Modeling and parameter uncertainties for aircraft flight control system design

The important f i e l d ofunsteady aerodynamics as related to structural res- ponse and f l u t t e r and control-configured vehicle design were not a part of this study b u t shouldnot be neglected i n future studies.

Section I 1 1 gives a l l o f the actualdata, while Section IV notes some possible applicationsof the data. Aerodynamic equations of motion and the definitions of the dimensional stability derivatives are given i n the Appendix.

I ' . .

SECTION 2 DISCUSSION OF DATA AND METHODOLOGY The d a t a i n c l u d e d i n t h i s r e p o r t a r e . from many sources and d i s c i p l i n e s and covertime spans r a n g i n g up t of o u ry e a r s .T h i ss e c t i o n w i l l discusssources o f d a t a and presentationformats, and i s d i v i d e d i n t o t h r e e p a r t s where d e t a i l e dd i s c u s s i o n so fa e r o d y n a m i cp a r a m e t e rv a r i a t i o n s , and s t r u c t u r a l modal analysisarepresented. Examples of d a t aa r eg i v e n i n t h i s s e c t i o n t o . .

s u p p o r tt h ed i s c u s s i o n ,b u tt h ec o m p l e t ed a t as e t i s g i v e n i n S e c t i o n 111.

AERODYNAMIC PARAMETERS The aerodynamic l o n g i t u d i n a l and l a t e r a l - d i r e c t i o n a ld i m e n s i o n l e s sd e r i v a - t i v e s g i v e n i n T a b l e 1 and 2 have been r e c o n s t r u c t e d as a f u n c t i o n o f t i m e , where time i n general goes back t o i n i t i a l c o n f i g u r a t i o n development f o r t h e p a r t i c u l a ra i r f r a m eb e i n gc o n s i d e r e d . The r e l a t i o n s h i po ft h e s ed e r i v a t i v e s t o a i r c r a f t dynamics i s shown i n t h e Appendix,where t h e a i r c r a f t e q u a t i o n s o f m o t i o n and t h e i r c o e f f i c i e n t s a r e d e f i n e d i n terms o f t h e d i m e n s i o n l e s s d e r i va ti yes.

D e t a i l e d Example o f S t a b i l i t y D e r i v a t i v e C a l c u l a t i o n An exampleof t h e t y p e o f d e t a i l e d c a l c u l a t i o n used f o r t h e d e t e r m i n a t i o n o f a s i n g l es t a b i l i t yd e r i v a t i v ei sp r e s e n t e dh e r e . The selectedparameter i s thefundamentalderivative, (CL ) k t h ea i r p l a n el i f t - c u r v es l o p e ,w h i c hc a n a A be c a l c u l a t e df r o mt h ef o l l o w i n ge x p r e s s i o n : Equation ( 1 ) where E cL (CL,)A = f o r t h e e n t i r e a i r p l a n e a n d i n c l u d e s s t a t i c a e r o e l a s t i c e f f e c t s .

TABLE 1 LONGITUDINAL AERODYNAMIC PARAMETER UNCERTAINTIES 1. Drag coef f i c i e n t ( CD) 2. Rate o f change o f d r a g c o e f f i c i e n t w i t h f o r w a r d speed (CD ) . u 3. Rate o f change o f d r a g c o e f f i c i e n t w i t h a n g l e o f a t t a c k (CD 1 a 4. Rate o f change o f d r a g c o e f f i c i e n t w i t h e l e v a t o r d e f l e c t i o n ( C D ) 6e 5. L i f t c o e f f i c i e n t (CL) 6. Rate o f change o f lift c o e f f i c i e n t w i t h .- forward speed (CL,,) 7. Rate o f change o f lift c o e f f i c i e n t w i t h a n g l e o f a t t a c k (CL ) a 8. Rate o f change o f lift c o e f f i c i e n t w i t h a n g l e o f a t t a c k r a t e (C,.)

a Rate o f change o f lift c o e f f i c i e n t w i t h p i t c h r a t e ( C L ) 9.

10. Rate o f change o f p i t c h i n g moment c o e f f i c i e n t w i t h f o r w a r d speed (C,,,,,) 11. Rate o f change o f p i t c h i n g moment c o e f f i c i e n t w i t h a n g l e o f a t t a c k Rate o f change o f p i t c h i n g moment c o e f f i c i e n t w i t h a n g l e o f a t t a c k 12.

r a t e (Cm&) 13. Rate of change o f p i t c h i n g moment c o e f f i c i e n t w i t h p i t c h r a t e ( C

. m q )

14. Rate o f change o f lift c o e f f i c i e n tw i t he l e v a t o rd e f l e c t i o n (CL6e) 15. Rate o f change o f p i t c h i n g moment c o e f f i c i e n t w i t h e l e v a t o r d e f l e c - t i o n (Cm ) e 16. Rate o f change o f d r a g c o e f f i c i e n t w i t h p i t c h r a t e ( C D ) q 17.Rate o f change o f d r a g c o e f f i c i e n t w i t h a n g l e o f a t t a c k r a t e (CD.)

a TABLE 2 LATERAL-DIRECTIONAL AERODYNAMIC PARAMETER UNCERTAINTIES 1. Rate o f change o f yawing moment c o e f f i c i e n tw i t hs i d e s l i pa n g l e (Cn ) B 2. Rate o f change o f yawing moment c o e f f i c i e n t w i t h r u d d e r d e f l e c t i o n (Cns,) 3. Rate o f change o fy a w i n g moment c o e f f i c i e n t w i t h a i l e r o n d e f l e c t i o n "a 4. Rate o f change o f yawing moment c o e f f i c i e n t w i t h s p o i l e r d e f l e c t i o n SP 5. Rate o f change o f yawing moment c o e f f i c i e n tw i t h yaw r a t e (C ) n r 6. Rate o f change o f yawing moment c o e f f i c i e n t w i t h r o l l r a t e (Cn ) P 7. Rate o f change o fs i d ef o r c ec o e f f i c i e n tw i t hs i d e s l i pa n g l e (Cy ) 8. Rate o f change o f s i d ef o r c ec o e f f i c i e n tw i t hr u d d e rd e f l e c t i o n( c ) y 6 r ) 9. Rate o f change o fs i d ef o r c ec o e f f i c i e n tw i t ha i l e r o nd e f l e c t i o n ( C Y s 10. Rate of change o fs i d ef o r c ec o e f f i c i e n tw i t hs p o i l e rd e f l e c t i o n ( c a ) Y 6 11. Rate o f change o f s i d e f o r c e c o e f f i c i e n t w i t h yaw r a t e (Cy,) S P 12. Rate o f change o fs i d ef o r c ec o e f f i c i e n tw i t hr o l l r a t e (Cy ) P 13. Rate o f change o f r o l l i n g moment c o e f f i c i e n tw i t hs i d e s l i pa n g l e (C 14. R a t e o f change o f r o l l i n g moment c o e f f i c i e n tw i t hr u d d e rd e f l e c t i o n ( CQj r ) 15. Rate of change o f r o l l i n g moment c o e f f i c i e n t w i t h a i l e r o n d e f l e c t i o n (%sa) 16.Rate o f change o f r o l l i n g moment c o e f f i c i e n tw i t hs p o i 1e r d e f 1 e c t i on (%ssp) 17. Rate o f change o f r o l l i n g moment c o e f f i c i e n t w i t h yaw 18. Rate o f change o r r o l l i n g moment c o e f f i c i e n t w i t h r o l l E l a s t i c = t h e r a t i o o f CL w i t h aero- a Tai 1 - o f f e l a s t i ce f f e c t st o CL o ft h er i g i da i r p l a n e ,i nt h e absence o f a t h e t a i l .

" L f o r t h e r i g i d a i r p l a n e i n t h e absence o f t h e t a i l .

(CL a I & = 7j7

FH = f a c t o r t h a t a c c o u n t s f o r e f f e c t o f a f t fuselagebendingonhorizon- t a l t a i l lift.

Tai 1 e l a s t i ce f f e c t st o C, o fr i g i ds t r u c t u r e ,o ft h ei s o l a t e dh o r i z o n - L a t a l t a i l .

1 t a i l i f t c u r v e s l o p e o f t h e r i g i d a i r p l a n e

(cLaH) = - a 'L h o r i z o n t a

a aH

f o r t h e r i g i d a i r p l a n e

(E)R = downwash g r a d i e n t

i n c r e m e n t t o a c c o u n t f o r a e r o e l a s t i c e f f e c t s o n downwash W E = g r a d i e n t The method o f d e t e r m i n a t i o n o f t h e v a r i o u s components i n t h e e q u a t i o n depends o nt h ep o i n ti nt h ed e s i g nc y c l e a t w h i c ht h ec a l c u l a t i o n i s made. F o rt h i s discussion,fourstages i nt h ed e s i g np r o c e s s w i 11 beconsidered: 1 . Early preliminary desigr.

2. Laterpreliminarydesign, b u t before wind tunneldata areavailable 3. After wind-tunnel d a t a areavailable 4. Afterflight-testdataareavailable I t s h o u l d be noted t h a t wind-tunnel testsareusually conducted priortothe official program s t a r t b u t for the f l i g h t conditionselectedinthepresent studythesetests must be of the high-speed variety i n order to correctly represent compressibility effects.

In Stage 1 of thedesign effortthe components of theequation which apply to the r i g i d airplane a r d f o u n d by relatively simple methods. For example, the R

derivatives ( C L )TO and (CL. I R can be determined through the use of the

1H USAF DATCOM (Reference l ) , from which Figure 1 i s reproduced. The aero- elasticcorrectionterms, ( C L ~ ) ; ~ ~ , FH, (CL,):’~, and (dk/aa)E, may be estimated based on values from previous similar productiondesigns.

During Stage 2 i n thedesign more sophisticated methods are employed. Typical o f these is the Weissinger liftingsurfacetheory, Reference 2 , which lends itselftotheestimation of b o t h r i g i d and elastic characteristics.

1.61 1 I I 1 1 1 . 1 I 1.4 \ - = cLa 2n ! - 1.2 1 .ow C 0.8 La A 0.6 (PER RAD?

0.4 i I i i i i I i I THE DETAIL EXPLANATION OF THE METHOD AND 0.2 NOTATIONS ARE GIVEN IN REFERENCE 1.

1 I I I I 1

0 1 I I I I I I I I I I I I I I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 5 1 6

A [(I - ~ 2 ) + TAN^ h,,211/2

FIGURE 1. SUBSONICWINGLIFT-CURVE SLOPE Highspeed d i g i t a l computers a r e a v i r t u a l n e c e s s i t y f o r t h e a p p l i c a t i o n o f t h i s and s i m i l a r complex methods. Other a e r o e l a s t i cc a l c u l a t i o n sf o rw i n g and t a i 1 s u r f a c e s a r e made by means of the Hedman v o r t e x l a t t i c e 1 i fti ng s u r f a c e t h e o r y f o r e l a s t i c w i n g s , p r e s e n t e d i n Reference 3.

The t h i r d s t a g e o f t h e d e s i g n p r o c e s s b e n e f i t s f r o m t h e a v a i l a b i l i t y o f wind- tunneldata. As p r e v i o u s l yn o t e d ,t h et e s t sm u s tb ea p p r o p r i a t ef o rt h e f l i g h t c o n d i t i o n o f i n t e r e s t ; i . e . high-speed wind t u n n e ld a t af o rc r u i s e f l i g h t c o n d i t i o n s f o r w h i c h c o m p r e s s i b i l i t y e f f e c t s a r e i m p o r t a n t , and h i g h Reynolds number t e s t sf o rh i g ha n g l e - o f - a t t a c kf l i g h tc o n d i t i o n s .A s i d e f r o mt h eu s u a lt e s t i n gc o r r e c t i o n st h a tm u s t be a p p l i e dt ow i n dt u n n e ld a t a , otheradjustmentsmustbe made t o account f o r t h e f a c t t h a t g e o m e t r i c d i s - s i m i l a r i t i e s o f t e n e x i s t between thewindtunnel model and t h e c o n f i g u r a t i o n being analyzed. For the example derivative, (CL,)~, w i n dt u n n e lt e s t sp r o - v i d et h ei n f o r m a t i o nf o rw h i c ht h e components (CL,)$~, (CL )R and ( a ~ / a a ) ~ arederived. Ground t e s t st oa s c e r t a i nt h ea i r p l a n es t i f f n e s sc h a r a c t e r i s t i c s a r ec o n d u c t e dd u r i n gt h i sp e r i o dt ov e r i f yt h ee s t i m a t e dv a l u e s . If any s i g n i f i c a n t d i s c r e p a n c i e s e x i s t , t h e a e r o e l a s t i c c o r r e c t i o n s a r e r e e v a l u a t e d u s i n g t h e methods previouslydescribed.

The f i n a l s t a g e o f t h e p r o c e s s f o l l o w s t h e f i r s t f l i g h t and i n v o l v e s v e r i f i c a - t i o no ft h ee s t i m a t e dv a l u e so ft h ed e r i v a t i v e s . Measurements made i n f l i g h t t e s t a r e o b v i o u s l y f o r t h e e l a s t i c a i r c r a f t , so theprocedure i s now reversed.

I n t h e caseofthepresent example, t h ec o m p l e t ee l a s t i cd e r i v a t i v e i s measured, t h e e l a s t i c c o r r e c t i o n s a r e assumed a c c u r a t e , a n d t h e e l a s t i c t a i l e f f e c t i v e n e s s [FH(CL ) E / R ( C L ~ , ) ~ ]can be measured, t h u s p e r m i t t i n g t h e r i g i d Q H a i r p l a n ec h a r a c t e r i s t i c st o be c a l c u l a t e d . Any necessaryadjustmentsare made t o t h e s e r i g i d a i r p l a n e d e r i v a t i v e s .

A t t h i s p o i n t some comments r e g a r d i n g s t a b i l i t y d e r i v a t i v e s and f l y i n g q u a l i t i e s may be a p p r o p r i a t e . P r i o r t o f i r s t f l i g h t i n t h e development o f an a i r p l a n e , s t a b i l i t y d e r i v a t i v e s a r e c a r e f u l l y e s t i m a t e d asdiscussed i n the preceding paragraphs. To t h es t a b i l i t y and c o n t r o le n g i n e e rt h e s t a b i l i t y d e r i v a t i v e s s e r v e l a r g e l y a s t h e b u i l d i n g b l o c k s w i t h w h i c h f l y i n g q u a l i t i e sa r ec a l c u l a t e d . The f l y i n gq u a l i t i e sa r et h e n checked f o r com- p l i ance a g a i n s t numerous c r i t e r i a . 3 f c o u r s e , t h e s t a b i 1 i ty d e r i v a t i v e s a r e a l s oi m p o r t a n tt oo t h e re n g i n e e r i n gs p e c i a l i s t s ,i np a r t i c u l a r ,c o n t r o ls y s - tem designersand,others who u t i l i z e a i r c r a f t e q u a t i o n s o f m o t i o n i n t h e i r work .

Once t h e f l i g h t t e s t program i s underway, many o f t h e a i r p l a n e f l y i n g q u a l i t i e s can be d e t e r m i n e d d i r e c t l y w i t h o u t t h e n e c e s s i t y o f k n o w i n g t h e v a l u e so fs p e c i f i cd e r i v a t i v e s . The s i t u a t i o n i s somewhat d i f f e r e n tt h a n b e f o r e f l i g h t t e s t ; it i s g e n e r a l l y t h e f l y i n g q u a l i t i e s i n f o r m a t i o n t h a t i s a v a i l a b l e and t h e e s t i m a t e d s t ‘ a b i l i t y d e r i v a t i v e s a r e c h e c k e d u s i n g t h e t e s td a t a .F l i g h tt e s tr e s u l t s , however, a r eo b t a i n e df o rd i s c r e t ef l i g h t c o n d i t i o n s and f l y i n g q u a l i t i e s a n a l y s e s o v e r t h e r e m a i n d e r o f t h e f l i g h t envelope require know1edge o f t h e s t a b i 1 i ty d e r i v a t i v e s . Therefore , t h e s t a b i l i t y d e r i v a t i v e s a r e e s t a b l i s h e d a t tk,e f l i g h t t e s t p o i n t s and i n t e r - p o l a t i o n so rf a i r i n g sa r e made t o developcontinuousvalues. I n t h i s manner, a complete bank o f aerodynamicdata i s compiled,fromwhichtheairplane f l y i n g q u a l i t i e s a r e g e n e r a t e d f o r t h e e n t i r e f l i g h t e n v e l o p e .

R e f e r r i n ga g a i nt oE q u a t i o n( 1 ) i t i s a p p a r e n tt h a ts e v e r a lo ft h et e r m sa r e n o t u n i q u e t o t h e e x p r e s s i o n f o r ( CL,)~; t h e y a r e i n v o l v e d i n t h e d e t e r m i na- t i o n o f many o f t h e o t h e r a e r o d y n a m i cd e r i v a t i v e s .T h e r e f o r e ,e r r o r so r uncertaintiesassociatedwiththesetermswouldcause a degree o f c o r r e l a t i o n t o e x i s t between many o ft h ed e r i v a t i v e s .F o r example, t h eh o r i z o n t a lt a i l l i f t - c u r v es l o p e C L ~ H , t h eh o r i z o n t a l t a i l e l a s t i c - t o - r i g i dc o r r e c t i o nf a c t o r (CL,)~’~,and t h e a f t f u s e l a g e b e n d i n g c o r r e c t i o n f a c t o r F H , areelements i n t h ee x p r e s s i o n sf o r numerous l o n g i t u d i n a ls t a b i l i t yd e r i v a t i v e s .I nt h ed a t a p r e s e n t e d ,t h ep r o l i f e r a t i o no f an e r r o ri nt h e CL e s t i m a t e i s r a t h e rd r a - aH m a t i c . The error(causedby an underestimateof Mach number e f f e c t s on C L , , , ) i s r e f l e c t e d i n t h e e s t i m a t e s o f sevenseparate s t a b i l i t y d e r i i a t i v e s : C L , , C L ; , CmG, CLq, Cmq, CLGe, and Cm6e. The p o t e n t i a lf o rt h i st y p eo fe r r o r c o r r e l a t i o n e x i s t s w i t h many o f t h e o t h e r p a r a m e t e r s , i n c l u d i n g t h e l a t e r a l - d i r e c t i o n a lv a r i e t y .

”- Sources o f U n c e r t a i n t i e s Thereare a number o f s o u r c e s f o r e r r o r o r u n c e r t a i n t y a s s o c i a t e d w i t h t h e estimatedvaluesofthevariousaerodynamicparameters. I nt h ev e r ye a r l y 1 1 stages o f d e s i g n t h e r e i s c o n s i d e r a b l e l i k e l i h o o d t h a t t h e a i r p l a n e c o n f i g u - r a t i o n w i l l be a l t e r e d b e f o r e t h e f i r s t f l i g h t o c c u r s . I n some casesthese m o d i f i c a t i o n s t a k s p l a c e w e l l i n t o t h e p e r i o d betweenprogram commencement and f i r s t f l i g h t . Dependingonthetypeofchange made t ot h ec o n f i g u r a t i o n , t h e r e can be a considerableimpactontheestimatedvalueoftheaerodynamic parameters.

A n o t h e r u n c e r t a i n t y a r i s e s f r o m i n a c c u r a c i e s t h a t a r e i n h e r e n t i n a n a l y t i c a l methods used t oe s t i m a t et h ea e r o d y n a m i cc h a r a c t e r i s t i c s ,i n c l u d i n ga e r o - e l a s t i cc o r r e c t i o n s . The degree o ft h eu n c e r t a i n t y , when r e l a t e d as a percentage o f t h e a c t u a l o f f i n a l v a l u e , i s m a g n i f i e db yt h e method o f c a l c u - l a t i n g m o s td e r i v a t i v e s as t h e sum o f two o r more increments. When two incrementsare o f o p p o s i t e s i g n and t h em a g n i t u d e sa r es i m i l a r ,s m a l le r r o r s i n e i t h e r i n c r e m e n t r e s u l t s i n a l a r g ee r r o ri nt h et o t a lv a l u e .F o r example, t a i l incrementsareoften added t o a i r p l a n e t a i l - o f f i n c r e m e n t s t o o b t a i n t o t a l a i r p l a n e d e r i v a t i v e s , anddependingon t h e p a r t i c u l a r d e r i v a t i v e , t h e two increments may be o f o p p o s i t e s i g n .

If t h e f l i g h t c o n d i t i o n s o f i n t e r e s t s h o u l d change f o r anyreasonduringthe designprocess (e.g., m o d i f i c a t i o n o f f l i g h t envelope),thecoefficientsof t h ee q u a t i o n so fm o t i o n( d i m e n s i o n a ls t a b i l i t yd e r i v a t i v e s ) w i l l change because, i n g e n e r a l ,t h e ya r ef u n c t i o n so fs u c hf l i g h tp a r a m e t e r s as Mach number, dynamic p r e s s u r e ,a i r s p e e d ,a l t i t u d e ,a n da n g l eo fa t t a c k .

Then, o f course,therearetheubiquitouscomputationalerrorsthatoccasion- a l l y go u n d e t e c t e df o rs i g n i f i c a n tp e r i o d so ft i m e .O b v i o u s l y , i t i s n o t p o s s i b l e t o p r e d i c t t h e o c c u r r e n c e o f u n c e r t a i n t i e s s t e m n i n g f r o m s u c h causes.

D i s c u s s i o n o f D a t a The f l i g h t c o n d i t i o n f o r w h i c h t h e s u b j e c t d a t a werecompiled was d e l i b e r a t e l y chosen t o be one which i s p a r t i c u l a r l y prone t o aerodynamicmodelingerrors.

T h i s p r o c l i v i t y f o r e r r o r a r i s e s f r o m t h e t r a n s o n i c f l o w c o n d i t i o n s t h a t e x i s t a t high-speedcruise and t o t h e r e l a t i v e l y h i g h dynamicpressurewhich has a m a j o re f f e c t on t h ea e r o e l a s t i cc o r r e c t i o n s . Any small misjudgements i n t h e e f f e c t s o f c o m p r e s s i b i l i t y canbeexaggeratedbecause o f t h e r a p i d v a r i a t i o n s w i t h Mach number t h a t o c c u r w i t h many aerodynamic c h a r a c t e r i s t i c s I' i n the transonic speed range. A typicalvariation w i t h Mach number of a primary stability derivative, C L , , the airplane lift-curve slope, is shown i n Figure 2. The symbol denotes the approximate cruiseflightcondition of this study .

I t should be observed.that i t is d i f f i c u l t to generalize on the aerodynamic parameter uncerfainties. A derivative w h i c h is estimated w i t h greataccuracy on one airplane may not fare as we1 1 on the next. For example, uncertainties that are attributed to configuration changes certainly cannot be generalized.

However, some generalizatio'ns associated w i t h estimationaccuraciescan be offered.Table 3 a provides an indicationof the r e l a t i v e impact and the estimationaccuracynormallyexpected f o r eachof the aerodynamic parameters.

I t should be emphasized t h a t the categorizing of the parameters is approximate and w i l l t e n d t o vary w i t h particular aircraft configurations and w i t h f l i g h t conditions. I t i s emphasized t h a t the estimationaccuracy shown includes only the estimating methods normally used and does notconsideruncertainties due toothercauses.InTable 3 b the probable risk i n flyingqualities analyses and control system design i s shown f o r eachof the nine categories ofTable 3 a. For example, a derivative w i t h secondary impac.t t h a t is esti- mated w i t h only f a i r accuracy is no more o r less 1 i kely to create problems than a primary derivative that normally is estimated w i t h good accuracy.

MACH NUMBER FIGURE 2. TYPICAL VARIATION OF LIFT-CURVE SLOPE WITH MACH NUMBER 1 3 TABLE 3 PARAMETER ESTIMATION ACCURACY AND POTENTIAL ERROR IN FLYING QUALITIES ANALYSES

I I. IMPACT ON FLYING QUALITIES

7 Secondary

Negligible cD" (b) Potential Error in Flying Qualities Analyses: Large Moderate Small Minimal

I Historicalvalues of each of the aerodynamic parameters are presented i n Section 3 . Most of thehistoricalvaluesare normalized by thefinalvalue i n some cases i n which thefinallevel o f the .

of eachparameter. However., parameter i s very small,the normalized value would be grossly distorted.

In thesecasestheincrementaldifference between each value and thefinal .

value i s deemed more meaningful and is thereforepresented. In thecase of

ha, thevalue depends on theselected moment reference cen- '

thederivative t e r , which i s anotherreason t o present an incrementaluncertainty. On each of the historical plots the abscissa is marked w i t h the time of f i r s t f l i g h t ( F F ) and the FAA certification date ( C E R T ) , i n additiontotheauthorityto proceed (ATP, or program i n i t i a t i o n ) a t the o r i g i n . Also noted on the,plots is the timeof the f i r s t complete dynamics analysis ( D ) of theairplane.

This date i s s i g n i f i c a n t because i t representsthe f i r s t firmrequirement for some of theparameters. Included w i t h each graphicalpresentationare brief comments regardingthe usual impact of theparticular parameter on the augmented airplane dynamics. A 1 so included are comnents re1 a t i ng t o the f i n a l value uncertainty. A typicalpresentation i s repeated here for conven- 3 .

ienceasFigure In viewing thedata,certain o f theparametersreveala relativelylarge degree o f uncertaintyearly i n thedesign program. I n most of thecases, t h i s i s a t t r i b u t a b l e t o configuration changesbeing made asthedesign i s

FIGURE 3. TYPICAL DATA FORMAT; C,,/C,, k,rJAL

f i n a l i z e d .H i g hl e v e l so fu n c e r t a i n t yn e a rt h eo u t s e to ft h ep r o g r a mi n some casescanbetraced t o t h e l a c k o f e a r l y a e r o e l a s t i c c o r r e c t i o n s ; t h a t i s , t h e a v a i l a b l e e s t i m a t e s o f t h e p a r a m e t e r a r e f o r t h e r i g i d a i r p l a n e . Had t h e r e been a firm requirementfortheseparameters a t t h i s t i m e some approx- i m a t ee l a s t i c i t yc o r r e c t i o n sw o u l d have been a p p l i e d .S i m i l a r l y , compres- s i b i l i t y c o r r e c t i o n were n o t a p p l i e d d u r i n g t h e i n i t i a l d e s i g n s t a g e s t o some o f t h e l e s s s i g n i f i c a n t parameters.

It w i l l be n o t e dt h a t , i n g e n e r a l ,t h ed a t ap o i n t s do n o tt e n dt oc o i n c i d e c h r o n o l o g i c a l l yf o rt h ev a r i o u sp a r a m e t e r s . The p o i n ti nt i m ea tw h i c h a d e r i v a t i v e i s e s t i m a t e d i s a f f e c t e d b y s e v e r a l f a c t o r s , among whicharethe importance o f t h e d e r i v a t i v e and t h e d i f f i c u l t y o f e s t i m a t i o n . When a con- f i g u r a t i o n change occursduringthedesignthe more i m p o r t a n t d e r i v a t i v e s a r e r e c a l c u l a t e df i r s t .I nf a c t ,c o n f i g u r a t i o n changes a r en o t made u n t i lt h e impactoncertainparameters and a i r c r a f t f l y i n g q u a l i t i e s i s a s c e r t a i n e d .

O f theaerodynamicparameterspresented i n Section 3 a l l except two a r e p a r t i a ld e r i v a t i v e s . These two a r et h ea i r p l a n e lift and d r a gc o e f f i c i e n t s whicharegivenbecausetheyappear i n t h e c o e f f i c i e n t s o f t h e e q u a t i o n s o f motion, sometimes r e f e r r e d t o as t h ed i m e n s i o n a ls t a b i l i t yd e r i v a t i v e s( s e e Appendix). The a i r p l a n e lift c o e f f i c i e n t i s anindependentvariable i n s p e c i f y i n g a p a r t i c u l a r f l i g h t c o n d i t i o n so no u n c e r t a i n t y can be assigned.

As d i s c u s s e de a r l i e r , however, i f t h e f l i g h t c o n d i t i o n o f i n t e r e s t s h o u l d change f o r any r e a s o n ,n o to n l yt h e lift c o e f f i c i e n t , b u t t h e e n t i r e s e t o f e q u a t i o n s o f m o t i o n w i l l be a1 t e r e d .

The a i r c r a f t d r a g c o e f f i c i e n t i s somewhat unique i n t h i s d i s c u s s i o n because i t i s t h eb a s i ca e r o d y n a m i cd e s i g np a r a m e t e rf o rc r u i s ef l i g h t . The i n i t i a l value shown i n S e c t i o n 3 represents a guarantee and normally i s n o t r e v i s e d u n t i l f l i g h t t e s t d a t a a r e a v a i l a b l e . E v e r y e f f o r t i s made t oa c h i e v et h i s d e s i g ng o a l ,i n c l u d i n gc o n f i g u r a t i o n changes which can rangefromminor refinements i nt h ed e s i g nt om a j o rd r a gr e d u c t i o np r o g r a m s .

The u n c e r t a i n t i e sa s s o c i a t e dw i t h t h ef i n a lv a l u e so ft h ev a r i o u sa e r o d y n a m i c parametersare d i f f i c u l t t o assess when t r a d i t i o n a l methods a r e used i n t h e a n a l y s i s o f f l i g h t t e s t d a t a , as i n thesubjectcase. The measurement accuracy o f c e r t a i n o f t h e p a r a m e t e r sr e c e i v e sg r e a ta t t e n t i o n because o f t h e r e l a t i o n s h i pt op e r f o r m a n c eg u a r a n t e e s . An example i s presented i n F i g u r e 4 FIGURE 4. TYPICAL FLIGHT TEST DATA; DRAG COEFFICIENT where d r a g c o e f f i c i e n t , CD, has been p l o t t e dv e r s u s Mach number a t v a r i o u s l e v e l s o f lift c o e f f i c i e n t , CL. The s c a t t e ri nt h ed a t a can be regardedas an i n d i c a t o r o f t h e u n c e r t a i n t y i n t h e f i n a l v a l u e ; f o r t h e s u b j e c t case, t h e s t a n d a r dd e v i a t i o no f CD i s 2 percent. The t o t a le r r o ra r i s e sf r o ms e v e r a l f a c t o r s ,t h em a j o r one b e i n gt h ee n g i n et h r u s t measurement e r r o r .

Thereareotherparameterswhicharealsocarefullycheckedagainst f l i g h t t e s t d a t a b u t a r e n o t n o r m a l l y a n a l y z e d w i t h r e s p e c t t o measurement e r r o r .

An example i s presented i n F i g u r e 5 where t h e h o r i z o n t a l s t a b i l i z e r e f f e c - t i v e n e s s , CmiH, i s shown as a f u n c t i o no f Mach number. This parameter i s s i m p l yr e l a t e dt ot h ee l e v a t o rc o n t r o l power term, C , so t h a tt h e measure- ““6, ment accuracy o f t h e two d e r i v a t i v e s i s comparable. I n F i g u r e 5, o n l yd a t a p o i n t s n e a r t h e s u b j e c t f l i g h t c o n d i t i o n a r e shown, and t h ec a l c u l a t e ds t a n - d a r dd e v i a t i o no v e rt h i s Mach number range i s 4 percent. An i n s i g h t i n t o t h e p o t e n t i a l measurement e r r o r o f C m i H canbe gainedbyconsideringtheexpres- s i o n used t oc a l c u l a t et h ei n d i v i d u a lp o i n t s : A i H where theincrementalvaluesrepresentthedifferencesbetween two trimmed c e n t e r so fg r a v i t y .E r r o r sc a na r i s ef r o ms e v e r a ls o u r c e s :t h e measurement o f t h e s t a b i l i z e r a n g l e ; t h e measurement o f t h e c g l o c a t i o n ; t h e measure- lift ment o f a i r p l a n e w e i g h t anddynamicpressure t h a t d e t e r m i n e t h e c o e f f i c i e n t ; and t h e a b i l i t y o f t h e p i l o t t o p r e c i s e l y trim t h e a i r c r a f t .

The l e v e l o f measurement e r r o r f o r h i H i s c o n s i d e r e dt y p i c a lo f a number o f s t a t i c d e r i v a t i v e s .

I n t h e case o f t h e dynamic d e r i v a t i v e s , t h e v e r i f i c a t i o n i s n o r m a l l y accom- p l i s h e db y a comparisonofresponsecharacteristics;i.e.frequencies, damping r a t i o s ,t i m et oh a l fo rd o u b l ea m p l i t u d e ,e t c . If the match between theestimatedandthemeasuredcharacteristics i s reasonable,theestimated d e r i v a t i v e sa r e presumed t o be c o r r e c t ; if not,adjustmentsare made t ot h e d e r i v a t i v e s i n o r d e r t o a c h i e v e a s a t i s f a c t o r y m a t c h o f t h e a i r c r a f t dynamics.

Forthemostpart, i t i s e x t r e m e l y d i f f i c u l t , i f n o t i m p o s s i b l e , t o i s o l a t e t h e u n d e r t a i n t i e s o r e r r o r s a s s o c i a t e d w i t h t h e measurement o f i n d i v i d u a l dynami c d e r i va ti yes.

WEIGHT AN9 INERTIAL PARAMETERS A i r c r a f t w e i g h t and i n e r t i a s a r e k e y q u a n t i t i e s i r , t h e a i r c r a f t e q u a t i o n s o f m o t i o n andhave a h i s t o r y o f v a r i a b i l i t y d u r i n g t h e a i r c r a f t ' s c o n f i g u r a t i o n development.Althoughthe a i r c r a f t can be weighed q u i t e a c c u r a t e l y p r i o r t o f i r s t f l i g h t ( t o w i t h i n 0.2 p e r c e n t ) ,t h em a n u f a c t u r e r ' s empty w e i g h t (MEW) and grossweightcan change s i g n i f i c a n t l y a f t e r program i n i t i a t i o n .

M a n u f a c t u r e r ' s empty w e i g h t changes occurboth because o f changes i n t h e s p e c i f i c a t i o n( i n c r e a s ei nr a n g e ,s a y ) and changes i nm a t e r i a l( e n g i n ew e i g h t change). If thedesigner'sconcern i s c o n t r o l system dynamics, he w i l l be more i n t e r e s t e di ng r o s sw e i g h t st h a n M E W ' S . Botharegiven, however, so t h a t t h e u s e r o ft h e s ed a t a will have some a p p r e c i a t i o n f o r t h e k i n d s o f f a c - t o r s i n f 1uenci ng h i sc o n t r o ls y s t e md e s i g n .

The growth o f MEW o v e rt h ef o u r - y e a rp e r i o d i s shown i n F i g u r e s 6 and 7, be n o t e d t h a t t h e i n i t i a l v a l u e was 7.7 p e r c e n tl e s st h a nt h e where i t will f i n a lv a l u e . The major jumps i n MEW areaccountedfor i n Table 4. O f t h e 7.7 p e r c e n t d i f f e r e n c e ,s p e c i f i c a t i o ng r o w t ha c c o u n t e df o r 4.7 p e r c e n t and n o n - s p e c i f i c a t i o ng r o w t h 2.0 p e r c e n t .F i g u r e 8 g i v e st h e changes i n o t h e r d e s i g nw e i g h t sf o rt h e same periodoftime.NotefromFigure 6 t h a tt h e o r i g i n a l maximum t a k e o f fg r o s sw e i g h t (MTOGW) was 90 p e r c e n t o f t h e f i n a l Val ue.

C o i n c i d e n tw i t ht h ew e i g h ti n c r e a s e sa r e changes i n moments o f i n e r t i a .

These a r e summarized i n Table 5. It will be n o t e dt h a tt h ei n c r e a s ei ng r o s s weights can be u t i l i z e ds e v e r a l ways. The o p e r a t i o n can keep t h ef u e ll o a d c o n s t a n t and increasepayload, i n w h i c hc a s et h ep i t c h i n g moment o f i n e r t i a I y , shows a r a t h e rl a r g ei n c r e a s e( T a b l e 5, I t e m 2 ) . When the payload i s h e l d c o n s t a n t and f u e ll o a di si n c r e a s e d ,t h er o l l moment o f i n e r t i a , Ix, shows a l a r g ei n c r e a s e , as shown i n Table 5, I t e m 3.

STRUCTURAL PARAMETERS Modal V i b r a t i o nA n a l y s i s The approach f o r d e t e r m i n i n g t h e u n c s r t a i n t i e s a s s o c i a t e d w i t h s t r u c t u r a l dynamic c o n s i d e r a t i o n s a r i s i n g f r o m s t r u c t u r a l f l e x i b i l i t y was t o t r a c k h i s - t o r i c a l l y t h e v a r i a t i o n s i n t h e normal modes o f v i b r a t i o n asdescribedby 1 9 p3 X c W a

1968 I 1969 I 1970 I

FIGURE 6. MANUFACTURER'S EMPTY WEIGH 1 GROWTHHISTORY I 1 . o 0.9: 0.98 -I U I L I .

5 0.97

I I c a e l I- I

s 0.9f

> t

E

I" a w 5 0.9: I- o U U = Y a ZE 0.94 0.92 TABLE 4 MAJOR WEIGHT CHANGE SUMMARY (see F i g u r e 6 ) A. Revised estimate for wing bending material, i n c r e a s e d a l l o w a n c e f o r i n s u l a t i o n and i n t e r i o rp a n e l sp l u sm i s c e l l a n e o u s changes.

6. Increasedengineweight,revisedestimate f o r wing and l a n d i n g g e a r t o p r o v i d e f o r an i n c r e a s e i n t a k e o f f g r o s s w e i g h t p l u s m i s c e l 1aneous changes.

C. I n t e r i o r d e s i g n changes, 0.102m ( 4 - i n c h ) f u s e l a g e s t r e t c h , a d d i t i o n o f a f t e n g i n e m a i n t e n a n c ep l a t f o r mp l u sr e v i s e de s t i m a t e s forfuselage,wingand t a i l .

D. Customerrequested i n t e r i o r changes p l u s r e v i s e de s t i m a t e sf o rm i s c e l l a n e o u ss t r u c - t u r a l andsubsystem i terns.

E. Increasedengineweight,revisedestimate forwing and l a n d i n g g e a r t o p r o v i d e f o r an i n c r e a s e i n t a k e o f f g r o s s w e i g h t p l u s m i s - cel laneous changes.

F. I n c o r p o r a t e d more r e p r e s e n t a t i v e w e i g h t s f o r g a l l e y s andpassengerseats.

frequencies, modal displacements, and t h er e s u l t i n gg e n e r a l i z e d masses. The method forcomputingthe modes was begun by theusuallumping o ft h ef u s e l a g e , empennage, and wing i n t o a f i n i t e number o f mass a n ds p r i n ge l ements. As t h e a s p e c t r a t i o o f t h e s e l e c t e d a i r p l a n e was l a r g e , it was j u s t i f i a b l e t o r e p r e s e n tt h es t r u c t u r e as interconnectedslender beams. The r o o to r connectingsprings were computed from a f i n i t e elementrepresentationusing a redundantforceprocedure.

The timeperiodconsidered i n t h i s s t u d y covered 35 months s t a r t i n g w i t h t h e f o r m a la u t h o r i t yt op r o c e e d (ATP) a n de n d i n gw i t ht h eg r o u n dv i b r a t i o nt e s t (GVT) 1 month p r i o r t o f i r s t f l i g h t . Preceding t h i s 35-month p e r i o dt h e r e was approximately 6 months o fp r e l i m i n a r yd e s i g na c t i v i t y .T h i sp r e l i m i n a r y design phase was notcoveredbythisstudy because s t r u c t u r a lu n c e r t a i n t i e s ( I ) MAXIMUM TAKEOFF GROSS WEIGHT 1.5 MTOGW 1 .o MTOGWFINAL 0.5 (b) MAXIMUMLANDING WEIGHT 1.5 MLW

-

1 .o M L W ~ ~ ~ ~ ~ 0.5 (c) MAXIMUM ZERO FUEL WEIGHT . . .

. .

. . . ...

. . . .

-10 ATP 10 20 30 40 50 MONTHS FIGURE 8. DESIGNWEIGHTHISTORY TABLE 5 MOMENT OF INERTIA HISTORY I n e r t i a Value a t Program S t a r t Assumed Growth F i n a l I n e r t i a Value 1. Manufacturers Empty Weight (MEW) Growth o f 8.8 percent 0.95 ' X ' I X F i n a l 0.92 I Y / ' Y F i n a l 0.93 'Z'IZ F i n a l 2 . Maximum Takeoff Gross Weight (MTOGW) Growth o f 11 percent and Constant Fuel : ' X / ' X F i n a l 0.96 0.88 ' Y / ' Y F i n a l 0.92 'Z/'Z F i n a l 3. Maximum Takeoff Gross Weight (MTOGW) Growth o f 71 percent and Constant Payload : 0.85 ' X / ' X F i n a l 0.93 'Y'IY F i n a l 0.89 'Z/'Z F i n a l F i n a lv a l u e sa r et h o s ec a l c u l a t e da tf i r s tf l i g h t (31 months a f t e r program i n i t i a t i o n ) . The product o f i n e r t i a ( Ixz) does n o t change s i g n i f i c a n t l y .

would have been distorted by thelargeconfiguration changes resulting from changes i n design specifications t h a t were made d u r i n g this phase.

During theevolutionofthedesign from ATP to GVT eleven changes were made whichwere judgedof sufficient significance to warrant computation of new modes. During thattime,the modes were used primarily f o r f l u t t e r analyses w i t h application t o control system effects being made lessfrequently. For t h i s studythe modes wererecomputedand the results compared to the final or certificationresults. The finalresults were computed based upon corrections as made from GVT results.

In particular, the dynamic representation of theairplaneconsisted ofa masswise representation composed of 50 1 umped bays. Each bay was chpable o f six “ r i g i d body” degrees of freedom, threetranslations and threerotations.

The entire mass of theairplane was lumped i n t o these bays by calculatingthe mass properties of each bay about a selectedreferencestationinthe bay.

and stiffness symnetry was assumed allowing an analysis of halfthe Mass airplane.Figure 9 shows the mass bay distribution used i n the modal calculations.

Theoreticalcantilevered v i b r a t i o n modes were calculated using thestiffness

and mass d a t a of therespective bays f o r the major components - wing, hori-

zontalstabilizer,fuselage and verticalstabilizer. All modes were calculatedusingstructuralinfluencecoefficients (SICS) which relate con- trol p o i n t staticdeflections t o appliedforces. Complex j o i n t structures such aswing-fuselage,fin-fuselage, and enginepylons, used structural influencecoefficientsobtained from f i n i t e elementredundant force type analyses.Figure 10 i l l u s t r a t e s thestiffnessrepresentation used i n the vi bra ti on analyses.

Rigid body modes wereused inconjunctionwiththe above component modes to releasetheairplane and generatefree-free orthogonal modes intosymnetric and antisymmetric s e t s . In allcases,sufficient component modes were used i n thefree-free modal calculationstoassure modal convergenceof the air- plane modes of concern, from zeroto 10 hertz. All modal calculations were performed using a digital computerprogram.

2 5 FIGURE 9. INITIAL-TO-FINALMASSREPRESENTATION It s h o u l d b e n o t e d t h a t f o r t h i s s t u d y a l l t h e v i b r a t i o n a n a l y s e s wereper- f o r m e du s i n gt h ef i n a ls t r u c t u r a lr e p r e s e n t a t i o n .T h a ti s ,t h ee a r l y modes w e r e r e c o n s t r u c t e d u s i n g t h e r e s p e c t i v e s t i f f n e s s and mass data,but with t h e f i n a l 50 bay s t r u c t u r a lr e p r e s e n t a t i o n .I na c t u a lp r a c t i c e ,t h ee a r l i e r . ' vibration analyses were performed using a much c o a r s e r s t r u c t u r a l . r e p r e s e n - t a t i o n .T h e r e f o r e , ,t h eu n c e r t a i n t i e s shown f o rt h e. e a r l yv i b r a t i o na n a l y s e s f o r f r e q u e n c i e s g r e a t e r t h a n 3 o r 4 h e r t z may be g r e a t e r t h a n t h e s e s t u d i e s i n d i c a t e .

As n o t e d ,t h es t r u c t u r a l dynamic u n c e r t a i n t i e s were l i m i t e d t o t h o s e r e s u l t - i n gf r o m mass and s t i f f n e s sv a r i a t i o n so n l y .I nt h ep r a c t i c a lc o n s i d e r a t i o n o f s t r u c t u r a l dynamic e f f e c t s ,w h e t h e rf o rt h eu s u a la p p l i c a t i o nt of l u t t e r o r g u s t l o a d s o r f o r c o n t r o l - c o n t i n u e d v e h i c l e a p p l i c a t i o n , t h e p r e d i c t a b i l i t y oftheunsteadyaerodynamicterms i s as i m p o r t a n t a s t h e p r e d i c t a b i l i t y o f t h e mass and s t i f f n e s s dependent modes o fv i b r a t i o n .E s p e c i a l l yi m p o r t a n t aretheunsteadyaerodynamics o fc o n t r o ls u r f a c e sw h i c hm u s t beincluded i n t h e a n a l y s i s o f a l l a i r c r a f t w i t h h i g h l y r e s p o n s i v e a c t i v e c o n t r o l systems i n o r d e r t o a s s u r e t h a t t h e r e q u i r e d f l u t t e r speedand gustloadmarginsare met and t h a t f l u t t e r and excessivegust and o s c i l l a t o r y l o a d s a r e a v o i d e d underany l i k e l y c o n t r o l system f a i l u r e o r m a l f u n c t . i o n .

S t r u c t u r a l Dynamic ParametersInvestigated Duringthedesign phase, t h ea i r p l a n ed e s i g nd a t aa r ec o n t i n u a l l yc h a n g i n g due t o c o n f i g u r a t i o n changes, r e v i s i o n s based upon t e s td a t a ,o rr e f i n e m e n t s t oe x i s t i n gd a t a based upon d e t a i l e da n a l y s e s . Many o ft h e s e changes a r e minor and do n o t s i g n i f i c a n t l y a f f e c t t h e a i r p l a n e v i b r a t i o n c h a r a c t e r i s t i c s .

Therefore,onlythoseparameterswhichsignificantlyimpactedtheairplane v i b r a t i o nmodes were i nves t i g a t e d . These parameters were as f o l 1ows : 0 Wing s t i f f n e s s 0 Fuselage s t i f f n e s s 0 Wing e n g i n e p y l o n s t i f f n e s s 0 A f t e n g i n ep y l o ns t i f f n e s s 0 Wing engine weight 0 Wing e n g i n e c g l o c a t i o n 0 A f t engine weight ~~~ - The chronology and r e l a t i v e magnitude of these parameterchanges are shown i n Table 6. Symnetric and antisymmetric vibration modes were calculated f o r both empty and fullfuelcordi t i o n s . These two fuelconditionsresult i n the highest and lowest modal frequencies of the system and therefore are typical o f the modal uncertainties which can be expectedover the entirefuelrange.

, I .

For both fuelconditions the payload was 'heldconstant. These payload and fuelconditions match configurations used duringtheairplane ground vibra- tion test. Thus, a d i r e c t comparison of theoretical and experimental modal vibrationdata was easily made which facilitated corrections t o the theoreticaldata.

TABLE 6 DESIGN PARAMETER SUMMARY " METERCHANGES DESIGN PAR - WING TIME AFT AFT

-

PYLON WING FUSELAGE PYLON ENGINE STIFFNESS M O N STIFFNESS STIFFNESS STIFFNESS WEIGHT

"" -

- ~~ ~ ~~~ ~ 0 BASIC BASIC BASIC IC BI 1SI BI +30% BEND +13%TOR +I

I T (

0.81111 (32 IN.) AFT 5 I R S -2% CG SHIFT

I

v

+5% BEND -3 5% +lS%TORS 24% WEIGHT +: INCREASE -2% BEND +6% T O R S +1% BEND -2% BEND +1% T O R S -2% T O R S -24 BEND -2% TORS 16 0 RS +5% 20 +I T 6% WEIGHT INCREASE

- - - I L L " _ L

NOTE: THE ABOVE CHANGES IN EACH PARAMETER ARE CUMULATIVE S t r u c t u r a l Dynamic Data The r e s u l t s o f t h e modal v i b r a t i o n u n c e r t a i n t y s t u d y a r e p r e s e n t e d as p l o t s o f modal frequency,generalized mass and displacement versus time. The h i s t o r i c a l e v o l u t i o n o f t h e s e p a r a m e t e r s i s shown f o r a selectedsubset o f t h e a i r p l a n e v i b r a t i o n modes which are important i n b o t h c o n t r o l s y s t e m design,gustloads due t ot u r b u l e n c e , and f l u t t e r c a l c u l a t i o n , a l l w i t h o r w i t h o u tc o n t r o l - c o n f i g u r e dv e h i c l et e c h n o l o g y . The chronology o f changes t o t h ei n p u td a t a i s shown on each p l o t f o r easyreference. The mode names, such as f i r s t wingbending,are chosen so as t od e s c r i b et h ep r e d o m i n a n t m o t i o n o f each mode a l t h o u g h ,o fc o u r s e ,t h em o t i o ni n v o l v e st h ee n t i r ea i r - p l a n es t r u c t u r e . Each parameter i s givenas a r a t i o o f t h e v a l u e a t a given t i m et ot h ef i n a lv a l u e . The f i n a lv a l u e o f each parameter i s takenfrom a t h e o r e t i c a l s e t o f modes whichare i n agreement w i t h g r o u n d v i b r a t i o n t e s t r e s u l t s .

The g e n e r a l i z e d mass and d e f l e c t i o n p l o t s f o r each mode were c a l c u l a t e d u s i n g a c o n s i s t e n tn o r m a l i z a t i o np o i n tf o r each p a r t i c u l a r mode. Forinstance, f i r s t wingbending was n o r m a l i z e d t o u n i t v e r t i c a l d e f l e c t i o n a t t h e w i n g t i p s t a t i o n sw h i c hr e p r e - bay. The modal displacements are presented a t s e l ec t e d be s e l e c t e dt o sent a v a r i e t y o f p o s s i b l e s e n s o r l o c a t i o n s whi ch m i gh t i o n s and d e f l e c t i o n s implement various control systems. The f o l l o w i n gl o c a t were selected : SymmetricAnalysis 1 . Fuselage nose p i t c ha n g l e (a) 2. F u s e l a g e c g v e r t i c a l d e f l e c t i o n ( h ) 3. Wing t i pv e r t i c a ld e f l e c t i o n( h ) 4. Wing t i pp i t c ha n g l e (a) 5. H o r i z o n t a ls . t a b i l i z e rt i pv e r t i c a ld e f l e c t i o n( h ) 6. H o r i z o n t a ls t a b i l i z e rt i pp i t c h ang1.e (a) AntisvmmetricAnalvsis 1. Fuselage nose r o l la n g l e (8) 2. Fuselagenose yaw a n g l e ( J, ) 3. Wing t i pv e r t i c a ld e f l e c t i o n( h ) 4. Wing t i pp i t c ha n g l e (a) 5. H o r i z o n t a ls t a b i l i z e rt i pv e r t i c a ld e f l e c t i o n( h ) 6. H o r i z o n t a ls t a b i l i z e rt i pp i t c ha n g l e (a) 7. V e r t i c a ls t a b i l i z e rt i pl a t e r a ld e f l e c t i o n ( a ) 8 . V e r t i c a l s t a b i l i z e r t i p yaw a n g l e ( JI The modal d a t aa r ep r e s e n t e di n . S e c t i o n 3. The t y p eo fd a t a shown f o r each a i r p l a n ec o n f i g u r a t i o ni ss u m n a r i z e d i n Secti,on 3.

D i s c u s s i o n o f R e s u l t s General. An e x a m i n a t i o no ft h e modal frequency,generalized mass and d i s p l a c e m e n tp l o t s show, i n g e n e r a l , t h a t t h e s e u n c e r t a i n t i e s can be d i v i d e d i n t o two d i s t i n c t phases. The f i r s t phaseoccursduringtheearlydesign stages,approximately 0 t o 1 0 months, where t h e modal p a r a m e t e r se x h i b i t l a r g ev a r i a t i o n s . These v a r i a t i o n sr e f l e c tm a j o rc o n f i g u r a t i o n changes such as engineweight,enginelocation, maximum grossweight and t h e i n t r o d u c t i o n of s t i f f n e s s c o n s t r a i n t s t o s a t i s f y dynamic c o n d i t i o n s such as l a n d i n g ,g u s t and f l u t t e rr e q u i r e m e n t s .I na d d i t i o n ,d a t ag e n e r a t e dd u r i n gt h i s phase a r e basedupon a c o a r s e i d e a l i z a t i o n o f t h e s t r u c t u r e .

The second phase occursfromapproximately10to 24 months, duringwhichtime

t h e modal parameters, i n general , e x h i b i to n l ys m a l l changes.Thisreflects

t h e f a c t t h a t t h e a i r p l a n e c o n f i g u r a t i o n has been determined and a l l major d e s i g nc o n s t r a i n t s have been introduced. Changes t ot h ed e s i g nd a t aa r e m i n o rm o d i f i c a t i o n s based upon a d d i t i o n a ld e t a i l e ds t r u c t u r a la n a l y s e s ,w h i c h produce smal 1 changes i n t h e a i r p l a n e modal c h a r a c t e r i s t i c s .

But however d e t a i l e dt h ea n a l y s i s ,u n c e r t a i n t i e ss t i l le x i s t .T h i s can be seenbycomparingthe modal parametervalues a t t h e 2 4 t h month w i t h t h e c e r t i f i c a t i o nv a l u e s . The 2 4 t h - m o n t hp o i n t sr e p r e s e n tt h ef i n a lp r e d i c t e d t h e o r e t i c a l modes b e f o r ev e r i f i c a t i o nb yg r o u n dv ib r a t i o nt e s t . The theore- t i c a l modes a r e a r e s u l t o f a h i g h l y d e t a i l e d s t r u c t u r a l a n a l y s i s o f a f i x e d c o n f i g u r a t i o n .T h e r e f o r e ,t h ed i f f e r e n c e between the24thmonth and c e r t i - f i c a t i o n v a l u e s r e p r e s e n t u n c e r t a i n t i e s i n a n a l y s i s t e c h n i q u e s i n i d e a l i z i n g t h ea i r p l a n es t i f f n e s s and mass p r o p e r t i e s . The i d e a l i z a t i o n sm u s tr e d u c e t h e c o n t i n u o u s a i r p l a n e s t r u c t u r e t o a f i n i t e number o f elements,making s i m p l i f i c a t i o ni n e v i t a b l e .A l s o . j the. number o f elements used t or e p r e s e n t t h e a i r p l a n e i s l i m i t e d b yp r a c t i c a lc o m p u t a t i o n a l and economicconsiderations.

An examination o f t h e modal frequency,generalized mass anddisplacement p l o t s does n o t show anydiscernabletrend due t o f u e l c o n d i t i o n o r s y m n e t r y .

The c o r r e l a t i o n i s e q u a l l y good forzero and f u l l f u e l and f o rs y m n e t r i c and a n t i s y m n e t r i c c o n d i t i o n s .

i Modal Frequencies. The modal frequency plots, Sect on 3 , show thatextreme v a r i a t i o n s o f -30 t o +84 p e r c e n t may o c c u r f o r some modes d u r i n g t h e f i r s t phase. However, t h em a j o r i t yo ft h e modal frequenc i es appear t o be i n t h e - +20 percent range. During the second phase, t h e m o d a1 frequencies show v a r i a t i o n s o f o n l y 3 t o 4 p e r c e n t on theaverage. The f i n a l t h e o r e t i c a l modes (24 month p o i n t ) d i f f e r f r o m t h e c e r t i f i c a t i o n modes by 4 t o 5 percent ontheaverage,although maximum d i f f e r e n c e s r e a c h +13 t o -20 p e r c e n t f o r a 5 p e r c e n t d i f f e r e n c e i n modal smallpercentageofthe modes. An average o f frequency i s extremely good c o r r e l a t i o n c o n s i d e r i n g t h e c o m p l e x i t y o f t h e s t r u c t u r eb e i n ga n a l y z e d .

Generalized Mass and Displacement. The modal g e n e r a l i z e d mass and d i s p l a c e - ment p l o t s ,S e c t i o n 3, show much t h e same t r e n d as do thefrequencyplots.

However, themagnitudes o ft h ev a r i a t i o n sa r e much more extreme. It appears, i n g e n e r a l ,t h a t changes i n d e s i g nd a t ap r o d u c el a r g e rv a r i a t i o n s i n g e n e r a l i z e d mass and d e f l e c t i o n s t h a n i n modal frequency.

However, caremust be exercised i n i n t e r p r e t i n g t h e g e n e r a l i z e d mass and d e f l e c t i o nd a t a . It i st r u et h a tt h ed a t a show t h eu n c e r t a i n t yo f each para- i t s v a l u ea t a g i v e nt i m et o i t s f i n a lv a l u e .B u tt h e meter as a r a t i o o f a c t u a lv a l u eo ft h ep a r a m e t e r may be smal 1.Therefore,seeminglylarge v a r i a t i o n s i n i t s v a l u e may s t i l l be i n s i g n i f i c a n t as t h e y a f f e c t t h e a i r - p l a n er e s p o n s ec h a r a c t e r i s t i c s and hence t h ec o n t r o ls y s t e md e s i g n .I n a d d i t i o n , t h e r a t i o o f any p a r t i c u l a rp a r a m e t e r , such as fuselage nose p i t c h a n g l e ,v a r i e sf o r each mode o f a g i v e nc o n f i g u r a t i o n .N o t et h a tt h ep a r a - meter r a t i o i s g r e a t e r t h a n u n i t y i n some modes and 1essthanunity i n o t h e r s i n whichthere i s u s u a l l y a b i a s i n one d i r e c t i o n o r t h e o t h e r .

E v a l u a t i o no fR e s u l t s . The importance o ft h es t r u c t u r a l dynamic b e h a v i o ro f t h e a i r c r a f t t o t h e c o n t r o l systemdesign depends upon t h ei n t e n d e df u n c t i o n o f thecontrolsystem and i t s gain/phaseproperties.Forlowfrequencycon- t r o l systemssuchasconventionalautopilot and yaw damper systems, t h e e f f e c t o f t h e a i r c r a f t s t i f f n e s s d i s t r i b u t i o n onthesteadyaerodynamic derivatives i s ofsignificance. For higherfrequencycontrol systems, particularly those required for g u s t load and flutter suppression, the stiffness andmass distributions become more significant and hence the vibration modes of the a i r c r a f t become meaningful measures o f potential control system interaction. O f course,in any controlloop w i t h significant gain and phase shift i n the e l a s t i c modal frequencyrange knowledge of the v i bration characteristics of the vehicle is desirable. Use of mode shapes todetermine nodal and anti-nodal p o i n t s forsensorlocations is desirable forpassivegainstabilizationprocedures, b u t f o r l a r g e f l e x i b l e a i r c r a f t significant shifts i n node lines can occur w i t h fuel and payload distribution changes.

The use of naturalfrequencies t o assessthe change i n the vibration modes d u r i n g the design cycle i s a useful ( b u t n o t u n i q u e ) measure of potential effects on the control system design. The naturalfrequencies may be iden- t i f i e d by t h e a i r c r a f t components which a r e the most directly involved i n the mode and some subjectiveevaluation will be made by theengineerasto the significance t o any givencontrolsystem. For example, a low frequency fuselage bending mode may have a very directbearing on the control system f o r a h i g h gain system and a low frequency is an indicationthat design significant data error from forward mounted sensors may occur i n turbulence condi t i ons .

Knowledge of amplitudes a t various stations on the a i r c r a f t and the general- ized masses i s considerably less useful. The controlsystem designer faced w i t h thetask of designing a system for response o r s t a b i l i t y i n t h e e l a s t i c frequencyrangeor a system w i t h significant gain and phase variation i n this frequencyrange must use thesedata i n his analysesalong w i t h the appropriate aerodynamic functions .

A reviewof the modal parametersvariations t h r o u g h the design cycle presented hereinindicates a reasonable convergence w i t h time on the final design valuesfor the modal frequencies. The generalized mass and modal deflections for various modes and locations on the a i r c r a f t , however, indicate large variations and an apparentlackofcontinuity. T h i s l a t t e r i s not surprising i n view of the fact that d u r i n g the design cycledataupdatesresulting from structuralmodifications and improved data(resulting from both quality and i d e a l i z a t i o n m o d i f i c a t i o n s ) a r e g e n e r a l l y made f o r one o r two a i r c r a f t components a t a time.For example, a s h i f t i n wingenginecgwithrespect t o t h e w i n g may n o t s i g n i f i c a n t l y e f f e c t t h e b a s i c f u s e l a g e and empennage modes b u t c a n c a u s e l a r g e v a r i a t i o n s i n g e n e r a l i z e d mass o f modes i n v o l v i n g s i g n i f i c a n te n g i n em o t i , o n . The f r e q u e n c yv a r i a t i o n however w i l l show s i g n i - f i c a n t l y l e s s v a r i a t i o n .

1 , The importance t o t h e c o n t r o l systemdesignfromthe above change w i l l be i n s i g n i f i c a n t i n t h e b a s i c l o w f r e q u e n c y a u t o p i l o t d e s i g n b u t o f m a j o r s i g n i - f i c a n c e f o r a ' f l u t t e r s u p p r e s s i o n system.

A more s i g n i f i c a n t measureof t h e s t r u c t u r a l dynamicparameteruncertainties may be f o u n d b y . o b s e r v i n g t h e v a r i a t i o n o f t h e a i r c r a f t open l o o p t r a n s f e r f u n c t i o n f o r s p e c i f i c c o n t r o l s u r f a c e i n p u t s a t s p e c i f i c p o t e n t i a l s e n s o r locations.Forsuchresponseanalysesto be v a l i di nt h ee l a s t i cf r e q u e n c y ranges,theymust use theunsteadyaerodynamicfunctionswhich more r e a l i s - t i c a l l y representthecomplexaerodynamicforceswhichoccur i n these frequencyranges.

To i l l u s t r a t e t h e above, a frequencyresponseanalyses was performed t o determine i f t h e modal d a t a h e r e i n w o u l d c o r r e l a t e w i t h t h e a i r c r a f t t r a n s f e r f u n c t i o n sf o r two p o i n t s i n t h ed e s i g nc y c l e . The t r a n s f e rf u n c t i o n s were e v a l u a t e d ' f o r 0 t o 10 H e r t z f o r a p l us-or-minus 1 -degree elevator osci 1l a t i o n using'unsteadyaerodynamicsfor Mach 0.88 and f l i g h t a t 7,315 meters (24,000 f e e t ) .T a b l e 7 shows a summary o f r e s u l t s f o r two dynamic a e r o e l a s t i c modes, t h e a i r c r a f t s h o r t - p e r i o d mode and t h e Wing EnginePitch/FuselageBending mode. The i n p u t modal displacement and g e n e r a l i z e d mass r a t i o s and o u t p u t a c c e l e r a t i o n r a t i o s f o r t h e s e modes a r e shown f o r s e v e r a l a i r c r a f t l o c a t i o n s .

F r e e - f r e e a i r c r a f t modes wereused i n t h ea n a l y s i s .

It i s apparentfromthese summary d a t at h a tw h i l et h eg e n e r a l i z e d mass and normalized modal displacements show l a r g ev a r i a t i o n s ,t h ea c c e l e r a t i o n responses f o r t h e s e l e c t e d l o c a t i o n s a r e s i m i l a r and i n d i c a t e r e l a t i v e l y s m a l lv a r i a t i o n s . These responsedataare o f s i g n i f i c a n t use t ot h ec o n t r o l s y s t e m d e s i g n e r u s i n g t h e e l e v a t o r f o r f o r c e g e n e r a t i o n andany o f t h e selectedlocationresponseparametersforsensorlocations.

TABLE 7 RESPONSE ANALYSIS SUMMARY SYMMETRIC ZERO FUEL MACH 0.88 7315M (24,000 FT) RESPONSEACCEL. RATIO 3.23 Hz (124'4) MODE INPUT MODEL RESPONSE DISPLACEMENT 0.45 Hz 3.23 Hz PARAMETER RAT1 0 SHORT PERIOD MODE FUSELAGE BENDING MODE "~ ~ "" FUSELAGE NOSE OL 0.80 1.02 1.18 FUSELAGE CG h 3.21 1.02 1.12 WING TIP h 0.94 1.02 1.04 WING TIP OL 0.44 1.03 0.85 HORIZONTAL STAB1 LlZER h 1.00 1.02 1.27 I L ABOVE RATIOSARE FOR 24THMONTH OATAVERSUS CERTIFICATION OATA (GM24/GM,) = 0.51 F O R THE 3.23 HERTZ FUSELAGE BENOING MOOE Foranygiven a i r c r a f t s t r u c t u r e t h e s t r u c t u r a l dynamic c h a r a c t e r i s t i c s v a r y s i g n i f i c a n t l yo v e rt h e normal f u e l andpayloadrange. As a m a t t e r o f i n t e r e s t thesymmetrical modal f r e q u e n c yv a r i a t i o nf r o mf u l lf u e l t o z e r of u e l was compared t o t h e maximum f r e q u e n c yv a r i a t i o nf o rz e r of u e lo v e rt h ed e s i g n p e r i o df r o m ATP t o GVT. A similarcomparison was made f o r t h e f u l l f u e l frequenciesoverthedesignperiod. These comparisonsare shown i n Table 8 and i n d i c a t et h a tt h e modal f r e q u e n c i e sv a r yw i t hf u e ll o a d i n g by t h e same l e v e l o f magnitude as t h e u n c e r t a i n t y v a r i a t i o n s and i n some casesmore.

Payloadvariations,which have n o t been considered,would show even a l a r g e r range i n t h ef r e q u e n c i e so ft h ef i n a ld e s i g n .

TABLE 8 . . .

. .

COMPARISON OF FUEL EFFECTS TO UNCERTAINTIES EFFECT SYMMETRIC MODAL FREQUENCIES UNCERTAINTIES RATIO .

r

fFULL FUEL 'MA: ~MIN .

FULLFUEL MODE DESCRIPTION ZERO FUEL ZERO FUEL .

. . .

FIRST WING BENDING 0.63 1.26 1.16 WING ENGINE YAW 0.98 1.20 1.16 FUSELAGE BENDING 0.90 1.13 1 .os HORIZONTAL STABILIZER BENDING/AFT ENG PITCH 1.01 1.72 1.08 WING INNERPANEL TORSION 0.90 1.39 1.35. , WING ENGINEROLL 1.01 1.16 1.15 SECOND WINGBENDING 0.7 1 1.30. , 1 .os WING FORE AND AFT BENDING 0.56 1.27 1.20 . .

WING TDRSION/ENGINE PITCH 0.94 1.13 1.17 Aerodynamic, i n e r t i a l , a n ds t r u c t u r a ld a t a are presented i n this s e c t i o n .

The d a t a a r e d i s c u s s e d i n d e t a i l i n Section 2 so t h a t this sectionprovides asconcisea compendium a sp o s s i b l e . ' The parametersaredefined i n the l i s t of symbols a t the beginningof this report. The equations of motionand dimensional s t a b i l l t yd e r i v a t i v ed e f i n i t i o n sa r eg i v e ni n the Appendix. The dataareorganizedasfollows: Longi tudi nal Aerodynamic Parameters :

- _ " - . " " . " I _ _ _

Figure11HistoricalUncertainties of CL , Cma, CDa, CLi

a Figure 12 H i s t o r i c a lU n c e r t a i n t i e so f Cm,, CDi, C L u , CmU

Figure 13 H i s t o r i c a lU n c e r t a i n t i e s o f C , CL , Cm , CL

Du q 9 6e

Figure 14 H i s t o r i c a l U n c e r t a i n t i e s of C , CD , C L , CD

"'6, 6, " L a t e r a l - D i r e c t i o n a l - " _ "" Aerodynamic Parameters:

Figure15HistoricalUncertainties of C , C n g , C k B , C

YB yP Figure16HistorlcalUncertainties of Cn ,

, c , c

p ' ~ p yr nr

Figure 1 7 H i s t o r i c a l U n c e r t a i n t i e s o f C, , C , Cn

r Y , r 'r

Figure 18 Historical Uncertainties of C , C , c

5 3 ' 6 a n 6

SP Figure 1 9H i s t o r i c a lU n c e r t a i n t i e so f C ."6sp I n e r t i a1 Parameters : Figure 20 Manufacturer's Empty Weight Growth History

F i g u r e 21 MEW Growth Hi s t o r y - Cabin, Wing , Empennage, Forward Fusel age

Table 9 Major Weight Change Summary Table . l o Moment of I n e r t i aH i s t o r y Figure 22 DesignWeights H i s t o r i e s S t r u c t u r a l Dynamic Parameters : Table 1.1 Symnetric Modal Analysis Sumnary Table12Antisymnetric Modal Analysis Sumnary Figures 23through 26 Mode Shapes and Mode Lines . .

Figures 27 through 60 E v o l u t i o n of Frequency Ratio(seeTables 11 and 12 f o r i n d e x ) Figures 61 through 90 E v o l u t i o no fG e n e r a l i z e d Mass R a t i o (,see Tables11 and 12 for index) Figures 91 through 196 E v o l u t i o n of DisplacementRatio(seeTables 11 and 12 f o r Sndex) 3 8 .

. . , LIST OF FIGURES PAGE . ' FIGURE NO .

EVOLUTION OF . FREQUENCY- RATIO SYMMETRIC ZERO FUEL

1st Wing Bending. 1 . 8 9 . H ~ . . . . . . . . . . . . . . . 61

Wing Engine Yaw. 2.03 H z 28 . . . . . . . . . . . . . . . 61 . .

3.23 H z . . . . . . 61' ' . .

29 Wing 'Engine Pitch/Fuselage Bending, .

Horiz.'Stab. Bending/Aft Engine Pitch. 3.55 H z . . . . 61 . . . .

. . .

31 Wing Inner Panel Torsion. 3.71 H z . . . . . . . . . . . 62

32 Wing EngineRoll. 5.05 H z . . . . . . . . . . . . . . ., . . 62 . . , .

2nd Wing Bending, 5.69 H z . . . . . . . . . . . . . . 62

Wing Fore and A f t Bending. 6.79 Hz . . . . . . . . . . 62 ...

35 Wing Torsion/Engine Pitch. 9.94 Hz . . . . . . . . . . 63

SYMMETRIC' FULL FUEL

1 s t Wing Bending. 1.19 H z . . . . . . . . . . . . . . . 63

Wing Engine Yaw. 1.98 Hz . . . . . . . . . . . . . . . 63

Fuselage Bending. 2.90 Hz . . . . . . . . . . . . . . . 63

Horiz . S t a b . Bending/AftEngine Pitch. 3.57 H z . . . . 64

40 Wing Inner Panel Torsion. 3.34 H z . . . . . . . . . . . 64

41 Wing Fore and Aft Bending. 3.81 Hz . . . . . . . . . . 64

42 2nd Wing Bending. 4.05 H z . . . . . . . . . . . . . . . 64

43 Wing Engine Roll. 5.08 Hz . . . . . . . . . . . . . . . 65

44 3rd Wing Bending/Aft Engine Pitch. 7.51 H z . . . . . . 65

45 Wing Torsion/Engine Pitch. 9.30 Hz . . . . . . . . . . 65

ANTISYMMETRIC ZERO FUEL

Wing Engine Yaw. 2.08 Hz . . . . . . . . . . . . . . . 65

47 1 s t Wing Bending. 2.28 H z . . . . . . . . . . . . . . .

Aft Engine Yaw. 2.56 Hz . . . . . . . . . . . . . . . . 66

Horiz . S t a b . Bending. 3.16 Hz

49 . . . . . . . . . . . . . 66

50 Vert . Stab . Bending.3.48 Hz

. . . . . . . . . . . . . 66

Aft FuselageTorsion/WingEngine Pitch. 4.25 H z . . . . 67

52 Aft FuselageLateralBending. 7.49 H z . . . . . . . . . 67

53 2nd Wing Bending. 7.82 H z . . . . . . . . . . . . . . . 67

2nd Vert . Stab . Bending.10.82 Hz

54 . . . . . . . . . . . 67

ANTISYMMETRIC FULL FUEL Wing Engine Y.aw. 2.05 Hz . . . . . . . . . . . . . . .

55 68

56 1 s t Wing Bending.1.62 Hz . . . . . . . . . . . . . . . 68

57 Horiz . S t a b . Bending. 3.00 H z . . . . . . . . . . . . . 68

3.35 Hz . . . . 68

58 Aft FuselageTorsion/WingEngine Pitch.

59 Vert . S t a b . Bending. 3.49 Hz . . . . . . . . . . . . . 69

60 2nd Wing Bending. 5.16 Hz . . . . . . . . . . . . . . . 69

LIST OF FIGURES (Contd) PAGE FIGURE NO .

EVOLUTION OF GENERALIZED MASS RATIO SYMMETRIC ZERO FUEL

1 s t Wing Bending. 1.89 Hz . . . . . . . . . . . . . . 69

Wing Engine Yaw. 2.03 Hz . . . . . . . . . . . . . . . 69

63 Wing EnginePitch/Fuselage Bending. 3.23 Hz . . . . .

Horiz . Stab . Bendlng/Aft Engine Pitch. 3.55Hz . . . . 70

Wing Inner Panel Torsion . 3.71 Hz . . . . . . . . . . 70

2nd Wing Bending. 5.69 Hz . . . . . . . . . . . . . . 70

67 Wing Fore and A f t Bending. 6.79 Hz . . . . . . . . . . 71

Wing Torsion/Engine Pitch. 9.94 Hz . . . . . . . . . . 71

SYMMETRIC FULL FUEL 1 s t Wing Bending. 1.19 Hz . . . . . . . . . . . . . .

Wing Engine Yaw. 1.98 Hz . . . . . . . . . . . . . . . 71

71 FuselageBending. 2.90 Hz . . . . . . . . . . . . . .

Horiz . Stab . Bending/Aft Engine Pitch. 3.57Hz . . . . 72

Wing Inner Panel Torsion. 3.34 Hz . . . . . . . . . . 72

74 Wing Fore and A f t Bending . 3.81 Hz . . . . . . . . .

75 2nd Wing Bending. 4.05 Hz . . . . . . . . . . . . . .

76 3rd Wing Bending/Aft Engine Pitch. 7.51 Hz . . . . . .

ANTISYMMETRIC ZERO FUEL 77 Wing Engine Yaw. 2.08 Hz . . . . . . . . . . . . . .

1 s t Wing Bending. 2.28 Hz . . . . . . . . . . . . . . 73

A f t Engine Yaw. 2.56 Hz . . . . . . . . . . . . . . . 74

80 Horiz . Stab . Bending. 3.16 Hz . . . . . . . . . . . .

V e r t . Stab . Bending. 3.48 Hz . . . . . . . . . . . . .

82 A f t FuselageTorsion/WingEnginePitch. 4.25 Hz . . .

A f t FuselageLateralBending . 7.49 Hz . . . . . . . .

84 2nd Wing Bending. 7.82 Hz . . . . . . . . . . . . . .

85 2nd Vert . Stab . Bending. 10.82 Hz . . . . . . . . . .

ANTISYMMETRIC FULL FUEL 1 s t Wing Bending. 1.62 Hz . . . . . . . . . . . . . .

Wing Engine Yaw. 2.05 Hz . . . . . . . . . . . . . . .

Horiz . Stab . Bending, 3.00 Hz . . . . . . . . . . . .

89 Vert . Stab . Bending, 3.49 Hz . . . . . . . . . . . . .

90 2nd Wing Bending, 5.16 Hz . . . . . . . . . . . . . .

LIST OF FIGURES (Contd) PAGE FIGURE NO, EVOLUTION OF DISPLACEMENT RATIO S Y M M E T R I C ZERO FUEL 1 s t Wing Wing Bending, 1.89 Hz

91 Fuselage Nose Pitch Angle ( a ) . . . . . . . . . . . 77

92 Fuselage C.G. Vertical Deflection ( h ) . . . . . . . 77

93 . . 77

Wing T i p Vertical Deflection (h) - - . . . . .

- . . 77

Wing T i p Pitch Angle ( a ) . . . . . - . . . .

. . . 78

95 Horiz. Stab. T i p Vertical Deflection ( h ) . .

. . . 78

96 Horiz. Stab. T i p Pitch Angle ( a ) . . . . . .

Wing Engine Pitch/Fuselage Bending, 3.23 H z

97 Fuselage Nose Pitch Angle ( a ) . - . - . . . . . . . 7 8

98 Fuselage C.G. Vertical Deflection (h) . . . . . . . 78

Wing T i p Vertical Deflection ( h ) - - - - - - . 79

100 Wing T i p Pitch Angle ( a ) - - - - - - - - - - - - 79

Horiz. Stab. T i p Vertical Deflection ( h ) . . . . . 79

102 Horiz. S t a b . T i p Pitch Angle (cl) . . . . . . . . . 79

Wing Inner Panel Torsion, 3.71 Hz

103 Fuselage Nose Pitch Angle ( 3 ) - . . - . - - - - 80

104 Fusel age C.G. Vertical Deflection ( h ) . . . . . . - 80

105 Wing T i p Vertical Deflection (h). . . - . . . . . . 80

106 Wing T i p Pitch Angle ( a ) . . . . . . . . . . . . . 80

107 Horiz. S t a b . T i p Vertical Deflection ( h ) . . . . . 81

Horiz. Stab. T i p Pitch Angle ( a ) . . - - - . . - . 81

2nd Wing Bending,5.69 H z

109 Fuselage Nose Pitch Angle ( a ) a - - . . . . . . 81

110 Fuselage C.G. Vertical Deflection ( h ) - - . . - 81

111 Wing T i p Vertical Deflection ( h ) . - - - - - - - - 82

112 Wing T i p Pitch Angle ( a ) . . . . . . . . . . . . . 82

113 Horiz. Stab. T i p Vertical Deflection ( h ) - - 82

114 Horiz. Stab. T i p Pitch Angle (a). . . . . . . . . . 82

. .

LIST OF FIGURES (Contd) .

FIGURE NO . PAGE

. .

. EVOLUTION OF DISPLACEMENT RATIO S Y M M E T R I C FULL FUEL 1 s t Wing Bending. 1.19 H z

Fuselage Nose Pitch Angle ( a ) . . . . . . . . . . . 83

CG Vertical Deflection ( h ) . . . . . . . . 83

Fuselage

Wing T i p Vertical Deflection ( h ) . . . . . . . . . 83

Wing T i p Pitch Angle ( a ) . . . . . . . . . . . . . . . 83

Horiz . Stab. Tip Vertical Deflection ( h ) . . . . . 84

Horiz . S t a b . T i p Pitch Angle ( a ) . . . . . . . . . 84

Fuselage Bending. 2.90 H z Fuselage Nose Pitch Angle (a)

. . . . . . . . . . . 84

Fuselage CG VerticalDeflection ( h ) . . . . . . . . 84

Wing T i p VerticalDeflection ( h )

123 . . . . . . . . . 85

Wing T i p Pitch Angle ( a ) . . . . . . . . . . . . 85

Horiz . S t a b . T i p Vertical Deflection

125 ( h j . . . . . 85

Horiz . S t a b . T i p Pitch Angle ( a ) . . . . . . . . . 85

Wing Inner Panel Torsion. 3.34 Hz

Fusel age Nose Pitch Angle ( a ) . . . . . . . . . . . 86

Fuselage C.G. Vertical Deflection ( h ) . . . . . . . 86

Wing T i p Vertical Deflection (h) . . . . . . . . . 86

Horiz . S t a b . T i p VerticalDeflection ( h ) . . . . . 86

Horiz . S t a b . T i p Pitch Angle ( a ) . . . . . . . . . 87

2nd Wing Bending. 4.05 Hz

Fuselage Nose Pitch Angle ( a ) . . . . . . . . . . . 87

Fuselage C.G. Vertical Deflection ( h ) . . . . . . . 87

Horiz . Stab . Tip VerticalDeflection ( h ) . . . . . 87

Horiz . S t a b . T i p Pitch Angle ( a ) . . : . . . . . . 88

.- . . . . . . . . .

.

LIST O F FIGURES (Contd) PAGE FIGURE NO .

EVOLUTION O F DISPLACEMENT RATIO ANTISYMMETRIC ZERO FUEL 1 s t Wing Bending. 2.28 Hz Fusel age Nose Roll Angle ( e )

136 . . . . . . . . . . . 88

Fuselage . Nose Yaw Angle (4)

137 . . . . . . . . . . . . 88

Wing T i p V e r t i c a l D e f l e c t i o n

138 (h) . . . . . . . . . 88

Wing T i p P i t c h A n g l e ( a )

139 . . . . . . . . . . . . . 89

H o r i z . Stab . T i p . V e r t i c a lD e f l e c t i o n( h ) . . . . . 89

H o r i z . Stab . T i p P i t c h A n g l e (a) .

141 . . . . . . . 89

V e r t . Stab . T i p L a t e r a l D e f l e c t i o n

142 ( t i . . . . . . 89

V e r t . Stab . T i p Yaw Angle (4)

. . . . . . . . . . . go

H o r i z . Stab . Bending. 3.16 !!z

Fuselage Nose Yaw Angle ( 4 )

. . . . . . . . . . . . go

145 Wing T i p V e r t i c a l D e f l e c t i o n ( h ) . . . . . . . . . go

H o r i z . Stab. T i p V e r t i c a l D e f l e c t i o n ( h ) . . . . . go

Wing T i p P i t c h A n g l e ( a )

147 . . . . . . . . . . . . . 91

148 V e r t . Stab . T i p L a t e r a l D e f l e c t i o n

( a ) . . . . . . g1

149 V e r t . Stab . T i p Yaw Angle (4)

. . . . . . . . . . . 91

V e r t . Stab . Bending. 3-48 Hz

. . . . . . . . . . . 91

150 Fuselage Nose RollAngle ( e )

. . . . . . . . . . . . 92

151 Fuselage Nose Yaw Angle ( $ ) 152 Wing T i p V e r t i c a l D e f l e c t i o n

(h) . . . . . . . . . 92

Wing T i p P i t c h A n g l e ( a ) . . . . . . . . . . . . . 92

H o r i z . Stab . T i p V e r t i c a l D e f l e c t i o n . (h) . . . . . 92

. . . . . . . . . 93

H o r i z . Stab . T i p P i t c h A n g l e (3)

Vert . Stab . T i p L a t e r a l D e f l e c t i o n

156 (E) . . . . . . 93

V e r t . Stab . T i p Yaw Angle ( $ ) - . . . . . . . . . . 93

A f t FuselageTorsion/WingEnginePitch. 4.25 Hz

Fuselage Nose Roll Angle ( e ) . . . . . . . . . . . 93

Fuselage Nose Yaw Angle ($) . . . . . . . . . . . . 94

Wing T i p V e r t i c a l D e f l e c t i o n ( h ) . . . . . . . . . 94

161 Wing T i p P i t c h A n g l e ( a ) . . . . . . . . . . . . . 94

162 H o r i z . Stab . T i pV e r t i c a lD e f l e c t i o n( h ) . . . . . 94

H o r i z . Stab . T i pP i t c hA n g l e (9.1 . . . . . . . . . 95

V e r t . Stab . T i pL a t e r a lD e f l e c t i o n ( a ) . . . . . . 9 5

V e r t . Stab. T i p Yaw Angle ( $ ) . . . . . . . . . . . 95

LIST OF FIGURES (Contd) . . .

FIGURE NO . PAGE

E V O L U T I O N OF DISPLACEMENT RATIO ANTISYMMETRIC FULL FUEL 1 s t Wing Bending. 1.62 Hz

Fuselage Nose Roll Angle ( e ) . . . . . . . . . . . . . g j

Fuselage Nose Y a w Angle ( a ) . . . . . . . . . . . . 96

168 Wing T i p Vertical Deflection (h) . . . . . . . . . . 96

Wing T i p Pitch Angle ( a ) . . . . . . . . . . . . . . 96

Horiz . Stab . T i p VerticalDeflection(h) . . . . . . 96

Horiz. Stab . T i p Pitch Angle ( a ) . . . . . . . . . 97

Vert . Stab . T i p LateralDeflection ( a ) . . . . . . . 97

Vert . Stab . T i p Y a w Angle ($1 . . . . . . . . . . . 97

Horiz . Stab . Bending. 3.00 H z

Fuselage Nose Roll Angle ( e ) . . . . . . . . . . . . 57

Fuselage Nose Y a w Angle ( J I ) . . . . . . . . . . . . 98

Wing T i p Vertical Deflection (h) . . . . . . . . . . 98

Wing T i p Pitch Angle (a) . . . . . . . . . . . . . . 98

Horiz. Stab . T i p VerticalDeflection(h) . . . . . . 98

Vert . Stab . T i p LateralDeflection ( a ) . . . . . . . 99

Vert . Stab . T i p Y a w Angle ( $ ) . . . . . . . . . . . 99

Aft Fuselage Torsion/WingEngine Pitch. 3.35 H z

Fuselage Nose Roll Angle ( e ) . . . . . . . . . . . . 99

Fuselage Nose Y a w Angle ( $ ) . . . . . . . . . . . . 99

Wing T i p VerticalDeflection

183 (h) . . . . . . . . . 100

Wing T i p Pitch Angle ( a ) . . . . . . . . . . . . . . 100

Horiz . Stab . T i p VerticalDeflection(h) . . . . . . 100

Horiz . Stab. T i p Pitch Angle ( a ) . . . . . . . . . . 100

Vert . Stab . T i p LateralDeflection ( a ) . . . . . . . 101

Vert . Stab . T i p Y a w Angle ($) . . . . . . . . . . . 101

Vert . Stab . Bending.3.49 H z

Fuselage Nose Roll Angle ( e ) . . . . . . . . . . . 701

Fuselage Nose Y a w Angle ($) . . . . . . . . . . . . 1G1

Wing T i p VerticalDeflection(h) . . . . . . . . . . 102

Wing TipPitch Angle (CY) . . . . . . . . . . . . . . 102

Horiz . Stab . T i p VerticalDeflection(h) . . . . . . 102

Horiz . Stab . T i p Pitch Angle (&) . . . . . . . . . . 102

Vert . Stab . T i p LateralDeflection ( 2 ) . . . . . . . 103

Vert . Stab . T i p Y a w Angle ($) . . . . . . . . . . . 103

44 . .

. c . , La PRIMARY AERODYNAMIC DERIVATIVE CONTRIBUTES TO DAMPING OF SHORT- PERIOD MODE 1 .o ESTABLISHES LEVEL OF ACCELERATION CLa p . : SENSITIVITY, n/u (NUMERATOR .......

. .

I . " : - PROPERTY OF TRANSFER FUNCTION) C FINAL VALUE VERY CAREFULLY CHECKED La i i . : FINAL 'o.5 IN FLIGHT TEST BECAUSE OF PERFORM- ANCE GUARANTEES ; . ' . , , C AC ma PER RAD -0.8 : 1.5 . .

!..._ ..

........

. .

,-I!- ._ " . .

C ,_ 0, FlNA L '-7 - 0.5 " .....

- ..

.....

- 0 - C % CONTRIBUTES TO SHORT-PERIOD DAMPING 0 FINAL VALUE CHECKED USING FLIGHT TEST DYNAMIC RESPONSE MATCH C 0 ; o NEGLIGIBLE NEGLIGIBLE, NOT PLOTTED o FINALVALUENOT CHECKED IN FLIGHT TEST; IMPACT TOO SMALL C LU SECONDARY EFFECT ON PHUGOID MODE FINAL VALUE CHECKED BY FLIGHT TEST RESULTS; NORMALLY NO -0.1 L ASSESSMENT O F ACCURACY MADE F.F. ' CERT """2 -10 ATP 10 20 30 40 50 MONTHS 0.2 * .

0.1 AFFECTS PHUGOIO MODE FINAL VALUE CHECKED BY FLIGHT TEST RESULTS;NORMALLY NO ASSESSMENT OF ACCURACY MADE -0.1 L 0 F.F. CERT " . " L " .... I -I;_. . I -10 ATP 10 20 30 .40 50 MONTHS FIGURE 12. HISTORICAL UNCERTAINTIES OF LONGITUDINAL AERODYNAMIC PARAMETERS cD" AFFECTS PHUGOID DAMPING , CONTRIBUTES TO FLIGHT-PATH STABILITY (NUMERATOR cD" PROPERTY OF h h g TRANSFER.

FUNCTION) CD "FINAL FINAL VALUE VERY CAREFULLY CHECKED IN FLIGHT TEST BECAUSE OF PERFORMANCE GUARANTEES c Lq C Lq 0 SECONDARY EFFECT ON SHORT- PERIODMODE CL 0 FINAL VALUE NOT CHECKED IN FINAL FLIGHT TEST; IMPACT TOO SMALL, " 1.5 I . .

. .

. .

. .

. . . .

" - - - - - - - " - - -. - .... .... .. .. . ......

..... MONTHS _._...

FIGURE 13. HISTORICAL UNCERTAINTIES OF LONGITUOINAL AERODYNAMIC PARAMETSRS C m6e c m6.

PRIMARY PITCH CONTROL DERIVATIVE C m6 FINAL VALUE CHECKED BY FLIGHT *FINAL TEST RESULTS; NORMALLY NO ASSESSMENT OF ACCURACY MADE ESTIMATE0 ACCURACY: +4 PERCENT o USUALLY NEGLIGIBLE NEGLIGIBLE, NOT PLOTTED o FINALVALUE NOT CHECKEO IN FLIGHT TEST; IMPACT TOO SMALL CL o PRIMARY FLIGHT CONDITION PARAMETER o AFFECTS PHUGOID MODE o AFFECTS FLIGHT-PATHSTABILITY NOT PLOTTED (NUMERATOR PROPERTY OF h h e TRANSFER FUNCTION) o FINALVALUEVERYCAREFULLY CHECKEO IN FLIGHT TESTBECAUSE OF PERFORMANCE GUARANTEES . . . . .

. i . I .

. . . .1_ . . L~;..C:." ; . . . . . . . :. .: : . .

j j i : T CD

- , -

1.0 . .

. , , . , .

. , .

I ..:. _:. ... Y- .. . I . . . . . . .: .. . . . .

, . . . . . . . . . . . . . . . - o PRIMARY PERFORMANCE ~.

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

. . 1 . : PARAMETER . . . .

- . I _ . . : . I:..: :.I . . i . ..:. - .- j . . : .. I .... : : j .

1 ,, : I " : ! o AFFECTS PHUGOID MOOE DAMPING . . .

CD ........ f " i+"+.._j".L "+". I . . . . . . . . . . . . . . . . .

. , ' o AFFECTS FLIGHT-PATH STABILITY - . . I... ..... !..-. L. I.:.-.. .1 ..... I ..: "4 ... :. . . . . . . . . . . .

'0.5 ! ~ ! . I : ' I ' ' CDFINAL . ! T . : " . . . LLL!";. 2 ... !.i . . . . . . (NUMERATOR PROPERTY OF h/6e ' ! ; ' 1 1 ' . j ! : I : I . . TRANSFER FUNCTION) . .

I . .

. . . . . . . . . . . . . . . . , . . . . . . . ,_.. ... : ..i i .. : , : . , . . , .

. . . . , .

-. _ . , .. - . . . - - j _TT".~.","~"l_i"_" .L.. .- . - o FINAL VALUE VERY CAREFULLY . ._ .

. .

. .

. . I .

. . . . . . . . . . . , . . ~ .,.. : : : . D...i'. . i j . F.F. CERT CHECKED IN FLIGHT TEST BECAUSE . .

- I t ' I

1 1 dk OF PERFORMANCE GUARANTEES ' 0 . c .

! .. i -10 , . ATP : . . 10 : 20 . . 30 ., 40

. .

. . . . 50 ESTIMATED ACCURACY: + 2 PERCENT :.. . ..i_ L " MONTHS ... ". . . . . . . .

. . . . . .

FIGURE 14. HISTORICALUNCERTAINTIES OF LONGITUOINALAERODYNAMICPARAMETERS MODERATEIMPORTANCE CONTRIBUTES TO DUTCH ROLL DAMPING C FINAL VALUE CHECKED BY FLIGHT Y P TEST RESULTS; NORMALLY NO C ASSESSMENT OF ACCURACY MADE 'PFINAL C RP PRIMARY LATERAL STAB1LlTY PARAMETER c!?

AFFECTS BOTH DUTCH ROLL AND P SPIRAL MODES c!?

FINAL VALUE CHECKED BY FLIGHl PFINAL TEST RESULTS; NORMALLY NO ASSESSMENT OF ACCURACY MADE FIGURE 15. HISTORICAL UNCERTAINTIES O F LATERAL-DIRECTIONAL AERODYNAMIC PARAMETERS . .

n n CI Cn P

co I

0 SECONDARY IMPORTANCE 0 AFFECTS DUTCH ROLL DAMPING 'Cn P -0.02 0 ' AFFECTS TURN COORDINATION 0 0 PER RAD ' . 0' CHARACTERISTICS FINAL VALUE NOT CHECKED I N D F.Fi CERT 0 FLIGHT TEST; IMPACT TOO SMALL

"0.04 1 "l.l"" 1 ' I I

-10 ATP 10 20 30 40 50 MONTHS c!?

P BASIC ROLLDAMPING DERIVATIVE FINAL VALUE CHECKED USING FLIGHT TEST DYNAMIC RESPONSE MATCHING NEGLIGIBLE IN MANY CASES FINAL VALUE NOT CHECKED IN FLIGHT TEST; IMPACT TOO SMALL 'nr BASIC YAW DAMPING DERIVATIVE IMPORTANT TO DUTCHROLL DAMPING 'nr ALSO AFFECTS SPIRAL MODE Cn FINAL VALUE CHECKED USING 'FINAL FLIGHT TEST DYNAMIC RESPONSE MATCHING . ... MONTHS ... ..

FIGURE 16. HISTORICAL UNCERTAINTIES OF LATERAL-DIRECTIONAL AERODYNAMIC PARAMETERS o NEGLIGIBLE IN MANY CASES NEGLIGIBLE, NOT PLOTTED o FINAL VALUE NOT CHECKEO IN FLIGHT TEST; IMPACT TOO SMALL I .

' 0

ooi.r

" a AC o SECONDARY CONTROL DERIVATIVE

0 -.-w

Q o AFFECTSTURN COORDINATION CHARACTERISTICS D F.F. CERT o FINAL VALUE NOT CHECKED I N 1 1 I I I -0.01 FLIGHT TEST; IMPACT TOO SMALL . -10 ATP . 10 20 30 40 50 MONTHS : I . : : . i . : .... : _ i . . : . . . . : . .

. . ! : -, 3 . . .

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. . , a ' : C- . . .

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. .

. .

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

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o PRIMARYLATERAL CONTROL . . _ :. 3?-.-+ . 43. ..+ - ... i . . . . . . . . .

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. . . . . . . . . . _ . .

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. . . . . . . . o FINAL VALUE CHECKEO BY FLIGHT C . . . .

- , - - - - - - - -. - . . . . . . . . . . . . . . . . . . . . . . .

I : I : ' ' . Q6 . . . ! . . I . . . . . . . . . . . . . . . . . . . . . . . , . . . . TEST RESULTS; NORMALLY NO . .

. .

i I I ' a~~~~~ . . . . : :. ASSESSMENTOF ACCURACY MADE . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . . . :..:.I D . j : . F.F. CERT " 6 SD NEGLIGIBLE, NOT PLOTTED 0 NEGLIGIBLE IN MANY CASES o FINAL VALUE NOT CHECKED IN FLIGHT TEST; IMPACT TOO SMALL FIGURE 18. HISTORICAL UNCERTAINTIES OF LATERAL-DIRECTIONALAERODYNAMICPARAMETERS I 5 2 C "alp C '6 SECONDARY CONTROL DERIVATIVE S P AFFECTSTURN COORDINATION C CHARACTERISTICS "6 FINAL VALUE NOTCHECKED I N FINAL FLIGHTTEST; IMPACT TOO SMALL . .

. : :. .

. . . . .

. .

:.. .- ............... .- .. .-L. :. .. : .

. ." . . .

MONTHS . .

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. .

. . .

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. .

. . . .

pb SD . . . . . . . , -.

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' o PRIMARY LATERAL CONTROL ' 6 !

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sp : . . ," . . : DERIVATIVE . . . . . .

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C , .

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. .

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: , , : .) ! , : . I . . . . . .

- . . . . . . . . . . . . . . . I . .:.. . .

$pFlNAL .'0.5 1 . : : ASSESSMENT OF ACCURACY MADE . .

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. . . . . . . . . . . . . . . . _ I . . . . . . . .

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CERT : i I : i : . . : ,.I . ' ' ! i . ; I 6 I e -1 0 , -10: i ATP,. -.;. : j 10 ' 20 - 30 ' 40 : 50 MONTHS FIGURE 19. HISTORICAL UNCERTAINTIES O F LATERAL-DIRECTIONAL AERODYNAMIC PARAMETERS ' 3 5 3 Y I 1971 1968 I 1969 I 1970 FIGURE 20. MANUFACTURER'SEMPTYWEIGHTGROWTHHISTORY

1 .o I - " +

0.99 ! ' . .

. .

. .

1 . . : - I

I ' I I , , I F M A M J J

1969 I 1970 197 1

FIGURE 21. MANUFACTURER'SEMPTYWEIGHTGROWTHHISTORY TABLE 9 MAJOR WEIGHT CHANGE SUMMARY (seeFigure 6) A. Revised estimate for wing bending material, increasedallowance for insulation and i n t e r i o r panelsplusmiscellaneous changes.

B. Increased engine weight, revised estimate for wing and landinggear t o provide for an increase i n takeoff gross weightplus miscellaneous changes.

C . Interior design changes, 0.1021~1 (4-inch) fuselagestretch,addition of a f t engine maintenance platformplusrevisedestimates forfuselage, wing and t a i l .

D . Customer requested interior changes plus revisedestimates for miscellaneousstruc- tural and subsystem items.

E . Increased engine weight, revised estimate for wing and l a n d i n g gear t o provide for a n increase i n takeoffgrossweightplus mis- cellaneouschanges.

F . Incbrporated more representative weights

for galleys and passenger seats.

5 6 i TABLE 10 MOMENT OF INERTIA HISTORY Inertia Value a t Program S t a r t Assumed Growth FinalInertia Value 1. Manufacturers Empty .Wei.ght ( M E W ) . . .

Growth of 8.8 percent 0.95 'X/'X Final 0.92 'Y'IY Final 0.93 'Z"Z Final . .

2. Maximum Takeoff Gross Height (MTOGW) Growth of 11 percent and Constant Fuel : 'X'IX Final 0.96 0.88 'Y/'Y Final 0.92 'Z/'Z Final 3. Maximum Takeoff Gross Weight (MTOGd) Growth of 11 percent and ConstantPayload: 0.85 ' X / ' X Final 0.93 'Y/'Y Final 0.89 IZ'IZ Final Finalvalues are those calculated at f i r s t f l i g h t (31 months a f t e r program initiation). The productof i n e r t i a (Ixz) doesnot change significantly.

(a) MAXIMUM TAKEOFF GROSS WEIGHT MONTHS (b) MAXIMUMLANDINGWEIGHT 1.5 MLW

-

1 .o M L W ~ ~ ~ ~ ~ 0.5 (c) MAXIMUM ZEROFUELWEIGHT - . . . . . . . .

. . . . . . . . . . . . . . . - . . . . . . . . . . . .

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D '^ . . . . . . . F.F. . -.. CERT I I1 . 1 . ' , I . I I I ' 0- -10 ATP 10 20 30 40 50 MONTHS FIGURE 22. DESIGNWEIGHTHISTORIES TABLE 11 . . , SYMMETRIC MODAL ANALYSES SUMMARY ~~~ FIGURE NUMBER OF TIME HISTORY ZERO FUEL FULL FUEL

T

GEN GEN MASS MASS FREQ FREQ OEFL MODE DESCRIPTION DEFL .~ P ..69: > 61 21 FIRST WING BENDING 91-96 36 115-120 - - WING ENGINE YAW 62 70 28 37 WING ENGINE PITCH/ 63 71 29 97-102 38 121-126 FUSELAGE BENDING HORIZONTALSTABILIZER BENDING/ - - 64 72 30 39 AFT ENGINE PITCH 65 73 31 WING INNER PANEL TORSION 103-108 40 127-131 - - - - 32 WING ENGINE ROLL 43 66 75 33 SECOND WING BENDING 109-114 42 132-135 - - 67 74 34 WING FORE AN0 AFT BENDING 41 - - - 68 35 WING TORSION/ENGINE PITCH 45 THIRO WING BENDING/ - - - - 76 44 AFT ENGINE PITCH " ~~ _.~____ TABLE 12 ANTISYMMETRIC MODAL ANALYSES SUMMARY . "" . ..

: TIME HISTORY FIGURE NUMBER I ~ ZERO FUEL FULL FUEL - _. ~ - .

GEN G E N MASS MASS DEFL FREQ OEFL FREQ MODE DESCRIPTION . ~ ~ ~ _ _ ~ __ - __ - - 46 WING ENGINE YAW 77 47 166-173 78 136-143 56 FIRST WING BENDING - - - 48 AFT ENGINE YAW 79 49 HORIZONTAL STABILIZER BENDING 174-180 80 144-149 57 50 189-196 81 150-157 59 VERTICALSTABILIZER BENOING AFT FUSELAGE TORSION/ 51 181-188 82 158-165 58 WING ENGINE PITCH - - - 52 AFT FUSELAGE LATERAL BENDING 83 - - 53 84 60 SECOND WING BENDING SECOND VERTICAL STABILIZER - - - BENDING 5.9 0 ws .. .

"-"- . . . " - ... . .

. .

FIGURE 23.. MODE SHAPES AND MODE LINES FIGURE 24. MODE SHAPES AND MODE LINES " . " " 1 ._.

..- - . " -. " - L " . , , " & " " " . . . - ." *.. -.

\-\ \ \ FIGURE 25. MODE SHAPES AND MODE LINES FIGURE 26. MODE SHAPES AND MODE LINES ~ Q\ ' ' .~ .

U FIGURE 2 9 . EVOLUTION OF FREQUENCY RATIO

I - ' "' : I '

I I Y I N G S T I F F H E S S AN0 AFT PYLON STIFFHESS I l l !

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!

FIGURE 32. EVOLUTION OF FREQUENCY RATIO FIGURE 33. EVOLUTION OF FREQUENCY RATIO .,FIGURE 35. EVOLUTION OF FREQUENCY RATIO FIGURE 36. EVOLUTION OF FREQUENCY RATIO .I .

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" FIGURE 38. EVOLUTION OF FREQUENCY RATIO FIGURE 37. EVOLUTION OF FREQUENCY RATIO w ....

....

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FIGURE 40. EVOLUTION OF FREQUENCY RATIO FIGURE 39. EVOLUTION OF FREQUENCY RATIO NG I 1.

FIGURE 41. EVOLUTION OF FREQUENCY RATIO FIGURE 44. EVOLUTION OF FREQUENCY RATIO FIGURE 43. EVOLUTION OF FREQUENCY RATIO !

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, I " 'i .. L

FIGURE 4 6 . EVOLUTION OF FREQUENCY RATIO

A FIGURE 47. EVOLUTION OF FREQUENCY RATIO FIGURE 48. EVOLUTION OF FREQUENCY RATIO L FIGURE 49. EVOLUTION OF FREQUENCY RATIO i 8, !

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i 4..

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m FIGURE 54. EVOLUTION OF FREQUENCY RATIO 4 FIGURE 53. EVOLUTION OF FREQUENCY RATIO B .. .

.: , . .

FIGURE 56. EVOLUTION OF FREQUENCY RATIO . .

FIGURE 55. EVOLUTION OF FREQUENCY RATIO !

FIGURE 57. EVOLUTION OF FREQUENCY RATIO FIGURE 58. EVOLUTION OF FREQUENCY RATIO I , I !

I . I m FIGURE 64. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 65. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 66. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 67. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 68. EVOLUTION O F GENERALIZE0 MASS RATIO

T

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FIGURE 69. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 70. EVOLUTION OF GENERALIZED MASS RATIO ~ ~~ FIGURE 72. EVOLUTION O F GENERALIZEO MASS RATIO FIGURE 71. EVOLUTION O F GENERALIZEO MASS RATIO FIGURE 74. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 76. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 75. EVOLUTION OF GENERALIZED MASS RATIO L .

I

t

FIGURE 78. EVOLUTION OF GENERALIZED MASS RATIO cr, FIGURE 77. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 80. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 79. EVOLUTION OF GENERALIZED MASS RATIO

k G L H E R A L I Z E O

FIGURE 82. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 81. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 84. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 86. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 87. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 88. EVOLUTION OF GENERALIZED MASS RATIO I ~ FIGURE 89. EVOLUTION OF GENERALIZED MASS RATIO FIGURE 90. EVOLUTION OF GENERALIZED MASS RATIO . ' , ' I . , .

I FIGURE 92. EVOLUTION OF DISPLACEMENT R A T I 0 . h DlSPLACElERT RAT1 0 Ih/h,,) ! i ' !

. . . I . . I . : I ! ' ' : - '

T I M c m n w --

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FIGURE 94. EVOLUTION OF DISP,LACEMENT RATIO 01 4 FIGURE 93. EVOLUTION OF DISPLACEMENT RATIO h i 1 .I I I !

1.2 1 1 ; I I I C b ' .I I I . 2 !

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TM .mms I; I 1 , . ' I ! . ~ I . .

FIGURE 95. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 96. EVOLUTION OF DISPLACEMENT RATIO a . . .

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Y l N G PYLON nlFFNLSS CYGlRf YEIWT M D FUSEUGC STlFFkSS . . . .

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HG STIFFNESS M O Y T PILI% STIFFNESS 4 Y NG EllGlNE YElCHT L H O YlNG PYLON STIFFNESS , : ' ~ Y NG STIFFNISS ' . i ' "

I

Y I IG STIFFNCSS M O FUSELAGL STlFFNESS

1 U1 G STIFFNESS

UING STIFFNESS

YIII PVLoll STIFFNESS !

AFT E N G l l f UEIWT AHD PYLON STIFFMES!

C.fRTIFIUTION (GYT!

1 l , , , , , , * . .

' 0 2 4 6 8 10 I2 14 I6 11 20 22 24 26 28 XI 32 y y

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FIGURE 97. EVOLUTION OF DISPLACEMENT RATIO C Y FIGURE 98. EVOLUTION OF DISPLACEMENT RATIO h I , .4 . . . .

, . . .

I I / , o 2 4 L I I O I Z 1 4 1 6 18 zo n 2 1 % a 1 0 3 2 Y x TIM-IOITM P i ' . . . .

FIGURE 100. EVOLUTION 0F.DISPLACEMENT RATIO 01 ' ' .

. I , . ..

FIGURE 102. EVOLUTION OF DISPLACEMENT RATIO Cll . .

SrlfFnfss' : .I I .

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: ! 0 2 4 I I 1 0 1 2 1 4 1 6 1$ 20 22 2 4 2 6 Z a 3 0 3 2 Y X !

. - TIM %m~r)15 -_I : . j . . I . ! I I FIGURE 106. EVOLUTION OF DISPLACEMENT RATIO 01 FIGURE 105. EVOLUTION OF DISPLACEMENT RATIO h I FIGURE 107. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 108. EWOLUTION OF OISPLACEMENT RATIO 01 m FIGURE 109. EVOLUTION OF OISPLACEMENT RATIO a +- FIGURE 110. EVOLUTION OF DISPLACEMENT RATIO h m Iu

-I-

- 1 .

- 1 .

. 1~ FIGURE 111. EVOLUTION OF OISPLACEMENT RATIO h FIGURE 112. EVOLUTION OF DISPLACEMENT RATIO 01 FIGURE 113. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 114. EVOLUTION O F DISPLACEMENT RATIO 01 FIGURE 115. EVOLUTION OF OISPLACEMENT RATIO a FIGURE 116. EVOLUTION OF OISPLAC€M€NT RATIO h FIGURE 117. EWOLUTION OF DISPLACEMENT RATIO WING h FIGURE 118. EVOLUTION OF DISPLACEMENT RATIO WING (I FIGURE 119. EVOLUTION O F DISPLACEMENTRATIO h FIGURE 120. EVOLUTION O F DISPLACEMENT RATIO a 8 ' I 1 - - 1 - I I . I-

t

FIGURE 121. EVOLUTION O F O!SPLACEMENT RATIO u FIGURE 122. EVOLUTION OF DISPLACEMENT RATIO h ~~~ FIGURE 124. EVOLUTION OF DISPLACEMENT RATIO a FIGURE.123. EVOLUTION OF OlSPLiACEPdlENT RATIO h CD VI FIGURE 125. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 126. EVOLUTION OF DISPLACEMENT RATIO a t ' . 6 I ' , 1,111. nlllsr , , , , . . , , . . . .

FIGURE 127. EVOLTUION O F FUSELAGE Q DISPLACEMENTRATIO FIGURE 128. EVOLUTIOPd OF DlSPLACEMEPdT RATIO h I .

FIGURE 132. EVOLUTION OF DISPLACEMENT RATIO a FIGURE 133. EVOLUTION OF DISPLACEMEBIT RATIO h FIGURE 134. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 136. EVOLUTION OF DISPLACEMENT RATIO 0 FIGURE 135. EVOLUTION OF OISPLACEMEMT RATIO c t FIGURE 137. EVOLUTION OF DISPLACEMENT RATIO $ FIGURE 138. EVOLUTION OF DISPLACEMENT RATIO h FIGUR€ 139. EVOLUTION OF DISPLACEMENT RATIO c t FIGURE 140. EVOLUTION OF DISPLACEMENT RATIO R

i

m FIGURE 141. EVOLUTION OF DISPLACEMENT RATIO a FIGURE 142. EWOLUTIORI OF DISPLACEMENT RATIOP a FIGURE 143. EVOLUTION O F DISPLACEMENT RATIO $ FIGURE 145. EVOLUTION OF DISPLACEMENT RATIO a FIGURE 146. EVOLUTION O F DISPLACEMENT RATIO h FIGURE 147. EVOLUTION OF DISPLACEMENT RATIO (Y FIGURE 148. EVOLUTION OF DISPLACEMENT RATIO II FIGURE 150. EVOLUTION OF DISPLACEMENT RATIO e

P

FIGURE 152. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 154. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 155. EVOLUTION OF DISPLACEMENT RATIO 01 FIGURE 156. EVOLUTION OF DISPLACEMENT RATIO L w .. - . . FIGURE,.157. EVOLUTION OFDISPLACEMENT RATIO + FIGURE 158. EVOLUTION OF.DISPLACEMENT RATIO e .: . . - : ..

FIGURE 161. EVOLUTION OF DISPLACEMENT RATIO C Y FIGURE 162. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 163. EVOLUTION OF DISPLACEMENT RATIO (Y FIGURE 166. EVOLUTION OF DISPLACEMENT RATIO e FIGURE 165. EVOLUTION OF DISPLACEMENT RATIO -4 FIGURE 170. EVOLUTION OF DISPLACEMENT RATIO h b :.I FIGURE 171. EVOLUTION OF DISPLACEMENT RATIO CY I FIGURE 174. EVOLUTION OF DISPLACEMENT RATIO e FIGURE 176. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 175. EVOLUTION OF DISPLACEMENT RATIO J/ FIGURE 177. EVOLUTION OF DISPLACEMENT RATIO a FIGURE 178. EVOLUTION OF DISPLACEMENT RATIO h

-

FIGURE 179. EVOLUTION OF DISPLACEMENT R A T FIGURE 180. EVOLUTION OF DISPLACEMENT RATIO rL i - . . 9 :- 9 . . FIGURE-l81..' EVOLUTION OF DlSPLACEMENT RATIO 0 :.

FIGURE 184. EVOLUTION OF DISPLACEMENT RATIO a . .

I.

FIGURE 185. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 186. EVOLUTION OF DISPLACEMENT RATIO a ...

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. , FIGURE 187. EVOLUTION OF DISPLACEMENT RATIO II FIGURE 188. EVOLUTION OF DISPLACEMENT RATIO J, . ! ; ; , I ' I . , . . . . 4 . .

, .

. .

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I.

I FIGURE 190. EVOLUTION OF DISPLACEMENT RATIO J, ' j $?

: ' !.

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. . .

STIFFIIESS I 2 1 6 d 10 12 14 16 I( 20 22 2 1 26 28 YI 32 Y 1 2 " I m m s ' : ..... I ,..I..

FIGURE 191. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 192. EVOLUTION OF DISPLACEMENT RATIO CY FIGURE 193. EVOLUTION OF DISPLACEMENT RATIO h FIGURE 194. EVOLUTION OF DISPLACEMENT RATIO CY . .

!. .

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FIGURE 195. EVOLUTION OF DISPLACEMENT RATIO II FIGURE 196. EVOLUTION OF DISPLACEMENT RATIO JI

SECTION 4 S O M E POSSIBLE APPLICATIONS A s noted previously,thisreport is intended to provide a data base forthe control system designer so t h a t he can estimatethelikelihoodthat an advanced system will provide i t s intendedfunction a t f i r s t f l i g h t i n the presence of uncertainties i n plant and control system dynamics. The specific application of these d a t a will depend on thedesigner'sfavoriteanalytical t o o l s and thegeneraldesign approach used by a manufacturer. T h i s section willsuggest a few ways of using thesedata. As will be apparent,theassess- ment of s t a b i l i t y margins will be reasonablystraight-forward, b u t the estimation o f flyingqualitiesisnotquiteas easy.Reference 4 gives some results of recent t r a n s p o r t simulatorhandlingqualitiestests.Itis apparent that conflict exists between various popular handling qualities measures, so the system designer will need t o use severalyardsticksto measure the effects of thesereportedparametervariations.

CLASSICAL SERVO ANALYSIS APPLICATIONS Normally the system designerusesclassicalanalysis methods t o a f i r s t - c u t definition o f asystem based on nominal aerodynamics and an exactspecifica- t i o n o f control dynamics. A second iteration then considersthe change in closed-loop dynamics and system performance due t o p l a n t and controller parameter v a r i a t i o n . The classicalanalyst depends on thevisibility.of systemcharacteristics v i a root-locus, Bode, and time historyplots t o a i d . h i m i n the system design, so i t would be desirable t o reflect the effects of parameter variabi 1 i t y graphical ly.

If w e consider a simple pitch a x i s problem, the aircraft nominal open-loop dynamics m i g h t appear as shown i n Figure 197.

The tuck/subsidence,short-period,actuator and f i r s t s t r u c t u r a l modes (nominal 1 are shown along w i t h thepitchtoelevator numerator. The control system designer may be interested i n an attitude-hold system for this air- c r a f t and would then make the usual rateplus p o s i t i o n closureas shown i n , Figure 198.

. .

FIRST .

, STRUCTURAL MODE . .

. .

. . I . .

ACTUATOR.DYNAMICS . ' .

X SHORT-PERIOD MODE X FIGURE 197. TYPICALLONGITUDINALROOTLOCATION The nominal closed-loop system i s well behaved and performs t o specification, 6 u t theeffects of variations i n the system parametersare not obvious.

Past work has establishedthe mathematics for assessing the effects of uncertainties on closed-loopcharacteristics (e.g., Reference 8) b u t these ' methods have been so tedious f o r higherordersystems that they have not been generallyapplied. Computer capacity and speed are now sufficient t o allow sensitivity methods such asthosedeveloped i n reference 8 t o be programined.

A Monte Carlo parameter variational approach could be used t o establish the, "envelopes" o f open-loop and closed-loop system parameters for expectedplant parametervariationsassketched i n Figure 199.

I / -h * W ............

. . FIGURE 198. TYPICAL LONGITUDINAL FIGURE 199. CLOSED-LOOP ROOT SYSTEM ROOT LOCUS . VARIATION . , - - ~ ~

I

The designer'stask will be t o design the systemsuch t h a t rmin < r, <max, , etc.,tosatisfythebasicflyingqualities and structuralcriteria. Where the modes being stabilized are potentially "flight critical" (e.g., the first- orderdivergence o r second-order low-damped modes shown i n Figure 199 could possibly result i n unacceptable characteristics), the designer will take a1 1 possibleassurancesthatthetotalpilot/vehicle system will perform reasonably well beforethe first flight opportunity arrives.

STATE-SPACE SYNTHESIS METHODS In an i n i t i a l design exercise,state-space methods are now often used i n the form of linear-quadratic-Gaussian synthesis procedures forquicklyidentifying candidatecontrol systems (References 5 and 6). W i t h estimates of plant uncertaintiesavailable,state-space methods can be employed t o determine eigenvalue s e n s i t i v i t i e s i n a d d i t i o n t o time response and frequencyresponse sensitivitiesalreadydiscussed.Sensitivities may be determined for individual as well as statistically selected combinations of parameter variations.Alternativecontrol system configurations which yieldapproxi- mately equivalent dynamic performance characteristics are easily synthesized using state-space methods. The sensitivity of these systems t o p l a n t param- eter varjitions . , o f f e r c r i t e r i a f o r selection of a particular configuration.

E i gehval ue sensitivities can be obtained from a pole Placement synthesis program which employs a f i r s t orderperturbation approximation t o relate chinges. i n . thecl'osed-loop system coefficientmatrix, A , t o changes i n the system eigenvalues , A.

The matrixequation employed i s : D A a = Where D is the- eigenvalue sensitivity matrix, A a is a vector consisting of the system parameter variations, and A A is a vectorconsisting of theeigenvalue changes. The s e n s i t i v i t i e s of theeigenvalues t o parameter variations from the nominal values may be determinedmerely by examining the matrix, D. D would be calculatedforvariouscontrolconfigurations, and would also be recalculatedforcases where the parameters arevaried from the nominal values. ' -[Note . t h a t ,Aa consists .of the non-zero elements of the matrix A A ) .

by c a l c u l a t i n g D using the The o f f - n o m i n a ls e n s i t i v i t y matrix is obtained the ,nominal matrix matrix (A + AA) r a t h e r t h a n A. 1 PARAMETER IDENTIFICATION AND ADAPTIVE CONTROL APPROACH- Yet another system designphilosophycould be t o conduct an exercise that shows the aircraft/controlsystemhasclosed-loopdynamicsmeeting the d e s i g n e r ' s s p e c i f i c a t i o n f o r the entire rangeofplantdynamicsdescribed by the plantparameters w i t h their expectedvariation. Such a systemmight be requiredtoconverge w i t h i n a fixedperiodof time (whichwould be much less than the time f o r c o n t r o l problems t od e v e l o p ) fromanyand a l l c o r n e r s o f the open-loopdynamicenvelopeof Figure 199,and a t a l l f l i g h t c o n d i t i o n s . The approach would represent a s i g n i f i c a n td e p a r t u r e from the previous two approaches where c o n t r o l system g a i n s a r e assumed fixed a t a s p e c i f i c f l i g h t condition.

Sucha systemdesign wouldemployparameteridentificationtechniquesto determine plantparameters and t o update the controllaws i n r e s p o n s et o on- line measurementsof the p l a n t s t a t e s . A comprehensivebibliography on the system i d e n t i f i c a t i o n problem is contained i n Reference 7.

POSSIBLE STATISTICAL USAGE OF STRUCTURAL DYNAMIC RESULTS The p r e v i o u s l yd e s c r i b e ds t u d yu t i l i z e so n l yo n ea i r p l a n ea s a b a s i s f o r parametervariabilitydetermination. The l a c ko fd a t af o r a l a r g e number of a i r p l a n e s is compensated t o a g r e a t extent by the considerationofparameter v a r i a b i 1 i t y w i t h i n a reasonably large group of different flexible modes.

Foreach modal parameter, ameasureof v a r i a b i l i t y can be e s t a b l i s h e d by bringing the results t o g e t h e r fromeachof the modes. Let rij ( t ) d e n o t e the value a t time t ofanyof the p a r a m e t e rr a t i o s expressed i n the p l o t s . The s u b s c r i p t s i n d i c a t e t h a t the result f o r the i - t h parameter is beingconsidered f o r the j - t h f l e x i b l e mode and r i j ( t ) + ' l a s t + T, where T denotes the d u r a t i o n st oc e r t i f i c a t i o n . If b i ( t )s i g n i f i e s the averageof the logarithms of the i - t h parametervaluesover the modes ( i - e . , o v e r j ) , then M

b i ( t ) = - z l o g r. . ( t )

j=1. 1 J M is the number of modes presented i n the plots i n a definedcategory (e.g.

symmetric, zero fuel ) and bi (t) is the bias tendency of the 1 og of the i - t h parameter a t time t i n thedesigncycle, Then thevariability can be measured by vi (t) , where Having thebias and variability estimates f0.r each parameter,combinations of parametermagnitudes can be selectedforinvestigatingthecontrol system design. In studyingtheairplane performance tocontrol and externalinputs, the parameters used fortheairplane can be selectedasfollows:If p . .(t) 1 J is thevalue determined a t t h e t for the i - t h parameter of the j - t h mode,

thena s e t of nominal parameter values n . . (t) can,,be obtained as

1 J *

log n . . ( t ) = log p i j ( t ) - b i ( t ) ; i = 1 , ..., N

1 J *

j = 1 , ..., M

(The number of airplane flexible modes M* used forthecontrol system design may be different t h a n the number M defined above. N* i s the number of parameters pertinent to the M* modes).

The determination of parametercombinations from the P = M* N* parameters can be done asfollows: In one combination a l l parameters can be s e t a t their

nominal values . For theother 2P combinations, ( P - 1 ) of theoarameters

a r e s e t a t t h e i r nominal valueswhilethe remainingparameter i s s e t a t each of i t s extremes. T h i s simple scheme gives 2P + 1 parametercombinations.

In apreviousdiscussion, i t is indicated t h a t a Monte Carloprocedure can be used i n selectingthe parameters. In this approach i t can be assumed t h a t the log r C t ) have independent normal distri butions w i t h mean bi (t) and s t a n d a r d i j

devi ation vi (t). From a table o f random numbers, values of log r. . (t) are

1 J drawn for eachof the P parameters. Now pi j(t) i s thevalue of theparameter ~ . .

. .

@ P w . . . ' - * . . ' determined \ . a t time ,t i n the design , . . I cycle, . a : . , L e t p r . ( t ) be a randomly deter- 1 J mined wlue -. ' f o r ,. , -the ~ sqrne. . . - parameter,. . .This. ..yal.ue.can be found as follows 'from the . .,:;_. , , , . .

randomly selected r. .(t) value: .. .

. . 1. J . > . 8 , . . . . .

. . .

. .

where. n1 j&) ds previouslydefined.. The. number ofcombinationsrequired. for a Monte Carloapproach will a t l e a s t be as- large .as that used..Qn the above simple scheme.

The usages o f the .plotted results t h a t have been discussed abovecan be further clarified by consideringthefollowingairpqane openloop differential equations . f o r the fqexible modes: The..equation for.thej-th,mode, i n s-mbolic . .

. , / . ' . , \ . . - , . i . .

form, is x . , . .

1 " I .

, , ' .

(6 .j = 1 f o r a = j ; = 0 otherwise) - j = .1; ..:., M*

1..

, I This second orderequation 'can be converted into two.first orderequationsfor thestate-spaceformulation. T h i s transformationis.straight-forward and,wi11 not be done here. The parametersexpressed i n .the above equationsrepresent . . . .

thefollowingforthe j - t h mode: , .

-

m - generalized mass

j - c r l t i c a l damping factor : .

. .

f j - naturalfrequency

? ' ' :

IBj - modal coordinate a t location

. .

- .

. !

- .

. . . , . .

.' - . .

a , v .Cja, 'ja:. C d , j a - 'genera1ized"unsteady aerodynamic c o e f f i c i e n t m a t r i x . . elements . . as.s,ociated,' respectively, with aerodynamic . . . ;.. . . . - . . forces produced, by a c c e l e r a t i o n s ,v e l o c i t i e s and . .

. , . di,spl acements .

. . ' B P I . .

The time-dependent q u a n t i t y q,(,t)' i s the" general i zed displacement f o r t h e C Y -th mode. d a ( t ) and q J t ) a r e t h e a s s o c i a t e d . f i r s t .andsecond timederivatives.

F g ( t ) is.-an external- time-varying' force acting on the mode shapes, a t the . ' number- 8 modal coordinate. ' .

Thenumber, A, of.modes over, which the left-hand side sumncit.ion ' i s conducted e q u a l st h en u m b e r . o ff l e x i b l ep l u sr i g i d bodyminodesbeingconsidered. The numljer . o f modal. coordinates B .for the right-hand side sumnation i s given by t h e s i z e o f t h e modal ' eigenvectors. Thenumber of parameters N* necessary t o define each modal equation i s a t most 3 + 3A + B . (A modal coordinate b i s n o t needed i f no f o r c e i s a s s o c i a t e d w i t h it).

The N* parameters f o r t h e . j - t h mode can be placed i n t h e j - t h column o f a matrlxarray. The i - t h rowelement o f t h i s column i s theparameterquantity , . I previouslydefined.Sincethe'parameterspertaintoaparticulartime t , - P i j i n t h e design,theyaredesignated as p i j(t). No data has been presented I n t h i s . r e p o r t f o r theparameterpij(t) = 5 . t o c a l c u l a t e i t s b i a s o r v a r i a b i l i . t y .

J I t s v a l u e s e l e c t e d f o r any c o n t r o l system study w i l l t h e n r e m a i n f i x e d a t i t s determinedvalue. A s i m i l a rs t a t e m e n ta p p l i e st o ,e a c h . o ft h ec o e f f i c i e n t s C& C j o V and Cja. Thus o f t h e p o s s i b l e . s e t of N* parameters f o r t h e j - t h w i l l provide'bias and v a r i a b i l t y modal equation,theresults i n thegivenplots i n f o r m a t i o n f o r a t most 2 + 6 of the parameters.

I .

CONCLUSIONS AND 'RECOMMENDATIONS: .. ' Base& upon the uncertainty investigation over the design period of a single l a r g e t r a n s p o r t a i r c r a f t t h e f o l l o w i n g c o n c l u s i o n s - a r e made: ? .

1. The l a r g e s t v a r i a t i o n i n modal 'parametersoccursduringtheearlydesign . .

'. phase, reflecting major .design changes.

2. .Analysesperformed a f t e r t h e ' e a r l y d e s i g n phase show l e s s v a r i a t i o n i n modal parameters, r e f l e c t i n g mi'nor m o d i f i c a t i o n t o . t h e a i r p l a n e d e s i g n data.

. , . . .- 111 , .

. . . . .

, . . .

. I 3. No discernable trend was seen i n the parameter uncertainties between fuel condition or airplane symnetry.

4. The modal generalized mass and displacementparameters show much larger variatfons than do the modal frequencyparameters and are of limited use exceptas i n p u t to further responseanalyses.

5. The magnitude of the i n p u t modal displacement r a t i o shows l i t t l e or no correlation to the magnitude of the output response acceleration ratio.

6. B y the time thebasic design characteristics have stabi.lized,thestruc- tural dynamic modal characteristics can be predictedadequatelyfor use i n thecontrol system design.

7. The generalized mass and modal amplitudes, by themselves, do notprovide i n s i g h t t o the airplane flight response characteristics.

The modal parameter uncertaintyresultspresentedhereinrepresent a starting point for furtherstudies. Recommendations areasfollows: 1. The preferredlocationofsensors is often a t nodal orantinodalpoints.

The uncertainty of thesepointsshould be determined toenable optimum sensor placement a t an early part of the design cycle.

2. The effect of variations i n airplane(i.e.plant) parameters on airplane f l i g h t controlshould be evaluated i n a follow-on phase. The parameters whose varlations produce the mostpronounced effects should be identified.

Controlsystem designs shouldthen be studied todeterminetheextentto which thesesensitivities can be reduced. The properpositioning of sensors and controlsurfaces would be p a r t of this studyalong w i t h the design of thecontrol system.

3. The effect of unsteady aerodynamics uncertainties should be investigated for the purpose of notonly improving control system functions b u t for theimnediate problem of f l u t t e r prevention.

. APPENDIX A . .

BODY AXES EQUATIONS OF"MOTION . ' . .

. .

Longitudinal Set: . 1 ir = - woq - ge COS e, + xuu + x q + \W + b t , + x 6 , 6 , q . .

. , .: ~

9 = Uoq - gQ sin 8, + Zuu + Z q + Zww.+ ZQa +..Z6e6ee' . . . I

q . . . I . .

_ . . . _ 4 = MUu + M q + MWw + M# + M A , & , . . . L , .

Lateral-Directional Set: + = - Uor + Wop + g + sin Q0 + G $ cos 8, + Yrr + , Y v v . + . Y p ' : . , P . .

+ Y 6 , 6 , + YQr + Y , 6 S P S P = i- + Lrr + L , v + L p + LGaSa + L6k6r + L6 6 I X P S P S P I i- = + NVv + Nrr + N p + N 6 , 6 , + N , 6 . + N 6 s p ~ s p P r r z

APPENDIX B DEFINITION OF COEFFICIENTS' OF EQUATIONS. OF HOTIQN (DIHENSIWALSTABILITYDERIVATIVES) PUSb' c = a . U

Yr P -

*O 0 0 h . Y r

- PUS .'PUS

Yv = -

CY ' m

xu - - (-% -

3- P U S c 4m D , . , xq PUS

xw = -

2 m (CL - C D , )

- P S F C

xu - - -

4m D b * pUSb = -

N" 212 n

PiJSb2 : Nr 4IZ 'nr .. I , .1 1 5 " P USb2 - NP 41z ' n p t ' , pU2Sb N, =- r 2 1 ~ 'n, r pU2Sb

"' , a = -"n 21Z

PU2Sb PU2SE =- - M6 -- 21z Ct16 N, SP e 2 1 ~ 'm6 e SP pU2Sb " L€i - S P 21x SP PU2S " - y€i - 2 m F, SP ..

SP REFERENCES 1. F i n c k , R.D.; E l l i s o n , D.E.; and Malthan, L.V., e t al.: USAF S t a b i l i t y And Control Datcom. October1960,revisedJanuary 1974.

2. DeYoung, J.; and Harper, C.W.: "Theoretical Span Loading a t Subsonic Speeds f o r Wings Having Arbitrary Plan Foim." NACA TR-921 1948. . ' 3. Hedman, S.G.: "Vortex Lattice Method forCalculationofQuasi_Steady S t a t e Loadings on T h i n E l a s t i c Wings i n Subsonic Flow." Report..l05, . .

The AeronauticalResearchInstitute of Sweden, October 1965. .

4. Rickard, W.W.: "Longitudinal Flying Qualities i n the Landing Approach."

Proceedingsof the 1 2 t h Annual Conference on Manual Control , May 1976.

6. Markland, C.A.: "Optimal Model-Following Control System Synthesis Techniques." Proc. I.E.E., Vol. 117, No. 3, March 1970. - 8. McRuer, D.J.; and S t a p l e f o r d , R.L.: ' S e n s i t i v i t y and Modal Response f o r Single-Loop and Mu1 ti-LoopSystems." ASD-TDR-62-812, January 1963.

9. Vandierendonck, A.J.: "Design Method f o rF u l l y Augmented Systems f o r Variable Flight Conditions." AFFDL-TR-71-152, January 1972.