Analysis of flexible aircraft longitudinal dynamics and handling qualities. Volume 1: Analysis methods

CHAPTER I a INTRODUCTION • Usually, the atti t ude dynamics and handling qualities of aircraft are defined in terms of rigid-body m odal characteri s tics. For example, the frequency and damping of the short-period and phu g oid m odes ar e used for handling quali t ies specifications of the longitudinal dynamics of aircraft [1].

i This is possible, not because the aircraft are actually rigid, but because they however, that this approach m ay not be very accurate for aircraft with are " rigid enough " so that structural effects can be ignored. It has been shown, • significant amounts of structural flexibility [2]. Since, in the future, aircraft will become larger and lighter due to d es i g n requirements of incre as ed payload a nd improved fuel efficiency, they m ay also become much more flexible. In i addition to t he rigid-body m odes, flexible aircraft have aeroelastic-structural • the dynamics of these aircraft, without c onsiderin g the contribution of the modes which ma y significantly affect their dyna m ic characteristics. Analysis of structural modes, would be inaccurate. An y use of such an a nalys is , in flight control designs for example, could produce poor, if not dis as trous r es ults [3].

At present there is no universally accepted way to predict the ha n dlin g characteristics of an aircraft in which structur a l flexibility is si gn ificant.

Further, there is a need to describe qualitatively, the significance of struct u ral effects. The g oal of this rese a rch, then, is to addre s s the qu es tions of when a nd ij_ how do structural effects (especi a lly dynamic aeroelastic effec ts ) si g nificantly !

affect the dynamic characteristics of aircraft? An swerin g th es e questions is the ,- first step in developing a systematic approach to analyzing flexible aircraft handling qualities and synthes i zin g appropriate flight control laws.

This report is divided into the following chapters that pr es ent the development and application of the analysis *'tools " . Chapter 2 uses pole-zero plots and transfer functions of flexible aircraft to provide background o n why structural effects can be si gn ificant and therefore explains why they n eed consideration. Chapter 3 pr es ents the family of vehicle configurations which will be used throughout the analysis. In Chapter 4 , an open-loop analysi s technique [s developed and applied to the vehicle confi g urations w h ic h are presente- d in Chapter 3. Chapter 5 presents the application of a closed-loop pi lo t / vehicle analysis m ethod to the vehicle con fi gurations fr o m Chapter 3 . In presents a su mm ary of the results a nd conclusions based C hapter conclusion, on those results.

t

q CHAPTER II BACKGROUND t When the vib rational frequencies of an aircraft structure are lar g e compared to the frequencies of the rigid-body modes, t h e effect of the flexible _- m odes on the overall dynamic response of the aircraft is small. This is the situation for m ost aircraft a nd, as will be seen, allows the dynamics to be accurately modeled by the ri g id- b ody modes only. However, as the frequencies of the structural modes b ecome lower, the effect of these modes on the dynamics c a n become significant.

For example, consider the attitude response of an hypothetical aircraft due to elevator defleet l on by study in g the transfer funet l on , or the pole-zero plot cor r espondin g to t h is transfer function. Figure 2 . 1 s hows a typical po l e- zero plot of the longitudinal attitude response transfer function where the poles and zeros of the ph ug oid and short-period modes an d the fir s t few structural modes are included. Four poles an d two zer o m a y be considered to be associated with the p h u g oi d and short-period [, 1 ode s . Typically, the pole s are complex conjugates and the zeros are real. Note also th a t there is a pole-zero " dipole " associ a ted with ea c h of the struet , ' - al modes. The poles and the zeros a re complex for these modes.

Althou g h a real aircraft, like any structure, has an infinity of vibrational modes, for ease of discussion the exa m ple used here will consider only one of the structural modes, To further s i m plify the discussion, the phu g oi d mode will also be omi tted, that is, the " short-period _ pproximation " is invoked.

The pole-zero plot simplifies to Fi g ure 2.2 when the above simplifications are applied. The transfer function as sociate d with this simplified c as e appears in Equation (2.I).

Y 1r seco n d 0 aeroel a sttc -_ m o de X i u J t _ f'!r s t 0 _ . (S ) aero e l a sttC m o de X

X

short period - m o d e X

^ - - - " - 0

X p hu q ol d mo oe

. X

flrst X

aeroelastl¢

mod e 0

second X aeroelasttc mo de 0 _ 1 .

Fil K ure t. I Pole- Z ero Plot of Typiesl Flexible Aircraf t _ r

, z _ o jtu

X

Pl

e(s:}

_ (s )

P

sp X _ Zsp .. _ 0

" X

. _ o

. r Fi l Pi t e l .t , _, Poie-Zero P lot of S i mpleExu n pl e " ,, 1. . _ .T 6 q i :! O (s) _ (s -z . p)( s -z l ) ( s - _ ' i ) _i 6 E l S ) S ( s - psP ) (S -PsP ) ( S--Pl ) ( S-pl ) ( 2.1 1 , - where (-) d enot e s the conju g ate of () . Equivalen t ly , the transfer function for pitch rate due to elevator i nput is, I | :i _ (s) _ (s-z , plls-z l )l s - _l ) (2 . 2) ! -- _( s ) (s-p _ p ) (s -_ 'sp ) ( s - Pl ) ( s -_ ! ) t - i The Fitch-attitude-rate response of the aircraft due to an i m pulsive input is th e r ef or e, t !

!

!

(s-z_p) (s-zl)(s-_'l) i 0 (s ) = (2.3 ) , (s--P s p) (S--Psp)(S--Pl ) (S--Pl) T h e f oll ow in g f or m of t he at t itude-r a t e r e sp o nse res ul ts fr o m th e partial fraction expansion of Equation (2.3) and transformation into the time domain. !

0 (t ) -- R spe p 'p t + Rspe p' _ t + R le p i t + R I ep ' t ( 2.4) Here R i is the residue associated with pole Pi, a nd Ri is its conjugate. For i co m plex poles, the resid u es tre co m plex n umb ers with m a gn itudes that l determine t h e degree to w h ich each m ode contributes to the overall re sp on s e , i m Therefore, the si g nificance of an individual mode in the dynamics of the l aircraft is represented by the residue ma g nitude of th a t mode. This c a n be _ illustrated by writing Equation (2.4) in the for m , | * b ( t)= 2 1 R , pJ e ' ' t co s ( _, pt + _, p) • .

+ 21R n l e _ 't cos( _ l t + 4 _ 11" (2.51 - where Pi - ¢ri + J _ i, t , R e {Ri) ' _i = t a n-l{ l m (R i ) } I R , I - [Rel R i )] 2 + [Im{Ri l] 2 .

{ )°

The resi du e m agnitudes can be interpreted geo m etrically in terms o f the pole s a nd zero s of Figure 2 2 by con s idering the following relation s [4 ] ,

(2.8 }

= I z .p-P.p ['!z,-P .p |i[ i _,-P.pl

IR , , I I I ! Pl - Psp [ , Pl - Ps p [ '

and, I z , - p, [ ' [ z l -p , I ' [ [I-P, I (2.7}

I Rs p I - I Psp-Pl I ' 1 Psp - Pl I ' l ff , - p, I

and from symmetry of the pole-zero plot , I Ri l - IR , l - Here I' 1 d enotes the : m a gn itud e of a complex number a n d so I zl-P ll is the dist a n c e f rom the point z I to the point P l .

If the hypoth e tical aircraft w a s fairly rigid, the pol es a nd zeros a ss ociated with the structural mode would be far fro m the origin when co m pared to the poles an d zero a s sociate d with the short-period m ode . This implies that, I zl - P sp [ "_ I Pl-Ps p I (2.8) a n d , I z " -P,p I " I P'-P, p l" ( 2. 9) , w .

In th i s case, t h e e xpression for the residue magnitude s associated with the , s ho r t- p erio d m od e ca n b e si m plified by th e effective ca n ce l lation of terms involving z ! and P l from the nu m era t or a nd the d enominator s o that the short-period residue m agnitudes are relatively independe n t of the s tructural mode, or,

I z,p-P . p I 12.1 01

I Rsp I -_ I p sp -p s p I Si m ilarly, for a fairly ri g id aircraft, the zero near a pole associated with the structural mode is much closer to that pole than any other pole or zero of the system. Also, the distance from that pole to its complex conjugate is s approximately equal to the distance to the zero associated with that conjugate pole (i.e. the zero of the pole-zero dipole). These two statements imply that,

I z,- p , I <<I Psp-P, I ' (2.11)

[ zI- p ! [ << I psp-p , I ' (2.12) and,

i_ ,- p , I "_ I n -p,I . I 2 .1 3 1 i

i ll e re , th e resi d u e ma g nitud e s a ss o c iated wi th t h e str u ctural mode c a n b e s i mpl i fied, us i ng E qu at ion (2 .1 3) , to obt ain th e foll o wi ng e xpr e ssio n , i !

i

I " ,r-p, Il zI-p' I << 1. ( 2.1 4) i

]i { I ] "" [ p._p -p lI'l IT, p - p, I i It i s c lear, by a p pl ying E qua t ions 1 2 .11 ) and ( 2 .1 2), t ha t t he s t r uctura l mode residm, m a gnit,de is much less t h a n u ni ty . Since th e short-pe r iod mode residue m.lgnitud t r is on the ord er of unit y , it is obviou s th a t

I._, I << Ir,.,, I • ( 2 ._._)

,ink As n re su l t (_ f l i fts di s (.ussion, lwo conclusio n s ra n be d rawn con cer nin g fa i r ly rigid a i r cr af t. Fi r st, tilt ' sl r uctu ra l mode s h av e Illlie aff ect on the d e g re e to w hich th e ri g id-bod y mod e s conl r ib u te to t h e d y n a mi c res ponse o f th e ve hi c l e. An d second, t he c ontr ib ut i on o f t he s t ru c tural mode s to the dyna mi c response of the vehi c le i s insignifi c ant compared to the affect of t h e rigid-body modes. T h eref o re, t h e l ong i t u d i n al attitude-ra t e impulse resp o nse o f t h e air c raft c an he ac c ura, e ly ap p r o xi m at ed by the f o ll o win g e xpr ess i o n.

0 ( t ) -- _ 2 I R s p I e e 'ptcos (_ spt + _ sp ) ( 2.16) i The i m plic a tion of t h i s expre ss ion is t h at , wh e n it i s v a lid, the dyn a mic s of the air cra f t ar e d et er m i ne d a l m o st ent ir e ly b y th e v a l ues of the rigid-body p ole s a n d ze r os.

l llo w ever , if th e a m ount of fle x ibility i s s ignificant eno u gh s o that ._ E quat io ns (2 .8) , (2 . 9), (2 . 1 I) and (2. 12 ) are no longer valid, t h e conclu s ion s above will no longer a pply. In this ca s e, the degree to which the rigid-body " mod e s contribute to the overall re s ponse will depend somewh a t on the ch a r a ct e ri st ics of the struc t ural mode s . In addition, the contribution of the st ru ctu r a l m ode s to th e re s pon s e ma y b e s igni fi cant.

I I r

CHAPTER [] EXPERIMENTAL DATA BASE L A set of aircraft dynami _ models, one of which is similar to the U _ I bomber, were available from a previous study [2]. The 1 3 -I is a lar g e aircraft with a reas o nable a m ount of structural flexibility. Fi g ure 3.1 is a sketch that 5 depicts the g eo m etry of the aircraft that corr e spond s to all the vehicle models to be considered. The models represent a family of aircraft similar to the 1 3 -1 that differ only in their amount of st r uctural rigidity, quantified in ter m s of the invacuo- s tructur a l vibration frequencies. The confi g urations can be described physi c ally as vehicles with identical g eo m etries but made of d ifferent ma terials s o that t il e vi b ration frequencies are changed while the vibration m od e shapes remain unc h an g e d .

The mathematical models of the aircraft include two structural modes which correspond to the first fuselage bending m ode and the second fuselage b endin g m ode. The m ode shapes w h ich correspond to these aeroel as tic- structural modes can be found in Appendix A.4. The family of configurations were g en e rated by para m etrically varyin g the invac u o-structural freq u encies of the t w o structural modes. Ta b le 3.1 su mm arizes the ei g ht confi g urations, listin g their ei g envalues and the invacuo-vibration frequencies of the two stru c tural m odes.

N otice that for co _ .fi gu rations 6, 7 and 8 the second aeroel as tic mode is sli g htly unstable due to ne g ative aerodyna m ic da m ping. Th e original si m ulation s tudy [2] involved considerin g the effect of neutrally stable modes on vehi c le dynamics. To study this effect, the very slightly unstable (i.e. o effectively neutrally stable} configurations 6, 7 a nd 8 were developed.

The co m plete m at h e m atical model of the eight configuration s in state J vari a ble form corresponding to Equation {3.1) c a n be found in Appendix A.3.

I Ficure 3.1 Geometryof D l t a B _ seCon ap r i tio u _.

_v = A _ x v + B _ u + D v w ( 3 . 1 ) - where x v is t he vec t or of vehicle st a tes and A _ , By and Dv a r e sys t em m atrices , u is the vector of control inputs and w is the vector of disturbances .

T a ble 3 . 1 L _" Summary of D a t a Ba s e Configurati o n s I NVACUO AEROELA S T I C VEH I CLE M ODE F REQ ' S (Htj ) M ODE EIG ENV A LUE S CONF I G M ODE MO DE P H UG OI D SH O RT M ODE MO DE 1 2 PER I OD l 2 I 2.18 3.37 -0 .0015 -1.5 -0.66 -0 . 4 6 +j0 . 0 6 7 +j2 . 3 7 +j13 . 3 +j21 . 3 2 1.46 3 . 37 0 . 001 -I .35 -0. 73 -0. 46 ., :I:j0 .0 53 -I-j2 . 2 "I" j 8 . T b +j2 1. 3 3 0.07 3. 37 -0 .08; -4).0 -!.11 -0. 46 0 . 095 -l- j l.$ +i S .7 I j 2 1 . 3 4 2 . 18 0.76 -0.1 3 ; -1. 0_ -0 .7 -0 .53 o. t s :t: j l.l +jt a. 3 +j s o S 1.86 1.86 -0 .001 -1 . 4 -0. 8 6 -0 .12 +j0.049 +j2.17 -I- i 11.7 +j I 1. 0 6 1.1 o 1.1 o -0.IS ; -0. 0 s . 1. 3 1 o. o s 7 0 .1 8 +j0 . 9 7 ±j7.16 + i 7 . 0 7 1 .63 1. 55 O .001 - 1. 32 - 1 . 1 0. 08 5 • j0 . 017 +j2 . 0 +j10 . 2 "4-j9 . 9 8 1.70 1.4 8 0.0013 -1.3 -1. 08 1 0. 08 5 . . d:j0.012 +j2.0 4 "ji0.3 , d:jg. 8 In the previous stu d y [ 2], t he above vehic l e c o nfig urations were used i n a i fi x ed b as e d, la bo ratory s imu l at io n inv ol ving l ongitudina l tra c k i n g of a low fre q uenc y c o m m a n d s ignal . A cath od e ray tube wa s used t o disp l ay the _ ' f ol l o w i ng varia b les ; comma n de d att i tude an g le, @C , a n d ve hic le att i tude angle, S T, meas u red at the c o ckpit locat i on. T he veh i cl e att i tude a ngl e i s the pit c h .

att i tude m easured, f o r e x ample , by a gyr o l oc a ted at the cockpit a n d d iff er s from t he rigid vehic l e pitch angle, P R , by the c o ntribution o f t h e lo ca l structural deflections. This effect m illustrated in Figure 3.2 and defined by Equation {3.2}.

J

' 8 T

1 8R

_.o r_ zon 1the i

\

b o d y

fl exed angent to r! g! d body It body cockp i t loc at ion 11x) i FiBre 3.'Ji • Rigid and Elutie Pite5 Angles

J

II 0 1 (t) - O R{t } - _ , 0 / {l_)qi{t) ( 3 .2) " i= l -, - where l x i s t he co ckp it l ocat i on measured fr om the cen t er of g rav i ty, 0 i ' is the mode sl ope of the ith ela s tic mode and t / i is th e g enera l ized c o ordinate of the it h e l a sti c mod e [ 5I .

The above informa ti on was di s p l ayed to the pi k) t by m ea n - _ of a v i sual di s play s im i lar to the one depicted i n Figur e 3.3. H i s task wa s si m ply to m i n imi ze th e error between the co m m a nded a nd the i nd i cat e d attitudes an g l es .

Three types of data were collected from the si m ulat i ons : 1) rms tracking error (taken over a 1 2 0 second run for eac h, case } ; 2) Cooper-H a rper [6] pilot ra ti ng i n t h e t rackin g t as k _; and 3 ) p i lot co m ments. A summary of these ; results ca n be found i n Table 3.2. Th es e results indica t t ha t flexible aeroel as t i c effects s ignificantly affected pi t ch atti t ude trackin g performance.

Ta ble 3 . 2 Summary or Tra ck in g Er r or , Pilot Ra t in $ and Pilot Co mm en ts CO N TIG RM S Track i ng RMS P i l o t Pilot Comment s " , Error I d a .g) Rating I ! . 1.5 1 . 6 Gc _ od ; no p r oblem 2 1.0 5 2 . 0 l i ttle osc i l la tio n; sl i g ht c o ntrol respon _ I n| / 3 5.67 5.9' d i fficu l t; PI O problem; extreme res po n se I n| 4 1 . 90 3 . 1 l i t t k more di fficult than CI ; sluggi s h attitude r e s po nse S i. , 5 1 2 . 0 pr e tt _ good: s ame u C2 6 7 . 57 6. 7 sev er e oscillat io n; v ; r - , tuall _ uneontrol la bk.

7 1 . 48 2.3 not di ffi cult; annoying oscillatio n 8 1 1 1 6 !.9" n ot di ffi c u lt; l i ttle osc i ll a tion but could ignore i t and I! I r i g i d bod I * note: Th ese results a re for 4 pi lo ts with 2 t uns per pilot.

B e fo r'.. i ntroducing the analysis t _ eth od s, *.hesi m ul a tion res u lts , which are summ a rized in Figur es 3.4 a nd 3.5, will be reviewed, it is clear that a e roelasti¢ The Cooper-Harper rating s _ ale is u s ubjective rating (I being best an d 10 worst) l ised " _ to descr i bev e h i cle handling qu a litk _ i n variou s tusk s . [6 J _ L I I G _AX RCP. A_ LINE _ _

NORI Z_ / V \ _L

_ -r t TPACKIII(I E RROR

T

S Yt_Q I .

FiL , ure 3 . 3 Simul at ionVisu a l D LspI _ 7 _ r _' 16 F

•-- 12 •

, - 1 (1

• 0 i .__

_ 8

, c 0

e _

_. 6

° o [ "

= [ *

0 ' e,l

2 3 4 5 6 7

configuration

Fil_ re 3 .4 i Simul a tion Tr J c kia | Errors , '

level 1 - good

level 2 - fair

level 3 - poor

. , _ 10 l evel 3

L_ Q .

_ 8

i ] , o ve, ,-

level 1

_ 4 I '

O_ n i ! I

i i3,i s _ _ 8

configuration

Figure 3.$ , Simulstiou Pilot Rstia@ effects si g nificantly affected vehicle dyn am ic s. B y merely varying invacuo- structural fre q uencies, the dynamics chan ge d so drastically that two confi g urations {3 and 6) received Level 3 ratin g s while the others recei w., d Level i ra t in g s. Once a g ain the question to be answered is, " when and how do these aeroelastic effects affect aircraft dynamics? " Note that the " ri g id-body " phu g oid and short-period ei g enval u es alone give little i ns i g ht into t he effec t of re du ci ng the invac u o- s tr u ct u ra l freq u e n cies (see Table 3.1). For example, Configuration 3 has a much higher {worse) Coop e r-Harper rating and lar g er trackin g errors than Configuration 4 de s p i te the fact t hat Configuration 3 h as a more stable phugoid mode and only a slightly higher frequency as sociated with the short-period mode. In addition, Configurations 3 and 4 have similar lowest-frequency aeroel as tic mode eigenvalues. Based on this, one might predict that Configuration 4 should be worse than Co n figuration 3, which is contrary to the simulation results. Thus, the e i genvalu e s alone do n ot c om p l e t e ly c a p t u r e th e act u al d y n a m ic s of a vehicle.

'i

CHAPTER IV OPEN-LOOP MODAL ANALYSIS . A mo dal analysis of t h e family o f a i r c raft presented i l_ C n apter 3 w ill b e pres e nted. The results of thi s analysis will b c used to help explain some of the findings which were obtained in the simulation of those dynamic con fig urations.

Th es e r e s.lts will a l so be used to attempt to answe r the questions posed earlier - na me ly, w hen and how do structural effects significantly affect the dynamics of aircraft?

Modal An a lysis ('o, _ i de r t he vehi c le modeled in the state variable form, = A _ + B _ + D_w (4.1) y_: C _ + E _ + Fw_ ll e r e ..'S. is a v ( , (.tor ()r ve hi c h , s ta te s, y. is a vec t or of outputs and u and w are v e ct()r s o f c .nt rol i n puts an d d isturl , anc e s, respectively.

(),e may (li: Ig () n aliz e th e s y s t em us in g the m odal t ran s for m ation [.I,5], w h e r e t h e m o( hl l m ntrix, T, i . _ f o r me d f ro m the eigenve c tors of A ( as s umin g (listi n ('tr _ ' i gt ' n v _ due s ), so t hnt T _ [_', ,_ ' 2,..., En]. (4.2) Here u i is the ei g envector associated with the ith ei g envalue of A. In terms of the m od a l states, the system dynamics are, = Aa + b a + Vw, (4.3) : C _ + E _ + F_w.

Here _ i s the vector of modal coordinate s , A _ T- I AT (diagonal), ]B _ _ T- I B, I ) _ _ T- l D and ( _ _ CT. The matrices 13, C and I) are called the modal controllability, di s turbability a nd observability matrices, respectively. With proper vehicle state definitions, elements of these m atrices indicate how controllable, disturbable an d observable each mode is, with respect to the inputs a n d outputs. E a ch e l ement of t h e moda l contro lla b ilit y ma trix , for e xampl e , is a relativ e measu r e of how much the as s o c iat e d c ont r ol input cont r ibutes to the r espon se of th e c o rr e s pondi n g mod e . If the magnitude of o ne e l e ment of t h e modal c ont r ollability mat r ix i s s mall c ompa re d to th e m a gnitud es of the othe r e lem e nt s of th e co n t r olla b ility mat r i x th en th e mod e i n question is " r elatively un c o n t r ollabl e " f r om that i n p u t.

At thi s poi n t, to simplify th e de ve lopm en t, o ne may re d e fin e th e i n put vector as, S u bs ti t uting thi s i nto E qua ti on (4 . 3) r es ul t s i n,

= A_ + B_

( 4.5 ) _ := _ + E _, - where _ 4- and _ 4 _ The diagonal property of A is especially us e ful when considering the system i n t he frequency dom ai n, or, " 21 OR i GiNAL PA G E f S l J _ POOR QUALITY

=

(4 .6)

) : (s) = c a(s) +

J Since tile i d entity m a t r ix, I, a nd A a re d ia g onal and squar e , the in ver s e o f [sl- A] is (li a gonal as well. By multiplying th e fi rs t o f Equations ( 4. 61 by [sl- A] t a nd substituting into the second o f Equ a tions (4. 6) , th e f ollo w in g matrix equation f or the outputs y _re s ult s .

_ ( s ) = (_ [ s l-A l q EI.E( s ) + l;:, _ ( s ) ( 4 . 7 ) llv wriling t he eh.n w n l s of ( 3 as ci j (i t ile ro w and j the column), th e _.lem('n l s _, f B a nd E __ as I )ijanti e ij, respectively, and by using the f act that [sl- A] I is diag(Jnal, the trans f er f unction f o r the ith output due to the jth input . _ ,ca n be w r it t en a s , .v,(s) n (', k ' l ) l ,i - _'_ + % . ( , I ._) ' q ( " ) I, = l s - > 'k I h ,r e n i .,,t he numbe r c) r sy . , ,l emst 'll es an( ! ) ' k is t he k l h eigenva l ue o f ti le _VsI('Ill.

From Equation (I.8 ) Ihe h npulse responseo f yi c an be obtained by assuming ui h) I )e an impu l se and taking t he inve r se l , ap l aee tran s f o r m. T he impu l se r espo n se o f Yi be c omes, n v i(l ) -- %, . I ) k j O X l ) l , _ k l ) + • - . a ( ' i k eli. (I . g) k=l N ote t ha t th e v al u es of e lk. b k i are th e va lu e s o f the re sidues ass o cia t ed with mod e k ( k=l,2, ....n) f o r th e ith outl) ut ( Yi), w h e n t he s ys tem i s e xeit e (I b y Ihe jth input (U i ) " T hi s r elationship r esults fr om t he d e fi n ition of a " re sidue" [ -I] and Equation s 14 . 8 ) and 14 . 9 ) s o t hat , I 0 22 f_ • R k (y, u , ) = Cik'bkj • (4 . 10 } ; Ther e fore, t h e in for m a ti on refl e c te d in th e c on trollab i lity, d istu r b ability a nd o bs ervab i l i t y of a m ode is also c o mp le t ely con ta in e d i n th e mod a l re si d u e s.

E quati on ( 4. 91 can t h e refore be writ t en a s , , rl y i l tl = _ Rk e x p l X k t } +e i i . 1 4 . 1 11 k--I - w h e re n is the nu m ber of s y s te m p ole s . By re p re s enting the re s idue i n te rm s of it s ma gnitud e a nd p h as e a nd combining t e rm s i n volv in g com p l ex c o nju g ates, Equati o n (4.11) can b e written a s, m

yi(t) = 2 I R k I e- ' _' tc o s (' -v kt+ @ k ) + e i j. ( , 1. 12 )

k=l - whe re m is t h e num be r of mod e s of t he syst e m ( i .e. m = n / 2 ) and all e ig e nvalu e s are a s su m ed to b e co m pl ex for e ase o f discu s sion . As discuss e d in C h apt e r 2 and cl e arly fro m t h e a b ov e e qua t ion, the magnitude of t ile r e sidue of a mode is a dir e ct m ea sur e of th e contri b ution of t h at mode t o t il e dynamic r es pon s e of t he v eh icl e. From Equation ( 4 . 12 ), t he r e lative importance of eac h mode to a giv e n r es pon se can be deter m ined b y in s pection . B y nu me ricall _ impl e m e nting t h is analy s is, it is possi b l e to investig a t e h o w h ig he r ord e r m odes directly affec t t he dyn a mic re s pon s e of a n a i rcraf t by compar in g i m pul s e re s idue m a gni t udes.

The r e sults of the modal a nalysi s as developed above in c lude t he eigenvalue s and ei g envector s of the s y s tem . The eigenvector s are used to d etermi n e which mode of t h e d yna m ics i s asso cia t e d with eac h g ene r a l iz ed i _ , coordina t e of t h e sys te m. It s hould be noted that when the term "r igid- b odv i t Q i mode " i s u s ed it m e a n s the s y s tem mode who s e eig e nvector reflec ts s ig n ifican t _: p a rti c ipation of the rigid-bod y s tat es {e .g . attitude , attitude r a te a n d angle o f i a ttack) . Simil a rly, the t e rm " el as tic m ode " i s u s ed to me a n a s y s te m m ode wh o s e e ig e nve cto r refle cts s ign i fi ca nt pa r t ic ip a t ion of the el asti c st a tes { i . e . r / i , // i ) . I n th e ap pl icati on t o flexible a ircr a ft, th ere a re no t ruly rig i d-body mod es or p ur e ly s tru ct ural modes d ue t o ae ro e la st ic c o up li ng.

T h e m a nner in w h i ch th e mod es of the dyn am i cs ass o c i ate d w it h th e gener a lized c oordin ate s a r e identifi e d c a n b est be ex p lained th roug h exa m p le .

r f L Co n . l der the data pr e s e nted in Table 4.1 which representsthe two rigid-body modes of the Navion aircraft [7]. The magnitude of the element assoc i ated with attitude rate, 0 , in the first ei g envector is larger than the elements associated with the other states. This indicates that the mode associated with ; t h e first ei g envector primarilycontains attitud e rate a nd therefore, corresponds , _: A to t he short-periodmode. Similarly, for the second ei g env ecto r, the m agnitude ; _ of the element associated with forwardvelocity perturbation,u, i s largerthan t he other elements and so, that mode correspondst o the phugoid m od e. This technique was used to identify the vehicle m od es for the configuratio n su s ed in _ . _ - t his study.

Table 4.2 Summary of Na vion Longitudinal Dynami c s . Flight Condition : Se a Level U = 270.0 ft /s ec W = 2.84 ft /s c¢ Stat e V e ctor: XT.,, _ _ [ u r w _0r 0 ] unit s ei g envalue s - 2.51 -0.017 -t-j 2. 5 0 :t.. A o.2 2 3 se c-_ -0.003 2.0 :t: jo.o2 o f t .se c - -0 .220 -0.059 eigenvector s 4 - _ 0.656 2 _ _ 0.002, f t's ec -I 1.0 0.143 -,-j0.00o i0__ ra__dd I L b e _ . -0 .293 -0 .004 4 - j0.299 4 - jO. _ 2 10-=rad Notice that, in the exampl e , the units o f th e el e m e nt s o f the e i g env ecto r are not the s ame. Two of the elemen ts have u n i ts of feet p e r s e co n d and th e other two have units of 10- 2 radi a ns per s econd a n d 1 0 - 2 radia n s, respe c tively.

i This set of units enabl e the mod es of th e s ystem t o b e r e adily id e ntified. Sinc e . th e magnitude of a n ele me nt o f an eig e nv ecto r i s d e pendent o n th e uni ts sele c ted for th e s y s tem stat e s, prop e rch o i c e o f th e uni ts can aid in identifying th e mod es o f th e s y s t e m. In ge n e ral, th e s t a t e s o f s syst em (and th e ref o r e th e e l e men ts of the eig e nv ec tor s )d o not have th e s am e phy s ical uni ts . Th e units c an be c huged, however, by applying a s i m ilarity tran s formation to th e stat e v ar i a ble rep ,e s e ntati o n o f th e syst e m. It can b e e a s ily sh o wn that a similarity transformation does not affect the ei g e n values or residues of a system. As a result, a s imilarity transformation can be applied to the system without altering the modal analysis results. Appendix A.I pr es ents the development of • the transformation used in this study to change the units of the states and e i g enve c tors to aid in identifying the system m odes.

In addition to the eigenvalues . _n d ei g e n vectors, the modal controllability, dis turbabi l ity and observability m a tric es are available from the modal analysis.

With proper selection of units (using a similarity transformation), these can be used to gain further information concerning dynamic relationships between the control and disturbance inputs and the system mo d es, and between the outputs a nd the system modes as described previously.

The l as t and most useful of the analysis r es ults are the modal impulse r es idues, Ri. The impulse residues are useful since, as noted p l 'eviously, they a re a direct combination of the observability and controllability (or d is turbability) of a particular mode. The magnitude of the impulse residues will be used extensively in this study.

Vehicle Model The vehicle models considered were those used by Yen [2]. They consist of linear equations of the form - -- + "t d -i- .w. 14.13) XE Ac ' A _ x E B E The vehicle state s include s o me state s which are rigid-body degrees of freedom, : x R , an d others whi c h are the s tr uc t u ral degr e e s of fre e d om , Xg. Together the se form the partitioned s tate vec to r. The s y s tem matrice s are al s o partitioned to b e con s i s tent with th e s tate vector partition. Th e s ub-m a trice s with t he sub s cript R are th os e a sso ciated with the rigid s t a te s an d the s ub-matrice s with th e s ub s cript E ar e th ose a s s o ci a ted with th e e la s tic st a tes. Th e sub- m atric es A c and Ac _ relate the cro ss - c o uplin g between th e rigid s t a te s a nd the el as tic st a tes.

Th e m o d a l analy s i s pr o cedure could b e acc o m pli s hed with th e flexible v e hicl e m o de l d es cr i b e d i n Equation (4.13) but th e r es ults w o uld n ot b e th e most m e a n in g ful. This i s due to the fact that interpretation of the modal analysis requires extensive use of ch a r a cteristic parameters that influence the i m pulse response of the system {i.e. modal controllability, m od al observ a bility, residues, etc.} and since an i m pu lso input is unrealistic, these characteristic pa r a meters are unrealistic. Since an impulse is physically unrealizable, impulse respon s e s of an aircraft are unrealistic and do _ ot reflect the dynamics o f a vehicle in actual flight. In order to obtain meaningful r es ults from the analysis, inputs should represent i m port a nt aspects of the actual pilot command s and at m ospheric turbulence.

An impulse has infinite bandwidth a n d c _ mnot be pr od uced by any physical syste m . A pilot m ay try to pr od uce a n i m pulse but, due to his limited bandwidth, cannot achieve it. What r es ults i . , _a type of " realistic pulse " input that can be approximated b y treating the pil o t as a low pas s filter (ie. a first order la g }. The input to the filter is an i m pul: _ ea n d the resulti ng output is a finite bandwidth pulse which approxi m ates wh a t a pilot is capable of producing. Equation (4.14) is the s tate space r eprese n t a tion of a low pass filter.

i p- = Apxp + Gp q p ( 4.14} The scalar xp is the " rea l i s tic pulse " , t / p is the impul s ive input, Ap - _ n , where r p is the time cons t ant of the filter, a nd G 0 is _ ..1so that the Bode gain

r p

of the filter is unity. By using xp a s th e input to the vehicle m odel a n d a n impulse as the input to the filter, a r ea l is ti c respo n s e is obt a ined for the pilot / vehicle syste m .

Th e i m portance of using the low p a ss filter to obtain m e an ing f ul r es ult s can be clearly shown by a si m ple example. Con s ider a s y s te m with t w o s tat es in modal c oordin a t es - f i = -I00 x + I u ,

1 0 , 0 1 Ill

{4.15 }

y- [ ! l]x .

\ The response o f this system to an impulse input, u " 6 (t), is, ? ' N o w con s ider the same s y s te m and let the input, u, be represent e d as, _ . _ ylt) I u= _ t) -" e -t + e" 100t 14 .161 I d =-lOu + lO q . (4.171 !

That is, u is the output of a l ow p ass fi l ter with a time r onstant, r p - 0 .I 0 s e c t.

The system can be reorganized as, : [.] -1 0 0 : = - 100 + q , 0 -10

[: : JI J [100]

(4.18) y =[I I O][u _ 1.

The response of this system to an impuls e input , _ = 6 (t), is,

Y(t)I'a - _ O= (l'll)e- t i

+ (1.00) e - 1 0 _ + (0.11) e - I_ . (4.19) !

Note th e difference between the two respon s es , Equation s (4.16} a nd (4.19}. Th e i ' _ " unrealizable r es ponse (Equation (4.16)) indicates that both modes contribute !

| e qually to the - ' re rall r espo nse, in term s of re s idu e m agnitud e . Th e realizable, 1 !

: filtered re s por,se (Equation (4.19)) i n dicate s that th e fa s t system m ode ( (k =-I00) ha s a much le ss significant contribution to the overall r es pon s e than i d o e s the other o riginal s y s t e m m ode (X =- I). Th e o bviou s conclu s ion i s that an impu lse input t o th e syst e m e xcite s th e f as t m o d e , but cann o t b e e xcit e d as much by th e limit e d bandwidth filt e r e d i m pul s e . Th e ref o re, th e m o dal an a ly s is A tim e c o ns tant o f 0.I0 tec o a dsb ¢ o m bte a t witk bu mu b a adwidtk li m it a tion s .

/

\

4' W sh o uld be p erform e d on t ile s y st em whi c h inc l udes the l ow p as s fi lt er tha t m ore a ccu ra t ely r e flects t he tru e inputs that are expe c ted. If th e, filter i s not u s._ d, t i l e modal analys i s may i n dic at e that certain high frequen c y modes sign i fi ca ntly contr i bu t e t o th e ve hi cle re s pon se w h en they ac tually m a y h a ve i nsign i ficant eff ect s.

An a r gum e n t simila r t o t ha t use d fo r t he pilo t command s can be used t o , j us tify des c r i b in g t h e d i stu rbanc e s p ro d uce d by th e at m osp h ere in a s imilar : way.._i n ce at mosph eric t u rb u lence i s an i m portant ei rcraft disturbance and t u rb uh ,n ce is rand o m in na tu re , th e s e dist u rbances are m o deled stochastically.

' O n e c ou n m only use d d i s t u r b an c e a lo del is t h e Dryden Gust Mod e l [7] . The Dryden M odel m ay be cxp rt _, ed in matrix form as , xT _ [o s ,, os ], ( 4.20) i s = Asx s + G d l s .

Th e g, n .. , t s t at e o s is the angle of atta c k induced by a verti c al g ust a nd o s j is a n _ a dd it iona l gus t state whi ch i s nece s sary to obta i n th e proper frequency cha ra ct e r o f th e gu st mod e l. The syst e m m atri ces A s a nd G s { g iven later} provid e the p r o per ch nl a c teristics of th e random g ust r es pons e when th e " whi te" n o i s e, qs' i s t he in p ut.

By c o mbin in g the " p i lot equati o n " a nd the Dryden Gust M od el equati o n, (Eq u ati o n s ( - t . l,t) and { . l . 20 i ), wit h t h e aircraft, (Equ a ti o n (4 . 3)}, an a ug mented flexible a ircraft mathe m ati c al m odel i s f o rmed. The r es ulting flexible air c r a ft m ode l is t he n c o m p ose d of a c ombination o f the syste m m atric es from the pilot eq ua ti o n, the g us t e quati o n and t he fl ex ibl e vehicl e eq ua tion, (Equation (4 . 22)).

, x"r= ! xp'r . x _ 'r , x , 'r I H. _n l

i = . _ x + %+ , , p , ( 4._ ) '_

I B , D, Finally, the proper choice of air c ra f t responses, y., is critical for obt; " _ ing meaningful results from the analysis. If the wrong outputs are selected, erroneous conclusions m ay be drawn. This point is e m phasized so that tbe reader i 3 aware that a great deal of engineering jud g e m ent mus t be used in c h oosing th e proper outputs. Understandin g the physics of the problem is necessary to obtain m eaningful results.

Once t h e outputs of interest are chosen, linear matrix output e quations are ' " formed so that , y. = Cx + E q p + F q s ( 4 .2 3) - where y. is a v ector of outputs. The modal analysis m ay now be performed, using Equations (4.21}, (4.22} and (4.23) as the complete sy s tem.

Applic a tion To Data B u e ConOsur a tlons . , The modal analysis method was i m ple m e n ted in a c o m puter program. A listing of this program appears in Appendix A. 6 . As a result of the m odal analysis , several quantities of intere' _ t are readily available. In addition to the i modal impulse r es idues, the m od al eigenvalu es , the modal ei g envectors, and the "modalcontr o llability, di s_ urb a bility and observabiiity matrices are all easily obtained.

The " ' ehicle m odels used in Yen's si m ulation were exten d e_l to include two additional structural mod e s . The additional mu de s were t,,e second and fourth low es t frequency mod es of ti l e b as eline vehicle (i.e . Configuration I), thu s i incr eas ing the model to include the fou r lowe s t frequency s tructur a l m od e s .

Th e m o de s hap es of th e a dditi , m a l m od e s indicate th a t they are primarily s ymmetric wing bending m ode s . The s h a p e s o f the s e m od e s c an be s eem i n Appendix • 4 . These mod es could be imp o rt a nt in the gust r e s po ns es and will b e c o n s idered later in th e an a lysi s .

The total s tate vector includ es the s tandard ri g id-body de g re ¢ of freedom (i.e. perturbed f o rward vel.o c ity, u ; a ngl e o f a ttack , o ; rigid-b o dy pitch -.ttitud e and attitud e rat e , #a and O R},tnd the ge nerali ze d c o o rdin a t es of th e t o ur structur a l m o d es , (i. _ . the g en e rtli: e_ defleetion s , q i ; and th e generali z ed rat es // i)- Tile tota l state vector is defi n e d as,

x7_ Ix_ g Ix_]

(4.2 4)

" _ ( _, , / TR , u, o R I ,_ ,, .. . , '7 , 1 '_ , ..-, 6 4 1.

• The sys t em ma tr i ce s of th e ve hi c l e con figu r atio n s usi n g th is state ve ct or can b e found in Appendix A. 3 .

Th e ou t put pa r am eters we re c ho s e n to inchld e r igid - bod y fligh t pat h a ngl e ( ' 1), ( E qu a t i (,n (4. 2 5 ) ), to tal - el asti c ( O T ) an d r i gid-bod y ( O R} p it c h attitude a ng l, ,_, {Eq uat i o n (-t .2 6) , F i g ure 4 . 1), t ot al- elast i c { 0 T ) and ri g id- b od y { 0 a) pit c h a r t i tude r at es, ( Eq u a t i o n { 4.27 )) , an d n o r ma l a cc eleration a t t he c o c kpi t (n z ) , ( l'_'( l uat i on ( 4 .2 8)) .

" / R = 0 R-°R , { rad ) {4 .2 5 ) : i n 0 T = O R- '%__ t l , {t) ¢t i { l x ) , (r a d) { 4.26) i= l n 0 'r : 0 R- _ // i( t )* ) ' i(i_) , (rad / s ec) (4.27) i = !

n ! [Uo_ 1 + I x _R- X__}i ( t}$i ( I x) ] { g. t s } ( 4.2 8 } y n z _ _ , i- !

-wh e re g = g r av i tat io n al a ccele r a t ion, (f t / s ec 2 ) • U o = cr uis e v e l o c i ty , 9 .1 9 (ft / see) _. t Ix = di sta n ce be t ween c . g . and cockpi t , fi t ) | | I ,

, l

L I

8T

8n

h o r | z on lin e f lexed i X ta n ge nt to r _ gtd ,' -- b ody at bo d y { ' cockp i t l ocatton 11x) I

• !

i

i

_ Figure 4 .1 Rigid and Elutic Pitch An g les l The above parameters constitute what was judged to be the significant responses in lon g itu d inal attitude dynamics. Tot a l-elastic and rigid-body pitch attitude a ng le and pitch rate are used extensively by the pilot to control the vehicle a n d evaluate it s p erfo rma nce. In fact, in the simulation study, the ' pilot's task w as to minimize the error between a commanded attitude, 0 .c, and the vehicle attit, l de, 0 T. This implies that 0 T and 0 T as well as 0 1 _ and _ are of extreme importance in pitch attitude tracking. Normal (or plunge) _" acceleration is another significant response of the vehicle from the as pect of " ride quality, but of course w as not a factor in the fixed-b as e simulation.

Note that the equation for n t , (Equation (4.28)}, is not a n explicit function of the stat es i n Equation (4.24), but is a function of the state derivatives. It is , therefore, a n implicit function of the system s t a tes a nd control deflections. By usin _ the state equations (Equatio n 4.1), n s can be written as an explicit function of the system stat es , as pr es ented in Appendix A.2.

The algebraic equations for the chosen output parameters were combined to obtain a matrix output equation in the form of Equation (4.23) _ sin g the output vector, The numerical values of matrices C, E and F for each c onfiguration appear in Appendix A.3.

The pilot parameters, the time constants a nd D.C. gains were c hose n to accurately describe the bandwidth limitation of th e hu m an pilot. A characteristic la g of 0.15 seconds w as cho s en to be consiste n t with other studies ! 8]. The res u ltin g pilot filter equation used in the a n alys is is,

r e r e (4.so)

- where l "p - 0.15 sees.

T h e gust parameters were ch os en to be cons is tent with a previous study u s ing the 13-1vehicle a nd the Dryden Gust Model [9 ] . The gu st equ a tion used in the analysis is , 'k N ? , ' 32 [%,, %i , (4 . 31) • _g = 5 -- 9 . 5 + . 0 0 5 6_s"

I=:°°1, [o '°1

- f o r _ i nr a d ia ns a nd r / s o f un i t i ntens i ty.

N u m erical Results The moda l ana l ys i s m etho d was app lied t o t he eigh t config u ration s of t he dat a base des c ribed in Chapter 3. The complete numerical re s ults can be fo u n d in Appendix A.3.

Con s ider the graphic a l results on Figures 4.2 - 4.9 which are the n o rm a lized relative m a gnit u de s of the modal impul s e residues for each mode of t h e veh icle d u e t o pilot inp u t s . The normalization w as done so that the re s id u e m a gn i tude s o f t he v eh i c l e m o de s (no t i nc l ud in g pilo t l a g , i . e . phu go i d , sho r t - i p e rio d an d a e ro e la st ic) s um to uni t y f o r each out p ut . The equation use d to a cc om pl i s h th is i s Eq u a t i on (4 . 32) . i !

i

Ri

l 1

I Ri I . o ,m= ( i - l ,...,m ) ( 4 . 32) _ "

m 9

21 Ri l

j-- I - whe r e m is ti l e n um b e r of vehicle m odes.

The a bsoh fle m agni tudes f o r each m o d e c an b e ob tai n e d fro m the nu m erical 0 •

!

re su l ts i n Append i x A. 3 . Sin ce p i tch a t_Rud e a nd pi t ch a t ti t u d e r a t e a re output s o f p r ima ry c once r n in a pitch tra ckin g ta sk , t h e r esidu e m a gni t u des assoc i ate d wi t h the r ig i d-bod y a n d tot al -el a s t ic p i t ch att i t u d e a ng l e s , ( 0 R and 0 T ) , an d r a t t , .%( 0 i t a n d 0 1 - ), will b e c onsid er e d first .. C l e a r l y,# T a n d 0 1 ` ha v e mo r e a e r oe las tic mode c on tr ibu t ion tita n do 0 R a nd OR, which is as e xpec ted sin ce O R is the r igid-bod y a t t i t u de a ngl e and 0 T i s t h e t o t al att i t u d e a ngle i n clu d ing e l a s t i c de fo r m at ion a t t he co c kpi t .

i i I

i CFISE ] -

• G R I,IM R N Z

• _ _ .5

_ . _: .o r - -n

PH S P El E2 E3 E q P H S P E l E2 E3 E q I. tO DES MO DES 1HETR-P, THETR- T .

-. .5- F - 1 _ . s

1.o - _ _ 1 .0 . . _

. o- F -- I! t .o F - 1

P_ .sP E _ E 2 E3 E. e H SPE _ E2 E 3 E .

KODES MO DES

THETR OOl - R I" H ET A DOT -T

us ] . 0 _ ] . 0 .5 _ .

"L PH 5P El E2 E3 Eq PH 5P El E2 E3 Eq HODE S MODES / • R[$ 1 OUE S FOR[I _ H OUIR III _ NO R M RL I ZE D SO 1HRI IHE I R SUH I S ! .0 Figure 4.2 Pilot Impulse Residue Matmitudes - C onfig. I

CASE 2 -

G A P$H R NZ • 5 I .5 .o - -- _ . o r- -,

P . SPE l E _ E3 Eq P. S PEl E 2 E 3 E 4

MO DES M ODES

1HF _ TR- R THFTR - T

, . uI .0 _ 1 .0

_ .o c --n , ---, _ .o

PH SP E] E 2 E3 Eq PH $P E] E 2 E3 E'I KODES I " ,ODES

THETR D O T - R THEIR DO T- T

: r--l . o [7

PH SP El E2 E3 Eq PH SP El E2 E3 Eq • P, ODE$ M ODES RESIDUES FOR EgCH OUIPUI lie I_ C_ P_ L ) ZI[D S O 1H _ I 1HEIRSUNIS !.0 r Filpare 4.3 i Pilot Impulse Residue Msznitudes -Co a 6g. S

J L I

: 35

i case a -

l

GR Y _ HR NZ

_

- _ .o.I,-lrnl-1 II _ . 0

PH SP El E2 E3 Eq PH SP El E2 E3 Eq I . _O DES t ' _O DES "[HEIR - R T HETR - T P H S P El E2 E3 Eq PH S P El E2 E3 E4 I I ODE$ I _O DES

T H ETI:: : I 001- R T H EI R OO T - T

. , .o. _ FI

.0 . _ ; .o •

• P H 5 P Et E2 E3 Eq P H S P El E2 E3 Eq I';ODE$ HODES _ zszouzs F r A z _ :. ou T e = n R_ N0. _ ]ZZ0 SO • I_ _sn s u . )s s .o FiKure 4 . 4 Pilot Impulse Residue MaKnitudes -Cou§ s . 3

CRSE 4 -

G AI4MA NZ

.s E . s

l l l-I

.o , _= .o- _ . ---'

PH SP El E a E 3 E4 PH SP E_ E 2 E3 E4

IIODES P , ODES I H E T R - R T H ETR - T

,,o 1

i '

. o _-- _ . o r-nl----I_--_ r--I

P. sP E,_ E 3 E 4 P. s P E, E 2 E 3E.

HODES M ODES TH E T R DO T - R I HE'I'R DO T - T t

• " [-7

• _ _ . . . o , _ . .

P H S P E l E 2 E3 Eq P H 5P E l E 2 E3 Eq " I MO DES MO DE S RES: : ,JES FOI l E ACH O UIPUI l il le N OA _ ' A LI E EO SO 1WI1 l l' f .l R SU M iS !.0 Fi&ure 4.5 Pilot Impulse Re s idue M apitudes -C o nfl ll . 4 • - I 3 7

CnSE 5 -

GQMHQ N Z

- _ .o _ .o ._ I - - 7

P H SP E l E2 E 3 E H PH S P E l E2 E 3 EH _ ; OD ES M ODES

rHEl n- R THIZ TF I - T

z _ z . 5

1 ,.o

. 0 - - ' _ . 0

P H SP El E 2 E 3 E q P H S P E l E2 E3 EY MO D E S M O D ES

THEIF:I 001 " -R IHEIQ DOT-T

_._

.o _ .o_L_ __ r_ : • P H S P E l E2 E3 Eq P H SP E l E2 E3 Eq f_ODES M O D ES R [ SIDU($ FOR EA _ .H { _ ,1111 _ A RE N ea J _ Rl . l Z EO S O . II'I R 11H[IRSU e t IS 1.0 Figure 4. 6 Pilot !re pulseResidueMa gu itu d es-Con§ g . 5 ?

CASE B-

' G R I. _N R NZ , L .5 .5 ,

. o _ .o r-hi--1 I -- 1

PH SP E l E 2 E3 Eq PH 5P E l E 2 E3 Eq P, O DES Y , ODES

IH E _R - R ]H ETR-T

: 1 !

PH SP [ ! E _ E3 Eq PH SP E l E 2 E3 Eq NO DES HODES

, THETI:I DO T- R TH E IR D O T - T

PH 5P El E2 E3 Eq PH SP E) E2 E3 Eq " HOOES HODES , { RE S IDUES FOIl IPOI ] 1 "I 1H I , IR _ IS I O,, I P MII I_l No I I r ,ALIZF.O S O l Fi p re 4.7 Pilot Impulse Residue Mtpitudes -ConSl . 6 t I 3O

CRSE 7-

GAMM A NZ i _ • - .0 PH 5P El E2 E3 Eq PH SP El E2 E3 Eq _O O E S I_ OOES THET R 00 1 ' - R I"H F .1R D O T - T .0 _ .0 - • P H S P El E2 E3 Eq P H S P El E2 E 3 Eq I',OOES I' II N3ES RtSIOU[S f _ [ llCI I OU1 P S l II All [ IO mA l .l l EO I0 . 1 HPl l1 H[IR _ I S l .O Fi lprt 4.8 Pilot Impale Residue Mspitvdes -Codli. ?

I

CRSE 8-

p G R I4M R NZ .5 _ .5

_ .o -- _ .o I- - 1 1 - - I c -n

P H S P El E2 E3 E q P H SP El E2 E3 Eq I - ; O DES M ODES 1HEIR- R THE T R- T T' .0 "- -- F .0 I- - - 1 . r--"l : PH S P El E2 E3 Eq PH 5P El E2 E3 Eq I_O DES I _O DES i

IHEI R DOT- R IHEI R D OT- T !

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PH 5P El E2 E3 Eq PH SP El E2 E3 Eq " i I , _O (S I ' _ )ES M ESIf)U(S Ion [ _ N O U lP _ I flit( N ORt _ LIZ(O SO .

1_ 1 ll _ ill _ IS 1.0 Filiure 4.9 r" Pilot Impube Residue Mqiaitudes -Co nl J l. 8 \ t Now consider the results for 0 R a nd _ a s the frequency o f the first ela s tic mode is reduced as in configuration s 1 through 3 ( s e e Table 3.1 f o r reference}.

The r es idue ma gnitude s of the fir s t a eroel as tic mode {El) monotonic a lly increa s es until, in Confi g uration 3, it is l a r g er than th e s hort-peri od mod al residue ! This indicat es that, for Confi g ur a tion 3, th e rigid-body attitude response is dominated by the first aeroel as tic mode ! It is obviou s that the use * of a pur e ri g id-body analys is would be wrong a nd a ny model not including th e effects of elastic modes wo u ld be inappropriate.

The r es ult s also explain why Configur a tions 3 and 4, while having similar eigenvalue characteristics, have very different simulation re s ult s -, ( s e e Table 3.2). The aeroelastic modes in Confi gu r a tion 4 do not dominate th e attitud e response (as they do in Configuration 3). The r es idue m a gn itude for the lowest frequency aeroelastic mode (E3 in th is c as e) is not larger than the short-period residue. In other words, Configuration 4 has attitude dynamics which are do m in a ted by a rigid-b od y mode and Configuration 3 ha s dynami c s which are dominated by an aeroelastic w .od e. S i nce Configur a tion 4 act s more like a " rigid vehicle " than Configuration 3 , the tr a ckir .g performance for Configuration 4 is better than Configuration 3. However, the aeroelastic mod e residue in Configuration 4 s til l contributes to s o m e d egradatio n in tracking performance.

This approach can al so b e used to relate the rest of th e tracking s i m , lation results to the effects of the aeroelastic m od e s . Th e tracking errors {Figure 3.4} and _ , neCooper-Harper ratings {Figure 3.5} of the s imulation s ag ree especially w ell with the trends in the m a gn itudes o f the i m puls e r es idue s for total pitch a t titud e angle ( 0 T). Th e confi gn , tio ns with large tra c king errors and p o o r pilot ratings have ae r oelasti c r es idu e m agnitu des which are larger than the rigid-body re s idue m a g nitudes in th e 01 ' r es po ns e. T h e c o n v erse is also tr u e; t,h e configurations with large aeroelasti¢ r es id ue m a gn it u de s tend to have large tracking errors and poor pilot ratings. ' T he graphical r es ults can be used to bring attention to other as pect s of the r W ve h icle dynamics as well. Take, for instance, the plunge acceleration a t the pil o t s tatio n {n z ). Th is p a rameter wa s , of cour s e, o f no im po rta nce in the . fixed-based simu la t i on, but would be of particular inter es t if the configur a tion s were to b y _ tudied u si n g a m ovin g -ba s e or in-flight s imul a t o r. Th e gr a ph ic al re s ults of the n , modal im pulse re s idu e magnitud es i n Figur es 4.2 - 4.9 indi c ate that ignoring a eroel as ti c a ffect s when c on s idering, for ex a mple, ride q ual i ty would b e i m proper. The c ont r ibction of the ae roela s ti c mod es is very import an t in th e n , r espo n s e of the vehicle for all confi gnr atio a s .

Con s! der also, the flig h t path angle ("i ) response for Configuration 3 .

Ignoring a ero e' , _ tic affe c ts in this c ase would giv e erron e ous results since *. he aero e la s tic residue m a g nit _ ud e s are signifi c ant compar e d to those of th e ri g id- body mod e s .

Finally, the insignifi c an c e of th e second and f o urth a eroe' :_ tie m ode s (E2 a nd E4) in th e pilot impuls e res po ns e i s cl e arly e vid e nt from _ , e graphical results. The exclusion of ti _ ese two m od e s in Yen's [2] simulation study wa s ' ther e fore valid. ' :_ The same type of trends i r e sidue magnitude o cc u r in the gu s t- " disturbance i m pulse r e sidue m a gnitud e s, Fi gu r e s 4.10- 4.17. Th e s e r es ults i indicate that the aeroelastic mod es contribut e , in varying degree s, t o , the x various vehicle r es pon s es due to an impuls e input to th e Dryden gu s t mode l, where an impulse input is the deterministic counterpart to " white " n o ise. Of particular interest are the r es ults for rigid*body pitch-attitude-rate { _ lt). For Configuration s 3, 4 and 6, the contribution of th e a eroela s ti¢ m o d es i s very si g ni f i c ant. One of the wing bu r .ding s m o d es , E2, h as s i g nificant r e sidue m agni t , i des compar e d to thos e of the rig; 4 -b od y m odes . Th is indic a t e s that attitud e tracking in turbulenc e would b e s imilar for e ach cf th es e config, r a ti o n s in that th e 0 it r es pon ses would b e do m inat e d by ae ro e l as tie- s trtl c tur a l m odes. This i m plies that even though Configuration 4 had a i sati. _ factory pitch attitud e r e s po nse in th e s imulation, added turbulence m a y i r e s ult in significantly different a nd degrad e d perfor ma nce.

Ti,t modal analy s is paints a different pictur e than th e e igenvalu e a nalys is pre _e n t ed in Ch a pter 3. Re c alling the di s cu ss ion in Chapter 2, one e _ n s ee that a . _ the frequen c i es of the structural m odes are reduced, th e inter ae ti _ m b e tween th e rigid-body mod es a nd th e aero ¢ , la. _t icmod es incr e a s es . The r e sul: is that t h ¢, r, _ idt w _ ass o ci at ed _ it h th e st r u c tu ral mod es and those a.*. s oeiatedwith the rigid-body m o¢l _ are m , , d ili e d a n d , as a r e sult, alter the vehi c le dyna m ics, if t he r _ i 4 tles of the a e ro e la, 4 i c mod es b e com e larg e enough to dominat e the v e h ic l e r _ pon s e, th e a ircr a ft no longer ac t s lik e a *'rigid a ircr a ft " . In oth e r t words , the vehicle attitu de re s po n s e is not domin a t ed by the ch a ract e r is tic short-peri od attitude dynamics.

Sinc e the r e s id ue mag nitude s a r e a m e as ure o f *he m oda l partit'ip a tio n , the a bov e a r g u me nt indicat e s that wh e n th e im pulse r e s id u e ma gnitud e s a ss ociat e d with a e ro e l,t s tic m od e s d o min a t e t h os e of th e s hort-p e r i od inod e , th e vehicle p e rformanc e d eg r a d es . Th e m odal a naly s i s r _ ult s s upport thi s argument.

J

CnSE 1-

GQFIMR NZ 1.0__ _. _ _ 1 . 0_

.o .o

PH 5P El E2 E3 Eq PH SP E l E2 E3 l- q I . ; nDES r '; ODE S tH E TP,- R TH E T R- T

¢= .o- - == .o

PH $P E l E2 E3 Eq P H S P Et E 2 E3 Eq M ODES MO DES T HE T R O O T - R 1HE I R DO T - T _ . 5 .5

.o _ . o ___r --- I _

P H SP El E2 E 3 Eq P H 5 P E l E2 E3 Eq I _ OD E S M O DES RES I O LRSFIR E AC HO U 1 POI ARE N 3RI _LI Z£O SO ll' _ l l t tElR _ ll _ I _ ! .0 FiKure 4.10 Gust I m pulse Residue M,=Kuitudes - Con§s , 1

CASE 2-

G RMMR NZ &

z ,--,rl I---II--I

_ .0 r--, , -----,r--! . _ .0 P H S P El E2 E3 E q P H SP El E2 E3 E q I '; O D E S I .; ODE S THE T R- R T HETR - T l . O- PH SP El E 2 E3 Eq P H SP El E 2 E3 Eq NO DES H O DES !

!

T HE I R DOT- _ T HE I R D OT- T i PH SP El E2 E 3 Eq PH SP El E 2 E3 Eq " MOD ES MO OES 1_ 1 I _] R S _ IS I .O Fig ure 4 . 11 Gust Impulse Res i due Magn i tude s - Conflg. 2

CASE 3-

GRI4MR NZ , &

I o, .o ] t .o _ l -- -l,--_ = :.o , ---_v-ql--][--I

P H SP El E2 E 3 EW PH SP El E 2 E3 E q 1 ", O DE S M OD ES

-IHETI::I- R THETPI- T

. o IFII - _ .o I F_I I__

P H s e El E2 E 3 EW P H s e El E2 E3 EW I _O D E S M OD E S THETR DO1- P , THEII::I DOT- T _ . 5 . 5 PH SP E l E 2 E 3 EW P H SP El E2 E3 EW I _ O DE S b ODES R[SIDUES FOR[ _ H GU T PdlkqE N _RM_ I.IZ E DSO lrkql 1HEIR _ IS 1,0 F'ilp re 4.1 2 Gust I m pulse Res i due U l pit u des - C On§l;. 3

CASE q -

GRI - IHR NZ , ' : t

q *"°I i

_ . o .__ ::.o _-._F-I_

; P H 5P El E 2 E3 E L I P H SP E l ... 2 E3 Eq i r _OOE 5 _O D ES i

THEIR- P , THEIR -T i

!

i PH S P E| E2 E3 Eq P H S P E l E2 E3 Eq ' -I b O DES KO DES . _ I t _

' !

! I HETA D OT- R "I' H EIA D OT- T .

.o I .o _ !--I I -- ]

t = PH 5P E l E2 E3 Eq PH 5P E 1 E2 E3 Eq I ',ODES N ODES RESIDUES FOIl [IT CH OU T PI II R ._ I _ RKR I . l l r(O 50 , , TflAl 1H(IR SUI'I IS !.0 Fii[ure 4.13 Gust Impulse Residue Map itudes - Con § z. 4

CASE 5-

r " GR H M R N Z , :

_ . o I _ -- _ _ 1 .o

-

z .5 z . 5_ I .o . --_ r -_ _ .o P H SP E l E2 E3 Eq P H S P El E2 E3 Eq I .;O O ES _ ; O DES " [H ETR- R T H E T R- T I.0-[ r___. I _ : _ 1.0 .5-_ J _ .5 BODES t ; O DES THE T R DO T - R THEIR DO T - T Z

"°I n li' • _ . 0 F - - ' 1 [- - '1__.__

.. _ P H S P El F..2 E3 Eq PH 5P El E2 E 3 Eq , BODES MO D E S $ RE S IOU £ $ FOR EACH O UlPgll M E NOR I' _ I . IZ£ O S O l l ' kqT I t E IR SUN IS | .0 FiKure 4. 14 _ .

Gust Impulse Residue MaKuitudes - Coa §K. 5

IH E _n- R THE T R- T

.5 _ .5

,. o] 1

_ _1 _ I--I I--I __ _ o .0 , - ----,- - -- , = : _ r-----.

P H S P El E2 E3 Eq P H S P El E2 E3 Eq . I _O D E S MO D E S t THEIR DOT- R THEI R DO T-T ;:

,.,., 1 .o _ ! .o i ] ]

.0 _= .0 F--I, _ .

P H S P El E2 E 3 Eq P H 5P E l E2 E 3 Eq KOD E 5 H O D E S ' _ RES I OLE$ FC A [RCH O U I PLff _ N S R e _ I:IL IZ ED 50 ' l t _l THE I R 5LIN I S I .O Figure 4.15 Gust I mpulseResidue MaKnitudes - C o uBg.6 l 4 9

:, CASE 7-

,_ GRI "IM R NZ . #

.o _ .o

P H 5 P E l E2 E3 Eq P H SP El E2 E3 Eq 1 4 0DES NO DES

T H ET R DOT- R I H EI R DOT- T

_ .5 .5

, ) .o .o

• PH S P E l E2 E3 Eq P H SP El E2 E3 Eq I _0 0£5 HODES RES I O Ur S FOR(RCN OLIIPUT M E NO. _ R L IZEIDSO " ll_:_l I HE I R Stir I S 1 .0 • Fig ure4.18 i Gust ImpulseResidue M atmitudes- Confl g .7 e , - _ " t ,

. CRSE 8-

G A HH A NZ

.s 2 .s

I.o ,__ _ I .o ]

. o . o , P H S P E 1 E2 E3 E 4 P H SP El E2 E3 E 4 I - ;3OES Y , ODES "[HEIR- R I "HETR - T : - .5 . I .5 :

. o . o c-I r-n r--n[--I

PH SP E l E2 E3 Eq PH SP El E2 E3 Eq Y, O lXS I ' IO DES

THEI R OOT- R THEI R D OT- T

. o [-I___ . o .I --._F1FI

P H SP El E2 E3 Eq PH SP El E2 E3 Eq " _ ODES NO DES fl(SlOU($ FOR (RCH OOlPOl m E NF _ ,RLIZ(OSO l t tgl l t 'W _ IR SUN15 I.O l I I Figure 4.17 ; } : _I Gust Impulse Resid ue Mspk u des - Co_ fi{8 _" J _ - I _ ., , ,,*'1 J I _ s umm a ry , th e m od a l a naly s i s m e th od des c ribed in th is ch a pter has been us ed to a tt ack the q ue s ti _ m of, " how and when do aeroelastic effects s igni fic a Il tly aff e ct aircraft dynamic s?" Th e a nalysi s indicates that when the m ag n itud es of the modal i m pul s e r es idues of the " aeroelastie modes " become the do m inant re s idue m agnitudes of the vehicle system for i m portant output s , t ile v c.hi c le d y nami c s chan g e s i g ni fi cant l y and m ay change i n s uc h a way as to re su lt in "u n-air c r a ft like " c haracter is tics. In addition, the trends in the rela t ive r es i d ue m a g nitude value s f o r so m e outputs are closely related to the pilot r at i ngs an d tra c kin g errors o f t he si m ulations.

• A d r a_s back of u sin g t h i s m o da l analy s is approach i s t h at it i s essentially op e n-l oo p i n nature. Ev e n though ti le modal analy s is procedure considers so m e a spe c ls {_ f the p ilot, s pecifi c ally hi s li m i t ed bandwidth, it is still an open-loop an a ly s i s metho d . Since t he pilo t / ve h icle system performance is really d etc , r m i ne d by th e dy n a m i c s of the vehi c le when the pilot closes the loop, a "r .l _._e4 -1oo p" or " pilo t -in- th e-loop " analysis m ay give more insight into the _,ff_ , ctsot t he aeroela., _ tic mo des. Tl w n ext chapter considers such an approach.

CHAPTER V CLOSED-LOOP ANALYSIS b Co mp l e te flight vehicle syste m dynamics are dependent, not only on the aircraft dyna m ics, but also on the d y na m ics of the pilot and on how he interacts with the aircraft dynamics. Though the modal analysis method did con s ider lhe bandwid t h li m i t ations of the pilot, the m ethod was still open - loop in na tu r o . This chapter w ill appl y a closed - loop analysis procedure to the configur at i o n s in the da t a b as e to study the e ff ect of aeroel as tic modes on c losod -l oop dynami c s.

The c l o sed - loop an a l v._ i sp r oc e du re that wi l l b e us e d h e r e i s an e x te nsion , of til e Ne aI- Smi t h p r oc e du re [ 10 ] which u se s an o p timal co ntr ol mod e l ( O C M ) of the pilot [ 1 1] i n a p itch tra c k i n g t as k . T hi_ appr oach has, i n t he p a st , b e en a pp l ie d t o study th e effe c t of fl i g h t c on tro l sys t em dy na m ics o n pi tch track i ng performan ce o f fi gh ter -typ e a i rcraft [1 2 , 13] • Since fl ig ht- co ntr ol s y s t em i d ynamics a nd aeroe l a st ic m o d e s are both exa mp l es of h i gh e r o rd e r d yn a mi c s, • there is re as on to b e l ieve t hat, t his p rocedure m a y b e u s ef ul in eva lua t i n g the eff e c ts o f a e roe lasti c modes on t h e p it c h track i n g perf o r m ance of th e d at a b as e "_ c on fig u ra t i o ns. t Be f or e usin g this pro ce du re to study the data base con fi gurations th e p ro c edu re mu st be ext end ed f o r app li c a t i o n to flex ib le a i rcraft . This enta il s 1 und ersta ndin g t h e N ea l - Smi t h m et hodolo gy a nd a ppl y in g t h e O C M t o t h e N e aI-Smi t h a pp r o a ch. Thi s w il l b e a c c ompli s h ed b y b r iefl y re vi e win g t h e wo r k do n e by Ne a l an d Smi t h [ 1 0] a nd b y Ba con a n d Schmid t [ 1 2 ].

o N eal-Sm l th / Baeon Methodology The inv e stigation performed by Neal a nd Smith in the ear l y 1970's resulted in a criteria developed to expose problem areas in aircraft where the i pilot w as to perform a given t as k. Their criteria utilizes a closed- loo p or " pilot-in-the-l oo p " analysis procedure. This pilot-in-the-loop ap p roach wa s 5 3 use d becaus e o f di f ticull ies enc ou ntered i n usi ng e xis t i ng open-l oo p han dli ng qualities criteria and be c aus e o f th e t r ul y c losed-loop natur e o f piloted v ehicles.

The analysis method was based on the f a ct tha t th e subj e ctiv e pilot ratin g ()f a l ) itch - attilud e t a sk is primaril y determin e d by how w e ll th e pilo t ca n ( .(ml n) l l he pi t ch atti l udc and ho w di ff icult i t is to do so. Sp ec i fic all y , th e , an : llx.,is was l w rf o r n w d u. - ing a compensatory trac k ing ta s k m o del ( i.e. the pil o t only I w rceives lh e differ e nc e b e tween I he attitude o f th e a ir c ra ft and th e conmmnded attitude ) , and by representing th e pilot by a d e s cr ibing f unc t ion - c_ m ,isling o f a time delay and a lead - l a g c o m p en s ato r ( see F ig u r e 5 . 1) . T h e time delay a c count s f or perc e p t ual delay s a nd n e uromu scu lar lag s and th e h , nd-lag c-ml w ns a tion is used as a first order app r oximation o f th e pilot' s dyn:lmi ( . ( '( )ml) c ' n _a l i( ) nin t he lask.

l lv ( ' , ) n_,id_vinglilt c loscd-h ) _ ) p frequency r esponse o f t he me ) del e d I,ih,I / :_ir('rafl .,.v.,lem , Nq'al and Smith were able to relat e the pilot's objective ra / in_, I_)fr_ , q,_ , nc ) r_ , sl , _ m se characteristics depicted in Figur e 5.2. The r_ , sulling .,I,_ , cilicalic m s also r_ , late Io the stated goals of actual pilots.

I"_,r g cJod i w r f ormance, a pilot wants to be abl e to a c quire the target quickly and pr_ , diclably and with a minimum of overshoot and oscillation.

"Quick :lcquisilion of lhc target" implies that the pilot wants to a c hieve a high l) :in ( l_i ( ll h. N ( ,al and Smi t h al to ) r e l aled minimizing () ve r sh_ x) tto minimizing It . , ( 'l-s_ ' d-l_q ) r esonan c e l) eak, ( _ 1. This in f e re n c e c ( m ) _ . s fro m t he III,I _ .

I °

r_ , h_li,,nsh i I) I H , !ween svsl o m ( lam l) ing and Ihe magni!ude o f t he r esonance l) o ak in n s o (.( m (l _ r( le r sys t em , l ly coral )t h i ng the two obje c tives , Nq , a l a n d Smi t h conc l uded that , "pi l o t r ating is a f unct i o n o f t h e com p e n sa ti o n r eq u i r ed to ach i 0 ve g ood Imv f r equency pe rf o r mance and the os c i ll ato r)' tende nci e sth at r esul t ." [1 0 ,1 2, 1 3] These obje ct ives we r e r ela t ed t o t h e ch ) sed- h ) o p a n alysis by defining t he syst e m l ) a n dwidth to be the fr e (i t w n e y a t whi c h the e l os e(l -l o _ I ) sys t em l ) hase l ag r eaches4 }0 (l e g .eesas i ll ust r a t ed in Fi gur e 5 . 2 . The p i l ot compensat i o n was delined as the phase o r th e r es ul t i n g l)i h ) t d esc r ibi n g 6 mel to n a t the • b a n( l_ i(Ith f r equ en cy , e x e lu (l i n g t he e ffec t o f t he p u re tim e d e lay , a. , _sh o wr, in Equatio n ( 5.1 ) .

I- } i .i l al o I. m .I " % I_ ' V l I -- LI.J I.iJ O (3 _ el:: ..J ..J L L

_ uE

_ j I

i

• /. i i

* V I g | • v g ' v - " • - o o 0 m |

.. _ . .I ._ _

' -V-

= j viiwTp_ + I (5 . 1) PC A / J_ VI ]wTpl + 1 - v : h_' r e | ' C it the p il o t c omp e nsa ti o n . Th e cl osed -l oo p r e s on,_nce peak i s

f

define d t o b e t he m ax imum valu e of t he ma g ni t u de of t he cios e d-h_p [ f r equency respons e . ( see F i gu re 5 . ? ) " !

Th e c ho ice o f para m et e r s in t he pil o t descri bing f u n c ti o n ( K p, r, T pl , Tp 2 } [ we r e m a de t o bes t s at i s fy a se t o f p erf o r ma nce re qu ire m e n ts . T h e req u ir ,-me nt s c o n sis t o f- l) a spec ifi ed ti m (. de l ay (r) , 2 ) a spec i fied bandw i dth c h a r ac t er i s ti : o f t h e t as k , 3 ) a m ax i mum a ll owab l e " l o w fr eque n cy d r( _ )p ", T p , . .- - t) c omp en s ati on (i .e. t h e va l u e o f 7 .1 , --_ . ! r equi r ed ° , a I nin i ll l ll m value o f r esonanc e peak.

Nea l a n d Sm tt h foun d t ha t t h e pa r ame t e rs which sa:is fi e d t hese r equi , 'emen t s r esulted in a pi l ot phase compensa t ion an d ck rs ed-loop r esonan t pea k t ha t co rr ela t e d wi t h pi l o t r a t ing.

l|y plo t ting t he va l ue o f r esonance pea k ag ains t P ( ' { i.e. pilo t temp t . an al ion ), N ea l an d Smi t h f ound re g i ons in w h i c h a i rcr a ft w i th s i m i l ar p il ot rati n g w e re g r ouped . F ig u re 5.3 sho w s the N eaI- Sm it h r esu lt and tilt.

r e g i .ns wh i ch corr e spond t o t h e t hr ee le ve ls o f h a ndl i ngs qu a l iti es descr i b e d i n MI I . -F- 8 7 8 5 C [I] . l . evel I i nc l udes a i rc raft h a v i n g pil o t r_ ti ngs (C(x) p e r- ll a rp er_ o f I. 0 - 3 .5, level 2 include s p il o t r a tings of 3 . 5 - 6 . 5 a n d level 3 ratin gs o f 0 . 5 - 10. 0 .

T h e r e are problems a s soci a ted with the Neal-Smith method h o weve r .

" l ' ht_e pr (:t4em s lie in t h e di l l ic ulti es as s oci a ted wit h ch oosing a ppropriate * fr eque n cy rt ._ p o .-_es pe c ifi c ation s. F o r instance , , it m a y b e very difl _ erUlt to d et e r m in e t he proper ba nd w idt h freq u ency for an air c raft in a p a rti c ul a r task wit h out e x pe ri m ent a l data . Anot h er pr o ble m lie s in t h e c h oi ce of a m axi m u m low f requency droop. Sin c e t h e droop i s only a relative measur e of low frequency tracking performance, th e c hoie _ of a maxi m um allowable value is rather arbitrary. Still fu _th,_ r , th e Neal-Smith metho..l u s e s a com pen s ator) ta s k that wa s not r e pre s entative of the ac tu a l ta s k u s ed in their flight te s t s .

B a con [13] e xt e nded th e work of Neal a nd Smith to try t o addre ss the s e - _ 57 Q J PILOTP, A T Ill6 1o 0 I - $.$ $.$ 6 .$

A

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e Z_" Ax A" A -

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, _ - w 4 0 o _ w w m

• , ,,-LAG L[ I O + PILOI¢0flPt[NSA T I01I te n ,) ,!

+

l Fi b re 5. 3 Neu _ ,-Smith Criter¼ p r o blem s . I ie app lied a n op ti mal con tr o l model o f t he pi lo t ( O C M, [1 1 ] ) t o the c l o s e d - loop a n aly s is m et hod. T he u se of t h e OC M pro vi d e s m ore fle x ibil i t y in conducting the analysis by allowing the Neal-Smi t h crite r ion to be applied to

i

-, othe r , more gene r al, piloting tasks, it also eliminates the r equi r ement of " choosing t he ar bi tra ry fr e qu ency response s p e c ifi c ations which a r e requi r ed t o de t e r mine t he pilo t desc r ibing func t ion. ; T h e o pti ma l co n tr ol m o d e l ( OC M} o f t he pilot i s based on th e as s ump ti o n : th at a we l l t r ained, h i g h l y mot i va t ed pi l ot t he se s his cont r o l input ( U p), sub je ct t o ph ys iological limitation s , in s uch a wa y t ha t a quad rat ic co st func t ion,

J p = E I f _ "°

T o x TQy_ + r u_ + g u_ dt , ( 5 .2) is m ini m ized, llere, Q and r are wei g htings chosen to reflect the task objectives and g is usually cho s en to reflect physiological limits.

A lthough de tai ls c o n c ernin g t h e O CM c an b e f ou nd in [ 11 ] , a b rief de s c ription wi l l b e i nc luded h e r e . Co n sid e r th e block diag r am o f the O C M i n Figure 5 . 4 . Th e huma n p er ce p tio n ch ar acte r istic s th at are mode l ed i n volve th e pi l o t o b servation s ( y_ p) ,pass e d through a pu re tim e del a y a n d co nta minat e d b y whi t e no i s e o f i n ten s it y V y . (see E qu a tion 5. 3)

Z p( t )= x (t - d + _vy (t - r)

: ( S. 3}

_ (t ) = C px(t ) Th e solu t i o n t o t h e s t ated o pti mal c o ntro l pro bl em y i e lds a Ka l man fil ter to e s t ima t e the dela y ed s tates and a h ' ast-m e an-squar e p re dicto r to obtain a cu rr en t e stim a te o f th e s t ates, (g(t ) 1 . T h e c ontrol la w , obt a in e d f rom m i t fimization o f the c ost function J p , for a sca l ar Up, can be e x pressed as, r . t ip = - K x f, - U p ( 5 .4) i 6O _ - whe r e Kx i s t he op t imal c o n tr ol gain ma tr ix. T h e n e u r o-mo t or l ag ( rn ) le_ull_ f rom including cont r ol rate (t ip } in t he ' ,'os t f unc t ion , / p and by • weighting it s<)as t o obtain a physically achievable value of r , , based ( m man- !| machine exp e riments.

' i : As discussed in [ 1 2, 1 3 ], w hen to OCM cost function is used to minimize : , t r acking e rr o r , ( 0 - 0 c 1 , the r esulting cont r oller au t oma t ica l ly minimizes low - f r e quency droop and r e sonance peak o f th e c losed-loop system f requen c y r e sponse. This is an altern a tive to speci f ying maximum droop and de t ermining t he compensation required to minimize r e sonance peak in the Neal-Smith approach. In addition, th e OCM will automa t ic a lly det e rmin e th e achievable bandy, tal l h o f th e c losed-loop sys t em. There f ore, t he N eal-Smi!h requirements " ar e compat:ible with the OCM. Th e task f or the analyst is now to properly apply tt e OCM.

The prope r a pplication o f the OCM involves, , 1) se l ect i n g a re alisti c pi lot obse rva tion v e cto r f o r t he tas k ( y_ p) , 2) definin g th e c ost f un ct ion to b e minimiz e d ( J p ) in th e task, 3) d e finin g the command si g nal to be track e d ( 0 c) , ,I } s pe cifying th e noise var ianc e s , o bse rv ati o n al t hre sh ol ds and d e lays co ns i stent with th e huma n o p era to r .

B y p r op er ch oic e o f pilo t o b s er va t ion s , c o st f u nct io n and comma nd signal, the .anal3 sis using t h e O C M c lo s el y r efle cts t he in fl ight tra c king t a sk u s ed b y Neal and Smi t h. i A subtl e ty d is c usse d by Ba co n [ 13] wa s ass o ci a t e d w i t h t he a lmost - guaran t eed s t ability o f the O C M solutio n . W i th e c os t f unc tion reflect s n finim iz in g trackin g err o r, the re su lt i n g c on tr o t r i e s t o m ak e t h e c los e d -loop system ac t li ke a pe r fe c t t r a ck e r , tha t is, a sys te m wi th a re sponse - to -c ommand tr ans f e r f unc t ion of u n i t y. As a r e s u lt t h ere i s a t r ade-off b et w e e n t he l ow f re quenc y d r oop a nd res on a nc e p ea k. Tha t is, a n a i r c ra f t th at would a ctu a ll y _- lead t o os c i l l at o ry t e nden ci e s an d a si gn ific ant re son ance peak in t h e N ea l - i S mith a na ly sis , woul d y i eld an OC M solution tha t would sac r ific e low " : : fre quency p erf o r mance for s ta bi l i ty . Th is i s a n alo g ous t o the p ilo t b e i ng less aggre ssiv e a nd tr acki ng th e tar ge t so t ha t th e oscill a tions would not occu r .

T h is p ilo t i ng strat e gy tends n o t to e x po s e the tenden c y of the a i r c raft to oscillate.

B acon ar g ued that, s ince o s cillati o n occurs fro m "s uboptimal " piloting strate gy , the O CM controller would do a better jo b of tracki hg tha n a real pilot. He further argued that by increa s in g the forward path gain, one could -" ap p ro x i m a t e an a gg re s sive pilot . An a gg ressive pilot would try to o b tain b etter low frequency perfor m ance at the expense of hi g h frequency oscillations. This type of pilotin g strategy would therefore expose the oscillatory nature of an aircraft . Hence, in Bacon ' s approach, the for w ard path gain (i . e. Kp in Fi g ure ' 5.5) w as adjus t ed so t h at each vehicle confi g uration led to a maximum low " freque n c y droo p , simi l a r to the N e a l- S m i th m ethod . The a dju s t me nt e x p o se d the o s c i ll at ory ai rcr a f t by incr easin g the r es on a nc e pe a k o f s uc h ai r c r a f t.

. F ig u re 5 . 5 illu s tr a te s t h i s e ff e c t.

By plotting the gain-a d ju s ted re s onance peak a gain s t the pilot ph as e com p en sa tion {obtained from the OCM) at the bandwidth frequency ( as illustr e _' _ edif Figure 5.6), Baco n obtained a p lot analogous to the one obtained by N eal and S mith. Figure 5. 7 pre s e n t s B acon's re s ults for the N eal- S mith Configurations which can be co m pared with Neal-Smith's or i ginal resul t s shown in Figure 5.3. Ju s t a s in the N eal-Smith criterion, aircraft with s i m ilar pilo t ratings gro u p togeth e r on Bacon's plot.

The choice of proper ban d width is not necessary in the Bacon method and is replaced by cho os ing a weighting in the co s t function which results in a rea s onable neu ro - m otor lag (rn), which is a n atural phy s iolog i cal li m it. B acon's [13] results al s o i n dicate tha t the closed-loop syste m band w idth, a result from the O C M analy s i s , correlate s with s ubjective pilot rating. I n fact, this relation s hip ha s been s ugge s ted as an additional criterion for m ea s uring the q uality of the vehicle dyna m ic s .J12,13] Figure 5. 8 illu s tr a te s the relation sh ip : between clo s ed-loo p ba ndw idt h and pilot rating for t h e Neal- Smi t h Co n fig u rat i o n s .

A d isa dv an t a g e of bo th me t h od s i s t he n e ed to c hoos e au arb i tr a ry m axi mu m lo w f req u en cy droo p. As an ext en s ion of th e Ne a l-S mith / B ac on m e th od, an a l t er nat e w a y of co n s i de ri, ,g o s c i llatory tendenc i e s will be _ pre s ented . T h i s ne w vari a tion o f t h e th e N eal-Smit h / Bacon met hod wi l l b e used t o con s ider how aeroel as tic modes a ff ect th e closed-loop chara , _ ter i stics of aircraft.

Exten s ion Of Neal-Smlth / Baeon Met h odology In an attempt to make the analysi s procedure independe _: t of an arbitrary c h oice of t he m aximu m low-frequency droop, an a lternate method i s proposed.

Pilot ind u ced o sc illation s {PlO' s ) u s ually occur wit h ag gre s s iv e pilot be h avior.

: I f t h e pilot " bac ks -o ff" {i . e . reduce s his aggress ive n e ss ) , th e o sc illatio n s

02 I

i $P E C IF " IE n t ' A X l r. q J H //

H(s) G(s) i

I I

PILOT PLANT , { Fi b re S.S Reso n ance PeakAdjustment I )

t

I .

: 6 3 .: freq.

.!

i oo l B ,w ....

. , _ ( iw ))

- ,00 4 ......... _ _%

freq.

BW

t i n j PC ......

, . p i lot

phase

i • F i p r _ 5.6 Nesl-Sm i_ b / Bscon Pilot Com p en= ¢ io =

i--i , A

i

I I

i Nest-Smith / Ba con C r ite r is f + g

_ S

& / I

§3

• v • @ I

2.0 2.5 3.0 3.5 q.O

' _ osso-Lom, _ ,.o, l o _ (p , x o / ssc)

Figure 5.8 Pilo t Rating versus B _ ndwidtbfor Ne a l-Smith l D a eonCon§ gu r a tio u d i s a ppe ar. A poor a ircraf t m a y have ch a r a ct e ri s t ics wh ich prod uc e Pin' s, w i t h j ust s lig ht i n cr eases i n a ggr ess iv eness. It i s th i s c ha r a c te ri s ti c w hi c h B a co n ' s g a i n a dj u st me nt exp o ses.

A fir st ord e r app ro xi m a t i o n t o pi lot a ggre ssi v eness i s t he DC g ain of t h e pi lo t d esc r i b in g f unc t ion. A sl i g ht i nc r ease in t his g ain fro m ti le O ('M app r oxim at e s a n in cr ease i n p ilot aggr ess iv eness. If t he in cr ease in clo se d-lo op re sona nce p e ak which r es ult s from _he incre as ed pilot gai n i s relativ e ly large, o ne co u ld c o n clud e t hat th e ai rcr a ft is sensiti v e t o p ilo t aggr essi v eness or , in o the r word s , has o sci ll at ory te n d encies.

T his a rgu men t imp li es t hat a g a in sensiti v it y p roc e d u r e can be use d t o exp o se ai rcr a ft wit h o sci ll a tory t e n d en c ies. Si n c e t he O C M o p t i m izes t h e c o nt r o ll e r d es ign i n such a way t h at low fre q u ency droo p is s acrifi c e d for r es o nan c e pea k , excessi v e dr oop r esu lts for config u r at ion s w ith o s c i ll at ory te nd en c ies. U si ng these i d eas , a g ain sensiti v it y pa r a m e t e r i s d e fin e d t o be, ITIU : * =t SP __DROOP x AK ( dB ) (5 . 5 ) The te r m, D RO OP, i s the m agnitude o f the " droop " f o r the c as e in q uest i on, obtai n ed di rec t ly from th e clo s e d -l oo p O CM analy si s {see Figu re 5 . 2) . The o th er te r m o n ti l e r i ght ha n d sid e is t h e rel ati ve gain s e n si t i v i ty wh i c h is determined by calculating the change in resonanc e peaks obtained u s ing the basic pilot {model} gain and that obtained using a perturbed gain, o K. The gain di ff e ren c e (AK) . t T o j ust ify the v a lidity o f u s ing SP a s a me asu re o f os cill a tory t endenc y , c o n s i de r Table 5 . 1 . Thi s table prese n ts th e resonance peaks o f t h e configuration s from t h e NeaI- Sm it h study an d t h e B ac o n st ud y a nd t h e valu es of t he SP ca l cu l ate d for the s a me co n fig u ra t io n s • t TheAK thatwasused to obtain t henu m eric a l results w a sde t er m ined b y per t ur b in g t he pilot DC g a in by approxim a tely ten pe r cent .

S _ I ' I Table 5. 1 . C o mparis on o f Resona n ce Peak a nd SP Values Resonan c e P e ak (dB) - Config . SP (dB) Neal-S m ith Bacon " IA 7.0 7.19 1 . 33 IB 0 . 5 1 . 8 6 0.49 IC 2.0 4.84 1.13 ID 0.0 1.83 0.3 9 IE 9.0 3.50 0.73 • t , • , 2A 3.0 4.97 1.50 2B 7. 0 11.37 2 . 44 _" 2C 2 . 0 1. 2 0 0 . 93 2D 2 . 0 1. 2 4 0.8 4 _ , 2 E 3. 5 3.28 1 . 40 • |, 2 F 2. 5 3 . 00 1. 2 0 2 G 6.0 9 . 25 2.09

2 H 3 .0 2. 5o 0 . 8 1

21 7 . 0 e .3 e I. e O

3 A -1 . 0 O'. 6 8 0 . 28

4A 1 o . o 1o . 17 2.2e

5A >1 2 18.21 3 .71

8 c i. 5 1.2 5 0 . 38

7C 0 . 0 0 . 77 0 . 17

• 8A 0. 0 0 . 64 8 0 . 1 1

The trends between the three parameters , for the most part, a g re e b ell with eac h o the r . Furt h ermore, Fi g ure 5 . 0 , _h en com p ared wit h F igure 5 . 7 . indi c ate s ] th a t SP i s a n a n al o g ou s m e a su re of o s ci ll at o ry te nden c i e s an d can theref o re b e " ,!

u s e d in ste ad of resonance p e ak in t he anal ys i s.

Applie a t l on Of The Nesl-Smith / Baeon Ana l y s is To Th e D ata Ba s e C o n figurations In o rd e r t o apply t h e N e a l -S m ith / Ba c on a n al ys i s met h od, a clear u n der st a n di ng c r th e simi l arities a nd differe n ce s b e t ween the type of confi g ura t ion s studied by Bacon an d the flexi b le aircraft of the da t a base of Ch a p t er 3 i s necessary. The aircraft used in Bacon's s tud) ' v l ere s o me of the _ confi g ur at ions u s ed in the Neal-S m ith stud)', a nd represent basic airframe d y n am ic s wi t h control sy ste m dynamic s added. The basic a ircr a ft d y n a mic s t h a t were ana}yzed included only the short-period m c :le. The added high order m o d e s repre se n t ing co n tr o l s yst e m d y n am i cs included a real po l e , a re a l zero a nd a s e c ond-ord e r, o s cillator)' m od e. Fig u r e 5.1 0 sh ow s t he b as i c airfr a me pl us flig h t c o n trol syst e m { FCS ) d y n am i cs in th e pi t c h - a ttit u de - r a t e -to -s t i ck- d efl e ct io lJ tr ans f e r f un ctio n. T he sh ort p e riod d y n a mi cs a re d e t e r m in e d b y T ez an d by w , p a n d £ p a nd the £ CS d y n am ic s ar e d ¢ '. e r m in e d by r I , r s, wa and fs .

T h e flexible aircra ft of t h i s st ud) " al s o have h ig h er order mod es b ut they corr esp o n d t o ae ro e l ast ic eff e c t s a n d n ot F CS e ff e cts . Th e d y n a mic s of the fl ex ible a ir cr af t ha v e al read y been discussed in C h apter s 2 and 4 a n d it will su ff i ce t o summa ri ze t h em wit h Figur e 5 . 11 . H e re t he rigid-body d y n a mi c i p ar amete r : , a re T e, , T e ,, _ p h , fp h, _ ,p and f,p , a nd the a eroe las tic effects lead to .

i #i , _ i (i=l .... , m) a nd , o i, _, ( i = l, ... ,m) wh e re m e qua ls t h e num b er of structural mod e s i nc luded in t h e v ehi c l e mod e l.

A n im po r ta nt ste p in t he an a lysi s is to d e c i de o n a n _ , p propria te co s t f unc t i o n { J p } . I n t he c ase of fl e xibl e ai rcraf t th e pilot s ees or s e n ses tot a l i l

i

L O 0

_ o

°

V

L O _ _ / 0

A_ V / ft. ne

___ _._ o .

. / _ "!

/ = '_ !, g .

.= // _-._[, i

• i |

1- _' d o, ,--

,!

( S P) cl9

7O ¢ 1; I M • | A ; | , : , _ m - ,q 7 L

+

Q,

!l "i

,_ .

| i i iii / * 7 2 l ' tr _c k in g error { _ T A 0 T_ 0 C ) , w i th s t ruc t ur a l effect s included . Ch o o s i ng to ta l I e rr o r as t h e mini miz e d pa r ame ter , ho w e ver, m a y not correctly reflect th e p ilot 's obj e c t ive s. T hp p ilo t co mm e nts f r om the simu l ati on of the d ata base co ml g u r a t i o ns su gg es t t ha t the pi lot t r i ed to tr a c k ri gi d . body err o r .

f T h e foll o w ir. g q uot es a re t y p ic a l of th e pi lo t comm e n ts that r esu l te d from th e sim ul a t ion s t u d y [2] . - ,ID For C o n fig u r ati o n 8 , t he p ilo t c omm ent s inc lud e d - " m ore o s ci l latior _ i nvol ved du e to el as ti c ity a pp a rently , b u t it wa s h i g h : enou g h fr eque n cy that it w a s easy t o i g nore that and sim p ly to fly the ' ri g id b o dy ..." [2 ] Fo r C onfi g u r atio n 7 th e p il ot c o mmen t s incl u d ed - " i t wa s n o t too d ifficult to i g n or e ( t he o s c illation } a nd to fly t he r i g i d ( bod y} ..." 12 1 These c omments indica t e that the pilot places mor e emphasis on keepin g the ri g id trackin g erro r low than on minimizing t otal (displayed } t r a c kin g e r ror.

{Al s o s e e 1 1 2 , 1. 1]. ) T h erefore , t he appro priate c o s t f unc t i o n f or t he fl e x ibl e air c ra f t is, T : . ! o J p( O R) = E _ fi + g u dt ( 5 .6 ) ,_ - w h e r e c R __A ( 0 a - 0 c) a n d g i s c ' h o se n t o obt a in t h e d es i re d & .

l )a co n h a s sho wn [ 1 2,13] tha t th e c hoi ce of r n ta ke s th e plac e of band w idth / i n the Nea l -Sm ith me tho d . The v alue o f r n i s c h o s en to refle ct p i l o t : a ggr e ssi v e ness in th e t rack i ng t a ._ k and d ete r mines the ba n d w i d th o f t he c losed -l o o p s y s t em. A s rn in cr ea s es, b y inc r e a sin g g , t h e bandwid t h dec re as e s.

: Low r,, which repre se nt s "a g g re ssive b eha v io r " , re s u lts i n a f as t c l o s e d-l oo p : syst e m and s o a hi g h band w idth. Th e le f o re , to obtai n th e max imu m po ss i b le I' a ndwid! h, rn sh o uhl be set a t ! .he lowest physically possible value , w h ich is usually c , I, !de r ed In be 0.1 se c onds. [8 1 The value o f g used in this study to obtain a rn of a ppru x imat e l y 0 . ! sec o nd_ was ,

y

_ g = 0.0040 .

_: The an a ly s is i nc l udes the v, hi c le dynamics, the pilot observations and the c om m a n d s ign al d yn a mics. These factors must be chose n to b e con si stent with t he ta s k. The complete pilot observation vector therefore include s _ T and i T, 0 T and 0 T. These four p a rameters are naturally displayed to the t p ilot in the tra c king ta s k .

: _ The intermediate output of the analysis consists of the controller gains, * clos ed -loop ei g envalues, r m s values of th e inputs, st a tes and output par a meter s , co s t function v a lue s , optim a l estimator gains and , m ost importantly for this ap p lication, frequency r es pon s es for selected transfer functions. By combining the transfer functions properly, the d es ired cl os ed-loo p transfer function frequency response, similar to th a t of N e a l and Smith, can be formed.

Consider the block diagram of the tracking task in Figure 5.12. The clo s e d -loop tran s fer function of intere s t in this study is 0 c (s---- _ , sinc e rigid -b ody attitude, 0 R, is what the pilot " cares about " in rating the performance of the aircr t tft. This transfer function can be written as ,

e R(s) H (s)' G l( s )

0 c(S) 1 + H(s)'Gl(sJ'G2(s) • (5.7)

0 R(S )

T (S)

0 T ( S) 1 4"_

T (S )

, O R ( s ) O c(s )

i ' T he t ransfer f un ctio n s o f interest are therefore _ T( S ) and CT(S ) Table 5.2 summ arizes t h e n um erical val u es u sed to obtain the d es ired results from the : _ ; . anal y sis.

t Thc v ehi c l e d y nam i cs ( Equa t ions 4. 21-4.23 ) a r e the sam e as th ose used in the o p en- l oop r r g da l a na l ys i s . T he s e cons is t of the A v and 1 3 matri c es of th e i t d at a bas e con fi gurat i o n s. The D m_ trix i s zero however sinc e gu st disturbance 1 dyna m ic s i s not con s ider e d i n th is an al y sis .

J

7 4 7 5 Table 5.2 Summary of Closed-Loop Analysis Inputs Obs erv at i o nal 0 R ' 0 T, c T 8 . 7 x 10-4 rad * Thr es hold s 0 R, 0 T, i T 3.1 x I0 -3 ra d , sec Q V a ri a nceof c T, iT , 0 T , 0 " r -20 dB ' ' Sensor Noise O R O R -6 dB :; 9 • Attention Allocation {T, _ T, 0 T9 0 T 0.245 " (_ A.A . i = 1 . 0 ) 0 R , 0 R 0 . 0 l _. i • consistent with pr e vious work [1 3 ] The r em aining r e quiremen t, is t h e c ommand signal dynamics . A command s ig n al ( 0 c), wh ic h acc u rat e ly repre se nts a ch a ll e nging pitch tracking t as k and a p pro x i mates t h e tr ac k in g tas k use d by Nea l an d Sm i th, i s d e fi ne d by E quati o n 5 .8.

0 ( : + 0. 5 0 ( .,. + 0 . 25 0 ( , = w(t) (5 . 8) Ih ,r _ ,. 0 C ,i _ _he ( . :) mnl au( h ,(I attit u d e a n d w( t ) is zero mean Gau ss ian wh ite n o i s e f c: f i nte n ._ ity V ,, . . T he in te n ._ i t y of the w h i le n o i s e w as ch osen to resul t, in an r ms val, e for 0 c of a ppro x ima te ly th r e e (3) d e grees .

T he r es ul t i n g, mo d e l- c o mp ati b l e , st ate variable representa t ion h a s t h e form , •X oc M + ._ + . w ( 5 . 9 ) • . i -o ' 1 = A, .

l°l H

t, w her o.

r I

: _ xoTcM = O C ' O C [ xT ' 1 5 " 1 01 f • ,r • o ?

Here , A C and E C are t he matric e s resulti n g from t h e state varia b le repre s e n tatio n of t h e command si g nal.

C los e d - loo p e v alu at ion o f the mod e l y i e ld s the des i re d f re qu e n c y res pon s e s and the p ilo t desc r ibi n g f un ct i o n fre qu ency response . The d e si , e d c l o s e d - loo p .

fr equt respons e , namely 0 R ( s-----_ ) i s ob ta ined by m a nipula t ing the f requen c y

0 c ls)'

responses accordin g to E quation 5 .7 at selected f requencies. That is,

O R (S)

0 R (j w )_ _T( s ) ( & ll l 0 c{j ' , _) 0 T( S ) 1 "4 " C T(S) s=jw The fre quen c y re sp o ns e s t h at r e s u l t f r o m th e cl ose d - loop ana l y si s o f th e e ig ht d a t a b ase co n fi g uration s , can be found in Appendix A. 8 . An exampl e of the fre qu e n c y re spo n s es is sh o wn i n F i gure 5. 13 . The "P ur du e Pil ot " freq u ency re s p o n s e c o rre s p o nd s t o I t ( s ) or _ and the " Aircraft (O .L.) " frequency

T( S )

re s po n se corr espo n ds t o Gi ( s ) o r _ a s d ep i c t e d i n Fig ure 5 .1 2 . The

6(s)

" A i r cr af t Plus Pilo t . ( O.L. ) " fre qu ency re s p o n s e c o rr es p o n ds t o H( s )Gl( s ) or ' 0 n ( s ) and " A i r c r a ft Plus Pilo t (C . L . ) " co rre s pon d s to the freq u ency response

O r (S)

f o r ?RIs )

0 c ( s ) •

Numer i ca l Resu l t s " T h e closed - l oop s y s te m fre qu e nc y respo n se pr o pert i e s ; b a n dwid t h , droop, " r e sonance peak and s e nsitivi t y pa r ame te r; p ilo t p has e comp e nsation a t th e ba n dwidth f r e qu e ncy and pilot rating a r e summ a riz e d in T a bl e 5 . 3 f or ea ch o f the data bas e configuratio n s.

F i r st e x a mi ne th v t ren ds in p i lo t ra ting w ith c los e d-l o o p b a nd w id t h.

Figu re 5 . 14 i s a plo t o f pilo t r a t ing (PR_ ver sus b _ndwi dt h fr equ e nc y ( _ V nw) f o r . : the eight data b as e c o nfig u rations. Tho u gh the n u mber of d at a p o i n t s is 7" / _ - _ ,.. . , _, _ .

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Table 5.3 S ummary of Clo s ed-! _ op " , naly s i s o f Data B a s e C onfigur a tions r_ d 0 R DROO P d[: _m d 8 SP dB P C d e l l PR Ca se _ BW see 0 C m ax 1 2.29 1 .0 2.17 0.67 -58.9 1. 6 2 1.80 1.51 1.12 0.40 -06.0 2.0 3 0.07 8.45 O.OO 7._ 9 -_ .02 5. 9 ..

4 1.77 1.93 -0.II 0.78 -16.08 3.1 $ 1 . 53 2 . 14 0.10 0. 49 -68 . 9 2 . 0 6 0. I0 6 . 94 9 . ,59 ,5 . 7 -72 . 3 6.7 7 1 . 70 1.56 1.02 0.48 -60.55 2 . 3 8 2.00 1.23 1 . 09 0.56 -, 5 3.3 1.9 limi t , . d , t he trend i s con s i s t e nt wi t h th a t of [12,13]. The confi gu ra t ion s wit h low c l()s e d- l oop bandwidth have poor pilot ratings ar.d th e configuration s with r e l nt i w ,ly hig h b a ndwidth ha v e better pilot r a tings.

Ni ,x t , c onsid e r t i le e l. se al-loop s y s tem param e t e rs or Neal a nd Smith.

_ t " b * " Fig, re 5.15 i, _a p l o t of N (a l-S mn t h like " criteri a for the eight dat a ba s e c- nfigur at i-n _ , ll.w e ver, in pla c e or re s onanc e p e ak, t he sensitiv i ty p a rameter ( S l ' _ i., u _( ,d a . _a m c ,n. _ ur , , of . sc ill at ory t e n de nci e s. Note that t he c_) , l igur nt i., s wit h s imil a r pi_ ot rating s (I'R) ar e grouped t o g e t he r an d the configur nf i.n s wi t h p c . ) r e r rating s (i.e. configurations 3,4 and 6) ar e di s tinctly s e p a r at e d from t he bette r a ir c raft. A l s o, two of t h e c onfi gu r a tion s (i .e . 3 an d 6} haw , r e l at iv e ly l a rg e val m _ of SP, indi c ating os c illat o ry t e nd e n c i e s. This -seill a t.ry n a tur e i s al so n o t e d in th e p ilot c o m m ent s from t h e s i m ulations ( s e e Ta b h . 3.2). N- li ce t h a t t h q , value of SP f o r Configuration 4 indi c at es that its l .. )r c 'r l) e rf-rm a n ce is not due to os c illatory te nden c ie s . T he l)il()t c - m l) ,. sa li. n (l'( : ). t h oug h , in,licat q_ t h , _ *,the pilot has to S Ul)l)l) n n-r e le ad f.r ll w I . ,. _ 1l)(,:f(,r z , _lan e t ,,-rin -l he r _ . v .r(I s , the ai r c r a ft re_ l . )n se i s s l ugg i sh. Th i s s luggi sh i l a t,i t ,, i s a l _ ) n ,t ed ;n t he pilot co mm ent s f ront t h e s i m u l a ti o n s ( s t,,.

T a bl e 3 . 2) .

m Therefore, the analysis not only groul) ed air craf t with s i m il a r p ilot ralin _ , but i t a l s o e xpo s ed resp(:as e c har a ct e ri st i c s th a t contribut e 1,o de grad e d !

I perf o r m a n c e. T ho u gh there i s not enough dat a to determine bound a r ie s defining the three handli' l g q u a lit ies levels, t h e tr'md s tend to imply th e ir l - exi s t e n c e . Th e i m pl, ea t L nn is th a t thh c l os e d-loop a n a l ys i s m ight l)e able t,) i" b ide n tify w h en ae:, ,: l _ , ._| i c eff ec t s ,:ignifi,'an t ly a ffe c t t he dy na mic s .f fle xil) h ' airc r a ft . T ha t i s . t t:¢ ,' - ' e al- Sm_ t h / I h, , a a n alysis, a pp ro p ria t ely utiliz e d , m a y

I I

I.

. _ 83 C6 ,

O

1 C1 C4

C5 { ; C8 •

•

. -80 -60 -40 -20 0

PC (deg)

_ 'il r d r eS.IS SP versus PC rot Dat a B Lo cCon61 _r atio n s " a ppl. _o l a r g e flex ib le a i r cra ft as _ell as sm al l air cra f t with a dded con tr o l sy.stem d ynamics. More s p eci fi c a l l y , the r esul t s indicate t hat the da t a base c'on fig uralions svith p o o r t ra c k in g perfo r mance rec e i v e d t he poo r r esu l ts b ec a us,, of sensitivity to for_ard path g ain ( us e d to app r oximate pilot ag gres siw.ness ) J an d indicates osci l lato r y tenden c ies.

Q

CHAPTER VI : ° o CONCLUSIONS : _ " The objective of this st u dy w as to i n vesti g ate w h en and how structural effects (especi a lly dynamic a eroelastic effects) affect the dynamics of aircraft.

Two analysis methods, an open-loop modal technique and a pilot-in-the-loop method, were used to see how a eroela _ tic modes affect the dynamics of a i rcraft in t he lon g itudinal axis. Both procedures were applied to a family of aircraft w hi c h exhibit co n , _ iderable aeroelas t ic effe c ts.

The re s ults of the modal a n a ly s is indicate that when th e m a g ni t u des of the m odal i m pulse residues of the aeroel as tic modes b e c_ me lar ge co m pared to - ,; the residue ma g nitudes of t h e ri g id- b ody modes for impor t ant outputs , t h e dyn am i c s c a n c ha n g e si g nificantly and in suc h a w ay t h at t he h andlin g qualiti e s of t h e ve h i c le ma y b e de g raded . In addition, the trends in impulse residue m a g ni t ud e s for some inpu t s are closely r e l a ted to th e trends in pilot ratings of t h e con figu rations from t h e fixe d b as e d simulation .

T h e pilot-in- th e-loop analysis verifies t ha t as t h e frequencies of t h e a eroela s tic modes d ecre ase, th e performance of t h e v eh icle t e nds to d eg r a de.

More speci fi cally ; as t h e s t ructural vi b ration frequen c ies w ere decre as ed, t h e sen s itivi t y of t h e closed-loop system to pertur b ations in for w ard pat h g ain increased . This effect w as demonstrated b y plottin g feed-forward gain • sensitivity (SP) versus pilot compensation (PC) in a trackin g t as k. It was also s h o w n t ha t th e b a nd w id t h of the closed-loop system correlates with the _ su b j e ctive pilot ratings and those confi g ur a tions with lower structural _ " fr e q ue ncies tend to have lower closed-loop b andwidths. These results indicate that reduced a eroel as tic mode frequencies can cause de g r a ded h a ndlin g qualities , w hic h ma y appear in the form of oscillatory tendencies and slu gg ish response.

In conclusion, dynamic aeroel as tic effects can definitely contribute to degraded performance o f aircraft in the longitudinal a xi s . The aeroel as tic m odes contribute to p oo r perform a nce primarily by, I} introducing dynamic effects of their own in the form of high frequency oscillations, and 2) modal interaction which alters the dynamics ef th e rigid-body modes. In addition, these effects can occur when the aeroelastic mode frequencies are still several ti m es higher than the frequencies of the rigid-body mod es! A s a c onsequence of these effects, aeroela s tic modes should be taken into account for vehicles that exhibit these dynamic aeroela s tic effects.

Future work in this area should include expanding the data base. With a lar ge r set of co n fi g urations to study, the analysis m etho d s developed here can b e ap p l ie d to o b t a in m ore con c lusive re s u l t s w h i c h may l e ad to qua nti ta t i ve _ i rules for specifying h a ndling qualities for flexible aircr a ft. For example, it m ay i be possible to define handling qualities boundaries in the SP versus PC (i.e.

sensitivity parameter versus pilot compensation) plot from the pilot-in- the-loop analysis. The bound a ries would divide the plot into three regions which correspond to the three handling qualities levels. Also, the analysis methods developed here should be extended to study lateral-directional dynamics in order to understand the problem m ore completely. Finally, since it ha s been shown th a t aeroel as tic modes can be important , fut u re work _ ho u l d b e ai m ed at develo p in g control synthesis techniques t h at utiliz e the m od a l i I .

techniques, either directly o r indirectly, to gain in _; ght into the consequences of _ .

aeroelastic effects. Suc h techniques might address rest o rin g exce l lent h a ndl in g i qualities to vehicles with poor handling due to dynamic aeroela s tic effects, i \ LIST OF REFERENCES ?

8 7 LIST OF REFERENCES [!] Mili t ar y ._ pe c i fica t i on - Fl yin g Q uali ties o f Pil oted Ai rcraft , MI I , -F87 85 C ( AS(1 ) , 19 80.

'i l ' ilot l{a l ing." Phd. Th e sis, Department o f Aeronautical and A ._:ron a utica! En g in e erin g , Purdue Univ e rsity, 197 7 .

[ 2] 'fen , W.Y. , " E ffe ct s o f !)) namic A eroelas t ieity on l tan d ling Qualities and t [3] ( ; ilbert, M.G., Schmid t , D.K., an d Weisshaar, T.A., "Quadratic Sy n thesis o f A ct ive Controls for an Aeroelastic F orw a rd-Sw e pt-Win g Aircr af t," , lour na l o f G u i dan ce , Con trol an d D ynam i c s, M arch - Apri l , 1984, pg . 190 .

[.I] D' A zzo, J., and l loupis, C., Lin ea r Con trol Syst em Anal ysi s an d D esi gn: ( ' on ven t io n al and Mod e rn . New York: Mc G raw-Hill, 197 5 .

[ .5 ] Meir ov itch, L ., El em ent s o f Vi bra ti on A nal ysi s . N ew Y ork : M cGraw-H il l , 1 9 75.

[ 6 ] C o o p er , G . E . , _ nd Harper, R.P., " The Us e of a P il ot Ra ti n g Sca l e i n the Ev aluat ion of Aircraft Hand li n g Qu alities, " NASA TND-5153, April, , 1969 .

[7] Teper , G . L., " Air c r a ft Stab ili ty and Contro l Data, " Systems Techno l ogy In c . Technical Report 176-I, Apri l , 1979.

[8] Mc Buer , D . T., and Kr e nde l, E . S ., "M athematica l M ode l s of Human P il ot Be h avior, " N orth At l antic Treaty Organization Advisory G roup for Aerospace Research and Development, AGARDo g raph N o. 188, J a n., , 1 974.

I [9] Ro b ert s, D . A. e t. a l ., " Effect s of Cont r o l L a ws and Relax e d Stability o n _ , Vert ic a l R i de Qua li ty of Fle xi b l e Aircraft, " NASA-CR - 143843, Apri l , 1977 .

[10] Ne a l, T.P., and Smith , R.E., " An Inflight Investigation to Develop Sys- tem Desi g n Criteria for Fighter Airpl a n es , " Fli g ht Dyna m ics Laboratory, WPAFB, Ohio, AFFDL-TR- 7 0-74, Vol. I, De c ., 1970.

[II] Kleinman, D.L., Baron, S., and Levison, W.H. , " An Optimal Control M odel of Human R es po n s e, Part s I a nd II, " A u to m a ti on , Vol. 6 , M ay, 1970, pp. 357-383. . , [12] Bacon, B.J., a nd Sch m idt, D . K., " An Opti ma l Control Approach to Pilot / Vehicle An alysi s a n d the Neal-Smith Criteria, " Jo u r na l of G u ida n ce an d Co n trol , Vo l . 6, No. 5, Sept.-Oct., 1983, pp. 339-347.

[13] Bacon, B.J., " A M ode r n Approach to Pilot / Vehicle Ana lysis a nd the Neal-Smith Criteria, " Master Thesis, Department of Aeronautics an d As- tronauti cs , Purdue University, 1982.

[14] Schmi d t, D.K., " Pilot Modeling and Clo s e d -Loop Analysis of Flexible .

Aircraft in the Pitch Tracking Task, " AIAA Paper No. 83-2231, Gui- dance and Control C o nfere n ce, Gatlinbur g TN, Aug., 1983.

!

8O ._I Appendix A.I -_ Seallns Transformation for Mod e IdentPleatl o n , 4" - The aircraft states are scaled so that the elements of the eigenvectors have comparable units. This is done so that the e igenvector s c an be used to simplify the task of identifying the modes of the syste m . That is , aid in determining one of th e aeroel as tic modes.

The scaling of the system states is accomplished by means of a similarity I which ei g envalu e s a re associated with, for example , the short-period mode or transformation applied to the vehicle state variable model of the for m , .t = AX + BU (A.I.I) = Cx + Du.

Consider a flexible aircraft in the longitudinal axis. The following state vector definition is representative of such an aircraft.

XT _ _ i n , _ , 0 , 0 , q , _ ] ] (A.I.2) In the longitudinal axis, pitch angle and pitch rate are two pertinent dimensions. The vehicle states can be scaled so that all of them are nondimensional or can be physically interpreted as angles and angular rates, (i.e. units of radians and radians per second). The forward velocity ' perturbation , u , can be divid e d by the cruis e velocity , U0. The g eneralized • e lasti c deflection , q, c an be multiplied by t he m ode slope , _w , which m akes th e the state physically analogous to e l as ti c pitch angle with units of r a dians. Th is is evident when considering the equation for tot a l-el as tic pitch angle , n

0 T = (A. l .3}

i =i Similarly, the generalizedrate, / / , c a n be multiplied by the mode slope. The re su lt i s that the state becomes an a logous to ela s tic pitch rate with units of radians per second. The rigid-bodypitch attitud e , 9 , pitch rate, 0 , an d angle of _ .

a t tack, e , are expressed in radi a ns and so do not need to be scaled.

For the model and the scalin g f ac t ors described above, t he similarity transformationcan be defined to be, _ 000 0 0 0 1o0 0 0 0 010 0 0 T = 0 00 I 0 0 (A.I.4) 0 000_ I 0 0 0 00 0 _l T he transformedstate vector is defined by, _ . - TX. (A.I.5) Ap p lying t he tr a nsformationto the vehicle model in E q uation ( A .I.I} results in the s c a l e d sys te m , i !

• _ . = TAT- Zi + TBxt,

(A. L 6 )

= CT-Ii + Du.

| An import a nt prop e rtyof a similarity tr a nsformation is that it h as n o affect on the ei g e n v a lu es or r es idu es of the origi na l system. This property a llows the scaling tra n sformationto be applied to the vehicl e model without alteri n g the r es ults of the moda l an alysis.

Therefore, by applying the scaling tr ans , r or m ation described above , th e units of the ei g envectors ca n be adjusted to make identifying the modes of the system easier. In addition, this c an be done without affecting the r es ults o f the modal analysis p rocedure.

r ! •

g2 Appendix A.S n, as • Function o f the Vehicle Stat e s Con s ider the longitudinal s tate variable model of a n aircraft, Equation ( A .2.2), with the following state variable definition.

X T _-- [U , Or, O , 0 , q , _] (A.2.1) = Ax + Bu (A.2.2) y. - Cx + Du Th e plunge acceleration of an aircraft (n s ) is des c rib e d by th e following | | expr ess ion, , n z( t} = 1_. I To _ (t } + nx_ t}_ 0i ( ix}_i i{t) ,(g a s } (A.2.3) i g i=n !

i - whe r e g = gravitational acceleration, (ft / s e c 2 ) U 0 = cruise velocity, (ft / sec) I x = distance from e. g . t o cockpit, (ft) * 0 i = mode shape of ith aer oelastic m ode, (ft} m = number of aeroel astic mod e s The other parameters in Eq u ation (A.2 . 3) c a n be e xpressed in t e rms of the s t a tes of the aircraft.

The flight path an g le is defined a s , , _ t) _ _ 0 (t) - _ (t). ( A._. ') Therefore, - b {t}.

Note th a t 0 (t} is a s tate of the vehicle but & (t) is the time derivative o f the ve h i _ 'le s tate o(t). Note also that the deriv a tive o f the a ng le of a tt a ck with r e s pe c t to time c a n be written as , b (t} - A b x + B _ u (A.2.6) . where A _ , and B _ , are the rows o f th e m at r i c e s A a nd B , respectively , associated with the scalar equation f or & (t).

Si m ila r ly , 0 (t) and _ (t) are the time deri v atives of the states 0 (t) and _ (t).

T h erefore.

_ t) = X _ ix + B j u (A.2.7) a nd , _ (t} = A _ ix + B , i u (A . 2.8) - w h ere A i a nd B i , and A . ,/ a nd B , i are the rows of A and B a s sociated with t h e scalar equ a tions for 0 {t) a nd i _ {t), respectiv e ly.

U s ing Equation s {A.2.5), (A.2.6) , (A.2.7) a nd (A.2.8), the exp r e ss ion for the plunge acceleration ca n be written as , Note also that 0 {t) can be written a s , 0 {t) = A i x 4- B i u (A . 2 . 10) ; - where A i and B e ar e the rows of A and B a s sociat e d with the scalar eqt ja t i on for O (t} .

Ther e fore,

: , { }, + i

I }

By group in g t he t e rm s mu l tip ly i n g x and u, s im p l e exp ress ion s for t he ro ws of C and D a ss_ .,c ia ted with t he s c al a r n, output e q uation c an b e for me d.

i=l !

- whe re O n, a nd Dn, a re the row of C and D a ss oc ia t e d w i th th e s ca lar n z _ i eq ua tion , i T h is met h od of determ i n i ng t he proper coe ffi cient s for the C a nd D I ma tr i ces a s s oci a ted wit h n. c a n be implement ed nu m erically very easily by depend s on the state vector fo r the system. For the st a te defin i t i on in u s i ng a tran s formatio n row vecto r, X . The defin i t i o n of the tr a nsfor m at i o n 1 E q u a tion { A.2. 1), t h e t ra n s format i o n v ec tor h as t h e following fo l m, t

i

I U o U o Ix 6 X 4 [0,---, -- 0,- _ " ] (A.2.!4) - g g' _ ' • Thus, C ,, , = X.A { A.2. 15 ) ¢ and, , " D.. = X'B. (A.2.i6) I R e pm t No. 2 . Gov _ nment Acceuio _ NO. I Reciple _ t ' s Ca u Jo0 No.

NASACR-177943 "4 T,t le _ Svl R , t_" " S. ne_ on Oou Anal y si s of Flex i ble A i rcraf t Long i t u d i nal Dyn amics a nd June 1985 Hand1 i ng Qua1 iti es - Volume I - Anal y sis M ethods L e ., _ . ,- ._ O, l . _ ,. ,_ .,a.a .

7 Av t hc w { s | O. Pe t fmm _ nl C _ roe n_la t _ on Peport NO .

Hartin R . W aszakand Dav i d K. S chm i d t 10. Work L k _ t NO .

9 Perfo, min9Organizatio n Nam e a nu Add re ss Pur due Univ e r s ity ' 11. Comlr cw c t or Grant NO .

Sch oolo f A er on a u tic sand A str o na u ti cs NAG-I-254 _.

We s t Lafa y ette,IN 4 7 90 7 13. Type of R e po _ a e 4l Period Cov e red ii 1 2. S p o _ o _mn 9 Ag e ncy N a _ a n d Address C o ntract o r Re po rt Nati o na l Aer o nautic s an d Space Administrati o n Wa s hingto n ,D.C. 2 0 5 46 1 4.s, , o, , o, _ ; A s, nc y Cod, q 505-34 -0 3 -0 3 IS. Supplementary Note s NAS A Tech n ica l M o nit o rs : Wi ll iamD . Grantha m an d Jarre ll R . El l i o tt LangleyResearchCenter 16 Abs t rac t A s aircraftbec om e larger and l ighterdue t o design requirement s f o r increa s ed pay l oadand improvedfuel efficiency,the y wil l als o becomemore flexible. F o r highly flexiblevehicles,the handlingqualitiesmay not be accuratelypredicted by c o nventional methods. This study appliestw o analysismeth o dst o a family o f flexibleaircraftin o rder to investigate how and when s tructura l (especially dynamic aeroelastic) effectsaffect the dynamiccharacteristic s o f aircraf t .

T h e first type of ana l ysis is an open- loo pm o da l anal y sistechnique. Thi s me *ho _, considersthe effect o f modal re s iduemagnitu d es o n determiningvehic l e han dl ingqualitie s . The second meth od i s a pi lo t-in-the- loo p analysi s pr o cedurethat consider s severa l cl os e d -l oo psy s tem characteri st ic s .B ot h analyse s indicatedthat dynamicaer o e l as t iceffect s cau s e d a de g radati o ni n vehicle trackingperf o rmance,based o n the evaluati o n o f so me s imulationre s ult s .

17. Key Woq ds |Sugg e st e dby Author{ , )) 18 . Oist rit)u t io m,. Statamenl ' _ F lex i bleA ircraft ) Fl ight Dynamics Uncla ss ified - Unlimited Han dl ing Qualities M odal Anal y s i s S u bjectC a t e g o ry 08 .,.,, .

IJ Secw ht y Cl a u l f. Iof t h e | r e p m t l _ lO . Secv e i t yCl a u i f . Iof this liiOlp) 2 1. No. o1 Pages _|. INk:e Unc l as si fl ed Un c l a s s ified 10 8 A06 ,,. J o _ F o e sale by t h e Naho n a l Tec hm c a l Inf oe mahon S e rwc e, Sprml _ held . Vir|i n la 22111