integrals are from=_ to where: m = wing mass per unit length fi = ith mode shape function Ixx,, = rotational inertia per unit span about the x" axis (slug-ft) b S T = structural semi-span Ii! _....

S' = spanwise location along the neutral axis I( ) ( ) w = undeformed wing inertia (slug-ft^2) _ _, _i _ i__ _.ii_, M = wing mass ' / : i; ¸¸ ;_._": "q( ) = mode shape amplitude A = C/4 sweep angle of semi-span = sweep of neutral axis relative to the C/4 Q1 =sin(A+ _) Q2 = cos ( A + _ ) >_18i:?i. !

m i':il/i/!i!! _!i [ XNA YNA ZNA]'= location of the undeformed N.A. at the wing root, relative to the e.G. in body axes i¸¸¸, _ : i/• The potential energy consists of the vertical displacement of the wing mass relative to the center of gravity and the strain energy due to bending deformation.

b$ N T N 02f.

(6.17) U.

wang =Mwg(Z+j_lfjTIj)+l_EI( x J Vlj 2 "= bs j=l_ ) ds' -T A Bemoulli-Euler beam model has been assumed for the wing structure in the formulation of the potential energy expression (Eqn. 6.17). Substituting the wing's contribution to the potential energy into the equations of motion and linearizing: generalized coordinate potential energy term (wing) (6.18) X I M w g cos(0 0) 0 + M w g sin(0 0) (6.19) YI

- M g cos(0o) ¢

(6.20) -M w g cos(0 o) + M w g sin(0 o) (6.21) (6.22) 0 (6.23) ¢ N _2f 02f.

(6.24) 1] i _ (EI j___ 1 J ......_L_) ds , bs 0S '=-'=_ rlj 0S '2 -"_2- bs + g (-cos(00) + sin(00) _ m fi ds' bs The non-conservative generalized forces acting on the wing are the lift, drag, and quarter chord pitching moment at each spanwise station. Expressions for the nonlinear generalized forces are given here, but these will be linearized after the fuselage and tail contributions

i !: I

have been included.

: ; !/!i ¸:: : associated nonconservative generalized speed generalized force !% :_' _ (6.25) u S Fwx ds' (6.26) v Fm, ds'

iiiiil

(6.27) w _i _i•_, (6.28) P j" (-Fwy Z/4 + Fwz Y/4 + M ) ds' (6.29) q (Fwx Ze/4 - FwzXe/4 -4- M ) ds' (6.30) r ( -Fwx Y/4 + Fwy Xe/4 + Mwz ) ds' Of. 3f.

(6.31) rl ( -fi F + M cos(A).---A.-_ + M sin(A) -..--2-1 ) ds' ,_ w,, 3s' "q bs' where: * all integrals are from-_ to -_ IF Fwy Fwz] T F-w = wx

[ z,,]"

:?!

/ i The rigid fuselage and tail contribution to the EOM is evaluated, linearized and combined with the wing contributions (Eqn. 6.10 - 6.31) to produce the intermediate form of the complete EOM's (Eqn. 6.32 - 6.38). Substitutions have been made for the undeformed aircraft inertia, mass and airloads (where appropriate).

i¸ _•i , i)', Contribution to the Linearized EOM Generalize41 Speed (6.32) u: M+ q w 0 M + M g cos(00) 0 = F x ; ,i,. _ (6.33) v: M+r U 0 M - p w o M - M g cos(00) ¢ = Fy N (6.34) w: _' M- q U ° M + M g sin(Oo) 0 + Z_i _-m fi ds' = F i=l z N 0f.

(6.35) p: rii f(-Q2 Ixx"_v'- YNA m fi ) ds' : M x Ixx + ClIxy + _"Ixz + _.=

S

(6.36) q: f_ Ixy + q Iy + _ I + _tli j'(m XNAfi- I x.,QI_-)ds' =M i !

(6.37) r:.

15Ixz + _ Iy z + r lzz = i f. 0f.

(6.38) _i: 15_(-Ixx" Q2 "2-l-0s, YNA fi -Ixx" Q1 1 m ) ds' + dl J'( _ + m XNA fi ) ds' N 0f. 0f.

+6'_-mfids' +qUo_mfids' + _(Ixx" 0s; 0s' j_l_j ' J +mfit])ds' N 02f. 02f.

+ 0g sin(0o ) _m fi ds' + _ 1]j fEI 1 J ds' Os ,2 Os ,2 Of.

= _[-fi Fwz + (M cos(A)+ -- o_-" Mwvsin(A))-'_'-'_ 1 ds' * all integrals are from--_ to -_ where: Inertia tensor of the undeformed aircraft about the Ixx, Ixy, Ixz, etc.

mass center Total aircraft mass M Net aerodynamic force acting on the undeformed _ = [ Fx, Fy, Fz] T aircraft Net aerodynamic moment acting about the wing c/4, M 14=[ i, i, M ] T on the undeformed aircraft The aerodynamic terms are now linearized and the equations of motion are re-arranged into a matrix form convenient for stability and control analysis.

] Matrix form of the linearized EOM , i //I¸ , (6.39) [A] i+ [B] x = [C] x + [D] u p = number of states (=9+N), m= number of controls where: [A] = inertia matrix (pxp) [13] = kinematics and flexibility matrix (pxp) [C] = aerodynamics matrix (pxp) [D] = controls matrix (pxm) T x = [ u, v, w, p, q, r, % 0, ¢P, _l, TI1, ..._ _IN] (pxl) T " ii • u = [_elev right, _elev left, _rudder] (mxl) Notice that the Fast (P) terms of the state vector are identical to the state vector for the rigid aircraft as derived in chapter 3. The explicit form of the matrices [A]-ID] are: m M 0 0 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 M 0 0 0 0 0 0 0 0 0 0 0--" 0 0 0 0 M 0 0 0 0 0 0 Iii0 I12 0 "'" IIN0 0 0 0 IxxIxyIxz 0 0 0 I410 142 0 ... I4N0 0 0 0 Iyxlyy Iyz 0 0 0 I610 I620 ... I6N0 0 0 0 Izx Izy Izz 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ... 0 0 [A] = (Mass Matrix) 0 0 0 0 0 0 0 0 1 0 0 0 0 ... 0 0 0 0 Ill I411610 0 0 0 Isll0 I8120 ...I81N0 0 0 0 0 0 0 0 0 0 0 1 0 0 ." 0 0 0 0 I12 I421620 0 0 0 I8210 I8220 ... I82N0 0 0 0 0 0 0 0 0 0 0 0 0 1 "" 0 0 0 0 I1NI4NI6N0 0 0 0 IsN10 Ism0 ...IsNN0 0 0 0 0 0 0 0 0 0 0 0 0 0-" 0 1 Equation 6.40 I0 0 0 O C) C) 0 O C) C) 0 C) O ..... O c)J

" " • • ! : i

o o oo o o oo o_o_o ...._o

o o oo o o oo oooo _,.....oo

o o oo o o oo o_o_o ...._o

o ooooo oo oo _, oo .....oo

%o

oooo oo o oooooo .....

0 0 0 C:) 0 0 _ C) 0 C) 0 0 0 ..... C) C) o 0 0 II

o_ oooo_o_oooo_ , ..... oo

0 0 a 0 0 0

o_ ooo o_, o_o_o .... _o

E !

0 0 0 0 0 0 0 0 0 0 0 0 0 ..... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ..... 0 0 r,

_2

i ._ .... : i ¸ i¸ i. _ Aerodynamics Matrix [C]= m 0F o_F OF 0F OF 3F X X X X X X OFx OFx OFx OFx OFx OFx 0 0 0 _F Z _ °°°..°°°°.*,,i......°°°°°°,,.,, i ...°....*..,...°,o..*...*.,..°., _u _ °..°°.°.°.l*,.,°°....I.°,°....., iii iI _ ..°.°°..°...°,.,°°....°,°°°...., °°,°°°°°°,°°°°,°°°°° ...... ,°,°0°°°°°°°°° ...... °°° _u • ° ............. ° ...... ........°,....° ....... ..°..

0 ,,°,°°°,°°,,,°,°,,,,,,°,,°°°°,°, 0 ...... ......°°°°°°.. .... °°°°°°°, °°°.°._.°..°.°...°. .... °..°°°°°°...° ...... .. .... ° ,,°,°,,,,°,,°°°°,°°,,,,°,,,_.,,, °.°..°° ........ ° .... ........°°°°..°°...°. ....... ° _Ilol °......°....°..°.°.°°..... ........... .,...°° .....

_ ,°,,,,.,°, ....... ...°..°° .......

......... ....°..... ......... ._°, °.°°..°°°._.°...°°.°°.°°.°.°...**° ........... 0°°° °°.°.°°.°°.°....°°...°°.°°°°°.°.. ..... .°....°°°..

T "''''°''''°°°''°°°°'°''''''''''' ,,,°,,,,°,,°,,,,,°,,,,,°°,..°,,, °°...°.° ..... . .... °°°°..°..°.....° ......... .° ....

_I1 ON i .... ,.°..,,.°°°....,.°.°*...,°,, Ou °°°°°°°°°°°°°°°.°°°°°..°°°°°°°_°,°°°0°°°°°° ..... ° J .... .°.°.°.°°°°.,,,..°°°°.....,, Equation 6.42 m m 3F x 3F x 3F x .,***°.m,.

?

381 382 3_ m 3F 3F Z 3M Controls Matrix [D] = 3M Z °°°°°°°°°°,°°°°°,° .... °°° °°,°°°**°°**°°°°,°°°°°,°° °°°°°°°°°°°°°°°°°°°°°°°°° 3Iloo ************************* 31102 °°°°°°°° .... °°°°°° .......

38_ ,°°°°,,°°°°°°°°°°°°,,,°°° 3110 N °°°°°°°°°.°.°°o.° ..... °°.

°°°°°°°°°°°o°°°°°o°°°°°°° Equation 6.43

The equations of motionincludetermsrequiredto account for thewing flexibility which are

weightedintegrals(I(...)(...))of mass,flexibility, andaerodynamic propertiesover the

wing span.Equations6.44- 6.48andfigure 6.3 definetheseintegrals.

X X'

/

I _ _ | _ y R Figure 6.3 wing flexibility spanwise integrals - undeformed wing, top view Definition of the Spanwise Integrals i = 1,2 .... p j= 1,2 .... p (6.44) Ili = j'-m fi ds ' _f.

(6.45) YNA mfi )ds' I,i= f(-QIx ., _f.

(6.46) I6i = _(m XNA fi - Ixx,, Q1 _-) ds' 3f. _f.

1 j (6.47) I8ij = f( Ixx" c3S7 _S' 4-m fi fj ) ds' _f.

(6.48) Iloi = _(-fi Fwz + ( _ Cos(A) +Mwy sin(A)) _-) ds' * all integrals are from--_ to -_ The integrals 1(1 , 4, 6, s, 9) involve the normalized assumed mode shape functions, fi, and the wing mass and stiffness properties. The integral (I100 is a function of the spanwise loading in the (z) body axis direction and quarter chord moment per unit span (i.e. Fwz & Mwc/n ). The spanwise load distribution as a function of the state vector is predicted by the vortex lattice method (LINAIR) which is also used to predict the remaining stability and control derivatives in the EOM.

The linearized EOM may be re-arranged into the following form: [A] m_ = {[C] - [B] Ix + [D] u or Kinematics & Mass & Inertia Aerodynamics Elasticity m m m u Rigid Rigid Rigid Aircraft Aircraft Aircraft °°°°,°°°°°°.°°.°.. °°°,,°°,°°°°°o°° °°°°°°o°°°°,°°°°° X (6.49) Wing Flexibility Wing Flexibility Wing Flexibility Terms Terms Terms m m ._a Jb..i ..,a h..l -_ v|-._ v I -.., =.q -., v I _, v! -I, v I P N P N P N Control m m Rigid Aircraft °.°°°°°°°°.°°°°,.°°°.o + u Wing Flexibility Terms m u v m The portions of the matrices [A]-[D] labeled "Rigid Aircraft" contain terms identical to those in the rigid aircraft 6 DOF linearized EOM derived in chapter 3. The influence of wing flexibility appears explicitly in the portions marked "Wing Flexibility Terms". The :i formulation of the EOM in this way allows the designer to analyze the influence of flexibili- ty on an aircraft whose rigid aerodynamic properties may already be well known. This separated form of the EOM also permits the designer to see the influence of flexibility expli- _iI citly as a deviation from the rigid aircraft's dynamics. Current aeroelastic analysis methods stress the influence of structural flexibility and unsteady airloads on the stability of the de- ./ i ¸ forming aircraft structure, but are not easily adapted to analyze unrestrained motion. Simi- , L ; ¸_ , larly, most stability and control analyses are not easily adapted to account for the dynamics associated with aeroelastic phenomenon. The equations presented here show how these two areas may easily be combined into a single formulation which may be used to solve problems spanning the fields of aeroelasticity and stability and control.

i ¸' , The linearized equations of motion are now reduced to their final form for the purpose of i ¸ • .... i , stability and control analysis (E,qn. 6.52).

[A] i = {[C] - [B] }x + [D] u (6.51) x = [A]-lttc]-tB]}x + [A]d[D]u • ? or (6.52) i = [,_]x + [I_]u i_ ..... '_ where: (6.53) [,_] - [A]-I {[C] - [B]} (6.54) [5] - [A] d [D] The system matrices [A] and [B] require that the following information be known to evalu- ate the equations of motion: 1) Aircraft geometry including the C.G. and elastic axis location 2) Flight condition- altitude, velocity 3) Aerodynamic stability and control derivatives of the rigid aircraft 4) Mass properties of the rigid aircraft 5) Assumed bending mode shape functions 6) Aerodynamic derivatives involving the spanwise variation of wing loading All of the aerodynamic derivatives for the cases considered in this work are calculated nu- merically using the vortex lattice method LINAIR (chapter 2). The mass and elasticity inte- grals are evaluated by numerically integrating the appropriate quantities along the structural wingspan. The process of generating the EOM for the flexible wing aircraft has been auto- mated in a FORTRAN code and linked to an eigensystem solver, dynamics simulation, affd the MIDSM synthesis code to permit detailed analysis of oblique wing aircraft with wing flexibility.

:i i_, • _; i_, _ _ i_" _

6.2 Aeroelastic Stability of Oblique Wing Aircraft

Flight tests of the AD-1 oblique wing demonstrator aircraft revealed that wing bending can ! i¸ S: , seriously influence handling qualities for oblique wing sweep greater than 30 °. Unlike a conventional symmetric aircraft, the bending deformation of an obliquely swept wing pro- duces large pitch and roll moments which can overwhelm the aerodynamic coupling effects present in the rigid aircraft. Figure 6.4 shows the influence of static aeroelastic wing defor- mation on various force and moment coefficients as load factor varies. These results were obtained from flight test results for the AD-1 [Ref. 6].

.005 AI deg ------ 30 AC t due to - .005 load factor - .010 I t I - .015 Z_Cn .005 dueto 0 --- __._.-_-- -_ factor _ _ load 005 I _. I I I | .5 1.0 1.5 2.0 2.5 3.0 Normal acceleration, g Figure 6.4 effect of wing flexibility on aerodynamic coupling, AD-1 flight test results Increasing load factor causes the wing to bend upwards which increases the local angle of attack on the forward swept portion of the wing and decreases it on the aft. This shifts the cenlroid of the span load towards the forward swept wing panel which produces a roll-left, pitch-up tendency as the wing bends under positive loading. If the load factor is changing due to angle of attack change, the aeroelastic effect on the aerodynamic coupling acts oppo- site the coupling present in the rigid aircraft over a wide range of the flight envelope: The influence of wing flexibility can be a dominant factor in trim, stability, and controlof an oblique wing design. Even so, many analyses of oblique wing designs (such as the VMS "_.i,_: _,i iiI . .- i, _ , handling qualities study for the F-80WRA) neglect this important phenomenon. In this • , - i '¸

chapter the influence of wing deformation is included in the design of an oblique wing air- •:_i:_ <i_i 'i, _ craft for improved handling qualities.

i •1 , ;'i_ i¸ When the oblique wing configuration was first analyzed, designers believed that the critical _L ¸ .

mode of aeroelastic instability would be a divergence of the forward swept wing panel at high sweep angles. Jones and Nisbet [Ref. 7] showed that this conclusion is incorrect be- cause the assumption of a clamped fuselage boundary condition is inaccurate for the free fl_dng oblique wing aircraft. Instead, a low frequency flutter characterized by a coupling of the wing-bending and rolling motion of the aircraft occurs. Most importantly, this mode becomes unstable at dynamic pressure which may exceed the clamped divergence value by more than a factor of 4. Rutkowski [Ref. 27] calculated in greater detail the critical flutter modes of the AD- 1 design using NASTRAN and FLUT numerical analyses. The results showed that for oblique sweep greater than 30 ° the roll-wing bend flutter mode is critical.

The frequency of this mode is low (4Hz for the AD-1) and its sensitivity to torsional stiff- ness was shown by Rutkowski to be very small.

In order to verify the aeroelastic analysis proposed in this chapter (section 6.1), the AD-1 configuration is analyzed using the proposed method and the flutter results compared to Rutkowski's calculations using NASTRAN and FLUT. The data shown in table 6.1 is used to model the AD-1.

1700 (lbs) Weight 32.3 (ft) Span 93 (ft^2) Wing Area 0.39 Taper Ratio 12% Section Thickness 0.5 Mach Number .002377 (slug/ft^3) Air Density 3.8e6 (psi) Modulous of Elasticity •16 (in) to .04 (in), linearly tapered Wing Skin Thickness Material Density Glass = 121 (lb/ft^3) foam = 2 (lb/ft^3) Number of Assumed Bending Modes 6 polynomial modes Aerodynamics Model Quasi-Steady Vortex Lattice (LINAIR) Table 6.1 AD-1 aeroelastic stability analysis input parameters Figure 6.8 shows the vortex lattice model for the AD-1 along with a 3-view of the aircraft.

The flutter speed, assuming only the wing contributes to the airloads (as per Rutkowski), _ i I ' is calculated for wing sweeps of 30, 40, and 60 degrees and is compared to Rutkowski's results in figure 6.7. The flutter speed is calculated by solving the eigensystem of the air- craft's linearized equations of motion at increasing airspeed until an instability is detected.

The eigenvector corresponding to the unstable mode defines the flutter mode shape. Simi- /_ 7 • _::i _' lar cases are studied where the fuselage is assumed clamped and divergence of the forward swept wing is critical. All of the flutter analyses assume that the aircraft is free to roll, pitch and plunge. The results show that the method presented here accurately predicts the roll - bend flutter mode. The flutter frequency for all sweep angles (except 60 ° ) is approximately , " i, _ 4 Hz, the same as Rutkowski's results. Calculated divergence speed differs by as much as 30% due to an overly stiff structural model of the AD- 1 wing. It was assumed that the skin thickness varied linearly across the span (in the absence of exact information of Rutkow- ski's model) and this is an overly stiff wing structural model. The flutter speeds compare very well (within 15%) for all wing sweeps except 60 o where the proposed method pre- dicts a switch in the flutter mode to a very low frequency roll-plunge motion instability. If this low frequency rigid body instability is ignored the next critical flutter mode is a roll - bend mode and this occurs at a speed of 700 (ft/sec), which is much closer to Rutkowski's results. It is not known if Rutkowski's work discovered a similar low frequency instability and ignored it as a "rigid body mode" or if this discrepancy is a valid one. It is typical to ignore unstable low frequency modes in an aeroelastic analysis where the aircraft is only given partial freedom of motion and the total aircraft is not modeled (i.e. no tail). The method proposed here is capable of predicting the free flying dynamics of the aircraft and this instability may very well be physical for the contrived case of a wing-only aircraft with roll, pitch, and plunge DOF. Another potential source of discrepancy is the doublet lattice aerodynamic model used in Rutkowski's work. For a thin lifting surface model a doublet lattice method is not capable of predicting the important leading edge suction forces which can strongly influence the rolling moment and sideforce of an oblique wing design. None- theless, the two methods show good comparison and ability to predict this unique a&oelas- tic instability.

I.

_ii_ _ _ Figure 6.5 3-view of the AD- 1 oblique wing demonstrator •i:?

300 400 500 60 ° Increasing Sweep Figure 6.6 AD-1 vortex lattice geometry, wing only Aeroelastic Instability Speed vs Wing Sweep roll-pitch-plunge DOF, AD- 1 configuration 800.

700.

/ 00.

°°°.°°°°°°°°°,.

Velocity °°°°°°°°°°°°°°° (f/s) 500.

i : " _¢00° ................................... ! ................ _.......... _d body m_ :+ Divergence Speed, Morris ] ----O-- Flutter Speed, Morris | 300.

...................... --.4-- Divergence Speed, Rutkowski I" ....................

Flutter Speed, Rutkowski | | | | | 200.

20. 30. 40. 50. 60. 70.

Sweep (deg) Figure 6.7 AD-1 aerolelastic instability speed vs. wing sweep The effects of full 6 DOF, complete aircraft aerodynamics, fuselage inertia, and tail size on oblique wing flutter will now be considered. First, the influence of the full 6 DOF on the flutter mode will be studied. Figure 6.8 shows the vortex lattice model for the AD-1 at 45 ° wing sweep. A root locus of the natural modes as flight velocity increases is shown in fig- ure 6.10. The action of the lowest wing bending mode and the roll mode can be clearly seen as these two modes become combined in the unstable flutter mode. The effect of wing flexibility and increasing dynamic pressure on the stability of the short period, dutch roll, and other rigid body modes is seen to be small. Also shown in figure 6.9 is a root locus for the same aircraft with roll degree of freedom and wing bending. The flutter speed and frequency for these two cases is nearly identical, indicating that the reduced degree of free- dom model is useful for studying flutter. Simulation results for the 6 DOF aircraft under- going flutter are presented in figures 6.11 - 6.13 along with a frame by frame pictorial sim- ulation of the aircraft flutter motion (front view) in figure 6.14. These results show that the flutter mode consists mostly of wing bending and fuselage roll even for the 6 DOF case.

The left and right wing tips move nearly in phase with each other and the right (forward) tip deflects almost twice as much as the left.

,?

_!!i_i _

! • i•" i iiiii_ •: Top Side

L

Front Figure 6.8 vortex lattice model AD-1 oblique wing research aircraft 45 ° sweep Flutter Root Locus AD-1, 45 (deg) Sweep, Roll-Wing Bend DOF Only 50.

i _' , : i _ L: ¸. _ _ _ ,"ii __ : 40.

30.

20.

Imaginary (rad/sec) 10.

i O.

-10.

-16. -14. -12. -10. -8. -6. -4. -2. 0. 2.

real (rad/sec) Figure 6.9 AD-1 flutter root locus, roll-wing bend DOF Flutter Root Locus AD-1,45 (deg) Sweep, 6 DOF, 6 Bending Modes 50.

i (lst " Wing Be!ding) ! 1

40.

30.

. il • 20.

Imaginary (rad/see) 10.

Oo -10.

.

-14. -12. -10. -8. -6. -4. -2. 0.

-16.

Real (rad/sec) i: Figure 6.10 AD-1 flutter root locus, 6 DOF and wing bend i ¸_ , > Translational Velocity vs Time AD-1 Flutter, 6DOF & Wing Bend, Uo = 624 (f/s) i: i! _ _ .

_i:_! _ ° . =.. ° °.. °°......._.**.°...-...i..°******o.°°° _.°*****. = °°. °*..°._* °....* °,. °.. .o,'''o'''oo • • ° ,i _ ,_" _ • ." . . ."

Z '_ i_ i Plunge velocity (w) Velocity 2.

if/s) 0.

? _ii • -2.

i_i ,_ -4.

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 Time (sec) Figure 6.11 velocity vs. time i • Euler Angle vs Time AD-1, 6DOF & Wing Bend, Uo=624 (f/s) 10.

• : • .

° ........ i........ i........ i ........ ........ i........ i........ ...... ii ........ i .......

Angle ° (deg) -5.

-10.

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1,8 1.9 Time (sec) Figure 6.12 Euler angles vs. time Tip Deflection vs Time AD-1, 6 DOF & Wing Bend, Uo=624(f/s) i_ __ .

__ right wing tip -- left wing tip .

IIo ° : ii _ -1.

:" { -2. i 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 T_e(sec) Hgure6.13 fipdefl_tionvs, dme

\ \ \ \

v w

/ / / f (

v _ ••_ili ¸ v Figure 6.14 6 DOF flutter simulation AD-1, 45 ° sweep (front view) The influence of wing flexibility on stability is shown in figure 6.15 where the natural modes are plotted as the wing stiffness is decreased by an order of magnitude from the ac- (:i ( i,?_ _i:_ tual AD-1 values. Notice that the AD-1, unlike the F-80WRA, has a stable dutch roll mode at 45 ° sweep. This plot shows that decreasing wing flexibility at a fixed flight veloci- ty will produce the roll wing-bend flutter instability and not significantly affect the stability of the other modes. The coupling of the aircraft's dynamic response is significantly affect- ed by changes wing stiffness and this will be discussed in the next section.

Root Locus on Wing Flexibility AD-1, 45 ° Sweep 14.

12. ....... °°°°° • °°°°°.,°°°° °.°°.°°°.°° 5.0E5 _I- 5.0E6(psi) l i i IT_'_'_II 10.

_ ._. ° . . ° . , ..... _, , ,,. ,m _.,., .................................................. i............ i............

8.

Imaginary 6.

(rad/sec)

' iNi

4. °**lo* iii*,, _ol°****li,,, ii **lilooill *,,+*,**lI*_*l**l°*°°*6," ,l+oe* * 2.

,°°°.°°°°°°, ............ •:.r q .... i............ ........... :. ............ ' ...........

0.

l"[&Phugc -2.

-12. -10. -8. -6. -4. -2. 0. 2.

Real (rad/sec) Figure 6.15 effect of wing flexibility on aeroelastic stability (6 DOF) i' Jones and Nisbet fast suggested that fuselage roll inertia can have a significant effect on • %•¸,¸¸¸ oblique wing flutter speed because large inertia causes the fuselage to behave as though it is clamped. This lowers the flutter speed to that of the clamped divergence value as the fuse- lage inertia becomes infinite. Figures 6.16 - 6.17 show this effect for the AD-1 with 45 ° + sweep, where flutter speed and frequency are plotted versus fuselage inertia. Most appar- ,, .. ; :(, ,, ent from these results is that the flutter speed initially drops off quite dramatically as fuse- i ¸ • •: lage inertia increases. For zero fuselage inertia the flutter speed is 3.1 times the clamped di- • / •i• vergence speed. The AD-1 flutter speed is only 11% greater than the clamped divergence value.

Flutter Speed vs Fuselage Inertia Ratio i ¸ AD-1,45 ° Sweep, Roll-Bend DOF, Wing Only 1800.

1600.

..................... i.................... )...................... i...................

1400. ,I,tlo _iiIo*weeeIIII_IIe_eIIIIIolIIQoQoe_o_II_aotII_IoIoem_gmIlB_oa*oIIaoI_II-_mII_I*oi 1200.

v_

(f/s) L 1000.

..................... i..... _ ........ i....... J..C! .am....P_l.. Dive.r.g.e.n._. I ..........

800.

ii!ii!i!i!

600.

400.

-4 -2 0 2 10 10 lO 10 10 L.

I XXO Figure 6.16 flutter speed vs. fuselage inertia ratio Flutter Frequency vs Fuselage Inertia Ratio AD-1, 45 ° Sweep, Roll-Bend DOF, Wing Only 14.

12.

°°°°°°o°°°°oo,,,°°°°;°.°°,,°°°°o°°.°.°°oo°i°°°o°°°,°°°°°°.,..,°°:o°°.°°°°o°° ..... °°°°, 10.

8.

f 6.

(ttz) 4.

2.

: ' " -" : I 0.

I -2.

' ' 10' ' 10 -4 10-2 10 o 2 10 I X XO Figure 6.17 flutter frequency vs. fuselage inertia ratio T , " , The horizontal and vertical tail size can also influence flutter speed because they affect the aircraft's overall roll damping. To demonstrate this, the AD-1 configuration with 45 ° sweep is studied with various size tails and the flutter speed and frequency is plotted as tail size increases (assuming no affect on inertia) in figures 6.18 and 6.19. Figure 6.18 shows that flutter speed initially increases with increasing tail size because the additional roll damping removes energy from the flutter mode. As the tail size continues to increase the roll damping becomes detrimental as it resists the rolling motion and effectively clamps the fuselage. Typically, the tail size required for this negative effect to occur is unrealistically large and in general the tail's influence on flutter speed is favorable.

Flutter Speed vs Tail Roll-Damping AD-I, 45 ° Sweep, Roll-Bend DOF 700.

680.

• .,....•°.,...•.•°°.o°,.._..°°°°••••°•.•.•••.••°•,•°°.._°o.•.°°...°°o°.,.•.° ..... °.°, 660.

640.

°•..° °°°.°•_••o°°°°°°°°°°°°°.°...°..°•°°i°..°o.°°.•..°°...o. ....... °°, Vf 620.

(f/s) 600.

580.

• o.°.•o...... •••°°..°•o.•o°.°.°...o.......°°•.°°°•.ooo.•oOo.•o°°°•••••°.•..••°°°•.°.0•.

560.

• .°°,•.

540.

520.

10 10 i 10 2

qp

Figure 6.18 flutter speedvs, tail roll-damping Flutter Frequency vs Tail Roll-Damping AD-1, 45 ° Sweep, Roll-Bend DOF .

3° °° _.*******°°°°°,.°,°**°,o,...,, iiiiiiiiiiiiiiiiiiiiiiiii!ii ..............

(nz) _ .i . ,°.**... .,.****** **i,.,,,i****,***°..,,,,.,°°**, O. ........ _ ..........

ir I Divergence I -1.

10 o 10 l 10 2 10 c_._, Figure 6.19 flutter frequency vs. tail roll-damping

6.3 Influence of Wing Flexibility on the Closed Loop

Handling Qualities of the F-80WRA

To asses the impact of wing flexibility on oblique wing handling qualities, the closed loop performance of the F-80WRA is studied, with and without an elastic wing. An RMF SAS for the nominal configuration OWRA with a rigid wing has already been discussed (chapter 3) and this synthesis is repeated here with an additional degree of freedom for wing bend- ing. A single parabolic wing bending mode is included in the aircraft's dynamics model to account for the affects of aeroelastic wing deformation on the airloads and the dynamic lag of wing bend during maneuvers. The dynamics model used in the synthesis is the Mil.

Spec. F-8785C model previously presented in chapter 3, but augmented to include dummy states which represent zero wing deformation. In the synthesis, the RMF SAS attempts to force the flexible OWRA's dynamic response to be most like that of the rigid model's.

This synthesis could be extended to include flutter suppression (in future work) if the flight "i condition is chosen so that flutter is critical, more bending modes are assumed for the wing, and aileron control surfaces are modeled. Because the dynamics model is rigid and stable, the SAS would attempt to stabilize the flutter and restore handling qualities most like that of the rigid decoupled aircraft. This differs from conventional flutter suppression de- signs where only flutter stability is the goal.

_ !i_ ¸'¸'' k The synthesis is carried out identically to the RMF SAS synthesis described in 3, except that the EOM are now calculated using the linearizecl EOM for the free-flying aircraft with an elastic wing (section 6.1). The structural model for the F-80WRA's wing is chosen to produce the same fast bending frequency and 1-g wing tip deflection as the preliminary de- sign results [Ref. 21] predict. Figure 6.20 shows the assumed mass and stiffness distribu- tion for the wing which meets these requirements.

?i: .

Wing Bending Stiffness vs Semi-Span F-80WRA 300 (fi^2) Wing x1076.

5.

4.

EI (lb_ft^2) 3.

2.

ii

o .

0. 5. 10. 15. 20. 25. 30.

Semi-Span(fi) Vaccum Bending Frequency = 5.73 Hz Wing Mass Distribution 1-g Tip Deflection = 9.6 inches 3.0 2.5 2.0 m (slug/ft) 1.5

i

1.0 .5 .0 0. 5. 10. 15. 20. 25. 30.

Semi-Span / Figure 6.20 F-80WRA wing structural model

Theweightingmatricesandinitial conditions assumed for the synthesis are identical to

those in section 3.3.2.3, except that no weight is placed on the wing tip deflection states (table 6.2).

Output error weighting gygzp q r t_ [Q] = diag [ 10, 5, 1, 1, 1, 10 ] Control effort weighting

8 _ 8

roll pitch yaw

[R] = diag [ 100, 100, 100 ]

Pilot command washout filter pole location

e=2.0

Actuator pole location s = 10.0 (1.6 Hz B.W.)

RMF SAS Initial Conditions T vertical gust Yo = [ 0, 0.5, 0, 0, 0, 0, 0, 0, 0, ] T lateral gust

yo = [0.5, 0, 0, 0, 0, 0, 0, 0, 0 ]

_o = [0, 0, 0, 0, 0, 0, 0.5, 0, 0, ]T roll input T pitch input

_o= [0, 0, 0, 0, 0, 0, 0, 1.0, 0 ]

T yaw input

'_o = [0, 0, 0, 0, 0, 0, 0, 0, 0.5 ]

Table 6.2 flexible wing OWRA RMF SAS synthesis input parameters The performance of the closed loop OWRA (with RMF SAS and flexible wing) perform- ing the 4-g pitch-up maneuver is shown in figures 6.21 - 6.27.

Lateral Acceleration vs Time Nominal OWRA, 45 ° Sweep .2 0.0 gy -.1

I h i I---_- Rigid Wing OWRA I

[ _ i 1_ Flexible Wing OWRA [ -.2 °°,,,,,.',,,,°,oe,allilIlle .,,,,,,°°,..,,.

3 : : -°4 .

0. 1. 2. 3. 4.

Time (sec) Figure 6.21 lateral acceleration vs. time Normal Acceleration vs Time Nominal OWRA, 45 ° Sweep .

O° ,,,,o****°,, °°°,,ll,al****llllqlolll,a,,°,,°°*°pt°°o°°°****olll*_-*o,,,,_°°°°°°,l, -1.

'"..........._M_i .................................. i................

gz -2.

i: ........ __Wmwg°_ ............... i................

-3.

-4.

0. 1. 2. 3. 4. 5.

Time (see) Figure 6.22 normal acceleration vs. time : !7 ' * 2'.

i Pitch Rate vs Time Nominal OWRA, 45 ° Sweep 25.

m 20.

,...i.°°°°°l%))..*****))°..)°o °.°°.°..o,..°.o)_,°).°.)°°...°.°..°°° .°°°...°.°°°°.° 15.

q ----/k--- Model r a ----D--- Rigid Wing OWRA (deg/sec) 10.

......... i_ ....... -'-__ ...... .: ...............

.

.

.

° 1. 2. 3. 4.

Time (sec) Figure 6.23 pitch rate vs. time Yaw Rate vs Time Nominal OWRA, 45 ° Sweep 12.

10.

8.

6.

r 4.

(deg/sec) 2.

0.

-2.

< -4.

i _, ,i_; • ,, ,5, ° 1. 2. 3. 4. .

ii, _, _ ; Time (see) , i:. • _,i _•: ?, Figure 6.24 yaw rate vs. time Roll Angle vs Time Nominal OWRA, 45 ° Sweep 30.

! /_/¸¸ ! i • i,_i • !, 20 ..................... _"""- .................. !..............

Ji I i

10 .......... _i.-.! ....... 1 _ Rigid Wing OWRA I .... i ................

:: / + Flexible Wing OWRA I :: ---_'...... ; ......... i ................... ; .........

, O.

-10 .........

-20. i__ _ AJk Akbl, JkJk & * /kJk JkAk _lk& &AI, Ark& Jk.Ji, A& _rkAk_k& AAht_& At "_ • •, * •, * I * * * * * * * *, • • * • r_ _ _ _ v.r _ _ ,_,_-v-.ir_r,v-wrv,_ ,wv _r._.,irlr-v,v._rw_,lr_rv _ - ,

:_i _!il i

-30.

0. 1. 2. 3. 4.

.

Time (sec) Figure 6.25 roll angle vs. time Tip Deflection vs Time Nominal OWRA With Flexible Wing, 45 ° Sweep 2.5 2.0 1.5 (5tip (ft) 1.0 .5 • .

.0 0. 1. 2. 3. 4.

.

Time (sec) Figure 6.26 wing tip deflection vs. time Control Surface Deflection vs Time Nominal OWRA, 45 ° Sweep 15.

10.

left elevator rigid OWRA fight elevator rigid OWRA rudder rigid OWRA .

left elevator flexible OWRA Deflection right elevator flexible OWRA rudder flexible OWRA (deg) O.

-5.

-10.

Figure 6.27 control surface deflection vs. time The excursions in lateral acceleration, yaw rate, and roll angle are still large for the nominal closed loop OWRA with a flexible wing, which indicates that wing flexibility has not sig- nificantly contributed to the aircraft's controllability and therefore, re-design of the configu- ration is still needed. The flexible aircraft shows reduced lateral acceleration during the pitch-up maneuver, but increased yawing motion. The elastic wing design is using exces- sive sideslip (8 o) to limit excursions in lateral acceleration, where as the rigid aircraft mere- ly rolled fight. A most noticeable difference between the response of the two aircraft is that the rigid one rolls right (+20 ° roll) during the maneuver, where as the elastic one rolls left (- 20°). The strong influence of wing bending, which produces a roll-left coupling with in- creased normal load, is dearly dominating the response of the OWRA with the elastic wing. These results show that it is a gross error to assume that a rigid wing model ade- quately describes the dynamics of the OWRA and furthermore, that a SAS designed for the rigid aircraft has little in common with an appropriate SAS for the flexible design. Wing flexibility must be included in even the preliminary assessments of handling qualities of /- oblique wing designs.

/

6.4 MIDSM Synthesis of the F-80WRA

with a Flexible Wing,

45 ° Sweep, M=0.8

The previous example has demonstrated that the F-80WRA with wing flexibility still has insufficient controllability to produce acceptable closed loop handling qualities and that wing elasticity significantly alters the aircraft's dynamic response. These results indicate that the optimal OWRA configuration with wing flexibility may be very different from the optimal rigid configuration. To test this hypothesis, MIDSM is used to re-design the flexi- ble wing OWRA at 45 ° sweep and M=0.8. AN RMF SAS is assumed and all of the inputs are identical to those in the previous example. The configuration design variables consid- ered are: = Wing bank angle relative to the fuselage X pw = Root chordwise location of the wing pivot Xpf = Location of the wing pivot along the fuselage dihedral = Tip deflection due to jig-built parabolic dihedral Table 6.3 design variables for OWRA synthesis with wing flexibility Notice that the dihedral now defines the jig built value, so that the aim solution automatical- ly accounts for the additional 1-g deformation that occurs at trim. The results from MIDSM are presented in figures 6.28 - 6.34. A comparison of the optimized rigid and flexible wing OWRA configurations with the nominal design is given in figure 6.28. The optimized flex- ible wing OWRA shows significantly more wing bank and different wing displacement than the optimal rigid wing design, verifying that wing flexibility alters the optimal oblique wing configuration.

_b =-8.5 (deg) Xpw -- 4.79 (ft) Xpf -- 38.81 (ft) dihedral = -.008 Table 6.4 optimal configuration variables 45 ° sweep OWRA, RMF SAS, flexible wing Nominal OWRA ii:i! ................ ........

MIDSM OWRA (flexible) Top Front Side > Figure 6.28 MIDSM F-80WRA with flexible wing Lateral Acceleration vs Time 45 ° Sweep OWRA i .3 :: J _ MIDSM OWRA Rigid Wing I [ ,2 ................. i................. [ + Ncxninal OWRA Flexible Wing [ ..... t : [ _ MIDSM OWRA Flexible Wing [ I m .1 gy 0.0 -°|

_.2 .......................... i ................. .............. .......... F

i/: -.3 i 0. 1. 2. 3. 4. 5.

Time (see) Figure 6.29 lateral acceleration vs. time Normal Acceleration vs Time 45 ° Sweep OWRA .

.

....................... Nominal OWRA Flexible Wing ...........

J_ MIDSM OWRA Rigid Wing J MIDSM OWRA Flexible Wing -1.

_°_°°°_.._°_°_.__°_°_°___*_ gz ° -2.

° ° °°°. o,...****,°.. ° ° °.°.-...°° ° o°°°°.,. ° ° °°°.. ° °° ° ° ° °. ° ° ° o ° ° _° -3.

4.

2. 3. 4.

Time (sec) Figure 6.30 normal acceleration vs. time n Yaw Rate vs Time 45 ° Sweep OWRA .::i' , '¸ , 12.

10.

,o.,oo,,o,o,Q joo,o,l_.,llooo.ooo**oo,oo_o,.**o,oo*oo,oooo_..oo.o.oo....o..

?

8.

+ Nominal OWRA Flexible Wing -,O,,-- MIDSM OWRA Flexible Wing + MIDSM OWRA Rigid Wing ]i ..........

6.

r (deg/sec) 4. ooo o j i o o o i g i $ I t I o i .............. ii 2. i ............ _.................. ._ ................. . ................

i _] 0.

;i | | | m -2.

.

1. 2. 3. 4.

° Time (sec) Figure 6.31 yaw rate vs. time / Roll Angle vs Time 45 ° Sweep OWRA 20.

+ Nominal OWRA Flexible Wing + MIDSM OWRA Rigid Wing I • -o.-. MIDSM OWRA Flexible Wing .....

10.

° i.

¢

(deg) -10.

-20.

• i: ¸ •• ,i • ° 1. 2. 3. 4.

/ Time (see) Figure 6.32 roll angle vs. time i , Wing Tip Deflection vs Time ! ,_i¸_ 45 ° Sweep OWRA 2.5 2.0 1.5 1.0 .5 .0 ° 1. 2. 3. 4. 5.

Time (sec) Figure 6.33 tip deflection vs. time o ° .

, Deflection (deg) ° -2.

-4.

O° ° ° 2. 3. 4.

Time (see) Figure 6.34 control surface deflectoin vs. time i _ i_ The impact of the design changes on the OWRA's handling qualities is shown in the simu- lated 4-g pitch-up maneuver in figures 6.29 - 6.34. Table 6.5 summarizes the peak excur- sions in the coupling variables for the nominal, optimal rigid, and optimal elastic designs.

MIDSM OWRA Nominal OWRA MIDSM OWRA % Improvement Rigid Wing Flexible Win S Flexible Win S i,):_ _ -.11 0.12 .07 42% gypeak ii _ • _peak 6.7 -6 68% -19 (deg) rpeak 11.5 -1.2 90% 1.1 (deg/sec) Table 6.5 optimized flexible wing OWRA performance summary These results show that the dynamic response of the flexible wing OWRA is significantly improved by the MIDSM re-design and its performance is equal to or better than the opti- mized rigid aircraft. Even though the optimal rigid and elastic wing designs have similar values of the peak coupling variables, their dynamic responses are quite different. Com- parison of figures 6.29 - 6.34 reveals that the optimal rigid OWRA rolls fight during the pitch-up maneuver, where as the flexible aircraft first rolls fight and then left. The elastic wing aircraft is designed with an excess amount of wing bank so that the initial roll motion is to the fight, but as the wing reaches its maximum deformation (1 second later) the coupling of wing displacement and wing bend causes a left roll which keeps the overall roll excursion very small. The initial right-roll also reduces the amount of lateral acceleration to the left. The improved response is obtained by a re-configuring of the aircraft that not only accounts for the effects of static wing deformation on the response, but is also dependent upon the dynamic deflection of the wing itself. For example, the correct amount of wing "overbanking" is highly dependent upon the time lag associated with the wing bending in order to produce the proper reversal in roll rate during the pitch-up. The optimized rigid OWRA designs obtained most of their dynamic response improvement by re-configuring the aircraft for reduced Cl0 t and Cy0t coupling. It is reasonable to assume that a similar process has occurred in the flexible wing design, except that the static aeroelastic values of the coupling terms are made small. Table 6.6 presents these coupling terms for the various designs so that this theory can be investigated.

/i ',_i= _ • Nominal OWRA Nominal OWRA MIDSM OWRA (Rigid Wing) (Elastic Wing) (Elastic Wing) -.2751 -.2417 .1882 -.1046 -.0114 .0574 -1.3112 .6428 .433 Cma -.0694 -.0419 -.0397 _:_ii _i __ Table 6.6 aerodynamic coupling of the optimized flexible wing OWRA (static acroclastic values) ,:_i, ; ii, _ The static aeroelastic value of Clo _ is significantly reduced in the optimized design, but the values of Cyo_ and Cno _are not made small. Notice also that the Cyo_ term is reversed in ,,.i _,I, i _ sign and made large by the wing "overbanking". These results demonstrate that the optimal design is not merely an "aeroelastic decoupling" of the nominal OWRA configuration, but is instead a highly integrated design which depends upon the detailed dynamic interaction between aircraft motion and wing flexibility. These results further stress the importance of wing elasticity as a significant phenomenon in the preliminary design of an oblique wing aircraft, perhaps more so than for a conventional symmetric aircraft.

6.5 OWRA Synthesis for Optimal Wing Flexibility

Because wing flexibility can produce changes in the aerodynamic coupling which oppose the coupling present in the rigid aircraft, perhaps there is an optimal amount of wing flexi- bility for improved handling qualities (in an oblique wing design). To test this idea, the i ; • ?i • previous synthesis is repeated with the wing modulus of elasticity (E) as an additional de- sign variable. The inputs are identical to the previous example, and the initial guess for E is 10xl06 (psi). The MIDSM results show that the optimized OWRA has the same values for the configuration variables as the previous optimized elastic wing design, and the optimal wing stiffness is 18x106 (psi), which is twice the nominal value. Figures 6.35 - 6.37 compare the dynamic response of this design to the optimized rigid and flexible wing i, I _, , OWRA's. Table 6.7 summarizes the peak excursions of the coupling variables.

" i_ _ ,, _ Elastic Wing Rigid Wing Elastic Wing Optimal (E) gype_ -. 11 .07 .045 t_P eak 6.7 -6 6.

(deg) rpeak 1.2 1.1 -1.56 J (deg/sec) Table 6.7 performance comparison optimized wing flexibility OWRA 45 ° Sweep, RMF SAS Lateral Acceleration vs Time 45 ° Sweep OWRA .3 M1DSM, Flexible Wing

[

.2 ............... ....6_. MIDSM, Flexible Wing, Optimized (E) .............

.1 : °...°°..*°°*°°°°°., .... ,.°,°.°.°°...°°...°..,.,.,,,,-'°,.°°°....,... °.*o..°°° °........°°

gy o.o [

-.1 -.2 -.3 .

O° 1. 2. 3. 4.

Time (see) Figure 6.35 lateral acceleration vs. time , i_ ill _¸ • , Roll Angle vs Time ) r 45 ° Sweep OWRA 10.

MIDSM, Flexible Wing MIDSM, Flexible Wing, Optimal (E) .

O.

(deg) -5.

-10. I t | | O° .

1. 2. 3. 4.

Time (sec) Figure 6.36 roll angle vs. time Control Surface Deflection vs Time 45 ° Sweep OWRA, optimized wing stiffness .

7.

right elevator .................

6.

rudder 5.

4.

Deflection 3.

(deg) 2.

1.

0.

" .............................. : ................. _................. !.................

:: :: :: : -1.

° 5.

1. 2. 3. 4.

Time (sex) Figure 6.37 control surface deflection vs. time :!if The optimized stiffness design is superior in performance to the other cases considered (by a small margin) and achieves must of its performance in reduction of lateral acceleration and control activity. The time histories show more clearly that the model following error of this design is less than the optimal rigid and flexible designs previously considered. These re- sults indicate that there is an optimum amount of wing flexibility for improved handling qualities in an oblique wing design and that an overly stiff wing can be just as detrimental as one that is too flexible. Furthemmre, this example demonstrates that MIDSM is capable iii _' i • of integrating many disciplines (structures, aerodynamics, and controls in this case) into a single design synthesis method which is capable of improving aircraft performance where sequential methods would otherwise fail.

i

Chapter 7

,i _ - •

Conclusions and Recommendations

for Future Work

7.1 Contributions

This work has demonstrated several new ideas and methods for the integrated synthesis of oblique wing aircraft and aircraft in general. The following list summarizes these contribu- tions.

1) A method for accurately analyzing the forces and moments of an arbitrary oblique wing configuration (in the attached flow regime) has been developed and tested (LINAIR). The airloads are calculated using a vortex lattice method which uses a nearfield flow solution to predict suction forces and loads normal to the panels. LINAIR accurately predicts the non- linear trends present in the aerodynamic coupling loads for the flow region where separa- tion and supersonic leading edges are not present. The coupling between angle of attack and yawing moment is not well predicted by this analysis.

?

2) The nature of the unusual coupling between angle of attack and yawing moment has been discovered. The mutual interference between the obliquely swept wing (with thick- ness) and the fuselage produces an asymmetric flow over the forward part of the fuselage.

The resulting pressure peak near the aircraft's nose produces a strong yaw moment (away from the forward swept wing) even at the zero lift condition. This effect can be captured by a panel code such as PANAIR.

3) A new model following SAS synthesis method has been developed. The reduced model following SAS (RMF) shows several advantages over previously implemented explicit and implicit model following controllers. The RMF controller attempts to match both the forced and the unforced response of the model's dynamics without any restrictions on the SAS ar- chitecture, except that it be linear. In the case where the controllability is reduced or the • _. -•,, _ , _• i_ _ L• ,_ control authority weighting is large, the performance of this controller degrades such that the linear quadratic cost function representing the model following error plus control effort is minimized. Because the RMF SAS attempts to match both the forced and unforced re- sponse of the model, the bandwidth of the feedback control can be substantially less than that required for an explicit model following SAS (which does not attempt to match the un- forced response of the model). The freedom to implement partial state feedback in the RMF controller permits the designer to enforce bandwidth restrictions on actuators and oth-

' i_'

er devices without excessive adjustment of the weighting matrices. Comparisons of the SAS designs have shown that the RMF SAS can achieve dynamic response performance equal to the EMF SAS but with less control effort, lower bandwidth actuators, and partial state feedback for an oblique wing aircraft SAS.

ii :!

4) The linearized equations of motion for a free flying aircraft with a flexible wing have been derived and tested. This approach combines the aeroelastic affects of dynamic wing deformation with the full 6 DOF associated with free flight, so that the interaction between aircraft flexibility and unrestrained motion is fully represented. Current methods for flutter analysis stress structural flexibility effects and are not well suited to model the dynamics of the rigid aircraft; similarly, the equations of motion normally used for stability and control studies often neglect all of the dynamics necessary to accurately predict flutter. The deriva- tion presented in this work does not compromise in either of these fields and is therefore capable of predicting unusual flutter modes such as the oblique wing aircraft's roll wing- bend flutter as well as the oblique wing aircraft's handling qualities. The equations can be reduced to study only the effects of flexibility or rigid aircraft dynamics, if so desired. The form of the EOM preserves all of the terms present in the equations for the rigid 6 DOF air- craft so that the influence of flexibility on the aircraft's dynamics can be seen explicitly.

5) Analysis of the 6 DOF oblique wing aircraft with a flexible wing has verified that the critical flutter mode (roll-bend) is still the critical mode of instability when all 6 DOF are in- eluded in the analysis. Previous work had only considered roll, pitch and plunge DOF , •_ ,i_ , ._' .

with wing isolated aerodynamics. This work has extended these results to include com- plete aircraft aerodynamics and 6 DOF motion.

6) An integrated design synthesis method (MIDSM) which permits the designer to trade handling qualities for other types of mission performance is presented and demonstrated.

Configuration design variables which determine the aircraft's layout, structure, etc. are in- eluded, along with (optional) control system gains, as design variables with which to mini- mize a composite objective function representing handling qualities and other specified types of mission performance. The MIDSM procedure finds the configuration and SAS which minimizes the total mission cost in a single integrated design synthesis. By varying the weight between handling qualities and other types of mission performance, an aircraft is synthesized for the greatest performance with a fixed level of handling quality. Examples presented in this work have shown that this procedure can design aircraft with performance that exceeds those designed sequentially (i.e. airframe first then SAS).

?

7) Integrated synthesis of an oblique wing aircraft and its control system for improved han- dling qualities was performed. Designs were synthesized for the aircraft at several flight conditions, rigid and elastic wing models, improved performance over a range of flight conditions, and with several types of SAS. In all cases studied the MIDSM method is able to significantly improve the controllability of the F-80WRA and therefore improve its closed-loop handling qualities. The results indicate that reductions in lateral acceleration and rolling motion during pitch maneuvers is large enough to raise the handling quality rat- ing of the F-80WRA to an acceptable level.

7.2 Conclusion Summary

The MIDSM approach to the integrated design of oblique wing aircraft is capable of im- proving the aircraft's controllability for improved handling qualities by making minor con- figuration changes to the aircraft. This work has identified that nonlinearities due to chang- ing flight condition and the influence of wing flexibility on the dynamic response are two factors which must be included in a final synthesis of an oblique wing design for a specific mission. A new type of model-following SAS (RMF SAS), which shows improved per- formance over other types of SAS architectures, shows great promise as a candidate con- troller in a future oblique wing design. Integrated designs of the F-80WRA with modeled wing flexibility and an RMF SAS have been presented and the method required to carry out this synthesis for a single aircraft at multiple flight conditions has also been demonstrated.

These examples provide the basic tools for the integrated design a final oblique wing air- , i !

craft configuration, once a mission specification is known. The aerodynamics, dynamics, and integrated design methods developed specifically to solve the oblique wing synthesis problem may also be applied to a host of other analysis and integrated synthesis studies in- volving non-oblique wing configurations.

7.3 Recommendations for Future Work

1) Improvements in the MIDSM synthesis: a) Incorporate the option for finite time integral evaluation of the dynamic cost as in reference 23.

b) Calculate the gradients of the dynamic cost with respect to the control gains using the i_ iii _ analytic expressions derived in reference 23.

2) Apply MIDSM in more detail to an oblique wing synthesis for a specific mission.

This would include the design of a single aircraft for improved handling over the entire flight envelope with wing flexibility modelled.

3) Application of MIDSM to integrated design problems in other fields: a) Aeroservoelastic flutter suppression and possibly composite tailoring for flutter stability and improved handling qualities with minimum weight. Use MIDSM to design the structure and the SAS of an aircraft for improved handling with minimum weight when aeroelasilc effects are significant.

b) Analyze the trade-off between handling qualities and trimmed drag (and/or) improved maneuverability for aircraft with relaxed longitudinal stability. Use MIDSM to calculate optimal smile margin and configuration.

c) Application to non-aircraft problems such as minimum weight space structures with maximum active/passive damping to disturbances.

4) RPV study of oblique wing handling qualities.

Use a small-scale, subsonic, remotely-piloted-vehicle with active control to verify the improvements in handling qualities that MIDSM calculates for oblique wing aircraft.

_••!i ¸• _,r!i_ : i•i_i•!_•, .,•_ _• _ k ¸•¸: _

References

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!?_,_i" _ "i _ 1986.

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[4] Kempel, R., McNeill, W., Gilyard, G., Maine, T., "A Piloted Evaluation of an Oblique-Wing Research Aircraft Motion Simulation With Decoupling Control Laws," :,i_ _II;! :!

NASA TP-2874, Nov. 1988.

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[12] Kennelley, R., Carmichael, R., Kroo, I., Strong, M., Wind Tunnel Test of a NASA • . _ •% • Designed 250 sq. ft. Wing for the F-80WRA, Personal Contact, NASA Ames Research Center.

:_ • " , .3• [13] Kennelley, R., Carmichael, R., Kroo, I., "Transonic Wind Tunnel Testof an Oblique Wing for the F-80WRA," NASA TM Draft, June 1988.

[14] Kennelley, R., Carrnichael, R., Kroo, I., 'Transonic Wind Tunnel Test of a 14% Thick Oblique Wing," NASA TM Draft, March 1989.

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[16] Kroo, I., "LinAir n A Discrete Vortex Weissenger Method for Rapid Analysis of Lifting Surfaces," Desktop Aeronautics, 1987.

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[18] Schlichting, H., Truckenbrodt, E., "Aerodynamics of the Airplane," McGraw Hill, 1979.

[19] Gill, P., Murray, W., Pitfield, R., "The Implementation of Two Revised Quasi- Newton Algorithms for Unconstrained Optimization," National Physical Laboratory Rpt.

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[21] White, S., "A Feasibility Design Study for an F-8 Oblique Wing Research Demon- strator - Final Report," Rockwell International NA-84-1135, July 1984.

[22] Tyler, J., "The Characteristics of Model Following Systems as Synthesized by Opti- mal Control," IEEE Transactions on Automatic Control, Vol. AC-9, Oct. 1964, pp. 485- 498.

[23] Ly, U. "A Design Algorithm for Robust Low-Order Controllers", Stanford Universi- ty, SUDAAR 536, Nov. 1982.

[24] Cooper, G., Harper, R., "The Use of Pilot Rating in the Evaluation of Aircraft Han- dling Qualities," NASA TN D-5153, April 1969.

[25] Golub, G., Nash, S., Van Loan, C., "A Hessunburg-Schur Method for the Problem A X + X B = C", IEEE Transactions on Automatic Control, Vol. AC-24, No. 6, Dec.

1979, pp. 909-913.

[26] Schweger, J., Sensburg, O., Berns, H., "Aeroelastic Problems and Structural Design of a Tailless CFC Sailplane," 2nd International Symposium on Aeroelasticity in Structural Dynamics, 1985.

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' _ _i i__! i¸I_ iii_!i _' _ ir_ _¸I¸