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Analysis of flexible aircraft longitudinal dynamics and handling qualities. Volume 2: Data

19850026890 · NASA · 1985

Public domain · NASATechnical Reports

Overview

Two analysis methods are applied to a family of flexible aircraft in order to investigate how and when structural (especially dynamic aeroelastic) effects affect the dynamic characteristics of aircraft. The first type of analysis is an open loop modal analysis technique. This method considers the…

Publisher
NASA
Document
19850026890
Year
1985
Pages
240
Chapters
2

Key points

  • This document is Volume II of a technical report analyzing flexible aircraft longitudinal dynamics and handling qualities.
  • The research was conducted by Martin R. Waszak and David K. Schmidt at Purdue University and supported by NASA Langley Research Center.
  • The document includes experimental data, modal analysis, and closed-loop analysis of various aircraft configurations.
  • It contains detailed appendices with vehicle configurations, modal analysis results, and frequency response data.
  • The report is unclassified and was published in June 1985.
Frequently asked questions
What is the focus of this document?

The document focuses on the analysis of flexible aircraft longitudinal dynamics and handling qualities.

Who authored this technical report?

The report was authored by Martin R. Waszak and David K. Schmidt from Purdue University.

What type of data does Volume II contain?

Volume II contains experimental data, modal analysis results, and closed-loop analysis of various aircraft configurations.

When was this document published?

The document was published in June 1985.

Is this document classified?

No, the report is unclassified.

Appendix A.3

9 6

Appendix A.3

Vehicle Configurations

• and

Modal Analysi s Results

Th e vehicle c on fi gu r a ti ons a r e pr e se n t e d i n s t ate va r iabl e fo r m u s ing t h e following stat e and output d e finitions.

Xv T _ [O _, _ , U ; O R , 7 1 , 7 2 , 7 3 , 7 4 , _ ! ' _ 2 ' _ 3 ' _41, (A . 3 .1_

y _ T __ A [ u , 0 R , _ , " _, 0 T, 0 T , nz ] - ( A . 3 . 2)

- w he re all an g l e s are e xpr e s sed in r a di a ns, all an g ul a r rates _ n ra dian s per sp , , , _ n , ._, t h e fo, 'w ar d velo c ity pertur b a t ion , u, h as units of feet per s _ , cond and .th e gener aliz e d e l ast ic d efle c tions, _ / , a r e dimensionless . T he matrices w hic h de s cri b e t he ve h i c l e confi g ur a tions are defined in t h e f ollo w ing equations .

_v : [AV] _ v + [BI Xp + [D] x s

(A.3 . 3)

• _ : [C i x v + IE ] Xp + [F ] x_ _g

4 Th e ve c tors, Xpa nd x g, a r e de fi ned Chapt e r 4.

Th e result s o f th e modal anal y sis c ot_sis t of s evera l para m eters w hich are ..

pr esented i n ma t rix f orm. The eq ua t ion s w h ic h f o ll ow pre se nt the d efin i t ion s _ of th es e mod a l a na l y si s re sul ts .

The similarity transformation which was used to scale the ei g envectors, as

d escr ibe d i n A p pen di x A . I , has the fo llo wing d efin i t i on .

' 97

_L

T : SCALING I_£6,TRIX

A diag[l 1, 1 _ r 0'

: ' _0"0' 1, 1, 2,

_ '3 , _ ' ,,, 0' _ ' 0' 0' l, 2 , 3, 4 ] 1 A . 3. 4) '

z , The s c aled system is therefore des c ribed by the f o ll o wing set of equations.

_-v = Tx v ( A.3.5)

__v - - T[AV ] T l_ v + T[Blx p + T[D]_

= [AV - SC A L ED ] _v

+ [ B- S CALED] _ _ v + [ D-SCALED]x s (A. 3 .6)

X : [ C I T q_ + [E ]xp + [ F] _

: [ C-SCALED ]_i v + [E]x p + [ F]x s (A. 3. 7)

The pi l ot l ag s tate s a nd g u s t s tates ar e de fi ned in the f o l lo w in g e q ua tio n s . .

_.p = [ AP] xp + [ Gp] / ] p (A . 3 . 8)

i s - { AG ] X s + [ G s ] r/s (A. 3 . 9 )

The complete a i r craft system stat e mode l used in the moda l analysis is d e fined

in the followin g equations.

,x_.. T -_ [ Z p T' .x s T'X vT ] ( A .3. 1 0)

9 8

- [AG] 0 x

[,_1 0 0 1

lIB-SCALED] [D-SCALEDl [AV-SCALED]

+ t / p + s T / _

Ii p ] I:l

= [AAIx + [GPlr l p + [GG ]_ 7 _ (A . 3 . 11)

y. = [[ E ] [ F ] [ C - SC AL ED ] ]x

= [CC] x (A.3.12)

Th e mod a l c o ntrolla b ility , dist u r bab ility a nd ob s e rv ab ility m atric e s ar e d e fined in t h e foll ow ing equations .

[EEl A _ modal mat r ix o f [ A.A I ( A . 3 . 13)

T he. m od al c o ntr o lla b ili ty m a t r ix,

[C O N] A [EEl I[ Gpl . ( A . 3 . 14)

" The m_)d a l di sl u r b abil ity m a t r i x ,

, {DIS T I A= [E El '[ GG] . ( A . 3 . 15 )

And, the modal obse r vability m a trix,

Io n s ] _ [C C l [E E [. ( A .3. 16 )

The r e s idu e s fo r th e c o m ph,te ai r c raf t m o de l ar e defined a s fol l ows. T h e I r e s idue s of the ve h icle due t o t h e i t h c on t ro l input a r e desc r ibed by,

[ C-IMP RESl i = A[[ 0 1 38 1 [C O N ]i ] r ( A. 3 . 1 7 )

- whe r e [ C O N ]i is the diagonal mat r ix f o r med by the ith column o f the

mod a l co ntrolla b ility m at rix a nd lOBS] i s t he modal o bse rva b ili t y mat r i x.

Ea ch k j e l eme n t of [C-IMP RE S ] i c orr es pon d s t o th e r es idue of t h e kt h mod e for the j t h outpu t d u e to an impul se in t h e i th c ontr o l inp u t . Similarly, fo r t h e d i st ur b an ce r es idu es,

[D- IM P RES]i _,-A [l O BS] [DIS Tli ] T (A . 3 . 18)

llere, [ D I S T ] i is t he di ag;o nal matrix fo rmed b y the ith c o lum n of the modal

d i s tur b a b ili t y matrix a n d s o e a ch k j e l e m e nt of [D - IM P RESi l c orr es p o nd s t o

t he res i due of the kt h mod e f o r j t h outpu t d ue t o a n i m pulse in t h e it h

d i s tur b an ce inpu t.

T he r e mai n d e r of App e ndi x A .3 c ontains the num er i c al r e sul t s of the

modal a n aly s is and t h e ve h i c le matri c es f o r eig ht dat a b a se c on fig ur at ions as desc ri bed a b ove .

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2 , 15

Append i x A.4

Aeroelast i c - Structural Mode Shapes

The mode shapes of the data base configurations are defined to be

( l ['fl( , ('ti ,, n_ in fee t n¢) r m_ . !i z ed so t hat fo r e ach mode t h e deflection at t h e n()s( , (i._'. F.S. 0) is (me f(),)t. The deflecti( , ns are <lelin_,<!relati v e to a set of

1,<)<1_ :_\( , s <leline<l in ihe f_,ll_)win g way. The x-axis lies along the longitudinal

nxi_ _,f th{ , v{ ' hi(' l e and is di r ( ' cl_ , (I 5) r wa r d. The ) '- axis lies in I l l( , plane of the _in g, ,. i_ (_rth. g _ , nal to the x -; ,xis and points out the ri g ht win g . The z - axis is orihc, g , m : )l i() the xy-plane and is di r ected downwa r d. The o r i g in of the xyz- c()()rdin:lt_ , sysl( , m is defined tt) be the cente r of gr avity of the ai r c r aft.

The defieciion definilions and their si g n conventi_)ns that a r e used in thi:_

siudy are defined in Fi g u r e A..I.I. The mode shapes fo r the fou r I()n g itudinal

mo(h , s u_( , d in this study a r e p r esenlcd in Fi g u r es A..l.2 th r ou g h A . .t.5.

.!

t

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l

X1ODL' sI I AI ' E sIGN CONV E N T IONS

+X

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n or mal i z ing po i nt

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O , - _ mod e d efle c t io r t ( fe et )

¢' , _ mod e _ lop e {f ee t / f ee t)

1 / ,_ _ g en e ra l ize d d e flec tio n (d i m e n s i on le ss )

ID

F i g u r e A.. t . l

M o d e Sha pe Sig n C o n ve nt ions

24 7

t

M OD E 1

" deSectio _ a r e nor m al iz edto a onefoot d i splacement a t the nose

(

Fi gure A,4. 2

A e roe l a s t i c -St r ucturaj M ode Shape. M o d e 1

MODE 2

" __.._ _ f us el a g e stat i on (in)

• 0o.! '*- ___ - __ " , _ oo , _

[ 200 4 00 ¢00 -- _ 80 0 _ - I{ _0"_ 1 2 0 0 ' - 2 -0 '

\

0 -- - *. I \

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; - I00

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- I _ 0 -

" deflecti on s a re normalized to a o n e fo o t d is placemen t a t t h e no s e

Figu re A. 4.3

Aeroela_ tic-Structural Mode Shspe - M ode 2

J

MODE 3

!0

0 -- fu,-_I z| t

D _ w ing

/ _ -- t,4

-\ ! t ! t t ! _. _ .]_ t

¢ m. _ O0 _ 0 _ _ 00 t OO 1106 !000 1200

-11 2 -

" 4 } ,_.

- 06

• deflections are normalized to a one foot displacement a t the nos e

Fig ure A . 4 . 4

Aer o e l a sti c- S truct u ra l Mod e Shape- Mod e 3

2 5 O

MODE 4

• de fl e ction s ar e norm a l i zed to a one fo ot di s placement a t the ne se

t

Figu r e A .4.5

A e r oe l asti c -St r uctural M o de Sh a p e - Mode 4

I 2 5 1

Appendi x 4. 5

OCM Frequency Responses

l J

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F izu re A .5. 1 (c o n 't)

OCM F re qu e n c y [ { e sponse s - Con l _Z.!

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ii tO'e " iO- t I0 o I0 * I0 e

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F i g u re A. 5 . 2( con' t)

O CM Fr e quen cy Re s p o nses - C on fig . 2

2 58

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OCM Fre qu ency Re s pon s e s - Co u55. 2

1985026890-177

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O CM Fr eq u en c y Respo n se s - Co n fig. 3

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1985026890-17

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; O CM F r e que ncy Response s - ConfilL 3

FiKure A. 5.3 (con ' t)

...... "" 1985026890-17_

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CFtSE 3"

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P ILOT C O_ P ENSRIION -96.02 0 E G

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F i gure A .5. 3 ( co u 't}

OC M Fr e quency R espons e s - Confi _ , 3

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W(RRD I SEC }

Figure A. 5 . 4

O C M Fre q u e n c y Re s ponses - C o t_fi g . 4

,o.= CFtSE q :

AIRCRAFT(O.L.) e_ ( s____ ] ._

6 is)

IO -_ I0-I 1 0 o I 0 I 10 _

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BAND W IDTH 1.7 ? RAD I SEC

PIL O T COMPENSRTION - 1 6.08DEG

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SENSITIVITY .78 DB

I

|

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F i gur e A . 5 . 4 ( t on ' t )

OC M F req u ency R e sp on s es - C on fi g . 4

I

:266

" CASE4 :

QIRCRRFT P LU S PILOT(O . L o ) _'ms_

_.O0'

• . -. -, 00 '

Q

: _- 20.00 -

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BRN OHI [}TH I. 'T T R_D / SEC

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D R OOP -1.93 DB

• ' , BENSI [IVITY . ' 78 DB

0 _ RE$0NRNCE PERK - .11 DB

(

. !

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Fig ai re A. 5 . 4 ( t on' t)

O CM F r equen cy Re s pon s e s. C o nfig . 4

"_ 26 7

: _._. CRSE 4:

: , RIRCRRFT PLUSPILOTiC.L.) O R(s)

0c ( , )

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N(RRO / SEC l

i-

OCM Frequency R es p onse s - Con fig . 4

F i gur e A, 54 (t on ' t) 1

1985026890-185

26 8

, /

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BANDWIDTH I .52RRD / SEC

PILO T COMP E NSATION - 69 .00D EG

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F i g u re A. 5 . 5 (t on' t )

OCM Fre qu enc y Res pon s es - C o nfi g. 5

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Fi g u re A. 5 .5

O C M Fre qu ency R e s p ons es - Co n l i _. 5

, 270

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O CM Frequency Re s p o n s e s - Config. 5

• , : '7"

27 1

• CASE 5: RIRCRRFT PLUS PILCITIC.L.) #R(s)

#c(_)

2 0 , 00 -

nn

r '_

w

rr

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N[ RAD / SEC ) I BQND WIDT H !.52RRO / SEC

RE S ONRNCE PE R K .08 DB

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10 " : 10 "s 10 o 10 1 10 _

W(RBD / SEC)

F i g ur e A , 5 . 5 ( t on't)

O CM Frequency R e spons e s - Conflg. 5

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_ 272

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1 0 "_ 1 0 "_ I 0 o

I A ( RRO I SEC

F i gur e A. 5 , @

OCM Frequency R e sponses "Confg , 6

,, 1985026890-19

: 273

Fi g u re-%. , _ S { to n 't )

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,I

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I0-2 I0-_ 10 o I0 I I0 _

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B AN DWID T H .10 RR D I SE C

PILO TCO MP ENSAT I O N -32 . 2 6DE G

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OCM f'requency Respon s es - Config. 6

27 _

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RIRCRQFTPLOS ?ILOT{C.L . ) C R(_----.D

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_. 0 0-

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' ! O CM F re quency Respo nse,_ - Co n flg . fl

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1985026890-195

_ 27 8

CRSE ? :

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. {

I0 -_ lO-t I0 o I0 I I0 e

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S E N$III V ITY , q 8 DB

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Figu r e A. S . 7 ( eon't)

OCM Fre q ue n cy Responses - Co n fig. 7

2 7 g

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Q

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BANDWIDT H 1.76RRD / SE C

P IL OT COMPE N SA T I O N -6 0 .55DE8

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.._ OCM Freq ,_ en e y Response _ - Config. 8

t

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co

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10 "= lO -t 10 o 10 t 10 e

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F i br e A . 5 . 8 ( c on ' t )

; , O C M Fre qu e n cy Respo nses - Con lig . 8

1985026890-19

2 82

._ . _ CASE8 : , _(_)

I:tIRCRt_F'[ PLUSPILI3Tt_3 . L.I 0R(..___)_)

-4 0. 00

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IqIRCRRFT PLUSPILOT[C.L.)

, O c(S )

m

o

Z

E - _ . _

_ 0. _

. .-,80. _ ' '' ..... I ........ 1 ........ I ........ I

I0-2 I0-1 I0 0 iO _ I0 _

N ( RnD / SEC )

BRNDWIDTH 2.00RRD / SEC

PILOT COMFENSRTION - 53.30DE8

. 0 - - RESON R NCE PERK 1.6 9 DB

DROOP -1,23 DB

SE N SI T IVITY . 53 DB

- 36{

10 - e t O " t t O o 10 I tO 2

H(R R D / $ E C )

Figure A. 5 . 8 (ton't)

OC M Frequeacy Resp o nses. Config. 8

Next: Appendix A. 6 →

Appendix A. 6

2 8 4 Appendix A. 6 Listing of Modal Analys i s Program The modal analysis c ompute r program which was used fo r the open - loop analysis method is presented here. This program is written in FORTRAN and makes use of a modular structure to simplify its use and to allow the reader to more easily i ' ollow the procedure. Documentation is integrated with the code and a variable name listing and description is included to further aide the read e r.

l : v(_n lh_uq_h the program w , '_ written to be flexible , it is based on a linear sysl_ , m analysis package of subroutines that is unique to tile Purdue computer sysl(,m. The pa c kage was developed at Purdue and is entitled LSLIB.

2 8 5 LIST OF UARIABLES USED IN MODAL AMALYSIS PROGRAM UARIABLE NAME t UPRIQBLE TYPE t DESCRIPTION ...................................................................................

Q t REAL t UTILITY UQRIQBLE USED IN SUBROUTIME PHAG t t AA t REAL ARRAY t AUC , :EMTED SYSTEM MAT R IX t t AG t REAL RR.RA '/ t GUST ECUATIOM SYSTEM MAFRIX t t AMASE t REAL ARRAY _ ARRAY USED IN FORMZM_ MAG / PHASZ FORM OF t t COi ' t P LEX MA T RICES t t AP t R EA L A RRAY t P ILOT E G UATIOM SY5 T EM MA T RIX t t A R RAY t REAl. ARRAY t UTILITY ARR A Y U SED IN S U BROUTINE PHAG t t A$ ? RE A L AR RAY t U TI LITY AR R AI / U SED IN 5 C ALII_G MA T RIX A U ? t A U t RE A L ARR A Y t UEH I C LE EO U A T XOh SY S' I E M MATRIX t t B t R E AL ARRAY t V EHIC LE EO U AIrIO ; ' ! CI2M T RGL MA T R I X T t B t R E AL t UT ILI T Y UA RIABLE US ED I14SUBRO U TIIIE PH A G t ?

lI B t REA L ARRAY t AUGMErlT E D COM T POL MATRIX - U SED OM L Y I F ? ? TH E PILO T CH ARA C T ERISTICS ARE ,_,OT BE I M G t t MODELED IM T HE ANALY_ .: I5 t t B_ t REAL ARR, W t U TILITY ARRAY U SE D I14 SC_LIM G MATRIX t t C t REAL ARRAY t OUTPUT MATRIX ASSOCIArE!] WITH THE t t UE H ICLE ST A T ES t t CA S E t RE A L A R R AY t ALPHA - MUM E RI C U SED TO TITLE OU T PUT t @ CC t RE A L ARRAY ? A U CMEM T ED OUT PU T P, AFRIX FORMED BY ? t COMBIMIMG C, E. AI_D F t t CMM t REAL _RRAY t UTILITY UECTOR USED TO DETERMINE ELEMENTS ? ? OF C _ SS OCIATEO Wi[H PLUrlGE ACCE t _RATIUM (M Z ) t t CO M ",, CGM P L EX ARRAY t COM T POLABI L ITY I IqTRIX FORMED FROM THE MOI3AL t t MATRIX (EE) AlIO ThE 5YSIEM COhTRO: t t M A TRIX (BB OR CP) @ @ C S t R EA l . A RRAY t U TILITY ARRAY USFO IM SCALIMG K, A IRIX C @ t D t RZ, Q -L ARRAY t UEHICLE E(IUATIOM DISYUR_AMCE MAT, _ ( t DIST t COMPL E X ARRAY t DISTURBABILITY _ATRIX FORMFn F col _ r FE M O DAL t t MATRIX (EE ) AMD TIIE S STEr_ IMCE ' ? t Mn F RIX (GC. ) t t J DS t REAL ARRAY t UTILITY A_RRY USED IN SCALIMG MATRIX D t t E t REAL ARRAY t OUTPUT MATRIX ASSOCIATED WITH THE CONTROLS t t , EE t COMPLEX ARRAY t MATRIX OF EIGEMUECTORS OF MATRIX AR (ALSO t t C A LLED TH E MO D AL M AT RI X) t t "i El t COM P LEX ARRAY t INUER S E OF MATRIX E E (INUERSE MOO A L MATRIX) t E IG t COMPLE X A RR A Y t UECTOR OF ETG E MU P LUES O F MATRIX AA t t EIGMRT t C OM P L E X ARRAY t DIACOM A L M A TRIX OF EIGEMUALU E S E I G ? t E II_ t C OM P LEX ARR A Y ? INU E RSE O F MATRIX EI G MAT t t EI_ t R E AL AR R AY t UTILITY UECTOR USED TO DETERMINE ELEMEMTS ' t t OF E ASS O C I A T E D W I TH P L U N G _ A CC ELE R ATIOH ( M I) t t F t REAL ARRAY t OU T PU T ;IA TR IXASS O CIATED WITIITHE _IS T URBANCES t t FI E LD t REA L ? US E _ IM P RIN T IN G REAL MA T R I CES (O;I L Y U S ED I H t t M P RIMT , A L5LI B R O U'_NE) t t FIEL D C t REAL t U S ED IN PRIMTIN G COM PLE X MATRICES (( } MLYU S E D t t IN C M P R M T , R LSLIB RO UTI ME ) t t FMM t R E A L ARRAY t U T I L IT Y U EC T OR US ED T O DE T ERI I INEE L EMEN T S ? t O F F ASSO C I AT ED W ITH P L U M C E A CC EI. E P _ T IOH ( M Z) t t F O RMAA t SUB R OUT I M E t FO RMS A UG MEN T ED SYS T EM MA T R I X , AA t t FORMB B t SUBR OU TINE t FORMS A U GMEN T ED COMTROL MA T RIX B B UHEH I_= 0 t t F OR MCC t S UBR O UTINE t F OR MS AU G MEN T E D O U T P U T MA TR IX C C t f F ORM GR t S UB RO UT INE t F O RMS A UG MENTED CONTR O L MATRIX GP A AMD t t A UG MEMTE D DIST U R D A_CE MATR I X GG A t t FORMNZ t SUBROI 3 T INE t FORMS PLUNGE ACCELERATIOM ELEMEMTS OF THE ? t O UTPUT M AT RICES C, E AMD F t t GG t REAL AR R AY t DIS TU RBANCE EQ UATI ON W H ITE MOI S E M A TRIX t t GGA t R E AL A_RAY t AUG M E NT E D SYSTEM WHITE NOI SE DIST U R E AM C E ; t t MATRIX " t t . GP _ R EA L AR RA Y t P ILO T EQU ATION WHI TE N O IS E M A TR IX t t G P A t RE AL A R RAY ' r A UG MEN T ED S Y ST EM _HIT E N O ISE CO N T R OL t tA TR I.X I ?

! GRAU _ R EA L t GR AUI T ATIONAL A C CELER AT IOM H B LA_I K _ R EAL t U S ED Ir lPRI N T IN G H_TR I LES ( ONLY US E D IM t t MPRINT A r i d CPPRMTo LSLIB ROU T INES) t I t I H T E G E R t DO L O O P COU N TI N G UARIABLE ? ?

IM P M AI t CO MPLEX ARRAY t UTI L I TY ARRAY F O R OU T PU T T IN G RESI DU ES t t ( IM P MAT = I_ PU LSE R E S ID U E MATRIX ) _| t t IP t INTEGER t DO LOOP COUNTIN G UARIABLE - t t IT_TLE t I NTE G ER t ALPHA-MUMERIC USED TO TITLE MATRICE S t t J f IN T E G E R t D O LOO P CO U M T I'_ GU ARIABLE t t d P t I I _TEGER t DO LOOP C OUN T ING U ARIABLE @ @ JTITLE t INTEGER t ALPHA-NUMERIC USED TO TITLE _ I ATRICE5 @ t K t INTEGER t DO LO C P CO UrI TINGUARI F _BLE t t X P t INTEGER t DO LOOP COU rI T INGUARIABLE t t L t IMTEGER t DO LOOP CQUhTING UARIABLE t t LINE t IMTEGER @ UARIABLE USED I rl IIEADER,A LSLIB R OU TINE t t US E D T O TI T LE OU T PUT t t LX t REAL t DIS T AN C E F RO M NOSE T O C . G . OF AIR C RAFT ( FT ) t t M t IN T E G E R t # O F CO L U MNS IN MATRIX ARRAY IN ? t S UBR O UTINEPH RG ? t MA TOU T t C O M PL E XH R P , AY t UT ILI T Y NAT R IX FOR OUTPUTT IM GC OM PL EX t t M ATR ICES Il l M I a G / P HA S E FO R rt t t MOrtAL t 5UDROU T IME t D E T ERMIIIESM OI ] P LMATRIX , IN U ERSE MO D AL t I MA T RIX AN n EI G EN U AL U E S O F A M AT RIX t ?

M t I M TEG E R t I # OF COL U MN S I N M ATR IX MA TOUT I _ t t SUB R OUTI NE PHRG t t rt qGC ? IN T E G ER t _ O F A G CO L U MN$ ? t NA GR. t INT EG E R t # OF A G RO W S t ?

N R t IERG t IM T E G E R ? ALPHA- N U M ER I C NAMIN G THE RG M AT RIX ? ?

M R M ER P t IM TEG E R ? AL PHA- N U ME R I C NAMIM G THE R P tt_ TR IX t t rt AME AU t IMTE G E R t A LPHA- N U ME R I C N R MI rtG THE RU M ATR IX ? t N AM ED t IM rEGER t RL P H F - M U ME R I C N P Z I IN G THE B M ATRI X t t N R M EC t IM T E G E R t P a .PHR- M U M ER I C N R M I N G TH E C M RTRI X t ?

rt N tLr D t IMT E G E R ? A LPH A - N U M ERICrI N II N G TH E D M ATR IX t t t _E [ t IPITEGER ? A L PHA- N U ME R I C NAMIM G TH E E M ATR IX t ?

M N I E F ? IN T E G E R ? RLPH A -PIU _ R I C PIR M IPIGTHE F M A TR IX t ?

rIR ME GG ? I t tT E G E R ? RLPHR-I _ J M ER I C M R MIN GTH E GG MATR IX t t M R ME GP t I t tTEG E R t RLPHR-1 9 J ME R I C t t_I rlG TH E G P M ATR IX ? t M RPC t IM TEGER t _ OF AP C OLU HH S t t MA PR t I r l TEG E R ' t _ OF _ R O W S 2 8 8 t t _ . _ U C t I N T E GER t _ OF A U EOLI J _ t t M AWR t I NTE GE R t _ O F I_U ROWS t t M B C t IH T E G E R t _ O F B CO L Ut _ S t t NBR t IN T EGER t _ O F B ROWS t t MC t IN T E G E R t _t OF C ON T ROL S I H U E H I C LE EO U A T IOM t t H CC ¢ I HT E C E R t ;F O F C CO L U M N S t t ; M CDU t IH T E C E R t N C + PID + H U t . H C P t I H TECE R t U SED IH PR I H TI H G MATRICES ( O NLY USED I H t t M P R I H T AN D C M P R N T, L S L I B ROUT INE S) t t rlC PC t I H TE GE R t US ED IN P R I H T I M G MA TR I C ES (O N L Y USED IM t t MPR INT A ND CMP Rr IT ,LS L IB ROUT INES ) t t : : flCR t IN T E G ER t _ F O F C ROW S t t H D t INT E G E R t _ OF D ISTU RB A h _E S IM VEH I C L E E QUA TIO N t t H D C t INT EGER t _ : _O F D COLU MNS j t t - 1 NO R t I NT E G E R t • O F O R O W S t t NDU t I NTEG ER t M D + NU t t MF ' , t I NTEG E R t _ O F E COLU M NS t t ME R ? INT EG E R ? _ OF E R OWS t t M FC t INTEGER t I _O F F COLUMNS t t M F R t IH TE GE R t _ O F F R OW S t I H T EG E R t j¢ OF G G COLUM N S l _ ? I tITEG ER t _ OF G G R OUS : _ t t M CPC t _M T EG ER t _ OF Cd D CO L Lg g l5 t t N G P R ? IN T E GE R t @ O F GI 3 RO WS • t t MM t I NTE G E R t _ OF"OUTPU T S I M OUT P UTEOU A T I O t t t t riP t I MT E G E R ? F L A G FO R U SIM G P IL OT EOU A T I O M 1 11 t t THE ANA LY SIS (I _ =0 ; HO PILOT - M P-tlC ; P I LO T ] t t tIP] _ J t IN TEG E R t riP + MD t t M R A t I ttTEG E JE t R OW DI ME rtSIO ; t OF tIRT R IX I _ TOU T I N t t SUB R OUTI NE P le _ G t t NU t I N TEGE R t _ " O F R I GID BO D YS L AT E S I M U E H I CLE EQU _ T I O t t t t M RO W t INTE G E R t R OW D IME NS ION O F t_ TR IX A RR A Y I M I' 1' S UBR O U T IM E P HAG ? ?

M S." 1" IN TE G E R 1 ' _ : OF S TRU C TURAL M O DE ST A T E S M OnE LED IN 1' 1" TH E UE HIE LE EC } U PTI O M 1' 1` ri U 1` IN TE GER 1' _,;OF ST A TES IN TH E UEH IC LE EG UA T I O M NW 1` IMT E GE R 1` ROW DIM E N S IO , _ 0F UE HI CLE SYSTEM MATRI CES : 1' t (TH IS IS TH E LA,o . CESTPCISS Z _ LE M O _EL W . _ICH 1' t AM B E A r lA L Y Z ED WIfHOUT ALTE ; _IMG T H E P 2 _GR_M ) 1' 1` r ; _ L X 1' IM T E GE R r h U + NX ? 1' NWXY 1` I r ITEGER 1'h_ + MY MUY 1' IN T E G E R 1' H W + MY ? 1` NX 1' IN TE GER 1` ROW _IM E N S I O ri C, / P I LO T ;;Y SI E M MA T P . _ C ES 1 ' 1' ( TH I S I S T HE L ARG EST PI L OT MODE L W H [ C. _I ' : 1' 1' T H E PR O GRAMCAM HAND L E W ITHO Ui P LTERA T IO PI ) 1' 1.

R Y t I I _ TE GER 1`RIgN DIM E NSI O N { IF D I ST UR_A t _CE S Y STE M M I _ T ,_I C E S t 1` (THIS IS TH E L ARGE S T GUST MOCE L WHICH TH E t t PI_OGRAM C A R HA r lD L E WI T HO U T AL T ERA T IOI_) t 1. - ri Z 1. Iri T EGER 1'ROW DIMEN S ION CF T HE O U T PUT MATRICES 1. 1' ( THIS I S T HE LAR G E ST P O SSI BL E NUMBE R _ F t 1. O UTPU T S W H I CH T HE PROGRAM C A H H AHJ D LE t 1. WITH OUT A LT ERA T IrN) 1' 1' OBS 1.C O riPLEX ARRAY 1'OB S ER U ABI L ITY MPvRIX FORMED FR O M T :iE MODA L 1' 1'MA T RIX (EL) AND iHZ O U T PUT MA TR IX ( CO ) 1' 1' PHAG 1' S UBR O U T INE t OUTPUTS A COMP L EX MA T RIX IN PHASE A_I) t 1' MHG H ITUDE FORM (PHA S E IN D E G REE S ) I' 1' P H I I 'REA L ARRAY 1' M O DE S H APE S FOR EACH OF TH E - , IRU CT URA L MODES I 1' P H I P • REA L AR R AY t MODE S SLO PE S F OR EA C W O F THE ST R U C T URAL M O DE S 1' 1' RZSID ? SUBROU TUI E 1' DE T ERMI t IE5 THe'.IMP U LSE R E SIDUE MPTRIX O F A 1' 1'S Y S T EM G I UE M H E C O r I T ROL#BI L ITY O R : t 1 ' DISTURBA_JI L ITY A r id CB S EPu F _BI L ITY " :ATRI C E S , 1` 1' S C A L E I 'S lJ BROU T IN F . . I "S C A L E S TH E PA T RI C [S C I F A SY ST EM ( IE. DYMAMI C , I' r C ONTROl_, DI S T U RBANCE A_D O U TP UT tlP T RI C E3) 1 ' 1' WITH A _IAG iJfI ALSCALIMG MptRIX 1' 1' 3T E P t S U BROU T T,w r. 1' FO RMS ST EP R E SID U E MA T RIX O I U EM IMPULSE 1' 1. R ESI DU E MA TR IX A, fl O U EC T O_ OF EI G EN U _LUE S 1' 1.

' . S T P r C O M P Ly'. . . e l 1' U T I L I T Y ARRAY F OR F OR MI f _G RESIUUE MA TR ICES I' 1' ( U S ED Ill S U BROLITI 'tE ST E P} 1' 1' 5TPMA T p C OM PLE X .A KK M', ° t UT I L I T Y A R RAY FO R O U T PU TT I ri G RESI D UES 1' t (S TPMAT - S _E P R E S IDU E MATRIW ) 1' 1' SUBT 1' R EAL FL R RAY 1' bARIAB L E US E D I M PRI_ IT I t IG THE H EADING I' t T _'E O UT PU v 1' 1' 2 90 T t REAL ARRAY t SC A L ING MATRIX F OR SCALIrIG T I I E U EHI C LE S TATES t t T ERfll t R EAL t INTERMEDIATE U A R IABLE IN S U B RO UTINE PHA G t t T ERM2 ¢ REAL t INTERMEOIATE UA R IABLE Itl SUBROUTINE PHA G @ t - T IN U t REAL AR R AY ? IN U ER S E O F M AT RIX T t t TI T LE t R EAL ARRAY t U ARIABLE USED FOR PRINT I _G THE H EA D IrtG O N t t T IlE O U T P U T t t U Z ERO t REAL t C RUISE U ELOC_ r Y O F AIRCRA FT t t W OR K ? S U BR OUT INE t DE T ERMINES T HE CO N TR OLA D ILITY , DISIUR D _31LI_Y t _ AN D O B SERUA_ILI T Y MATRI C E_ O F A SY3 T EM C IU E M t t THE SYS T EM MATRICES AND T HE M O DHL MA T RIX t t U K ? REAL ARRAY t U T ILI T Y U E CT O R USED IN SE U ERAL S U BROUI'IhE S t _ X t REAL ARRAY ? TRANSF O RMQTI G II U E CTO R U S ED IM _E T ERM ( N!M G t t ELEMENTS O F TH E C MATRIX A SSO CIA T ED WI TH t t T H E P LUN G E A CC ELERA T ION ( MZ ) i C ..........

_ ///I/////II/III//I/I/IIII//I///III//II/I///I/I//////////////////////////IIII C ....... • C - t C ANALYSI S OF FLEXIBLE AIRCRAFT DYNAMIC5 USING MODAL ANALYSIS METHODS t ......................................................... t C t C THIS PROGRAM U SE5 A MO D AL AMA!_YSISTECHMIOUE TO DETERMINE THE t C IMPULSE RESIDUES OF A STATE U ARIH B LE MqTH MODEL OF A F LEXIBLE t , C AIRCRAFT. IT DOES THIS AHALYSIS BY FORMIN G At!AUGMENTED SYSTEM ?

C FROM THE MATRIX EQUATIONS DESCRIBING THE U EHICLE DYUAMICS AND t C THE FRECUENCY CHARACTE R ISTICS OF A HUMAM PILOT AND ATMOSPHERIC ?

C DIST U RBANCES. ?

C t C .............................................................................. t C t C UEHICLE MATRIX EQUATION - t C t C X-DOT = [ A U] *X + [ B ] , U + [O ] *U ?

C t C Y = [ C_*X + [E] - U + [ F]mU t C t C t C PI L OT MATRIX EQUATIOM - ?

C t C U - D O T = [AP]* U + [ G P] wETA ?

C f C t D ISTURBANCE MATRIX EQ U ATION - ?

C ?

C W - DO T = [ A6 I * U + [ GGI * ET A ?

C t C t C /// /// _/ / ' / ////////////////////////////I//I/// / / /////// / / //////// ////////I// C .... _ .............

C P R OGR AM AMAFLE X( IN PUT , O U TPUT, T APES = IN PU T, T APEG = GUTPUT , I T A P Et, T AP E2 , TA PE3 ) C C ...........

C::::: DIMENSION STATEMENTS ::::: C DIMENSION A U (2 0 ,20_ , B( 2 0,5), g( 2 0 , 5) D IMEM S IOM C( 15 ,20) , E (15,5), F(15,5 ) C DIMEN S IOM AP ( 5 , 5 ) , G P(5,5 ) C DIMENSIOR AG(5 , S), CG( S , 5) C D IMENSION AA( 30 ,3 0) , BB( 2 5 , 5 ) , C C(1 5 , 3 0) _IMENSION GP A (30 ,5 ) , GG A( 3O tS) C C OM PL EX E1 6 ( 30) , EE ( 3 0 ,30 ), EI( 30 , 30 ) CO M P LEX CO M (30, 5 ) , 0B 5(15, 30) , DIS T ( 30 ,5) C CO M PL EX IM P MAT(3 0 ,15). STP MA T (30, IS ) C 51MENSI O M T (20 , 2 0 ) , T IM U (20,20 ) C DIMENSI E H TITLE(7). CASE(4), SUBT( 2 ) : DIMENSION PHI(4). PHIP(4) D IMENSION X(1,2 0 ), CMM(I,20) , ENM(I,5)o FMM(I, 5 ) C DIMEN5IOH A5( 2 0,2 0 ), BS(2 0 , 5 ), C 5(15 , 2 0 ),D5( 2 0o5) C DIMENSI O N AMASE(30,2), I T ITLE(2), JTITLE(2 ) DIMENSION ICMAME(2), IOHAME(2 ) , IONAME(2 ) C COMPLEX MATOUT(3 0 ,30 ) COMPLEX EIGMAT(30,30 ) o EINU(30,30 ) C DIMENSION A R( 20 , 20), AR T( 2 0 ,20 ) , A TR(20, 20 ) , A T(2 0 , 2 0 ) OIMEN510M BR(20 ° 5 ) , BT(20,5), 0R(20.5 ) , DT( 29 ,5) DIMENSIOR CA(IS,20) , CT(15,2 0 ) C LOGI C AL TE S T ( 20 ) C D I MErlSIOM A 3 (2 0, 2 0) , B3 ( 2 0, 5 ) , B 3( 2 0, 5 ) DIMEN S ION C 3 (I_ _ 0 ), E 3(IS,5)D F3(15,5) C D I MENS I ON AT I (20,2O) C DIMENSION ( 2 0 , 2 0) , BM(20°5 ) o DM(20,5)° DIMENSION C , :15t2 0), EM(I5 , 5), FM(I5 , 5) C REAL LX C : EX T ERNAL PH A C C C ....... _ .........

C: :::: C OMM O N B LOC KS ::::: C ........

Ct*_ THESE UARIABLES ARE USED IN PRINTI I _ THE HEADINGS OM T H E O U T PUT C COMMOM/MPRINT/ NCP,FIELD , NCPC . FIELDC,HBLAN K CO; I M O N / H EAO/TITLEpCASEoLINE C * C**m T H ESE U A R IABLES ARE TH E A CTU AL W ORK ING MATRIX D IMENSI O NS Cm * Q US E S I M MO ST OF THE S UBR OUTIN ES C : C OMM O N / D IM / N U , N C, M D, N M , NP C C::::: S E T ROW O IME rlS IO N S ::::: Ct,. T H ES E U ARIABLE 5 AR E T H E R O W DIM E NSI O N5 O F T H E S YSTEM MATRI C ES C C R O W DIMEN S I O N = S I Z E O F MATRIX SET IN DIMEN S I O N STATEMENTS C C**Q THE S E ROW D IMENSION S MUS T BE AT LEAST ' AS LAR G E C*** AS THE WORKING DIMENSIOHS C C A U( NWoNW ) o B ( MWoNX ) , D( MW° HY) C A P(NX ,NX ) . G P( NX,N X) C A G( NY , NY ) D G G (NY, N Y ) C C(N Z, N X ) t E (N Z ,MX ) t F ( MZ ,HY) C AA (H WXY, N _ XY) , _ (r qU X , H X ), C C (NZ o N _X Y) C GP A (H WX Y, N X ), G GA (H W X Y oH Y ) o C NW = 20 MX = 5 MY =5 M Z = 15 NW Y = N N + M Y NWXY = NW + NY C C ..... _ ...........................

C t C PHYSICAL PARAMETERS PARTICUL A R TO THIS FROBLEM t C /l/lll /ll/tl//lll ll///////lll///////////////////ll ///t ' //ll / _ C N U MBER O F ST R UCTUR A L M ODES MODE L E D M SM = 4 C S T R U CT UR A L M O DE ::: S H A P E S ( F T ) : :: S LOP ES ?

C=_ l _ PH I ( 1 ) = . 3 4 9 $ PH I P (1) = -. 0 2 S4 0 P H I ( 2 ) = .40 0 $ P H IP ( 2 ) = -. 03300 PH I( 3) = . 1 8 0 $ P HI P(3 ) = - . 03456 PH I ( 4 ) = .140 $ PHI P(4) = - °0 3 6 3 6 C C R UI S E VELOC ITY (FT / SEC) t UZERO = 949 . 0 C c¢ _ .j _ nl T L (2E A I" _ ON( FT ) t LX = G3.4 C G RA V I T A T IONA L A CC EL ERA T I O N (FT / S EC / S E C) ?

GRA U = 32 .1 7 C C I///I LLLLL//I LI /I LLI/I /ILL LI/ILI //lltI/ ILLLI ///I/I//Il LL //// C C::::: REA D T I TLE S ::::: C*** TITL E / CA S E / SUBTITL E C R E A D( 5 , 5 0 1 ) TI T LE WRITE ( 3.501 ) TITLE 5 0 1 F ORMA T( ? A IO ] READ(5 , 50 2 ) CA S E NRIT E (3 , 50 2 ] CA SE 50 2 FORMA T( 4AIO) REA D (S,5 0 3 ) S UBT WR ITE (3 , 5 03) S U BT 503 F ORMAT ( _AIO ) C C**m [HI S SU SRO UT I_ E P RI HTS r H E H EAD E R O_ T _E O UTPU T C CA LL HE A D ER(S U B T} C C:::;: R E A D MATRICES ::::: C *_* M_M E / D IME M S I O_ ( R O thC O LUM M ) l ELE _ EM T S C REA D( 5_504 ) £1 AMEAU i WRITE(3.504 ) _AMEA U REA D (5o* ) NAUR , NAUC WRI T E(3p* ) HAURINAUC DO I I = I , MAUR READ(5,* ) (AU ( I , J ) ,J=L , M fl UC ) 1 WRITE ( 3 , * ) (AU ( I , J) , J= I , MA U C ) R E A D (5 , 504) MAMEA G REA D (5 , * ) NACR , NAGC D O 2 I=I . MA G R 2 REA D( 5 . *) ( AG(I , J) , J= I , MA G C ) REA D (5 , 5 04) NAME B WRI T E ( 3 , 504 ) NAME B REA D( 5 , *) HBR , HBC W R I T E (3 ,_ ) NB R .HBC DO 3 I=I , N B R REA D (5 , * ) (B (I, J ) . J= I.H B C ) -; 3 WRITE( 3 ,*) (B(I, J ), J =I,NBC) REA D ( 5 ,50 4 ) NAMEC . _ _RITE( 3 , 5 04 ) NAMEC READ( 5 ,*) HCR,NCC " WRI TE ( 3, _) i i C R , M C C ,: DO 4 I =t ,N CR ! READ (5 , * ) (C( ! ,J) , J= I ,HCC) ; 4 WRITE (3, * ) ( C( I, J ), J = t ,N CC ) !I R E AD(5,5 04 ) NAM ED _ I RITE(3 , 504 ) MflME D 14 R I rE ( 30 o ) P I D R,ND C DO 5 I = I.ND R I R_AD(5, *) N O R, tl D C • R E A D( 5 , *) (D (I ,J } ,J= I,NDC ) READ(5,5 0 4) HAMEE _RI T E (3 . 5 04) NAM E E i 5 _RITE( 3, w) ( D (I, J),J = I , N D C ) :_ REA D(5 , ") NE R, NE C _! W R I T E( 3,*) HE R,NE E DO 6 I = I ,NE R ' REA D ( 5 ,*) ( E(I , J ), J f I , HE C ) : e WRI TE(3, * ) (E(I,J ),J ffiI, H E C) READ(5, 5 04) NRME F |4 RI TE(3,504 ) H AME F , _ READ(SI* ) N F R,N FC W R I T E( 3 , * ) NF R, N FC DO ? I = I,NF R READ( 5 , _) (F(I, J),Jf l , N FC) 7 W RI T E ( 3 , *) (F (I ,J),J= I ,H F C ) RE AD(5,5 0 4 ) NAME GG 50 4 FORMA T (A4) READ( 5 , * ) M G GR, N GGC DO 8 I = I , N G G R 8 READ(5, * ) (G G( I ,J ) ,J =I , N GGC) C READ(5.5 04) H AMEAP READ(5,*) MAPR,HAPC IF(NAPR.EQ.O) GO TO I0 C DO 11 I=t,NA P R A1 READ(5,*) (PP(I.J),J=I.NAPC) PEAD(5,504) HAMEGP REAO(5,1) NGPR.NGPC D O 1 2 I 'I , N GP R r ' 2 9 5 12 READ(S1m) (GP(I,J),J=I , NGPC) C C::::: SET WORKING DIMENSIONS ::::: C-- - C*** THESE UARIRBLES ARE THE ACTUAL D IMEMSIOM OF THE SYSTEM MATRICES - C C AU(MU , NU) , B(NU,NC), D(NU , M D ) • C C(N M ,H U ), E(NM , NC), F(NM,ND) C AP(NC,NC), GP(NC,NC) C AG (ND , ND ) , CG( NO , l) C AR(NCDU,HCDU), BB ( NDU , NC), CC ( N i l, r l CDU) C I0 NU : MAUC NC : NBC ND : MDC NM : MCR NDU : HU + N D F _CD U = N U + FrO+ NC C C_ ..............................

Cwe* I t lCLUDEPILOT MODEL IN ANALYSIS - NP = MC ?

Cm-* IGNORE PILOT I t lAHALYSIS - MP : 0 ?

C .................

C tiP: HC NPDU : NU NP C C ?

C TH IS SZCTIOM OF T H E PROGRAM IS P ROBLEM DEPENDENT ?

C lllll///////////l///////l///////////////l/////l/////l///l// ?

C IT IS DUE TO THE CHOICE OF USING PLUNGE ?

C ACCELERATION AS A SYSTEM OUTPUT UARIABLE ?

C_ ............

C C:::':. FORM UECTOR-X FOR II Z E O UATION ::::: C D O 13 I-I,NU 13 X( I, I) = 0.0 C X(I, I ) = -U Z ERO / G RAU X(I,2) = LX / GRAU X(I,4) = I|2ERO / GRRU X( I , 9) : -P H I ( 1 ) / G R AU X(1,1 0 ) : -PI41(2)/GRAU X(l,ll) = -P HI( 3 ) / GRA U X (I * I2 ) = - PH I ( 4 ) / GR AU C C ; :::: CO MPLET E C M A TRIX WIT H N Z EQ IJ ATION ::::: C CALL F O_MMZ( RU , B , C ,B,E,F,NW,H X , N Y,M Z ,X , CNM,ENM,FHM) C C //lll/1/l/l///lll//lllllllllllllll///ll/l/1/llllll///ll/ll C C C: :: :: OUTPUT SYSTEM MATRICES : :::: C CALL MPRIMT(NW,AU , HU.N U , S,G,HAMEAUol) CALL MPRINT(MY , AG.NB , N D ,N D ,G,MAMEAG,1) CALL MPRINT(NW,D,HU , NC,HC , G,NAMEB,I ) CALL MPRINT(NZ,C,NM,NU,S,S , HAMEC,I' CALL MPRINT(MN, B ,NU,N B ,ND.S,NAME_,t) CALL MPRINT(HZ,E , HM,NC,r_C,S , MAMEE, I) CALL MPRIMT(NZ,F , NMoND,ND,S, H AMEF,I) IF (HP.E Q .O) GO TO 14 CALL MP_INT(NX,AP,MC,NC,NC,S,NAMEAPol ) t4 CONTINUE C C ......

• ::: , .

C : :.:: F ORM S C ALING MATRIX "" C C ..........

C THE T MATRIX IS PROBLEM O EPENDEMT C I/l/ll///illl/ll/l//l/lillll////iI//llll/il/lJiJ /i /lli //ll C .........

C*** THE ACTUAL SCALING MATRIX, T, IS PROBLEM DEPENDE H T Cmm* B UT THE SCALING PROCESS IS ALWAYS RECOMMENDE D C C 'w* HU MBER OF RI G ID B OD Y S T A T ES C NRB = H U - 2. 0 * M SM C C* * * IM TH _S CASE, THE S CA L IMC MATRIX, T , IS BASEO OH MODE S L O PE S C C [I. 0 ] C [ 1. 0 ] C [ I / U ] C [ 1.0 ] C [ Y / X( 1) | C [ Y / X(2) ] C T = DIAG { Y / X(3 ) ] C [ Y / X(4) ] , C [ Y / X ( 1 ) I C [ Y / X(2 ) J C C Y / X ( 3 ) ] C [ Y / X(4 J I _ C C - W HE R E Y / X ( I) IS TH E HODE SL I C E AS SOC IA T E g WI TH C T HE I TH M O D EO F THE S TR UCTU RE C C - A MD- C C - W H E R E U I S S OME CH A R ACTERISTIC U EL OCIT Y C ( IM TH I S C A SE U = UZ E RO ) C C DO 15 I=I ,IIU DO IS J= I, N U ' _ t 1 6 T( I ,J) : O.O 15 T(I,I ) = 1.0 C T( 3, 3) = 1.0 / UZ E RO C DO 1 7 I =I, N SM T( I + NR B , I + N RB ) = P H I P (I) 1 7 T( I + NRB + NSM,I+NRB + NSM ) = PHI P (1 ) C C l/ll/illiilli///li/llillli/iill/ /I/li/i /ii//lliil ilii/ili l : C ::::: S CALE VEHICLE SYS T EM STA T ES ::::: C CALL SCALE(RU , D,C, D ,N IJ, NX,NY°NZ_T ,T INU , AS,BS, CS0 DS ) C C::::: CU T P dT SCALE D MA T RICES ::::: C C A L L M PR IN T( h_ ,T,I I U ,H U, G,S, _S C A L IN G _ , ; ) C CAL L M P RINT ( N IJ * AU,_ / ,N U , S, G,_A U ' SC AL E D_ , I ) CA LL M F RIH T( N W ,B , N U, H C ,N C ,S , _B - S C ALE D _pl ) • C ALL MPRIHT ( H Z .C,NM,N U,S , S, _C-S C AL E D_ , I ) C ALL M P R IH T( NW , D,N U, N D,6 , S ,_ D- S C AL ED _,I ) C C ::::: SET UP TEST A RR A Y ::::: C C *mt NR = T H E N U M B ER O F R ETAINED STA T ES C *** H T = TH E N U MBER O F T R U N C A T E D S T A T ES C H R = 1 2 NT = N U - N R C C**e I N I T IA L I Z E TH E T ES T A RR A Y T O . TRU E . .

C D O E S I =I, ; ?U 2 9 TES T ( 1 ) = .TR UE .

C I F ( N T . EO.O) G O TO 4 2 C C ome I F TEST( I ) = . FAL S e, T HE.I TH S T A T E I5 C *_o TRU N C A TED I N t HE R ESID U AL IZAT I O N C TEST(S)= ,FAL S E .

T ES T(?) = . F A L SE.

TEST(8) = .F A LSE .

TE S r( 1 0) = . F A LSE .

T E S T ( It )" .FA LSE .

T ES T ( 1 2) = .F AL SE.

C C ::::: A PP LY ST A T E M ODE L R ES IDU A LIZ A TIO M :::;: C C A LL SORT(A U, B ,C,O, AR,RR T * A TR,A f , BR , BT,DR, DT t CR , CT, T _ ST , • NM , NX ,N Y,N Z, N R, N T) _ : 298 OI_ IN AL PAGE IS OF P OO R QUALITY C C // ///// // O UTPU T PAR T ITIOM E B MATRICES ..... - ,-_ , _ CALL MPRIMT(MW , AR , _R , NR,NR , S,_AR_ , + I ) ' CALL MP R I_ T( MW , A R T , NR , M T, N T , S , _AR T _, + I) CA L L MPRIN T (MW , A T R . N T , NR , N R .S , #fl T R_ , + I ) !

CALL MPRINT(HW o AToMT.NT.HT,S,_AT_, CAL L MPRINT(MW , B R, N R , NC , N C ,6, _BR_ , - C ALL M PR IN T ( NW , BT , N T , N C , M C ,6, x B T _ , + I) C ALL M P RIN T ( HH , DR.HR,N B ,N D,S, _ D R_ , +I ) CALL MPRIN T ( NW ,DT,r i T, N D ,N D , S ,_ D T_ , +I) C ALL MP R IN T (N Z, C R ,r| M ,t l R , NR, S, * C R ._ , +I ) CALL MP R IN T( M Z , C T, NM , M ToHT,S, _C T _, + I) C CALL HR R (ARoAR T , ATR,A T , B R , B T , D R. D T, CR ,CT,E,F,AM , BM,CM, D M,EM, F M, m N W , MX ,N Y , M Z, NR ,NT, A3 , B 3 , C3 , O3 , E 3 , F3 , A T I ) C G O T O 41 C Cw** IF N O TRUNCA T ION SE T TH E U EHI C LE MA T RI C ES T O THE MO D IFIE D MAT R I C ES C 4 2 CALL HEQ ( AU , AM , N U, NU , N W,MW ) CAL L r_E O ( B , BM , N U , N C , _ I W , MW ) C ALL MEO ( C , CM , MLI , NU , N Z, MZ ) C ALL MEO( D,D M , N U ,ND. NW , NWJ C A L L MEQ ( E , EM ,M M , _ C , N Z , M Z ) C ALL MEO ( F ,F M , H M , M D , M Z , M Z ) C 41 C O N T I M UE C C *_ H _ R E S E T _ U SO TH A T T H E U E H IC LE MO D E L IS C **m RE P RE S EN T E D BY T H E S MAL L ER. R E S I D U ALI Z E D MO D EL C MU = H R N DU = N D + I ' K / ' ; h P D U = H P + _0 C::::: F O R M A UG MEN T E D W HITE M O ISE MA T R IC E S ::::: l C ( 0 I C C,P J C CC A = { GG ] GP A = [ 0 ] • i C (0 ] (0 J 1 C , C IF t ip = 0 '; C C C GG A = [ GG ] GP A - [ 0 ] C (0 ] C C I _ i . L FO RM G A (GG,GP , H _ i X Y , If , ,( _MY ,C CA ,CP A ) C C::::: OUTPUT N OI S E MA TR IC E S ::::: C CALL P;PR I MT (l _ lY, GG A , M PDU, 1 , 1 , G , _AM EGG , 1 ) I F ( N P, E O.O) GO TO 39 C ALL M P RI HT(H W X Y, GP A ,HPDU, K C,rlC , 6,rI AME GP. 1 ) 3 _ COMT I t IU E 2 99 C C C .....

C ::::: FOR M A U GME HT ED S Y ST EM MATRIX ::::: i C , C [ APtO t O ] C AA = [ 0 ? AG T 0 ] C [B t D tA U] " C C I F NP = 0 ; " - C C AA=[AGt O ] C [ D TA U ] C C .........

c I'

CALL FORMAA(AM,AG,_P,BM, D M,AA,MN,NX,MY,NWXY) I C I C ////// , // O U TPUT MATRIX AA .............

CALL MPRIN T (_NXY , AA , FIP DU, NPD U, 8 , S,_AA_ , I) I C IF (KP.HE.O) GO TO 1 8 C C ......................................

C _ : :: : FORrl,_ UC MEN T EDB MATRIX ::::: C .........

C G H LY IF MP = 0 ; C C BB = { O ] C [ BI C C CALL FO RMBB(BM , B B, MW , MX , MW Y) C C/I/It / It/ O UT PUT MA T RIX BB .........

C ALL MP R IM T (M_X, BB ,MD U ,MC, G ,S, _B B _, I ) C C_ ....

C::::: FORM A U C r l E _TEDC MA TR IX ::::: C C C C= t Et Ft C ] ,r C C IF H P =O C : C CC = CFtC1 C 18 C ALL FOR hC C(CM ' E M , FM ,C C oMW, M X,MY, M Z,MWXY) C ///////// _ TP U T MA T RIX CC ............

CALL M PR IM T (H Z , C C,MM,M P O U , 5, G , _C C W,I) C C ::::: DETERMIME E- U A L UES AM D E- UE C T OR S OF MATRIX AA "_::: Cmt, T HIS 5UBROUTI hE USE5 GEI C EH WHI C H OUT P Ur5 T IIEF-twLI,_5 c CALL MOI 3 AL(AA, E I C , EE, E I,NWXY ) C C / / / /// / // OUTPUT E - UECT(]R MATRIX ...........

CALL CMPRNT (NW×Y _EE, NPDU, NPDU, G,S, _EEx, 1 ) C C ........

C:::: : DETERMINE COMTROLABII.IT', ' ,OISTUPBABILITY ,qHI] ::: :: C: ::: : OBSER U ABILITY MATRICES ::: :: C ...........................................................

C C CCN = MIMU * B C DIST = MINU * D C OBS = C " M C -WHERE M = MODAL MATRIX : : C C ............................

C IF (NP.E O .O) GO TO 21 C CALL I , JORK (GPA, CC, GCA, EE, E I,i'fX, t 4Y, N Z , NWXY, CON, OT I S, D IST ) CO TO 22 C 21 CALL WORK (BB. CC, GC,q, EE, El ,NX, NYI,"I Z , MANY, CON° OBS, O IST)

i c

, _2 CONTINUE , C C // // /////// / OUTPUT CON, OIST, OBS MATRICES .............

-i c

i DATA ICNAME/_ CONTPOL_. _ABIL ITY _ / DATA Ini'4AME / _ DISTURBS, _ABILITY _ / I DATA IONAP, E / _ OBSERUA_,._B ILITY _ / CALL CMPRHT( t i_JXY, CON, rIPDU, NC, G, 6 , *CON*, I ) CALL PHAG( CON, NLJ',.(Y, hPDU, #' 4 C, ISNAME° F',MASE, #'IATOUT, #'4WXY) : CALL CMPRNT (N_ I XY,I) IS T, NPDU, I,G, G,_ 015T , ,,I, 1 ) CALL PHAG( I I IST, N_JXY, t IPDU°!,IDNAME, AMASE, MATOUT, I "I I JX' / ) C O LL ,:MPRMT(NZ,OBS. rIM,NPOI J, B,S.XOBS_, I] CALL PHAG (OB S,HZ. AM, t ip DU, IOMAME, APIASC°MA TOUT, IiZ ) C C ...........

C::::: DETERMINE MOn h L IMPULSE AND STEP RESIDUE MATRICES ::::: C ...............................................................

C C RES R AT -- OBS ) ( CO I '_ ( O_ D IST ] ; C C .......................................

C _ R I TE(S , GO I) G O 1 FO R MA T( /// , _ . OX ° _ e * eee * ee lm e m ee o e* lwm eeee m eee m e N e Qm e mm_,/ , " fOX.Xm e RC'SIDUE MA' I RICES ,PmW, / , ^ LOXo ; _Ommm 'e me t l l eue _ e m_ e m e m oe _m e m_ e _m_ , e _ e W_/_ ^ lOXo_e _ e Rl} I J5 CO_ R FqP I}I 'I II ,'3MODES emma ,/ , ^ IOX,_"*_ C OL U MNS C OP R ESFON n TO O U TP U T S .,o,_./.

^ I OX_ , _ e,_ m e em.m eee m ee _ e mm e . e m ee m m e m._m e m _ eeee _) C C**m I m _ . _O F " WHI TE H OI S E COD I T R OL INPU T S C lid IS I=I,NC C CALL RE_ID( C O I I,ODS, NZ, I 'I X , I ' I W,NY, IMPMAT, I) 3 01 C C ///// OUTP U T IMPULSE R E S I DUE MATRIX C _ ITE ( G, GO 2) I G0 2 FORMAT( / , 5 X , _ . , e_ / .

,., SX,_m IMPULSE R ES I D U E MATRIX F[_ CONTROL - m, 13, / , • . 5X, _m ............... ._ ) C Cf4LL 09PRNT(MI, JXY,IMPMAT, .F_IgU,MM, G,G,wr.-IMP ._ES re .l) C DATA ITI FLE / _ COMTROL x . m El_SIDLE _ / CALL PHAG( 1MPMAT. M tJ XY. FIPI_J, _M. ITITLE. _MA_,E. MATDUT. M_X ' Y) C CALL STEP (IMPMAT. EIG. H L;XY, WZ , STPMAT. EI C MAT. E I H U ) C C , ' // // ObTPUT STEP RESIDUE MATRIX ......

C 64RITE(G,GO4) I G O 4 FORMA T ( /// . 5X, _ #t, / , , , 5 X,_ STEP RFSI D UE MATRIX FOR CI_'TV.OL- _.1 3, / .

5X, mm , _, / ) C CALL _MPT_ST( r r3XY.STPMAT._DU.MMoSo G o_C-STP RE5 re .l) C CALL P HAG (Sl _gAT. N J XY. MPD U . MM ° IT ITLE, AMUSE °MAT OUT °MIJXY) C L 5 CO r ffI ra E C Cm_- I = • OF WHILE MOISE IgI ST UR D AMCES C I'l C CALL RESID(DIST° OBS. r lZ° MY. _ flJ XY. IMP_ H 4T. I ) C C ///// OU T P U T IMPULSE _SII]UE MATRIX -_--._ C WRITE ( So S03) I k_03 FORMAT ( /// . 5X, m ,am, • _ ° / , SX,_ IMPULSE RE S IDUE MAT;_IX r ( '_ _I S TU2_A_E - _. 13, / .

" 5X, # _ .- .° / ) C C A LL CMPRHT( r _XY , IMPM_T, rP DU, r lM°G , G , _e D -IMP RES * , X ) C D_TA JTITL E / _DISTURBAMC_',_E REST T %UE _ / CALL PHAG ( [ MP_AT •H WXY. rlPDU . _IM° JT_ T LC. P H ASE. MA rOUT. _WXY ) C CALL STEP( IM_MATo EI G , _UXY. MZ° SIT'MAT° E l _MAT. E I r IU) C C /I ll/ O U T PUT S T E P RESIDU E MATRIX .....

C W_ IT Et S ,_0 5 ) I S05 FORMA T ( /// . 5X. _ mm , / , • " 5X._ S TEP RE S II] U E MATRIX FOR _ ]S T_'_!_ E - ram. T3. / .

"" 5 ) 4 . # ................... _ e o / ) C E l=iLL EI " I P_ , " _ IT (f "_ , _XY,_ _T P ? " I AT . _ ',' P : IU, H M , G . G,m_ ) -S T p laE_ _. 1) C E,,I _ LL P H AG ( ST P Pk _T, HW Xy , f'I P 'LIU , r_ , j T _ TLF..,I qMAS F, MA ' T _ , _ , , , ) C _EW I M D 1 t _J Jr

c

"_ S T OP END ,L I CI' : I* ?|t| II |I ZlII*- I :*- I : I :* III *-:IZ _ I : |Z_ [ ISZ| * ZZ_SZ :|I C SUBROUT IriEFORMAA C C THIS SUBROUTINE F ORMS THE AUGMENTED MATRIX - C C I APt 0 t 0] C AA = [ 0 t AG 9 0] C [ B _ D?A O I C C - F ROM THE HAT R ICES A P , AG , AU. B AND I] C I F _P --- HC C C AMD ',EAUGME t ITED MATRIX - C C AA = [ AG ? O I C [ n t AU ] C C - FROM THE MATRICES AG, AU AND 0 IF t ip = 0 C C ......................................................

C C CALL STATEME_IT ARGU t IE t ITS - _" C C AU : f lu X r iu MATRIX OF ROW DIMEHSION NW C AC : _ ' ID X ri d MATRIX OF _0_ DIhEtISIO; 1 NY C AP :rlC X tIC MATRIX OF R OW D IMEt(SIOrl f ix C B : f lu X tiC MATRIX OF RO_ DI t lEhSION t i&4 C I) : NO × r id MATRIX OF _OW nIMENSI(] t l H H C AA : NCOU X NCDV MATRIX OF ROW DIME r _SIO t l M_XY C C ........................................ _ ................

C 5UBROUT INE FORMAA(AU, AG, AP, B, D,AK, NW, MX, NY, riWFY) DIMEFISION AU(NW, M_4), _)G( t IY, i ; _ ' ), AP(NX, t 4X) D I;'IEN5 ION B(NW,NX) _ D(NW,NY), AA(NW×Y, NWXY) C COMMO_ / DIM / l'_J, MC, NI), r i M , t i p C t _PI]V = NU • t i p C C INI T IALIZE MA TR IX AA TO ZE RO C DO I0 I = I,NPI)U 00 2 0 J = I ,M PDU 2 0 NA(I,J ) = 0.0 1 0 C{]NTIN U E C I F ( N P . EQ.O) GO T O 3 5 C C FORM U PP F . R L EF T P AR T I TIO M O F h ATRI X P _ I F riP = C DO 30 I=I,M P D_ 4 0 J = I , I " I P 4 0 AA( I.J) = A P (I, J) 3 0 COMT I M UE C C F OR M L O W ER L F' FT PA R T ITI O rt O F MAT R I X I :_ IF / " iF ' • i'tC C DO 50 I :, I , t c J DO GO J=I,MC G O AA(I+N P+ M D , J ) = B(I,J) 50 CONTINUE : C C F_RM CEMTE R PARTITIION OF MATRIX AA IF NP = MC, C OR THE UPPER RIGHT PARTITION OF MATRIX AA IF NP = 0 C 35 DO 7 0 I = I,ND DO 80 J=I,N D 80 _A ( I+NP,J+NP) = AG(I, J ) , ' 7 0 CONTINUE C C FORM L OI4 E R RIC H T P ARTI T ION OF MATRIX AA C DO 9 0 I= I , N U B O I00 J = I,N U I00 AA(I+riP+MD, J +NP+ND) = AU(I,J) 9 0 CONTINUE C C FORM LO_E R CE H TER P ARTIFION OF MATRIX AA IF N P = NC C OR T H E L OHER LEFT PARTITION OF M A TRIX AA IF MP = O C DO 110 I=I,N U DO t 2 0 J=I,N D l@O AA(I_NP+ND, J +NP) = g (I, J ) II0 CONTINUE C RETURN END < , c C SUBROUTINE FORMBB C C THI S S UBROUTINE FORM5 THE AU G MENTED MATRIX - C C B B = [ 0 ] C (B ] C C - FROM THE MATRIX B AND T H E ZErO PAR T ITION C H AS H D R OL JS C C .....

C C CALL STATEMENT ARGUMENT5 - C C B : N U X iICMATRIX OF _OW DIMENSI_flN_ C B B : N DU X NC M ATRIX OF ROW DI r IEHSIOMMWY C NX : COLUMN DIMEN S ION OF MATRICES B Afl£BB C C S UBROUTIME FQRMBB(B,BB , MW,MX,NWY) £ INE_SIO N B(NW , NX), B B(MWY, M X) C C OMMON / D IM / MU , NC , N D , HM, N P C _ C U = NU + M D C C INI T IALIZE MATRIX B B TO Z E R O C D O 1 0 I= I. MD U D O 2 0 J=I,MC 2 0 BB(I , J ) = O.O 1O CONTINUE C C . F O R M LO_ER P ART IT I O N O F M A T RIX BB C DO 30 I = I ,N U ., D O 40 J=I , H C 4 0 B B (I + M D, J ) = B( I , J ) 30 COM T INUE C RE T URM E N O r \ ::::::::::::::::::::::::::::::::::::::::::..... ..... ":':'|:. • :..-.,.

C S U BRO UTI NE FOR M CC .. ,. • • • :,'.:'':''" : . :::''::. ...

C C TH IS S U B ROU TINE FOR MS T H E M ATR IX - C C C C = [ E 1 ' F I ' C ) C " C - FR O M TH E MA T RI C ES C, E AN D F C C ............

C C CALL STATEMEN T AR GU MEN T S - C C C : NM X F IU MA T RIX O F" R OW DIMENS I ON NZ C E : HM X N C M AT RIX OF RO U D IME I qS IO N tiZ C F : N M X rid MA T RIX OF RI]' , I DIMEr_S I ON MMZ C CC : NN X N C D U MATRIX OF RLI U DI_IE I ' ISI L] M MWXY C I ' IW : COLUMN I ]IMENSION CF MA T RIX C C MX " C OLUMN D I MENSION O F MA T RIX E C tiY : C OLUMN ] ]INEN510H OF MATRIX F C C ..... Q ....

C SUOROUT ItiEFORMCC ( C, E° F °CC, Fl U . N ' ¢. tiY. NZ, MWXY ) _IME I ISIO r ! C(tiZ, N U), E(NZ . f iX ) , F (NZ,MY) , C_ .( NZ , N U XY ) C COMMON "D IM / M U , NC, ti ll , MM, l i P C ft °Di ; = ti U • rio 4- t iP C C INI T IALIZE MATRIX C C T O Z ER O C I ) 0 1 0 I=I.MM nO 20 . J= I,ti P DU 2 0 CC(I , J ) = 0.0 I O CO N T IN U E C IF ( RP.E Q.O) GO TO 25 C C FO RM LE F T P A RT I T I O ti OF M ATRIX CC I F ti p = tiC C DO 30 I= I, LI M ," no 40 J = t,tiC 40 C C ( I, J) =' E El ,J ) , 30 COF ITIti UE C C FO RM CEtiT ER P A RT I T I O fl O F M A T R I X CC I F M P = ti C C OR F ORM LE F T PAR T I T I O N O F MR [ NIX CC IF PIP = 0 C 2 5 DO 5 0 I = l.tiM DO 6 0 , J=l, M O GO CC(I, J* NP ) = F (I, J ) 5 0 COl ' IT INUE C C FORM RI C H T PA Rr I T I O M 0 ;: MAT R I X CC .. .

C f IG ? 0 I = 1.tiM DO 80 J :- l , ti U 80 C C ( I, ,J+I'IP+ ti [)) = C( I ,J) :i :' 7 0 CONTINUE C RETURM END !

,I

:I

l

i

_j

3 0 8 :::::::::::::::::::::::::::::::::::::::::: o°. °, • ° ° , ° • o , ° . , ..o _Z_ C SUBROUTINE FORMGA ¢ ::::::::::::::::::::::::::::::::::::::::::::::::::::. . ,.° C C THIS SUBROUTINE FORMS THE MATRICES - C C [ 0 ] [G P] C GGA = [ GG ] ; G P A = [ 0 ] C [ O] [ OI C C - FROM T HE MATRI C ES CC AND G P C IF NP = 0, WHERE THE LO!JER C ZERO PARTITIONS HAUE flU @OUS C AND THE OTHER ZERO P_RT I T I ON S C ARE C O MPATABLE WI TH T HE C CORRESPONDING POPULATED PARTITIONS C C AN D , THE MATRIX - C C GGA = [ GG l C [ O ] C C - IF MP = h e , _HERE THE L OWER C ZERO PARTITIO N HAS NU ROWS C C .....

C C C ALL STA T EMENT ARG U hENT5 - C C _G : ND X I KAT R IX O F RO W DIMEN S ION NY i C CP : NC X NC MAFRIX OF RO W DIMENSION NX i C GG A : NPDU X 1 MATRIX OF RO W DIMENSION hU XY , C (IF NP = O; NPD U = NO i C IF NP = HCf NPDU = NP + NU) : C CPA : NPD U X N U MA T RIX O F RO W DIMENSION NWX Y C C SUBROUTINE FORMGA(GG.GP,NUXY,HX.NY,GGA,GPA) DIMENSION GC(NY, I). GP(NX.MX). GGA(NWXY. I), GPA(NUXY,NX) C COMMON / DIM / N U , HC, N O , NM, N P i NP D U = N P • ND C INI T IALIZE MA TR ICES GG A AND GP A T O Z ERO " D O I I=I, NP D U £ GG A ( I.I ) = 0. 0 DO 2 J = I,NC 2 GP A ( I °J) = 0 . 0 I C O N T IN U E C C F OR M UPPER P AR T I T IC .' I OF M A TR IX CG A IF NP = 0 C OF FORI I CEN TER P ARI I TION OF M A T RIX GG A IF HP = NC C DO 3 K = I,ND 3 G G A ( K + N P) = G G( K ) C I F (N P .E Q . O) RETU R N C C F OR M UPP E R P AR T I T I ON O F M A TR I X C, P R = C DO 4 L= X , M C DO 5 M = L, M C 5 GP A (L, M ) = GP(L pM ) 4 CON T I NUE C R ETUR N E N D ,!

J

\ • CZ_Z|:::SlS_|=_:=Z_:::IZ:_:_U:_; ' $:_Z::==_=:IZSS::I| t C 5UB R O U TI _ F CR MM Z C C TH IS S U BROUT IME U E TERM I M E S TH E ROll O F "ELE ME M T S " C O F T H E MATRICES C, E AM D F WH IC H CORRE S P OND C T O T ERM S O F A N OUTP U T EQUATIOM WHIC H DES C RI B ES C A M _UT PU T UAR I A B LE WH I CH IS PR O P O R TIOMAL TO C THE F IR ST DERI U ATI U E OF THE STAT E S , W H ERE- C C Y = CX + FM .

C C TH E R O W Wr l I CHCOR RE SP O r 4D STO T H E T H IS R E S P ON S E C I5 ASSUMED T O BE TH E LAST ROW O F T H E MA T RICES.

C C T HE OPERAT I O N OF DETERM I NIN G THE X -gOT C PROPORTIONAL ELEME H T F COM_I S T 5 C OF E × TRA C TI NG CERTAIh ELEME N TS OF THE MATRICES R , C B A_ D D WHICH A PP EAR IN THE OUTPUT EO U ATIOM.

C C IM TH E S P E C I F IC P R OBL EM DO ME I M T HI 5 STUD Y , C T H E X-DO T PROPORTIONAL OUT P UT IS P LU HC E ACCELERR T I OH .

C THE ACTUAL N Z E Q UAT I OM CAM B E F OUMD I N T H E T EXT C D E SCRI B IMG THIS S UB R OU TINE.

C C C C ALL ST A TE ME NT A RGUMEN TS - C C A : MU X N U MA T R IX O F ROW D I MEMS IOM C B : NU X M C M A T RIX OF ROW DIM E MSION MW C C : NM X N U M A T RIX b ¢ ROW D I M E N S IOfl N: C D : M U X M D MATR I X OF R O W DIM EMS ION N g C E : NM X NC M A T R I X OF ROW D I ME MS IO M M: C F : MM X N D MATRIX OF RO W DI M E N SI OM M Z C MX : COLUMM D I ME M 51 0M O F MAT RI CE S B AN D E C MY : CO LUM N D I MEM S I O M OF M A T R IC ES O AN D F C X : I X N ) TR AMSF O RMATIOM R O W UE CTO R OF C CO LUMM D I M EM SI OM N W C C M M : I X NM UTILITY R O W UE C T O R O F MZ E L EM ENT S C FOR T HE MA TR I X C OF C OL UMM DIM E MS IOM M I4 ' C E H M: I X NC U T ILI TY ROW U EC T O R OF NZ E L EM E M TS C F OR TH E M A TR IX E OF COLUM M DIMENSION MX C FMM : I X N D U TI I" TY RO W U E CTOR OF r lz EL E M E NT S C FO R THE _ , _ lX F OF COLU MM DI M EMS I O MMY C C S UB ROU TIN E FORMMZ( A ,B,C _ D, E , F ,M W , M X,MY,NZX, C NM , ENM, FM M) D IMENSIO N A ( MW , MW ) , B( M U, M X) , C( M Z,N W ) D I MEMS I OM D(rl W ,N Y )° E ( M Z, N X ) , F ( NZ , N Y ) D IME NS I O _ X ( i o r l g ), CN M (I , N g), E H M ( I , NX ) , F NM (I ,N Y ) C COMM ON / D I M / MU, M C, NO , 1111, liP " C C S U B R O U TINE MU LT ( A , B, A D, L, M , N, rLq, M8, I ' kqB) C C COMPUTE S THE M AT RI X PR ODUCT : A B = R * B C A : L X M MA TRI X WITH ROW DIMEN SI O N H A 3]2 c _ : _ : :: :::: _ ::: ; ::: l :: **** _ : _ : _ : _ : •• _; :: .• * * _ : * _ * _ :: _ : _ ** ****•,, ** •,** . . .•: _ | C S UBROU TINE M O DAL • * • • w e * e• • • • • • C C T HIS SUB ROU TINE DETERMINES T HE EI G EI IU EC T ORS , C MODAL MA T RI X AND INVERSE MODAL MATRIX OF A C G ENERAL IIATRIX.

, C C .......

C C CALL STATEMENT AR GU MENTS - C C AA : NPDU X H PDU MATRIX OR R OW DIMENSIO N MWXY C EIG : NPDU UECTOR OF EIGEhVALUES OF MATRIX AA (I c C DIMENSI O N NWXY C EE : NPDU X NPDU MATRIX OF EIGENUECTORS O F AA C CF ROW DI_E H SION NUXY C El : NPDU X NPDU MATRIX IHUERSE OF EE OF ROW C DIMENSION NWXY C C .............

C SU B ROUTINE MODAL(AA,EI O ,EE,EI,NWXY) DI;IENSION AA(NWXY,NWXY), UK(IO00) COMPLEX EIG(NUXY) , EE(NWXY,MUXY), EI(NWXY,NWXY) C CCMMON / DIM / NU, NC , HO , NM , NP C NPD U = N U • N P C C C ........

C C SUBRO U TINE G EIGEM ( MR, AA, N, JOB, UK, KPR, FROM, C EIG, NE, EE, EI, _AME ) C ' C COMPUTE5 EIGENUALUES AND, OPTIONALLY, NODAL MATRIX OR C NORMALIZED MODAL MATRIX AND IN V ERSE MO D AL MA T RIX OF A C R E AL GE NERAL MA T RIX C C I N P UT ARGUMEN T S- C AA(N,N): REAL G ENERAL MA T RIX WITH ROW C DIMENSION NA C N : ORDER OF AA C J OB : OP T ION CO N TROL C = 0 : EI G EN U AL U ES Ot ILY C = I : EIGENUALUES PND MO_AL MATRIX C = _ : EI C E r4 UALUES, MODAL MATRIX AN D , , C I_ U ER S E MO D AL t XATRIX C = 3 : S ANE A S I WITH NORMALIZE D C MODAL MATRIX C = 4 : SANE AS _ WITH hORMALIZED C MODAL MATRIX C WK : R E _ L WO RK VE CTOR O F L EN GTH G R EA TE R C T H A_ 2 *N_ N + 3 m N C K PR : P R INT C C N TRO L C = 0 : PRIN T NOTHIN G C = I : PRIN T EI G EHUALUES C = 2 : P RIN T E V E R Y T H IN G C F R O M : IO -CH ARAC T E R NAME OF C ALLING PR O GR AM \ C MA, h _ : R OW D IMENSIOMS C NAME : I O -C H AR A CTER N A ME F OR MATRIX RR C C OUTPU T AR G UMEMT5- C EI C( N ) : COM P LEX U ECTOR O F EI C ENUALUES O F A R C EE( H .N) : M O D Al - MA TR IX WITH ROW D IMEMSIO R ME C EI(M,N) : EE IMUERSE WITH ROW _IME_SIOM NE C C CALL GEICEM(NWXY,AA , NFDU,4,WK, I,_MODAL*,EIG o NWXY,EE,EI , #A-AUG*] C RETURN END \ t • • • ! . B e ii • e Q • C SU B ROUT I NE P H RG • eo • C C THIS ROUTINE CALCULATES _N D PRINTS COM P LEX MA T R I CES C IN MAGNITU D E AND P HASE FORM.

C G -- ' C C CALL S T ATEME N T ARG UM EN T S - C C ARRAY : M X M COM P LEX MATRIX O F ROW DI M E M SIOM NROW C ITITLE : T_O fIOR D LEr , GTHL I TERAL _R_AY CO M TAIMI r lGTHE C NAM E OF ARR A Y IM T HE MAI M PRO G RAM C AMASE : NRA X 2 ARRAY CONTAI N ING T _ ; EM_GM I T UD E AND C PHASE FORM OF THE ELEMENTS OF THE COLUMNS OF ARRAY C MATOUT : M X M COMPLEX MATRIX OF ROW DIME t IS [ GMrlRA C CONTAINS ARRAY TRA;i S FORME I } [ N [ O HIICr l ITU3L AND P HA SE C FOR M C C- C S UBROUT I NE F HAG (ARRAY , NROW , M , M .I TIT L EpAMASE . MA T OU T, MRA ) C CO MP LEX A R RAY(MROW,M) , MAT O UT(MRA ,M ) DIMENSIO N AMASE(M_A , 2 ), I T I T LE(2) C C NROW : ROW DIMENSION OF T HE INCOMI NG MATRI X C NRA : ROW DIME N SION OF THE O UTP UT MA T RI X C C P RIMT H EADER FOR OUT P U T MA T RI X C B O T H WOR D S OF ITI T LE CAM BE SET IN A DA T A S T A T EMEN T O R C INDIVID U ALLY USING DIREC T ASSIGMMEM T .

C C I .E. I T I TL E (1 )=I OHCO N TR O LLA B C ITITLE(2)=IOHILITY_ C C NO ERRORS WILL BE FLA GG ED I F ITITLE IS P ASSED AS A S INGLE U ARIA B LE; C HOW E U E R, T HE REMAIMING WOR D WILL E E P RIN T E D A S J UI4KCHARACIERS I N C tH E H E RD ER.

C W R I T E( G , t O0) WR I T E ( G , t l O ) I TITL E , M . M WRITE(S , 12 0 ) C C PH A O L OO PS TH RO UGH ARRAY COL U MN BY CO LU MN C OM U ER TI N G C EACH ELEMENT IN T O MA G MITU D E AND P HASE F ORM .

C DO l O JP =I ,M • C C REIM ITI A LIZ E AMAS E C D O I K P= I , M R M A SE (KP, I )=O . O AMA S E( K P , _)= O o O I C ON TI N U E DO 5 I P =I . M C C SEPARATE REAL AND [ MAGIN P RY PAR T S O F COM P LEX NUM B ERS C R E A L AN D A I MAG A R E A_ SI 5 T A N D ARO F OR T R AN FU _ CT I O _$ 3 1 5 C TERMI=REAL(ARRAY(IP,JP)) TERM2=AIMAG(ARRAY(IP,JP)) AMASE(IP, I)=5ORT(TERMI**2+TERM2_*2) IF(ABS(TERMI).LT.I.OE-15) TERMI=I.OE-15 C C ATAM2 RETURNS ARGUMEMT IN RAOIAN5 ( -PI ( THETA < PI ) C AMASE(IP,2)=ATAN2(TERM2,TERMI) j C C CONUERSIOM TO DEGREE5 C AMASE(IP,2)_St.295 7 8*AMASE(IP,2) C C LOAD INTO OUTPUT MATRIX C C CMPLX 15 ANSI STANOARD REAL TO COMPLEX EONUER510M C MATOUT(IP, JP)=CMPLX(_MASE(IP, I), AMASE(IP,2)) C 5 C8MTIMUE 10 CONTINUE C C USE COMPLEX MATRIX OUTPUT RRUTINE TO PRINT MATOUT C HEABER PRINTING 5UPRE5SEO (LPRINT= -t) C CALL CMPRNT (NRA, MATOUT, M,M, S, G,_OUTPUTM, - l) C I_}0FORMAT( l/z/x/ , fOX, 50(_-_) ) IIO FORMA T ( I OX, 2AIO,_ MA T RIX (_ , 12,_ BY _,12 , _) _ , /, 15X , _O U TP U T F ORM: ( MAGHIT U B E , PIA S E(O EG .) )_3 120 FORMAT(IOX,5 0 (_-_)) RETURM " EMO o 31 6 C ::::::::::::::: : ::::::::::::::::::::::::::::::::t::::::: 1 ::: C SU B ROU T INE RESI D C C T HI S S U BRO U TIM E b E T E R MIHE S T HE OUTT E R P ROD U C T • C O F TH E MATRIX OB5 An P A COLUMn O F T H E MA T RIX COM .

C TH IS MA T RI X RESI I AT . _ P RESE HTS AM IM P ULSE RESIDUE C MA T RIX.

C C T HE ELEM E MT S OF T H E RESMAT MATRIX ARE U RI T TE H TO A TAP E C 50 THAT PO 3 T FXECU_IO M A N ALYSIS CA H BE PEPFORMED. SOME C EXA_IPLES OF THIS MICHT BE II,IR_IALI Z ATION O R REOROERI MG C OF THE RESIDUE S A S SOCIATE D WIT ) % F ARTICULAR MODES.

C ' C CAL L S TATEM E MT ARG U MEMT S - C C CON : _ P _ U X ME MATRIX O F ROW DIMEN S ION t _ U XY C OBS : _M X tiPDU i l A T RIX OF RO!I DIIIEM S IO M _Z C M X : CO L UMn DIMXtI S ION OF MATRIX CON C RESMAT : MATRIX FOR_ED F_OM OUTER PROD U C T OF C OB5 F_IO A C O L UM_ O F CON C I : C OLUMN OF C OM US E D IM O ETE R M I_I MG RE SM A T " C C SUBRO U TIME RESID(CO M, OB S ,MZ , MX,MUXY, R ES MAT, I) CO M P L E X CO N (t _ U XY , NX ) , O BS (HZ , M W X Y), RESMA T (M W XY , M Z } C C OMM O_ I / DIM / M U , MC_ _ 9 , M M , C _ PDU = M P + MV C DO 1 0 J= I , _ M DO _ 0 K= I,M PDU RESt I A T(K,J ) = OB S(J , K ) * COM ( K , I ) _ 0 WRI T E(I,IO0 ) RE S MA T ( K,J ) 100 FORMA T (_ FL O.5) 1 0 C OM TI I A. . q E R E TU RM , E MO i

!

• :.1| • " :" : :::| : ::::I S I: : :.** C -r4 J B ROUT IME SC A LE C ::::'-:''::*:,- • . ° -.::::::::::::::: : :::::: : ::: :: ::::::: : ::: : :::: : ::: : . . - .- . . o', C C THIS SUBROUTINE SC A L ES T H E M ATR : C ES A, B , C AN D O WITH C A DIA GOI _AL MA T RIX T. TH I S P RO C E DUR E IS AIIR L OCQU5 TO C A CH AMGE OF UR RIAB LE TRP Z L S F OR M R TI O N .

C C X -D O T = A X + BU + O W C Y = C X + F W C C A P P LY TRP MS FOR MA T IOM X = TZ C C _ - I ]OT = ( TI M U tAm T )Z + (TI M U m D) W C Y = (C m T) X + E U + F¼ C C SUB R OUT I NE S C A L E ( R, B_ C • O • MW ,M X i I1 Y• M Z P T , T I ttU oA S, BS t CS p DS ) DIMENSI O n R( MW, M W). B( M U .NX]m CiM Z , M W ) , O(; I W. MY ) O lr l EN_l Oll AS ( MW, MW )• B S(MW . N X). C S(M Z, M W), D S(MW , MY ) DI MEN SIOM T( N W . H W ), T I l g U( _ il . l , N W ) , W 2 K ( S O0) C C OM MOM / D I M / MU D IIC, M n, I_, M P , C C C C S UBR C UT IME R I IIU ( TT• TI, 141 ( , I _R , 11 ) C C C G ;_d TE 5 TH E In UE RSE OF A REAL M A T RIX C C IN P UT ARG UM EM T S - C TT( M ,M) : R E I _ . MAT R IX WITH ROW DIMEMSIOPI M R C Ul ( : R E AL W ORK UECT O R WIT H L E M GTH C GRE AT E R TH An M * f1 _2m M C M R : ROW DIflEMSIO H C C OUTPUT RRGUMEM T S - C TI (M , M ) : I MU E R SE OF TT WI TH R[_ . i DI I fl E fI SI O M fIR C C,.o- C C C FORM IM UER SE OF 5C ALI I _ MA TR I X T C CALL R IM U( T, T IMU, W R K,MI4,f ' IU) C C ....

C C 5UBROUTI ME MP3 ( MR , R R , R, MB, BR . B, MC, CA , CC , C, !_ , P ) C C COM P UT E THE M A TR IX P RODU C T: P = A *IB'C C C A I _ X B R REAL M A TRI X W ITH GO_ l DI_EM S I O M I _ C B : BR X CR R EAL M A TR IX W I T H R O W D IMEN S IO N MB C C : CR X CC RE AL M A TR IX W IT H R O W O I;IE f1 51 0 n_ .

C P : Al e X C C R E AL M A T RI X W I TH R O W D IM E ; 1S I O _ I PIP C C c C FO R M : _CAL E D DYNAM I C M ATR I X A S C CALL MP 3 ( NW,MUD T,MW, MU , A, MW, M U , M U i T I MUI M U , AS ) C C ...........

C C 5 UB ROUTIM F MED ( AIM , _ OUT, M R , M C , t _ R !, NR O ) 1 C C 5TORE . qA T RIX AIM ( MR,tiP.) IM THE MA T RIX AOUT ( NR, H C) C WHERE A I M t _M O AO U 1 HA V E R CU Ol t I E / _S IC i45 t i RI AN L I N RO C C- C C C E OU A T E S C AL ED D YNAMI C MATRIX AS T O D YNAMI C C MAT R IX _ F O_ RETURN TO MAIr l P RO G RN t C C ALL elE G (AS,A, N U . f lU. MW, r_; ) C C ............

C C S UB ROUTIME MULT ( A. D . AB , L, M , , ' I. MA D NBo t l A B ) C C CO R I P U T E MA T R I X P R OI J LJ C T :A_ = A* B C C A : L X 1 1 MA T R IX W I T H RI _W I i IM E N S IO H C B : M X M MAFRIX WIT H RO W Dt _ IE NSIOM MD C A 3 : L X I'I MA T RI)( WITH ROW DI H ENSI L) ' I fl A 8 C A , _I, A B R EAL MA T R ICES C C ...........

C C C F O RM SC ALED CONTR O L MA T R IX BS A I _D EOUA T E IT T O THE C CO MT R OL MATRIX B F O R R E TU R;t TO MAIM PR L I G R I :::_ I C CALL M U L T( T , B, B 5, IIU , M U , M CoI IW, I ' ¢ h I i &4 ) C A LL M EO (I )5 . B , r lV I H C, MW , ;IW ) C C FO R M S C AL E D D I STURB A H C E MA TRI X D 5 AM ]] EQU A T E I T TO C THE DI STRU B AI 'tC E MA TR IX 1 ] F O R R E T U R M TO MAIM _ R O G R AM C CALL H I _ .T ( T. D. DS, t l U, M U. MD . M U. fl lJ, N IJ) CALL M E Q( D S . D . NU , M / ). N .h M IJ) C F O R M SC ALEI] OUTP' " MA TR I X C5 A ri d EOUA T E IT TO T HE C O U T P UT MA TR IX C F J R R E TUR M T O MAIM P R OCRAM C C A LL M U L T ( C . T Ir l U . CS, i _ , t _ , ftU. M Z. N& I, t l Z ) C A L L ME O{C5, C° M ,'I, N U. H Z, M Z ) C R E TUR M , ; EMD _i_ 31g Ct C SUBROUTINE STEP C_ .............

C C THIS SUBROUTINE D ETERMIHES THE MA T RIX UF C STEP RESIDUES FOR A GIVEN IMPULSE RESIDUE C MATRIX AMD A SPECIFIED SET GF EIGEH- C VALUES.

; C C THIS IS DOME BY FORMING A DIAGONAL HAT?IX ._ C OF THE EIGEMUALUES AHD P OST--;IULTIPLYIHG ITS C INVERSE BY THE IMPULSE RESIDUE MA T RIX.

C C .....................................................

C C CALL STATEMENT ARGUEMENTS - C C IMP : HPDU X rimMATRIX OF ROW C DIMENSIOH MW×Y (IIIPUL_E RESICUES) C EIG : NPDU VECTOR OF DIMENSION MWXY C (VECTOR OF E-VALUES) C STP : MPDU X NM MATRIX nF RO H C DIMENSION NWXY (STEP PESIDUES) C EIGMAT : NPDU X MPDU DIACOMAL MATRIX C OF ROW OII_EMSION MNXY C (E-VALUES ON DIAGOMAL) C EINU : HPDU X NPDU DIACOMAL MATPIX C OF ROW DIMENSICN NWXY C (INVERSE OF EIGMAT) - C '_ C ........

C SUBROUTINE STEP ( IMP,EIG. HWXY, NZ, STP. EIGM A T, EIMU) : C COMPLEX _MP(M H XY,MZ), EIC(NWXY), STP(NWXY, HZ) COMPLEX EIGMAT(N H XY,MWXY), EINU(HWXY,NWXY) REAL HK(IO00) C COMMON / DIM / _U, MC, MB, _M, NP C HPDU = MP + HD * MU I C IMITIALIZE MATRIX EIGMAT TO ZERO C D O 5 J= I,M P O U gO _ K = I,N P D U G E I GMA T ( J , K ) = 0 . 0 5 CONTINUE C C F ORM DIA G ONAL MATRIX O F E- V ALUES C O0 10 I=I,MP BU I O EICMA T (I,I) = E I O (1 ) C C 5U B ROUT I HE CI HU ( TT _ T I , W K , M R _ M ) " , C C C O M PUT E T HE IHUERSE OF A CO MPL EX MA T RIX C C INPUT AP G UMEM T S - I i C TT(NoN) : COMPLEX MATRIX WITH ROW DIMENSION NR C UK : R E AL W O R K VECTOR WI T H LENG T H i C GREATER THAN NwM*2 + 3mM : C MR : ROW DIME H SIO M C M : O R D ER OF T T AND TI C ; C OU TP U T ARG U MEN T S- _I ' C T I(N. M ) : C OMPLEX I N V ERSE O F TT WI TH C R O W D I MEMSIOH M R !

I c

C--

4 c

' C IN V ER T D IAGONAL MATRIX O F E- V ALUES I c : C ALL C IM U (EI G MA T, EIN U,U K ,HW XY,HPDU ) ,i C C_w C C SUBROUTINE MULT C C ( A, B. AB, L, M, I4, NA, MB, HAB ) C C CO MP UT E MA T RIX P R ODU CT: AB = A * B C C A : L X M C OMPLEX MATRIX WITH ROW DIMENSI O N MA C B : M X M CO MPLEX MA T RIX W I T H RO W DIMEri S I O M N B C AB : L X N C OMPLEX MA T RIX WIT H ROW DIMENSION NA B C C C F ORM S TEP RESI D UE MATRIX C CALL M U LT C C(EIH U , IMP,STP,NPD U , M P D U ,HM, I ' I WXYINWXY, HWX Y) C C OUTPUT STEP RE S I DU ES TO A TA P E C D O 30 I=I.NM D O 20 J= I,N PDU | 20 U RI T E (2 , 20 0 ) STP( J ,I) 30 C ONTIN U E . t C _I 200 FOR MA T (2 FI O. 5) C I RE TUR M , EM D I !

% - , ' _ % , ?

::::::::::::: :::::::::::: : ::::::: : ::: : ::.. . .. ." .... ..: : ''....': : :.:: : C SU BROUTINE WORK ee C:======= == ====================== = = = - ".'.

C C THIS SUBROUTINE F 3 RM S THE COMPLEX MATRICE S C CON, ogs AMD g IST 9Y MULTIPLYING THE MATICE S C B, C AND D BY A COhPLE X MATRIX M.

C : C THIS COMPLEX MATRIX MULTIPLICATION REPRESENTS A } C TRANSFORMATION THROUGH THE COMPLEX MATRIX M, WHEM C THE MATRICES B, C AND D ARE RELATED THROU G H THE C STATE U A R IABLE EO U ATIONS - C ' C X-DOT = AX + BU + g _ C Y=CX C C 30 THA T - C C COM = MIMU * B C OB5 = C • M C gIS T = MIMU * D C C ....

c C CALL STATEIIEMTARGUMENTS - C C g : MPDU X NC MATRIX OF ROW DIMENSION NHXY C C : MM X MPDU MATRIX OF ROW DIMENSION MZ C D : NPDV X ND r lATRIXOF ROW DIMENSION MWXY C M : NPDU X NPDU MARTIX OF ROW DIMENSION NWXY C MIMU : IMUERSE OF MATRIX M C MX : COLUMN DIMENSION OF MATRIX B C MY : COLUMN DIMENSION OF MATRIX C COM : NPDU X N_ MA T RIX OF ROW DIMENSION NWXY C OBS : NM X NPDU MA T RIX OF ROW DIMENSION NWXY C gIST : NPDU X ND MATRIX OF ROW DIMENSION NWXY C C_ .....

C SUBROUTINE WORK(B , C , D,M,MINU,NX,NY,NZ , NWXYDCOM,OBS,DIST) nIMENSIOM B(NWXY,MX) , C(NZ,NWXY), D(NWXY,I ) COMPLEX M(NWXY,NWXY), MINU(N_JXY,NWXY ) COMPLEX CON(NWXY,NX), OBS(NZ,NWXY), DIST(NWXY , I) C COMMON / DIM / NU, NC, N O , HM, NP < C , C NPDU = NU + NP + N 9 C C 5UBROTUINE MULTCR ( A , B° AB , L, M, N, NA, NB, HA 8 ) C C COMPUTE MATRIX PR O DUCTs AB = AmB C C A : L X M COMPLEX MA T RIX WITH ROW DIMENSIOM NA C B : M X IIREAL MATRIX WITH ROW D IMENSION N8 C AB : L X M COMPLEX MATRIX WITH ROW B IMENSION NA B C I

i

i C FORM MATRIX COM C CALL M U L T CR(MINU,B,CON,NPDU,HPI]U, MCPMWXY,MWXY,I'B_XY ) C C ...........

C C SUBROUTINE M U LTRC ( A, B, AB, L, M , 1 1 , MA , Mg, NAB _J C C COMPUTE MATRIX PRODUCT: AB = A*B C C A : L X M REA l_ MATRIX H ITH ROllDIMENSION NA C B • M X M COMPLEX MATRIX W IT H ,_OWBIhErISIOH_B C AB : L X N COMPLEX MATRIX WITH ROW DIMEMSIO R I_AB C C C FORM MATRIX OB S C CALL MULTRC(C,M,OBS,MM,;_=I] U , N P DU,MZ, t IWXY, NZ) C C FORM MATIX B I ST C CALL ;'IULTCR (MINU, B ,DIST,MPI]U, MPI]U, I,MWXY,NWXY,NJXY} C RETURN END _h i I Repo, t N, _ F _" G _ *,,,, _ r , _ Acceu;on No. 3. R* _ ;p,ent's C 4_ log NO.

NASA CR-117943

I

4 T, Il e e , ' _ Subtill e _ Rei:x)rt Dete Andly s is o f Flexible A ircraft Lo ngitudinal Dynamicsand J une 1985 Hand l ing Qualities - V olu me II - Data 6 . r,,_,_o,_ .;, ,,_o_co , 7. A ulhor|$) | . P e rf o( mi r_ i O rg*n; / a tio n Repo w rt NO .

; Martin R. Wa sz ak and David K. Schmidt ....... 10 .W_ k U ni t N o .

9 . P e rf _r ming O rI Hn; z,m fion N e m¢ end Addrt.m Pur du e U niver s ity i, . c,,,_, , ,.:_ = , ' O , ,., N,, .

Sch ool o f Aeronauticsand Astronautic s We_t Lafayette IN 4 7 90 7 NAG-I-254 , ) 13. Ty _ of Repo r t a _ d Pai od C ov er ed 1 _ . $1:x)m, or;n9 A ge n cy N a m e and A_ re m4_ Nationa l Aeronauticsand SpaceAdministration ContractorReport Was h ington, DC 2 0 5 46 1 4 s_ , _,_ , _Ag, .¢ v c _, , 505- 3 4-03-0 3 iS. SupGWeme nt m ry Note1 N A S A Tech n ical Mo n itors : Wi ll iamD. Grantham an d Jarre ll R. Elliott LangleyResearchCenter 16. Abstra¢ l A s aircraft become l arger and l ighter due to design re q uirement s for increased pay l oad and improvedfue l efficiency, they wi ll a l s o becomemore f l exible. For high l y flexible vehicle s ,the handling q ua l itiesmay not be accurate l ypredicted by conventiona l method s . Thi s study applies two analysi s method s to a family of f l exib l e aircraft in order to investigate h o w and whe n structura l (especia ll y dynamic aeroelastic)effect s affect the dynamic characteristic s of aircraft.

T h e first ty pe o f a n a l y s isi s a n open-l o o pmo d a l an a l ysis te chn i qu e. T h i s met ;rod c o nsiders the effect o f m od al re s i du e ma g nitu d e son d etermining vehicle handling q ualitie s . The s ec o n d meth o d i s a p il o t-in-the-l oo p analysi s pr o ce d ure that con s ider s severalclose d -l oops y s tem characteri s tic s . B o t h analy s e s indicatedthat d ynamicaer o ela s tice f fect s cau s e d a d e g radati on i n ve h icle tracking p erf o rmance, ba s e d on the eval u ati o n o f so me s imulati o n r e s ult s .

¢ i; . Key Words #Suwested by Auth o r(sil tO. Ois t rib _ ;on St. t .n, _ nS F l e x i b l e Aircra f t F light Dynamics Unclassified- Unlimited H a ndl i n g Q u a l itles M od a l Analysis Sub jectCat eg ory 0 8 "

1 ,. o.. , . ' c o, s" , . , ,, tJ, , w 2, . N o . o , , . ,,, .. ;, . , i

U n c l a ss ified ,l Unc l a ss ifie d 2 4 0 All i , , ) , . ) _ F o r s ale bythe Nahonal Tech n ical I n f u mtio n Se r vice . S p a i n |fi e ld . Vir l ini a 2211;1 i

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Document details

Doc number
19850026890
Publisher
NASA
Year
1985
Pages
240
File size
6.0 MB
Chapters
2
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