Study of flutter related computational procedures for minimum weight structural sizing of advanced aircraft

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C J TECH LIBRARY KAFB, NM 3. . . . OObL54b ..

1. Rbport NO. 2. Governnmt Accession No, " NASA CR-2607 " "~ 6. R O W [ktb 4. Titlb and Subtitle March 1976 StudyofFlutterRelatedComputationalProceduresfor Minimum 6. Morrning Orpnization Cod.

Weight S t r u c t u r a l S i z i n g o f Advanced A i r c r a f t ~ " ~ . . - - - -. - .. " 8. Performing Organization Report No.

7. Author(sJ R. F. O'Connell. H. J. Hassig, and N. A. Radovcich LR-26650 10. Work Unit No.

Q . Performing Organization Namb and Address 11. Contract or Grant No.

Lockheed-California Company P.O. Box 551 NAS1-12121

Burbank, C a l i f o r n i a 91503 -

13. Typr of R e p o r t andPeriod C0ver.d '12. SponsoringAgencyName and Address C o n t r a c t o r 14. SponsoringAgency codc NationalAeronauticsandSpaceAdministration Washington. D.C. 20546 I __I 15.' SupplementaryNotes Technical Monitor: 5. Carson Yates, Jr., Corqwter Aided Methods Branch, S t r u c t u r e s E Dynamics Division, NASA LangleyResearchCenter, Hampton, V i r g i n i a '23665 F i n a lR e p x t $8. Abstract Resultsof a studytowardsthedevelopmentoffluttermodulesapplicabletoautomatedstructural design of advanced a i r c r a f tc o n f i g u r a t i o n s ,s u c ha s a s u p e r s o n i ct r a n s p o r t ,a r ep r e s e n t e d . In t h i ss t u d y ,a u t o m a t e ds t r u c t u r a ld e s i g n is r e s t r i c t e d t o autornatedsizingcftheelements o f a g i v e ns t r u c t u r a l model, It i n c l u d e s a f l u t t e ro p t i m i z a t i o np r o c e d u r e ; i.e.. a procedurefor a r r i v i n g a t a s t r u c t u r e w i t h minimum mass f o r s a t i s f y i n g f l u t t e r c o n s t r a i n t s . Methods o f ' s o l v i n g t h e f l u t t e r e q u a t i o n and computing t h eg e n e r a l i z e da e r o d y n a m i cf o r c ec o e € f i c i e n t si nt h e r e p e t i t i v ea n a l y s i se n v i r o n m e n to f a flutteroptimizationprocedurehavebeenstudied,and recommended a p p r o a c h e sa r ep r e s e n t e d .F i v ea p p r o a c h e st of l u t t e ro p t i m i z a t i o na r ee x p l a i n e di n d e t a i l and compared. An approach t o f l u t t e ro p t i m i z a t i o ni n c o r p o r a t i n g some o f themethods discussed i s p r e s e h t e d .P r o b l e m sr e l a t e dt of l u t t e ro p t i m i z a t i o ni n o r e a l i s t i c d e s i g n environment a r e d i s c u s s e d and an i n t e g r a t e d a p p r o a c h t o t h e e n t i r e f l u t t e r t a s k i s p r e s e n t e d .

Recormendations f o rf u r t h e ri n v e s t i g a t i o n s are made. Resultsofnumericalevaluations,applying t h e f i v e methodsof f l u t t e r o p t i m i z a t i o n t o t h e same d e s i g nt a s k , are presented.

17. KeV-bVords (Suggested bq Authoris) ) 18. Distribution Statement Dcsig,, f o r F l u t t e r ;S t r u c t u r a lO p t i m i z a t i o n ; U n c l a s s i f i e d - U n l i s i t e d A e r o e l a s t i c i t y SubjectCategory 32; StructuralMechanics 19. Security aJSlf. (Of thir report) 20. Security Classif. (of this pagel U n c l a s s i f i e d 1 2 7 Unclassified For sale by the National Technical Information Service, Springfield, Virginia 22161 SUMMARY R e s u l t s o f a studytowardsthedevelopmentofflutter modules a p p l i c a b l e t o a u t o m a t e d s t r u c t u r a l d e s i g n o f a d v a n c e d a i r c r a f t c o n f i g u r a t i o n s , s u c h as a s u p e r s o n i ct r a n s p o r t , are p r e s e n t e d .I nt h i ss t u d ya u t o m a t e ds t r u c t u r a l design i s r e s t r i c t e d t o automated sizing of the elements of a g i v e n s t r u c t u r a l model. It i n c l u d e s a f l u t t e r o p t i m i z a t i o n p r o c e d u r e ; i .e ., a p r o c e d u r e f o r a r r i v i n g at a s t r u c t u r e w i t h minimum mass for s a t i s w i n g f l u t t e r c o n s t r a i n t s .

Methods o f s o l v i n g t h e f l u t t e r e q u a t i o n a n d c o m p u t i n g t h e g e n e r a l i z e d a e r o - dynamic f o r c e c o e f f i c i e n t s i n t h e r e p e t i t i v e a n a l y s i s e n v i r o n m e n t o f a f l u t t e r o p t i m i z a t i o n p r o c e d u r e h a v e b e e n s t u d i e d a n d recommended approaches are pre- are e x p l a i n e di nd e t a i la n d sented.Fiveapproaches t o f l u t t e r o p t i m i z a t i o n compared. An approach t o f l u t t e r o p t i m i z a t i o n i n c o r p o r a t i n g some o ft h e methods d i s c u s s e d i s p r e s e n t e d .P r o b l e m sr e l a t e dt of l u t t e ro p t i m i z a t i o ni n a r e a l i s t i c d e s i g n e n v i r o n m e n t are discussedandanintegratedapproach t o t h e e n t i r e f l u t t e r t a s k i s p r e s e n t e d . Recommendations for f u r t h e ri n v e s t i g a - t i o n s are made. Resultsofnumericalevaluations,applyingthefivemethods o f f l u t t e r o p t i m i z a t i o n t o t h e same d e s i g n t a s k , are p r e s e n t e d .

iii TABLE OF CONTENTS

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

SYMBOLS AND DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . i x

1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives of Study . . . . . . . . . . . . . . . . . . . . . . 3

2 . OVERVIEW OF THE FLUTTER OPTIMIZATION TASK . . . . . . . . . . . . . . . 3

3 . SOLWON OF THE FLUTTER EQUATION . . . . . . . . . . . . . . . . . . 5

3.1 The Generalized Flutter Equation . . . . . . . . . . . . . . . . 5

3.2 . Types of Solution Sought . . . . . . . . . . . . . . . . . . . . 6

3.3 Methods of Obtaining Point Solutions . . . . . . . . . . . . . . 8

3.3.1 Bhatia Method . . . . . . . . . . . . . . . . . . . . . . 8

3.3.2 Phoa Method . . . . . . . . . . . . . . . . . . . . . . . 9

3.3.3 Lockheed Program 165 . . . . . . . . . . . . . . . . . . 11

3.3.4 Desmarais-Bennett Method . . . . . . . . . . . . . . . . 11

3.3.5 Two-Dimensional Regula F a l s i . . . . . . . . . . . . . . 12

3.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Minimum Damping i n Hump Mode . . . . . . . . . . . . . . . . . . . 13

3.5 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 . MODALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 . 1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Types of Modes . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Number of Modes . . . . . . . . . . . . . . . . . . . . . . . . 18

4.4 Updating of Modes . . . . . . . . . . . . . . . . . . . . . . . . 19

4.5 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . 20

5 . AERODYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1

5.2.1 Analytical Integration . . . . . . . . . . . . . . . . . 23

5.2.2NumericalIntegration of theProduct of Displacement

and Pressure . . . . . . . . . . . . . . . . . . . . . . . 23

5.2.3 Numerical Integration of t h e P r e s s u r e s . . . . . . . . . 24

5.2.4 Finite Element Approach . . . . . . . . . . . . . . . . . 24

5.3 Basic Formulation . . . . . . . . . . . . . . . . . . . . . . . 25

V . .

I 5.4 F a c t o r sA f f e c t i n g The Efficiencyof The Numerical Evaluation of The Matrix of Generalized Aerodynamic

Force Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 26

5 . 4 . 1 Matrix Population . . . . . . . . . . . . . . . . . . . . . 27

5 . 4 . 2 I n t e r p o l a t i o n f o r A r b i t r a r y k Value . . . . . . . . . . 27

5.4.3 Number o f k I n t e r v a l s . . . . . . . . . . . . . . . . . 30

5.4.4 Sequence of Multiplications . . . . . . . . . . . . . . . . 31

5.4.5 Form ofInputtingtheAngle-of-AttackGenerating Matrix . 33

5.5 Summary of Comparisons . . . . . . . . . . . . . . . . . . . . . . . 33

5.5.1 Input Storage Requirements . . . . . . . . . . . . . . . 33

5.5.2 CoreSpace . . . . . . . . . . . . . . . . . . . . . . . 33

5.5.3 Read-in . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.5.4 Number of Computational Operations . . . . . . . . . . . 34

5.6 Conclusions and Recommendations . . . . . . . . . . . . . . . . 34

6 . METHODS OF OPTIMIZATION FOR FLUTTER . . . . . . . . . . . . . . . . . 36

6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

-6.2 Rudisill-Bhatia Approach . . . . . . . . . . . . . . . . . . . . 36

. . . . . . . . . . . . . . . . 37

6.2.1 Velocity Gradient Search

6.2.2 Mass Gradient Search . . . . . . . . . . . . . . . . . . 38

6 . 2 . 3 G r a d i e n t P r o j e c t i o n S e a r c h e s . . . . . . . . . . . . . . 40

6.2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . 44

6.3 The Weight Gradient Method of Simodynes . . . . . . . . . . . . 44

6.3.1 The Method o f . Simodynes . . . . . . . . . . . . . . . . . 44

6.3.2 Discussion of the Method . . . . . . . . . . . . . . . . 46

6 . 3 . 3 A Modification of t h e Method of Simodynes . . . . . . . . 47

6.3.4 Discussion of the Modified Method . . . . . . . . . . . . 48

6.3.5 Assessment of the Method . . . . . . . . . . . . . . . . . 50

6 . 4 An I n t e r i o r P e n a l t y F u n c t i o n Method . . . . . . . . . . . . . . 50

6 . 4 . 1 Description of Method . . . . . . . . . . . . . . . . . . 50

6.4.2 Discussion of the Method . . . . . . . . . . . . . . . . 52

6.4.3 Assessment of the Method . . . . . . . . . . . . . . . . 54

6.5 A Method o f F e a s i b l e D i r e c t i o n s . . . . . . . . . . . . . . . . 55

6 . 5 . 1 Description of Method . . . . . . . . . . . . . . . . . . 55

6 . 5 . 2P r i n c i p a lC h a r a c t e r i s t i c so ft h e Method . . . . . . . . . 58

6.5.3 Assessment of the Method . . . . . . . . . . . . . . . . 6 1

6.6 A n Optimization Method Using Incremented F l u t t e r A n a l y s i s . . . 62

6.6.1 Main Features . . . . . . . . . . . . . . . . . . . . . . 62

6 . 6 . 2 P r e s e n t Form of Program . . . . . . . . . . . . . . . . . 63

6.6.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . 67

v i

6.7 Comparison of Optimization Methods . . . . . . . . . . . . . . . 68

6.7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.7.2 Arbitrary StepSize Procedures . . . . . . . . . . . . . 68

6.7.3 Defined Step-Size Procedures . . . . . . . . . . . . . . 69

6.7.4 Formulation of a Resizing Procedure . . . . . . . . . . . 72

7 . CONSIDERATIONS RELATED TO A REALISTIC DESIGN ENVIRONMENT . . . . . . 73

7.1 S t r u c t u r a l Model . . . . . . . . . . . . . . . . . . . . . . . . 73

7.2 M u l t i p l e F l u t t e r Speed Constraints . . . . . . . . . . . . . . . 77

7.3 Damping Constraints . . . . ' . . . . . . . . . . . . . . . . . . . . 78

7.4 Mass Ballast . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.5 I n t e r f a c e With Strength Optimization . . . . . . . . . . . . . . 82

8 . COMPUTATIONAL ASPECTS OF THE FLUTTER TASK . . . . . . . . . . . . . . 84

8.1 The Complete F l u t t e r Task . . . . . . . . . . . . . . . . . . . 85

8.1.1 F l u t t e r Survey . . . . . . . . . . . . . . . . . . . . . 85

8 . 1 . 2 I n i t i a l S t r u c t u r a l R e s i z i n g . . . . . . . . . . . . . . . 88

8.1.3 Flutter Optimization . . . . . . . . . . . . . . . . . . 92

8.2 Aspects of t h e Computing System . . . . . . . . . . . . . . . . 95

9 . CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . . . . . 98

APPENDIX A . NUMERICAL EXAMPLES OF RESIZING PROCEDURES . . . . . . . . . . 100

A . l I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . 100

A . 2 S t r u c t u r a l Model . . . . . . . . . . . . . . . . . . . . . . . . 101

A . 3 . Method of Simodynes . . . . . . . . . . . . . . . . . . . . . . 104

A . 4 Gradient Methods of R u d i s i l l and Bhatia . . . . . . . . . . . . 109

A . 5 I n t e r i o rP e n a l t yF u n c t i o n Method . . . . . . . . . . . . . . . . 112

A . 6 Method of Feasible Directions . . . . . . . . . . . . . . . . . 113

A . 7 A n Optimization Method Using Incremented FlutterAnalysis . . . 114

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

v i i SYMBOLS AND DEFINITIONS

square , r e c t a n g u l a r matrix

t r a n s p o s e o f a m a t r i x column m a t r i x row m a t r i x diagonal matrix aerodynamics matrix ( f u n c t i o no f k and Mach number) , modalizedaerodynamicsmatrix generalizedaerodynamicforcecoefficients [ A I C ] , [AIC(k)] basic aerodynamics influence coefficients (function of k and Mach number) defined by equation(5.13) amplitudesofsuccessivecycles a a n y n + l

C a r b i t r a r y c o n s t a n t ( e q u a t i o n s ( 7 . 1 6 ) , (8.1))

r a t i o between m and Pi : m = CiPi ; elements of a b a s i c i i ‘i r e s i z i n g column ( e q u a t i o n ( 6 . 5 3 ) ) C reference chord D( ) f l u t t e r d e t e r m i n a n t

[ D l viscous damping matrix

m a t r i x r e l a t i n g c o n t r o l s y s t e m d i s p l a c e m e n t s t o s t r u c t u r a l

L.51

displacements [DX1 i n t e r p o l a t i o n and d i f f e r e n t i a t i o n m a t r i x r e l a t i n g s l o p e s a t downwash c o l l o c a t i o np o i n t st od i s p l a c e m e n t s a t s t r u c t u r a l nodes(Section5.3) C D Z I i n t e r p o l a t i o n m a t r i x r e l a t i n g t r a n s l a t i o n s a t downwash c o l l o c a t i o n p o i n t s t o d i s p l a c e m e n t s at s t r u c t u r a l nodes ( S e c t i o n5 . 3 ) E M e q u i v a l e n t a i r s p e e d E1 b e n d i n g s t i f f n e s s ix combination of modal displacement matrices ( e q u a t i o n (5.14) ) g e n e r a l modes ofdisplacement s t r u c t u r a l damping, 2 7 t o r s i o n a l s t i f f n e s s at lumpedaero- i n t e r p o l a t i o n m a t r i x r e l a t i n g d i s p l a c e m e n t s dynamic l o a d p o i n t s t o d i s p l a c e m e n t s at s t r u c t u r a l nodes ( S e c t i o n 5 . 3 ) t r a n s f e rf u n c t i o no fa u t o m a t i cc o n t r o ls y s t e m ( f u n c t i o n o f p ) column matrixofdisplacements at aerodynamic l o a d p o i n t s c o n s t r a i n tq u a n t i t y( e q u a t i o n( 6 . 4 3 ) ) unmodalizedaerodynamicsmatrix(functionof k and Mach number) ( e q u a t i o n ( 5 . 1 5 ) ) s t i f f n e s s m a t r i x , m o d a l i z e d s t i f f n e s s m a t r i x base s t i f f n e s s m a t r i x i n c r e m e n t a ls t i f f n e s sm a t r i xp e ru n i td e s i g nv a r i a b l e i.

c o n s t a n t ( e q u a t i o n ( 6 . 2 2 ) ) c o n s t a n t ( e q u a t i o n ( 6 . 2 1 ) ) O C reduced frequency k = - V k n o t se q u i v a l e n ta i r s p e e d p o l y n o m i a l m u l t i p l i e r s u s e d i n L a g r a n g e ' s i n t e r p o l a t i o n f o r m u l a mass matrix,modalized mass matrix b a s e mass m a t r i x i n c r e m e n t a l mass m a t r i xp e ru n i td e s i g nv a r i a b l e i t o t a l mass a s s o c i a t e d w i t h t h e d e s i g n v a r i a b l e s d e s i g n v a r i a b l e a s s o c i a t e d w i t h s t r u c t u r a l mass ( i n mass or w e i g h t u n i t s ) resi zing column X d e s i g n v a r i a b l e o f R e f e r e n c e 14 m o d i f i e d o b j e c t i v e f u n c t i o n ( S e c t i o n 6.4.1) p r e s s u r e - k e r n e l i n t e g r a l matrix ( e q u a t i o n ( 5 . 4 ) ) P aerodynamic l i f t i n g p r e s s u r e d i s t r i b u t i o n c o r r e s p o n d i n g t o d e f l e c t i o n mode j aerodynamic l i f t i n g p r e s s u r e d i s t r i b u t i o n mode modal degreesoffreedom(modalparticipationcoefficients) modalcolumn c o r r e s p o n d i n g t o s o l u t i o n o f c h a r a c t e r i s t i c f l u t t e r e q u a t i o n r p e n a l t y f u n c t i o n w e i g h t i n g f a c t o r modal row c o r r e s p o n d i n g t o s o l u t i o n o f c h a r a c t e r i s t i c f l u t t e r

Lrl

e q u a t i o n V s p e e d , f l u t t e r s p e e d f l u t t e r s p e e d vf most c r i t i c a l f l u t t e r speed ' m c r e q u i r e d f l u t t e r s p e e d (minimum a l l o w a b l e f l u t t e r s p e e d : vR 1.20V for commercial, 1.15V f o r m i l i t a r y ) D L d e s i g n s p e e d a c c o r d i n g t o F e d e r a l A v i a t i o n R e g u l a t i o n s vD design speed according to m i l i t a x y s p e c i f i c a t i o n s vL MIL-A-008870A ( U s A F ) W t o t a l w e i g h t a s s o c i a t e d w i t h t h e d e s i g n v a r i a b l e s t o t a l w e i g h t o f b a s e s t r u c t u r e wO a r b i t r a r i l y c h o s e n t o t a l w e i g h t r e d u c t i o n w1 a r b i t r a r i l y c h o s e n t o t a l w e i g h t a s s o c i a t e d w i t h p o s i t i v e w2 component o f r e s i z i n g v e c t o r x i weighting matrix d i f f e r e n t i a t i n g a n d w e i g h t i n g m a t r i x c o o r d i n a t e i n a f o r e - a n d - a f t d i r e c t i o n c o o r d i n a t e i n a lateral d i r e c t i o n column matrix of displacements of s t r u c t u r a l nodes m a t r i x of modalcolumns ofdisplacements of s t r u c t u r a l n o d e s a n g l e of a t t a c k , p a r a m e t e r i n o n e - d i m e n s i o n a l m i n i m i z a t i o n g e n e r a l d e s i g n v a r i a b l e normalized real p a r t o f p = ( Y + i ) k maximum v a l u eo f Y i n hump mode c o n s t a n t( e q u a t i o n (6.10)) air d e n s i t y aerodynamic velocity potential c i r c u l a rf r e q u e n c y ,r a d / s i n d i c a t e s d e r i v a t i v e of a one-variablefunction i n - f l i g h t mode: modal column c o r r e s p c n d i n gt o a c h a r a c t e r i s t i cs o l u t i o n of t h e f l u t t e r e q u a t i o n f l u t t e r mode: i n - f l i g h t mode t h a t becomes u n s t a b l ew i t h i nt h ev e l o c i t y rangeconsidered hump mode: i n - f l i g h t mode w i t h a minimum damping p o i n t w i t h i n t h e v e l o c i t yr a n g ec o n s i d e r e d xi i STUDY OF FLUTTER RELATED COMPUTATIONAL PROCEDURES FOR M I N I M U M WEIGHT STRUCTURAL SIZING OF ADVANCED AIRCRAFT R . F.O'Connell, H . J. Hassigand N. A. Radovcich Lockheed-California Company

Burbank , C a l i f o r n i a

1. INTRODUCTION 1.1 General One o f t h e f a c t o r s c o n t r i b u t i n g t o t h e p r o f i t a b i l i t y o f a n a i r p l a n e i s i t s payload/rangecapability.Givencertainsafetyandperformancerequirements, t h e r e is a d i r e c t t r a d e - o f f b e t w e e n s t r u c t u r a l w e i g h t a n d p a y l o a d , a n d it i s t h e ideal o fe a c ha i r p l a n ed e s i g n e rt or e d u c es t r u c t u r a lw e i g h t .A l t h o u g ht h e i d e a l minimum weight design may beexpensivetoproduce,overshadowingany payload/rangegains, it provides a good s t a r t i n g p o i n t f o r a p r a c t i c a b l e designand a good b a s i s f o r comparing d i f f e r e n t d e s i g n s .

S t r u c t u r a lw e i g h tm i n i m i z a t i o n ,o fc o u r s e , i s not a new i d e a . It is one o ft h ea i r p l a n ed e s i g n e r ' s most c r i t i c a l t a s k s . It now has come t o t h e f o r e - f r o n t as a r e s u l t o f two developments.

F i r s t , it has become e v i d e n t t o the s t r u c t u r a l d e s i g n e n g i n e e r t h a t t h e combinationoffiniteelementmodeling,highspeedcomputercapacity,and mathematicaltechniques makes it p r a c t i c a b l e t o do detailed s t r u c t u r a l s y n t h e s i s aimed a t minimizingweight.

Second, t h e need f o r a comprehensiveanddetailedapproachtostructural d e s i g n o p t i m i z a t i o n h a s s i g n i f i c a n t l y i n c r e a s e d w i t h the adventofthesuper- s o n i ct r a n s p o r t .T h i sf o l l o w sf r o mt h ef a c tt h a tf o r a s u p e r s o n i ct r a n s p o r t t h e r e t u r n i n terms of payload/range per pound of structural weight saved i s c o n s i d e r a b l yl a r g e rt h a nf o r a s u b s o n i ct r a n s p o r t . For i n s t a n c e , a oneper- c e n t s t r u c t u r a l w e i g h t s a v i n g on a t y p i c a l s u b s o n i c t r a n s p o r t m i g h t r e s u l t i n an i n c r e a s e d p a y l o a d c a p a b i l i t y o f o n e t o two p e r c e n t ; onanarrowwingsuper- s o n i c t r a n s p o r t , r e c e n t l y s t u d i e d b y t h e L o c k h e e d - C a l i f o r n i a Company, a one p e r c e n t s t r u c t u r a l w e i g h t s a v i n g r e s u l t e d i n a f o u r p e r c e n t i n c r e a s e i n pay- l o a d c a p a b i l i t y f o r t h e d e s i g n r a n g e .

The s u b j e c t o f t h i s r e p o r t i s f l u t t e ro p t i m i z a t i o n ; i.e., s t r u c t u r a l w e i g h tm i n i m i z a t i o nw i t hf l u t t e rc o n s t r a i n t s . The need f o r a systematic, p o s s i b l y a u t o m a t e d , a p p r o a c h t o f l u t t e r o p t i m i z a t i o n a l s o hasincreasedsig- n i f i c a n t l y .S u b s o n i ct r a n s p o r t s , as t h e y are known, a n dt r a n s o n i ct r a n s p o r t s , as shown i n a r t i s t ' s sketches,can be representedbysimple, beam-type s t r u c t u r a l models t h a t are s a t i s f a c t o r y f o r o p t i m i z a t i o n w i t h f l u t t e r con- s t r a i n t s .F l u t t e ro p t i m i z a t i o nf o rs u c hd e s i g n sc a nb ed o n e ,a n dh a sb e e n done,withavailablemethods. The s u p e r s o n i ct r a n s p o r t st h a t are f l y i n ga n d thosebeingstudied,however, a l l have l i f t i n g s u r f a c e s t h a t cannot be repre- s e n t e ds a t i s f a c t o r i l yb ys i m p l e beam-type s t r u c t u r a l models.Thisfactalone makes t h e t a s k o f f l u t t e r o p t i m i z a t i o n a n o r d e r o f m a g n i t u d e more complicated.

Althoughadhocapproaches t o f l u t t e r o p t i m i z a t i o n s t i l l c o u l d l e a d t o a s a t i s f a c t o r i l yo p t i m i z e ds u p e r s o n i cd e s i g n ,r e f i n e d methods t h a t t a k e f u l l a d v a n t a g e o f t h e c a p a b i l i t i e s of t h e p r e s e n t c o m p u t e r s , i n r e g a r d t o a u t o m a t i o n as w e l l as i n t e r a c t i o n w i t h t h e e n g i n e e r , become a t t r a c t i v e and p o s s i b l y mandatory.This i s e s p e c i a l l y t r u e i n viewoftherapidlyincreasingcapa- b i l i t y f o r fast a n a l y s i sa n ds y n t h e s i si nt h ea r e a so fs t r u c t u r a lm o d e l i n g and a n a l y s i s , s t r e s s o p t i m i z a t i o n , andperformanceanalysissupportedbyimproved c o n f i g u r a t i o nc o n t r o l .F l u t t e ro p t i m i z a t i o n must keep abreastofthesedevel- opments. A balanced improvement i n c a p a b i l i t y i n a l l d i s c i p l i n e s w i l l make p o s s i b l e , w i t h i n a p r a c t i c a b l e time span,truein-depthcomparisonsbetween a l a r g e number o f c a n d i d a t e d e s i g n s .

The preceding paragraphs present generally well known j u s t i f i c a t i o n f o r a c o n c e r t e d e f f o r t i n i m p r o v i n g methodsof s t r u c t u r a l o p t i m i z a t i o n w i t h f l u t - t e r c o n s t r a i n t s . Work performedduringthesubjectstudy i s partofsuchan .

e f f o r t Work t o w a r d s t h e g o a l o f a g e n e r a l l y a v a i l a b l e a u t o m a t e d or semi-automated s t r u c t u r a lo p t i m i z a t i o ns y s t e m ,t h a ti n c l u d e s items such as o p t i m i z a t i o nf o r s t r e s s , f l u t t e r and c o n t r o l l a b i l i t y , m u l t i p l e f l u t t e r s p e e d and modal damping c o n s t r a i n t s , i s s t i l l i n a state of development. The presentstudyhas con- t r i b u t e d t o t h i s g o a li nt h ef o l l o w i n ga r e a s . Methods ofcomputingtheaero- dynamics p a r a m e t e r s t o b e u s e d i n a f l u t t e r o p t i m i z a t i o n programhavebeen compared i n d e t a i l w i t h r e s p e c t t o c h a r a c t e r i s t i c s which areindependent of a specificaerodynamicstheory. A method f o re f f i c i e n t l y and r e l i a b l ys o l v i n g t h e f l u t t e r e q u a t i o n f o r r o o t s o f i n t e r e s t i n a f l u t t e r o p t i m i z a t i o n module hasbeendeveloped.Fivemethodsofflutteroptimizationhavebeen compared i n d e t a i l and themechanicsoftheoptimizationprocesshavebeenexamined; numericalexampleswith a l l f i v e methodshavebeengeneratedforthe same a i r c r a f td e s i g n . Recommendations f o rf u r t h e rs t u d y and f o r the designof a f l u t t e r o p t i m i z a t i o n modulehavebeen made.

The p r i n c i p a lr e s u l t so ft h i ss t u d ya r ep r e s e n t e di nt h i sr e p o r t . Back- grounddiscussionsandsupporting material are p r e s e n t e d i n a companion r e p o r t (Reference 1) .

1.2 Objectives of Study The objectives of this study are: To survey and evaluate methods of representing unsteady aerodynamics 1.

parameters and make recommendations for a general, accurate and efficie formulation that minimizes the computational effort during the optimiz tion process. The assessment of aerodynamics theories, however, falls outside the scope of this study.

To survey and evaluate methods of determining the flutter characterist 2.

and make recommendat2ons for a method that is reliable and efficient suitable for the optimization process.

To evaluate and compare a number of methods of structural optimizatio 3.

with flutter constraints and make recommendations for further evaluati in a realistic design environment.

4. To make preliminary recommendations for the design of a flutter optimization module.

2. O V E R V I E N OF THE FLUTTER O P T I M I Z A T I O N T A S K Structural optimization with flutter constraints is both an extension of the structural optimization task related to strength and an extension of flutter analysis task. Being an extension of two tasks that traditionally are considered to belong to different disciplines, flutter optimization must tak into account requirements of both disciplines. Structural optimization requires that a structural model is used that incorporates sufficient stru tural detail, in terms of distribution of structural material, to aid the designer in defining hardware. Similarly the flutter analysis that is incor- porated in the optimization process must be of an accuracy comparable to used outside of flutter optimization. The latter refers to methods of representing the unsteady aerodynamics and methods of solving the flutter equation, since the more detailed a structural model is, the more accurat from an idealized theoretical point of view, is the flutter analysis. From a practical point of viciw, structural sizing for strength requires more detail in the structural model than is required for adequate prediction of flut characteristics.

Thus,the flutter optimization task starts with the definition of the structural model. This is one of the most crucial aspects of flutter optimiz tion, and it involves a serious conflict between simplicity of approach a computer cost. Present computer technology, or methods of structural analysis, or both,may not permit a structural model with sufficient detail for a stres analysis to be used in flutter optimization; computer cost could be exorbitan due to the repetition of operations during the design process. Section 7.1 . . . . ._.... .... .. .

d e a l sw i t ht h i sp r o b l e mi n more d e t a i l . S u f f i c e h e r e t h a t a s s o c i a t e d w i t h t h e c h o i c e o f s t r u c t u r a l model i s t h e s e l e c t i o n of a p r a c t i c a b l e number ofdegrees offreedom f o r t h e v i b r a t i o n a n a l y s i s t h a t h a s t o p r o v i d e t h e modes f o r t h e modal r e d u c t i o n o f t h e f l u t t e r e q u a t i o n , which is u s u a l l y r e q u i r e d t o l i m i t computer c o s t , If thedegreesoffreedomfor the v i b r a t i o na n a l y s i s are a s u b s e t of thedegreesoffreedomofthestructuralmodel,complications arise if a n o n l i n e a r r e l a t i o n s h i p b e t w e e n t h e s t i f f n e s s m a t r i x a n d t h e d e s i g n v a r i a b l e s r e s u l t s (see S e c t i o n 7.1).

W i t h o u t s e r i o u s r e s t r i c t i o n on scope or a c c u r a c yo ft h ea n a l y s i st h e mass matrixcanbe assumed t o b e a l i n e a rf u n c t i o no fd e s i g nv a r i a b l e s . It i s t h e sum of a b a s i c m a t r i x a n d as many elementarymatrices as t h e r e a r e d e s i g n v a r i a b l e s a s s o c i a t e d w i t h a mass change,eachproportionalto a design v a r i a b l e .

During t h e f l u t t e r o p t i m i z a t i o n t h e r e i s repeatedneedfordetermining r o o t so ft h ef l u t t e re q u a t i o n ,e a c h time t h a t a s t r u c t u r et h a th a su n d e r g o n e a r e s i z i n gs i n c et h ep r e v i o u ss o l u t i o no ft h ef l u t t e re q u a t i o n . For many, i f not a l l , o f t h e s e s o l u t i o n s a remodalization i s necessarybased on v i b r a t i o n modes o ft h ec u r r e n tc o n f i g u r a t i o n . It i s found t h a tf o rt h eo p t i m i z a t i o n p r o c e s st op r o v i d er e l i a b l e ,c o n v e r g e dr e s u l t s ,c o n s i s t e n tw i t ht h ec a p a b i l i t y o f t h e s t r u c t u r a l m o d e l , moremodal degreesoffreedomarerequiredinthe f l u t t e r e q u a t i o n t h a n f o r a r o u t i n e f l u t t e r a n a l y s i s (see S e c t i o n 4 ) .

I n c o r p o r a t i o no fs t a t e - o f - t h e - a r tl e v e la e r o d y n a m i c si nt h ef l u t t e r optimizationprocessdoesnotprovidesignificantproblemsbeyondthose e n c o u n t e r e di nt h eu s u a lf l u t t e ra n a l y s i s . For a givenexternalgeometrythe basic aerodynamics formulation i s i n v a r i a n t w i t h s t r u c t u r a lc h a n g e s . The repetitiveformationofgeneralizedaerodynamicforcesforsuccessive,updated m o d a l i z a t i o n o f t h e f l u t t e r e q u a t i o n i s s i m p l ea n dr e l a t i v e l yi n e x p e n s i v e .

I n viewof t h e o b j e c t i v e s and t h es c o p eo ft h ep r e s e n tp r o g r a m ,t h i s r e p o r td e v o t e sm a j o rs e c t i o n st oi m p o r t a n ta s p e c t so ft h ef l u t t e ro p t i m i z a t i o n procedure.Section 3 d e a l sw i t ht h es o l u t i o no ft h ef l u t t e re q u a t i o n . Sec- t i o n 4 deals w i t hm o d a l i z a t i o n .S e c t i o n 5 p r e s e n t sp a r to fc o n s i d e r a b l e work devotedtotheaerodynamics,withtheremainderbeingpresentedinReference 1.

I nS e c t i o n 6 , m e t h o d so fo p t i m i z a t i o nf o rf l u t t e r ,e v a l u a t e dd u r i n gt h i ss t u d y , arediscussed.Numericalresultsobtained by applyingthesemethodsto a s i m p l i f i e d o p t i m i z a t i o n t a s k a r e p r e s e n t e d i n Appendix A .

Againstthebackgroundprovidedbythesesections,Section 7 p r e s e n t s d i s c u s s i o n so fs e v e r a la d d i t i o n a lp r o b l e m s and c o n s i d e r a t i o n s t h a t n e e d t o b e s t u d i e d i n o r d e r t o c h o o s e a r a t i o n a l a p p r o a c h f o r f o r m u l a t i n g a f l u t t e r optimization module.

I nS e c t i o n 8, c o m p u t a t i o n a la s p e c t so ft h ec o m p l e t ef l u t t e r task are d e l i n e a t e d . This t a s ki n c l u d e sf l u t t e ra n a l y s i s as w e l l as s t r u c t u r a l syn- t h e s i s o f a d e s i g n t h a t s a t i s f i e s t h e f l u t t e r r e q u i r e m e n t s .

S e c t i o n 9 summarizes theconclusionsofthepresentstudyandpresents recommendations f o rf u t u r ew o r k .

r

3. SOLUTION OF THE FLUTTER EQUATION 3.1 The Generalized Flutter Equation When using the k method the flutter equation can be written as:

a [ K ] - p [ A ( i k ) ] ] (q] = 0

V2 One of several possible methods of solving this equation is to determine the A = for seyeral values of the reduced frequency characteristic value V wc k=- keeping all other quantities in the equation constant (Reference 2).

v ’

In the p-k method the flutter equation is and solutions p=(Y+i)k are sought for selected combinations of values of V and p (Reference 3). The p-k method formulation is convenient for the inclusion of viscous damping and control system transfer functions. This is accomplished by writing: where Hj(p), j = l , 2 . . , represents transfer functions of the control system

and pj] relates the control system displacements to the structural dis-

placements; [ D l is a viscous damping matrix (Reference 3).

A further generalization of the flutter equation can be made by making the stiffness matrix and the inertia matrix functions of design variables m which is the standard procedure f o r structural optimization. In addi- i’

tion, other quantities, such as k] , [Dj] , as well as transfer function

coefficients in H (p), may be made functions of design variables.

J Equation (3.3) i m p l i e s t h a t t h e d e t e r m i n a n t o f t h e s q u a r e m a t r i x on t h e l e f t hand s i d e i s zeroandthus, i n a very g e n e r a l f o r m , t h e c h a r a c t e r i s t i c e q u a t i o n c o r r e s p o n d i n g t o t h e f l u t t e r e q u a t i o n c a n be w r i t t e n as: (Y+i)k,g,V,p,mi (3.4) D is c a l l e dt h ef l u t t e rd e t e r m i n a n t . For a r b i t r a r yv a l u e so ft h ev a r i a b l e s it has a complex v a l u e . Thus e q u a t i o n( 3 . 4 )r e p r e s e n t s two e q u a t i o n sa n d ,i n p r i n c i p l e , c a n b e s o l v e d f o r two unknowns f o r g i v e n v a l u e s o f t h e o t h e r v a r i a b l e s .

L e t t i n g Y=O a n d s o l v i n g f o r g and V c o r r e s p o n d s t o t h e t r a d i t i o n a l k method o f s o l v i n g t h e f l u t t e r e q u a t i o n . S o l v i n g for Y and k corres- ponds t o t h e p-k method. L e t t i n g 'Y=O and solving f o r k and V l e a d s d i r e c t l y t o t h e f l u t t e r s p e e d f o r a givenvalue of t h e s t r u c t u r a l damping, g.

Solvingequation (3.4) f o r k andone o ft h ed e s i g nv a r i a b l e s , assuming

a l l o t h e r variables f i x e d , i s a new u s eo ft h ef l u t t e re q u a t i o n . It is c a l l e d IncrementedFlutterAnalysis(References 4 and 11, a p p l i c a t i o n so f which a r e i n c l u d e d i n S e c t i o n s 6.2.3 and 6.6.

3.2 Types ofSolutionSought A f l u t t e r a n a l y s i s i n t h e t r a d i t i o n a l s e n s e is t h e d e t e r m i n a t i o n o f t h e f l u t t e rc h a r a c t e r i s t i c so f a g i v e ns t r u c t u r e . It i n c l u d e st h ec a l c u l a t i o n of any f l u t t e r s p e e d t h a t may occur a t speeds up t o o r somewhat beyond a s p e e dc o r r e s p o n d i n gt ot h er e q u i r e df l u t t e rm a r g i n . It a l s oi n c l u d e st h e g a i n i n g o f i n s i g h t i n t h e v a r i a t i o n o f f r e q u e n c y andaerodynamic damping at s p e e d s b e l o w t h e f l u t t e r s p e e d f o r s e v e r a l i n - f l i g h t v i b r a t i o n modes of i n t e r e s t .C o n s e q u e n t l y , s u f f i c i e n t modal s o l u t i o n s are o b t a i n e df o rt h e con- s t r u c t i o no f f-g-V diagrams (Figure 3-1) f o rs e v e r a lf l i g h tc o n d i t i o n s .

A p r o c e d u r e f o r s t r u c t u r a l o p t i m i z a t i o n w i t h f l u t t e r c o n s t r a i n t s w i l l most l i k e l y start withsuch a survey-typeanalysis. However, d u r i n gt h e p r o c e s s o f r e p e a t e d r e s i z i n g , l e a d i n g t o t h e optimum d e s i g n ,t h e r e is no need fordeterminingcomplete f-g-V diagrams at e a c hr e s i z i n g ;o n l yp o i n ts o l u - t i o n s are r e q u i r e d .P o i n ts o l u t i o n sf o u n di nt h e l i t e r a t u r e are of two types: 1) d i r e c t l ys o l v i n gf o rt h ef l u t t e rs p e e d( t h ec o m b i n a t i o n k,V i n equa- t i o n ( 3 . 4 ) ) , and 2) determining the value of o n e d e s i g n v a r i a b l e n e c e s s a r y t o s a t i s f y a g i v e nf l u t t e rs p e e dc o n s t r a i n t( t h ec o m b i n a t i o n k, m j i n equa- t i o n (3.4) where m i s one of the design variables m. 1.

j 1

A m O W WING SST CONFIGURATION - COIKCRACT NASI--12288

Symmetric Flutter Analysis - 20 Vibration Modes

Two Rigid Body Modes Not Shown Mach Number = 0 . 6 Weight = 145,600 kg 6.0 5 . 0

N 4.0

X n h 3.0 a , a , k Frr 2.0 1 . 0 100 200 3 Velocity, m/s EAS Velocity, m/s EAS - 3 Figure 3-1: Example of Complete f-g-V Diagram One a d d i t i o n a l t y p e o f p o i n t s o l u t i o n h a s b e e n f o r m u l a t e d d u r i n g t h i s c o n t r a c t , r e s u l t i n g i n t h e d e t e r m i n a t i o n o f t h e minimum damping p o i n t of an i n - f l i g h t mode. Such a p o i n t , if it e x i s t s , is o fi n t e r e s t i f t h e minimum damping p o i n t l i e s w i t h i nt h es p e e dr a n g ec o n s i d e r e d . The a s s o c i a t e d mode is c a l l e d a hump mode (SeeFigure 3-1). T h i sp o i n ts o l u t i o nr e q u i r e st h es o l u t i o n of equation ( 3.4) and the equation 2.l = (3.5)

a v

f o rt h et h r e e unknowns k , 7 ' and V . Details o ft h ef o r m u l a t i o n are g i v e ni n S e c t i o n 3.4. No n u m e r i c a le v a l u a t i o no ft h e method hasbeen made t h u s far.

. - . . ., The followingsectiondealsmainlywithmethodsofobtainingpoint s o l u t i o n sf o rt h ef l u t t e rs p e e d . It s h o u l db ek e p ti n mind t h a t when such solutionsareneededinanoptimizationprogram a s o l u t i o n f o r a similar s t r u c t u r a l c o n f i g u r a t i o n i s u s u a l l y a v a i l a b l e as a first approximation t o t h e r e q u i r e d s o l u t i o n .

3.3 Methods o fO b t a i n i n gP o i n tS o l u t i o n s S e v e r a l m e t h o d s f o r o b t a i n i n g p o i n t s o l u t i o n s h a v e b e e n c o n s i d e r e d a n d terms o f computa- e v a l u a t e dt ov a r i o u sd e p t h s .T h e i ra p p a r e n te f f i c i e n c y ,i n t i o n a l e f f o r t , i s an important, p a r t of t h ee v a l u a t i o n . However, t h ed e g r e e o f c e r t a i n t y w i t h which a d e s i r e d s o l u t i o n c a n b e f o u n d i s even more important.

The l a t t e r c o n s i d e r a t i o n refers t o convergenceproblemsand t o problems a s s o c i a t e dw i t hr e l a t i n g modal s o l u t i o n s a t one v a l u e of V or k t o modal s o l u t i o n s at another value of V o r k .

The r e s u l t s o f e v a l u a t i o n s o f t h e f o l l o w i n g methods are presented: 0 B h a t i a method (Reference 5 )

0 Phoa - Boeing method (Reference 6 )

0 Lockheed's Program 165 (p-k method, Reference 3) 0 Desmarais-Bennett method ( R e f e r e m e 7) 0 Two DimensionalRegulaFalsiand Newton Raphson (Reference 8)

3.3.1 BhatiaPethod - InReference 5 B h a t i ap r e s e n t s a methodofsolving

d i r e c t l y f o r t h e f l u t t e r s p e e d . Numericalevaluationsofthe methodhave beenperformedusingdatafromthearrowwingstudythat Lockheedhas con- ductedundercontract NAS~-12288.

I n B h a t i a ' s method,which i s based on t h e k-method a p p r o a c h ,t h es t r u c - tural damping, g ,r e q u i r e df o rn e u t r a ls t a b i l i t y i s computed as a f u n c t i o n of l / k =- by means of a Laguerretypeextrapolation. It i s a ni t e r a t i v e O C

a

method t h a t i s i n i t i a t e d bychoosing a trial v a l u el / k and computing t h e a s s o c i a t e dv a l u eo f g and i t s first a n ds e c o n dd e r i v a t i v ew i t hr e s p e c tt o l / k . The Laguerreextrapolation leads t o a f i r s t approximationofthevalue of l / k f o r which g=O ... The process i s r e p e a t e d for t h i s new v a l u eo fl / k i s reached.

u n t i lc o n v e r g e n c e The method as p r e s e n t l y programmed usesonlyaerodynamicmatrices at p r e s e l e c t e d v a l u e s o f k , r e q u i r i n g a l a r g e number o f p r e s e l e c t e d k v a l u e s .

The method a l s o r e q u i r e s i n p u t t i n g the first andsecond derivatives of a l l aerodynamicsmatriceswithrespectto l / k . The method couid be improved by u s i n gi n t e r p o l a t i o nw i t hr e s p e c tt o k todeterminetheaerodynamicsmatrix and its d e r i v a t i v e s a t a r b i t r a r yv a l u e so f k frommatricesgiven at a moderate number o fp r e s e l e c t e d k v a l u e s . Care must b et a k e nt h a tt h e i n t e r p o l a t e d results are d e f i n e du n i q u e l yo v e rt h er a n g eo f k o fi n t e r e s t a p a r t i c u l a r s o l u t i o n t o p r e v e n t t h e s o l u t i o n fYom o s c i l l a t i n g between f o r two v a l u e s ( "hunting" ) .

Numericalevaluations were performed as p a r to ft h i ss t u d y .D i f f i c u l - ties wereencountered i n t r a c k i n g t h e p r o p e r mode and i n c o n v e r g i n g on t h e l o w e rf l u t t e rs p e e do f a hump mode. There i s uncertaintywhethertheprogram can be m o d i f i e d s u c h t h a t t h e p r o p e r modal s o l u t i o n is alwaysobtained.

A t e a c h s t e p i n t h e i t e r a t i o n t o w a r d s t h e s o l u t i o n a c h a r a c t e r i s t i c valueproblemmustbesolved.This may p r o v e t o b e c o s t l y i n terms of C P U time.

3.3.2Phoa Method - I nR e f e r e n c e 6, Phoa p r e s e n t s a formulation of t h e

f l u t t e re q u a t i o nf r o m a c o n t r o l st h e o r yp o i n to f view. Although it i s r e c o g n i z e d t h a t c o n t r o l s t h e o r y c o u l d p r o v e t o b e o f a s s i s t a n c e i n i n t e r - p r e t i n g t h e f l u t t e r phenomenon, i n t h e caseofReference 6 it leads t o an e q u a t i o nt h a t is e s s e n t i a l l y t h e same as equation ( 3 . 4 ) . Phoa'smethod,based on t h e k-method approach, i s i n u s e a t t h e Boeing Company. Discussions w i t h Boeingpersonnelindicate that i n t h e a c t u a l a p p l i c a t i o n t h e e q u a t i o n ( D ( 0 , V ) . -1) = D(w,V) = -1 i s s o l v e d f o r V and w.

The s o l u t i o n i s accomplished i n two s t e p s .C o n s t a n tv e l o c i t yl i n e si n - t h e complex p l a n er e p r e s e n t i n g D ( w , V ) are i n t e r s e c t e dw i t ht h e real a x i s .

The v a l u e s of t h e real p a r t s a t t h e i n t e r s e c t i o n s , as a f u n c t i o n o f t h e v e l o c i t y , are used to determine an estimate of t h e v e l o c i t y f o r which t h e real p a r to f B(w,V) e q u a l s -1. I na n iterative process the accuracy of t h e s o l u t i o n i s improved.

I n n u m e r i c a l e v a l u a t i o n o f t h i s a p p r o a c h it w a s shown t h a t t h e c o n s t a n t

-

v e l o c i t yl i n e s may have two i n t e r s e c t i o n s w i t h t h e real D ( w , V ) axis; t h i s can be a sourceofproblems(Figure3-2).

The method i s a sequenceof two i n t e r p o l a t i o n s r e q u i r i n g many determinant e v a l u a t i o n s . It i s expectedthatvery f e w , p o s s i b l yn o t more t h a n two or three, steps i n t h e i t e r a t i o n p r o c e s s are r e q u i r e d .

I

"L -3 -2 -1

lD(w,V) I 1 Corresponding t o Im,D(w,V) I 1 = 0 as a Funct:ion of Velocity

Figure 3-2 : Value of Re 3 . 3 . 3 LockheedProgram 165 -'Program 165 ofLockheed'sFlutterandMatrix Algebra System (FAMAS) is based on t h e p-k method approach. It i s designed s o t h a t com- t o g e n e r a t e many p o i n t s o l u t i o n s , a s s o c i a t e d w i t h i n - f l i g h t modes, p l e t e f-g-V diagrams can be c o n s t r u c t e d . The program solves equation (3,3) f o rp = ( ? + i ) kg i v e na ni n i t i a l trial s o l u t i o n .F o rd e t e r m i n i n gt h ef l u t t e r speed, Y is e v a l u a t e d at s e v e r a lv a l u e so ft h ev e l o c i t y .F l u t t e ro c c u r s at t h es p e e df o rw h i c h Y=O.

' The programhasbeenusedsuccessfullyinnonautomatednumericalevalua- t i o n sd u r i n gt h i sc o n t r a c t .A u t o m a t i o ns h o u l db er e l a t i v e l ys i m p l ea n dc o u l d bebased on t h ef o l l o w i n gs t e p s . A t t h ee s t i m a t e df l u t t e rs p e e da ne s t i m a t e d frequency is u s e d t o s t a r t t h ep r o c e s s . Both a r eo b t a i n e d from t h e s o l u t i o n f o r ' ap r e v i o u ss t r u c t u r a lc o n f i g u r a t i o n .D e t e r m i n a n ti t e r a t i o n (see Reference 3 ) w i l l l e a d t o t h e a c t u a l v a l u e o f ' Y at t h ee s t i m a t e df l u t t e r speed. A t a s l i g h t l y p e r t u r b e d v e l o c i t y , u s i n g t h e dampingandfrequency already found as t r i a l s , Y i s a g a i n e v a l u a t e d . The two p a i r s o f V and Y v a l u e s t h u s f o u n d a r e u s e d t o i n i t i a t e a One-DimensionalRegula F a l s i p r o - c e d u r et h a tl e a d st o a value of V f o r which ?=O. The approach i s expected t o b e q u i t e e f f i c i e n t , e x c e p t f o r t h e p r o b l e m of a s s u r i n g t h a t s u b s e q u e n t s o l u t i o n s b e l o n g t o t h e same i n - f l i g h t mode.

3 . 3 . 4 Desmarais-Bennett Method - Reference 7 p r e s e n t s a fast andeconomical

automated procedure t og e n e r a t e f-g-V diagrams,including the proper con- n e c t i o no fp o i n ts o l u t i o n so ft h ef l u t t e re q u a t i o n . The procedure i s based on t h e k-method approach.

Reference 7 shows t h a t t h e method is q u i t e p o w e r f u l i n p r o p e r l y con- n e c t i n gp o i n ts o l u t i o n s . The samplecases i n Reference 7, however, are o b t a i n e d by p a r t i a l d e f l a t i o n o f t h e f l u t t e r d e t e r m i n a n t a f t e r e a c h modal s o l u t i o n i s found. Thus u s i n g t h i s method would r e q u i r e s o l v i n g f o r more r o o t st h a na r eo fi n t e r e s t i f o n l yt h ef l u t t e rs p e e d is r e q u i r e d . Or, a l t e r n a t i v e l y , i f o n l y t h e r o o t o f i n t e r e s t i s determined,there i s uncer- t a i n t y w h e t h e r t h e method w i l l b e as s u c c e s s f u l i n following modal s o l u t i o n s as shown i n R e f e r e n c e 7.

A p p l i c a t i o n o f t h i s method t o d i r e c t l y s o l v i n g f o r t h e f l u t t e r s p e e d could be programmed a c c o r d i n g t o t h e f o l l o w i n g p r o c e d u r e .

The known s o l u t i o n f o r a b a s e c o n f i g u r a t i o n is considered a r e a s o n a b l e estimate o ft h es o l u t i o n for a s l i g h t l ym o d i f i e dc o n f i g u r a t i o n . Two k values,closelyspacedaccordingtotheDesmarais-Bennettapproach,are c h o s e ns u c ht h a tf l u t t e r i s e x p e c t e dt oo c c u r at a lower k v a l u e . Modal s o l u t i o n s at t h e s e two k values are o b t a i n e d . The repeated sequence of linear e x t r a p o l a t i o nt ot h en e x t k v a l u ea n dt h eL a g u e r r ei t e r a t i o n described in Reference 7 i s performed for the mode that i s e x p e c t e d t o g i v e t h e f l u t t e r c r o s s i n g andone or more a d d i t i o n a l modes on e a c h s i d e o f t h i s mode inthefrequencyspect.rum. The a d d i t i o n a l modes a r ei n c l u d e dt oa s s u r e t h a t a f l u t t e r c r o s s i n g i s o b t a i n e d , i n the e v e n t t h a t an e r r o r i n judgment is made i n s e l e c t i n g t h e p r i m e c a n d i d a t e mode f o r a f l u t t e r c r o s s i n g .

The preceding conceptual evaluation defines the problems that need to b resolved when adjusting the Desmarais-Bennett method for use in a flutter optimization program and no numerical evaluation was considered necessary.

3.3.5 Two-Dimensional Regula Falsi - The concept of solving the two equations

implied by equation (3.4) for two unknowns is not new. However, using this concept for directly solving the flutter equation for the flutter speed is relatively new. The need for such a solution arose with the advent of struc tural optimization with flutter speed constraints and, to the knowledge of the present authors, the first published record of solving directly for the flutter speed is Reference 9 .

In that Reference the Newton-Raphson approach is used in two dimensions to determine flutter speed and, as a byproduct, flutter frequency. The Newton-Raphson approach is based on determining the value of a function and its derivatives for an initial set of trial values and extrapolating linearly to an estimate of the solution. In Reference 9 , the derivatives are deter- mined by a finite difference technique. The Two-Dimensional Regula Falsi approach uses three trial sets of the unknowns to construct two planes. The common point between those planes and the plane D(w,V)=O defines the next estimate of the solution.

Table 3-1 compares the essential characteristic of the two methods. In the Newton-Raphson method with analytical evaluation of the derivatives, the formation of two derivative matrices is time consuming. In ell methods the determinant evaluations are the most time consuming. Other operations, Pro- related to solving two linear equations with two unknowns, are trivial.

visions to assure convergence are comparable for the two methods. Numerical experience with the Two-Dimensional Regula Falsi has indicated that problems with convergence on a solution are more easily solved than with the Newton- Raphson approach. It is concluded that the Two-Dimensional Regula Falsi approach is the more preferable one of the two.

It should be noted that both methods can be used for combinations of unknowns other than frequency and flutter speed. The Two-Dimensional Regula Falsi has been used successfully for solving for the value of one design variable, required to meet a given flutter speed, and the associated frequen The method does not require the computation of derivatives. No interpolation or extrapolation of converged solutions is required, unless nonconvergence is encountered and an intermediate configuration is analyzed to assist in obtain- ing a better initial estimate of the solution for the configuration f o r which the original nonconvergence occurred. Finally, the solution sought is a combination of real values of the unknowns, rather than a series of complex modal solutions associated with in-flight modes. The equivalent of converging on the wrong mode, as may occur in seeking modal solutions, usually leads t nonconvergence, and a recovery procedure that is described in Reference 1.

Thus mode switching to a non-flutter mode does not occur o r , at worst ,leads to nonconvergence. The chance of converging on the wrong flutter speed and frequency would seem to be quite small in view of the relati\-ely small number of solutions within the region of interest of the unknowns. It has never occurred in the many test cases that have been run during this study.

TABLE 3-1. COMPARISON OF NEWTON-WHSON METHOD AND TWO-DIMENSIONAL REGULA FALSI METHOD Newton-Raphson ' TWO- Analytic a1 Dimensional F Regula Falsi Operation Number of initial estimates 1 1 Yes Yes Yes Interpolation of aerodynamics matrix required?

No No Yes Derivative of aerodynamics matrix required?

No Trivial Yes Formation of derivative matrices required?

Number of complex Value of 3 3 determinant First determinant evaluations per step iterative step 0 0 2 Derivative Each Value o f 1 1 fol- determinant lowing Derivative 2 0 0 step

3.3.6 Conclusion - On the basis of overall engineering evaluation, supported

by numerical experience with all methods except the Desmarais-Bennett and Newton-Raphson approach, the Two-Dimensional Regula Falsi approach was con- sidered most promising and chosen for further development (see Reference 1).

3.4 Minimum Damping in Hump Mode Sufficient modal damping within the speed envelope can be assured by requiring sufficient damping in all modes at "all" speeds below the minim required flutter speed or by requiring that the minimum damping in each mode, in so far as it occurs below the minimum required flutter speed, is equal to o r larger than a given value.

To initiate exploration of the latter approach a method to determine t minimum damping in a hump mode was formulated.

The p o i n t o f minimum damping i n t h e hump mode is d e f i n e d b y t h e c o n d i t i o n ay " av - 0 , where Y d e f i n e s t h e real p a r t of t h ef l u t t e rr o o t , p=(Y+i)k, i n terms of the reduced frequency k. The q u a n t i t y Y i s a form of t h e logarithmicincrement: a 1 n + l 7 = -in- (3.6) 2 ~ r a n where a and a are amplitudesofsuccessivecycles.

n n + l An e x p r e s s i o nf o r - is found as follows.

av

Considerthe p-k method formulationof the f l u t t e re q u a t i o n( e q u a - t i o n ( 3 . 2 ) ) and t a k et h ed e r i v a t i v ew i t hr e s p e c tt o V:

[$k] p2 + 2$[M] C

Choose a v e l o c i t y V 1 f o r which - i s e s t i m a t e dt ob ee q u a lt oz e r o .

av

The s o l u t i o no ft h ef l u t t e re q u a t i o n at V 1 is : p = pl, I . } = (ql} andthe c h a r a c t e r i s t i c v e c t o r o f t h et r a n s p o s e de q u a t i o n : [ r) = ( rl} . S u b s t i t u t i n g t h i ss o l u t i o ni n t oe q u a t i o n ( 3 . 7 ) g i v e s :

where p'( i k l ) ] = a [ . ( i k ) ]e v a l u a t e d at k = k 1'

With p = (?'+i)k: Substituting equation ( 3 . 9 ) into equation (3.8) leads to a complex equa- ak

-

tion, and thus two equations in the two unknowns and from which

av

can be determined.

The process can be repeated for V2, leading to (%)*. A one-

dimensional Regula Falsi approach will lead to the value of V for which " - 0 .

av

In the above approach two characteristic value problems must be solve

a?

for determining the first iterated value of V for - = 0. Each following

a v

step requires solution of one characteristic value problem.

It should be noted that damping versus speed curves may be rather fl and for practical purposes a converged value of Y may not define a converged value of V. This causes no problem since the most likely application of these procedures is in connection with an inequality constraint such as:

' h u m p top ' ' m a x allowed

Determining the minimum damping in a hump mode can ?e combined with solving for the value of a design variable satisfying the constraint: - - - Y = Y hump top ' m a x allowed For V = V and Y = 7 , equation (3.7) is solved for k and the value of the design variable m Then equations ( 3 . 9 ) and (3.8) are used to compute i'

-

as before. In general f 0. and a one-dimensional Regula Falsi process

av av

I .

is initiated by repeating the process for another chosen value V = V 2' Numerical evaluationof the approaches outlined could not be accomplish within the scope of this study.

3.5 Recommendation The two-dimensional Regula Falsi procedure is recommended f o r inclusion in the Flutter Optimization Module for providing point solutions of the flutter equation., The procedure is more direct than any of the other pro- cedures considered. It aims at roots of the flutter equations, either flutter speed and frequency or design variable and frequency, of which for every flight condition there are considerably fewer present than there are in-flight modes representedin the problem formulation. As a result, convergence on the wrong root would seem to be less likely than when modal solutions are sought.

That the same procedure can be used for solving for different pairs of unknowns is considered an added advantage.In addition, it is equally appli- cable to the p-, the k- and the p-k method of formulating the flutter equation. A preliminary program is available that has shown good convergence behavior under a wide variety of input data.

Since it seems likely that the capability of directly solving for the

point for which E = 0 will be a factor in developing methods of flutter

av

optimization, numerical test cases should be conducted to evaluate the methods related to determining the minimum damping in the hump mode. The results may influence the development of methods of optimization that take into account damping constraints.

4. MODALIZATION

4.1 General

Modalization is the reduction of the number of degrees of freedom by establishing modes of displacement in which the original degrees of freedom (usually point displacements) have a fixed relation to each other.

Let [ z ' ~ ) ) define a relation between the discrete structural displace- ments z. The arbitrary column matrix of displacements { z } can then be approximated,by linear combination of several linearly independent columns or in short notation: The modalized flutter equation is: '.Modalization is desirable whenever the total number of initial degr freedom is sc large that solving the unmodalized equation becomes uneconomic and is necessary if the number of initial degrees of freedom exceeds the capacity of the available computer program to solve the original charact tic value problem. Since, in general, the flutter equation is solved more frequently than the vibration equation and, in addition, the flutter equati must admit complex numbers, modalization is usually associated with the flutter equation. However, when using all the structural displacements of a detailed finite element structural model as degrees of freedom, modalization may be desirable or necessary for the vibration analysis as well.

In any discussion of modalization, the type of modes and the number of modes to be used must be considered. When used in an optimization procedur the question of "updating" must be considered. Updating in this context means the adjustment of the modes after resizing the structural elements in the course of the optimization procedure. These three aspects of modalization will be discussed separately in the following sections.

4.2 Types of Modes Before the advent of the high-speed computer, modalization (e.g., Rayleigh-Ritz method) was required even for vibration analyses. Relatively few and simple modes were used. With the increasing capacity of computers, the need for modalizing the vibration equation has all but disappeared. Th present practice is to determine natural vibration modes of the entire a from an unmodalized vibration equation and to use a certain number of mo associated with the lower range of natural frequencies, to reduce the ord the flutter equation. For special investigations, such as the inclusion of or the entire automatic control actual control-surface-actuator impedances, system, additional control surface modes may be necessary.

In several instances in the literature (e.g., Reference 101, the use of component modes has been described. Component modes define the relations between discrete displacements of airplane components such as the wing or fuselage, and are obtainedby a v i b r a t i o n a n a l y s i s i n w h i c h o n l y d i s p l a c e m e n t s of a p a r t i c u l a r component are used as degreesoffreedom.Complications arise when theconnectionsbetween components involve many s t r u c t u r a l d i s p l a c e m e n t s .

Reference 11 shows, w i t h a simple beam as anexample, t h a t t h e u n j u d i c i o u s u s eo f component modes c a n g i v e i n a c c u r a t e r e s u l t s f o r e v e n t h e l o w e s t f r e - quencyof t h e e n t i r e body. The useof component modes i s only recommended f o r t h e d e t e r m i n a t i o n o f n a t u r a l v i b r a t i o n modes o f t h e c o m p l e t e v e h i c l e ; andthenonly i f it i s n e c e s s a r y t o r e d u c e t h e o r d e r o f t h e v i b r a t i o n equations.

A n a l y t i c a l modes,such as definedbypolynomialsand modes corresponding t o s t a t i c d e f l e c t i o n s , w o u l d o b v i a t e t h e n e e d f o r r e p e t i t i v e v i b r a t i o n a n a l y s e s d u r i n gt h eo p t i m i z a t i o np r o c e s s i f t h e ya r eu s e d as f i x e d modes. However, u s u a l l y a c o n s i d e r a b l yl a r g e r number ofsuch modes i s r e q u i r e d ,f o rt h e same a c c u r a c yo ft h ef l u t t e rs o l u t i o n ,t h a n when v i b r a t i o n modes are used. N o a d v a n t a g e so f f - s e t t i n gt h a td k a d v a n t a g eh a v eb e e ne n c o u n t e r e d .

S p e c i f i c a l l y , a n a l y t i c a l modes h a v e b e e n s u g g e s t e d f o r e f f i c i e n t g e n e r a - t i o no fg e n e r a l i z e da e r o d y n a m i cf o r c ec o e f f i c i e n t s , as d i s c u s s e di nS e c t i o n 5 .

The u s e o f a n a l y t i c a l modes may p e r m i t t h e a n a l y t i c a l i n t e g r a t i o n o f t h e p r o d u c to fd e f l e c t i o n and p r e s s u r e modes. It makes it p o s s i b l et o compute i n v a r i a n tg e n e r a l i z e da e r o d y n a m i cf o r c ec o e f f i c i e n t st h a tc a nb e combined l i n e a r l y t o formgeneralizedaerodynamicforcesforanyarbitrary mode. To takeadvantageofthisfeature,however,the number of a n a l y t i c a l modes must b el a r g e ,s i n c e it must b ea d e q u a t ef o r a l a r g e number o f s t i f f n e s s and i n e r t i ad i s t r i b u t i o n s . The a n a l y t i c a l modes thuscanserve as r e f e r e n c e modes t h a t a r e t h e d e g r e e s o f freedom f o r a l l v i b r a t i o n a n a l y s e s *om which a smaller number o f v i b r a t i o n modes i s o b t a i n e d f o r u s e i n f l u t t e r c a l c u l a t i o n s .

However, a l a r g e number o f v i b r a t i o n modes of a b a s i c c o n f i g u r a t i o n a l s o c a n beused as r e f e r e n c e modes andonewouldexpectthat fewer r e f e r e n c e modes are needed i f t h e y a r e v i b r a t i o n modes t h a n i f t h e y are a n a l y t i c a l modes.

It was t h o u g h tt h a tu s i n gt h e( c o m p l e x )f l u t t e r mode of a baseconfigura- t i o n mightreducethe number of modes r e q u i r e d f o r a n a d e q u a t e f l u t t e r s o l u t i o n of a modifiedconfiguration. Some preliminary work d u r i n gt h i ss t u d y was done,but was n o tc a r r i e d far enough f o r anyconclusiontobe drawn.

4.3 Number of Modes The number o f modes used i n t h e f l u t t e r e q u a t i o n i s o f i m p o r t a n c e f o r t h e accuracyofthe-computedflutterspeedandflutterspeedderivativeswith r e s p e c tt od e s i g nv a r i a b l e s . A t p r e s e n tt h e r e seems t o b e no r e a d i l ya v a i l a b l e g e n e r a l c r i t e r i o n f o r d e t e r m i n i n g t h e number of modes needed f o r a d e s i r e d accuracy.

When t r y i n g t o economizeby r e s t r i c t i n g the number of modes t o b e u s e d i n f l u t t e r c a l c u l a t i o n s , t h e r e i s a need t o f r e q u e n t l y c h e c k w h e t h e r t h e number of modes i s s u f f i c i e n t f o r a c c u r a t e p r e d i c t i o n o f t h e f l u t t e r s p e e d f o r a r b i t r a r y c o n f i g u r a t i o n s . Thus t h e r e i s an a d v a n t a g ei nu s i n g f l u t t e r analysispro- cedures t h a t a l l o w a large number o f modes even i f t h a t raises t h e c o s t o f eachindividual f l u t t e r s o l u t i o n .

It has been pointed out in Reference 12, and it was confirmed by limited n u m e r i c a l a n a l y s i s d u r i n g t h i s s t u d y , t h a t more modes axe needed f o r a c c u r a t e l y computing f l u t t e r s p e e d d e r i v a t i v e s t h a n f o r computing f l u t t e r s p e e d s .

I n d e c i d i n g o n t h e number o f modes t h e computerenvironment may be an i m p o r t a n t f a c t o r t o b e j u d g e d b y t h e analyst i n a d d i t i o n t o t h e a c c u r a c y r e q u i r e d . Even t h e method ofcomputercostappropriation may i n f l u e n c et h e d e c i s i o n .

4.4 Updating of Modes As r e s i z i n g s t e p s a c c u m u l a t e d u r i n g t h e o p t i m i z a t i o n p r o c e s s , t h e v i b r a - t i o n modes o f t h e i n i t i a l c o n f i g u r a t i o n become less s u i t e d t o a c c u r a t e l y r e p r e s e n tt h er e v i s e ds t r u c t u r e .I d e a l l y ,t h e r e f o r e , after e a c hr e s i z i n gs t e p a new v i b r a t i o n a n a l y s i s s h o u l d d e t e r m i n e new modes f o r m o d a l i z i n g t h e f l u t t e r equation. The need f o rs u c hu p d a t i n g i s c l o s e l yr e l a t e dt ot h e number of modes usedandthetypeandmagnitudeofstructuralchangesincurredbythe r e s i z i n g . The u s eo f a l a r g e number o f modes t e n d s t o r e d u c e t h e need f o r frequentupdating. However, i n s u f f i c i e n tu p d a t i n gc a nc a u s et h er e s i z i n g s t e p s t o f o l l o w a zig-zagpaththat,intheextreme, may notconverge.

The p h y s i c a le x p l a n a t i o nf o rt h i s is thefollowing. L e t t h eo p t i m i z a t i o n p r o c e d u r e i n d i c a t e a l o c a l s t i f f e n i n g as t h e optimum r e s i z i n g f o r r e s i z i n g s t e p j . Then t h e v i b r a t i o n modes f o r s t e p j + l would show a d e c r e a s ei nl o c a l deformation. If t h ev i b r a t i o n modes f o rs t e p j are u s e df o rs t e p j+l, t h e e x c e s s l o c a l d e f o r m a t i o n t e n d s t o r e i n f o r c e a n d o v e r e s t i m a t e t h e b e n e f i c i a l e f f e c to ft h a tl o c a ls t i f f e n i n g .T h u s , i nt h ea b s e n c eo f modal updating, material t e n d s t o b e addedwhere t h e f i r s t r e s i z i n g s t e p , w i t h t h e modes used, i n d i c a t e s where it i s most b e n e f i c i a l .

Modal updatingmustnotbeconfusedwith making t h e modal m a t r i x a func- t i o n of thedesignvariables.Thisaspectofmodalizing w a s r e c e n t l yi n t r o - ducedbyReference 1 2 and it is formulatedinReference 1. Determiningeach r e s i z i n gs t e pu n d e rt h ea s s u m p t i o no fc o n s t a n t modes ( i . e . , independentofthe a t e a c h r e s i z i n g s t e p , may under- d e s i g n v a r i a b l e s ) , b u t u s i n g u p d a t e d modes estimate t h e amount of material t o b e a d d e d l o c a l l y f o r a c e r t a i n amount of a p a r t i c u l a r s t e p , b u t it i s n o t e x p e c t e d t o c a u s e a n e r r a t i c o r s t i f f e n i n g i n nonconvergent resizing path.

As important as thefrequencyofupdating is on t h e e f f i c i e n c y o f t h e Optimizationprocedure,the number o f modes u s e d t o do t h e f i n a l f l u t t e r a n a l y s i s is more importantfrom a generalpointofviewsince it p r o v i d e s t h e f i n a l checkon t h eo p t i m i z a t i o np r o c e d u r e .I nt h eo p i n i o no ft h ep r e s e n t i n v e s t i g a t o r s , a check f l u t t e r a n a l y s i s u s i n g a p r o v e n . s u f f i c i e n t number o f v i b r a t i o n modes o f t h e f i n a l c o n f i g u r a t i o n s h o u l d c o n c l u d e a n y o p t i m i z a t i o n p r o c e s s . If f l u t t e rr e q u i r e m e n t s are n o t m e t , t h e n a new optimizationprocess c a n b e i n i t i a t e d a n d , p r o b a b l y , more modes or more frequentupdating, or b o t h , s h o u l d b e u s e d .

4 . 5 Recommendations I n view o f e x p e r i e n c e d u r i n g t h e p r e s e n t s t u d y , a n d as a r e s u l t o f e x p e r i e n c e w i t h f l u t t e r a n a l y s e s o f a c t u a l a i r p l a n e d e s i g n s , t h e p r e s e n t i n v e s t i g a t o r s recommend t h e f o l l o w i n g : 1. A f l u t t e r module s h o u l dp r o v i d et h eo p t i o no fi n p u t t i n ga r b i t r a r yi n i t i a l modalizing matrices or o f g e n e r a t i n g i n i t i a l m o d a l i z i n g m a t r i c e s b a s e d on a v i b r a t i o n a n a l y s i s of t h e i n i t i a l c o n f i g u r a t i o n .

2 . The number o fv i b r a t i o n modes t ob eu s e df o rt h ef l u t t e rc a l c u l a t i o n s should be an input option.

3. The f r e q u e n c yo fu p d a t i n gt h ev i b r a t i o n modes s h o u l db ea ni n p u to p t i o n .

4. An o p t i o ns h o u l db ei n c l u d e dt op r o v i d et h ea n a l y s tw i t hi n f o r m a t i o nt o d e t e r m i n e w h e t h e r h i s c h o i c e o f number o f modes andfrequencyofupdating h a sl e dt os a t i s f a c t o r yf l u t t e rc h a r a c t e r i s t i c s . Suchinformationmight beprovidedby a v i b r a t i o n and f l u t t e r a n a l y s i s o f t h e f i n a l c o n f i g u r a - t i o n w i t h more modes than were used throughout the resizing process, a c h e c k o n w h e t h e r t h e o p t i m a l i t y c r i t e r i a f o r f l u t t e r are s a t i s f i e d , or other check procedures.

5 . AFRODYNAMICS 5 . 1 I n t r o d u c t i o n One o f t h e o b j e c t i v e s o f t h i s s t u d y i s t o d e v e l o p g e n e r a l , e f f i c i e n t a n d accuratecomputationalproceduresforevaluatingtheunsteadyaerodynamic p a r a m e t e r s n e c e s s a r y f o r u s e i n a f l u t t e r o p t i m i z a t i o n module,without, how- ever, evaluatingaerodynamicstheories.

The procedureshouldbegeneral. That i s , it s h o u l db ea p p l i c a b l et o a l l p r e s e n t , a n dh o p e f u l l yf u t u r e ,t h e o r e t i c a lf o r m u l a t i o n so fu n s t e a d y aerodynamics.

The procedure should be efficient. In the context of application in a flutter optimization module, this implies a minimum of computational oper required to recompute the generalized aerodynamic force coefficients each time a modal updating occurs.

The procedure should be accurate. This implies it should be able to accommodate the most sophisticated formulations of the aerodynamics, such that the aerodynamics used in the flutter optimization module have the accuracy asthe aerodynamics used in a flutter analysis module.

In the following section general background for a matrix formulation that allows a procedure satisfying these requirements is presented. It is followed by the definition of the formulation and a discussion of how dimensions of the matrices, the method of interpolation for modal deflecti and arbitrary values of the reduced frequency k, and the number of reduced frequency intervals to be considered determine the sequence of operations that is most efficient. Conclusions and recommendations regarding the aero- dynamics subroutine in a flutter optimization module are presented.

5.2 General The elementsof the matrixof generalized aerodynamic force coefficients are defined by: Here p.(x,y) is the lifting pressure distribution associated with an J angle-of-attack distribution., Q.(x,y) , which is defined by: J

Z aZ

which expresses as the sum of - and - terms in the case of harmonic

ax

v

ffj motion with reduced frequency k in a mode defined by fj (x,y).

Expressed in the form of equation (5.11,the evaluation of A requires ij evaluation of the surface integral each time new modes fi are used. In the usual flutter investigation many different sets of modes are used, corre- sponding to different weight and stiffness distributions. In addition it is expected that frequent,remodalization is required in an optimization proc Thus it is advantageous t o develop a method i n which the generalized aerody- namic f o r c e c o e f f i c i e n t s are formedfrom a mode-independent p a r t t h a t c o n t a i n s as many o f t h e n u m e r i c a l o p e r a t i o n s as p o s s i b l e , and a r e l a t i v e l y s i m p l e mode-dependent p a r t .

Four d i f f e r e n t a p p r o a c h e s are r e c o g n i z e d i n s e p a r a t i n g mode-independent operationsfrom mode-dependent o p e r a t i o n s . One method r e l i e s e n t i r e l y on a n a l y t i c a l e v a l u a t i o n of t h e s u r f a c e i n t e g r a l ( E q u a t i o n (5.1) ) . A second method formulates a n u m e r i c a l e v a l u a t i o n o f t h e s u r f a c e i n t e g r a l l e a d i n g , e f f e c t i v e l y , t o "lumped"aerodynamic f o r c e s at a g r i d o f i n t e g r a t i o n p o i n t s .

I n a t h i r d method p r e s s u r e d i s t r i b u t i o n modes are a n a l y t i c a l l y i n t e g r a t e d over small areasand combined intoelementaryaerodynamicforcesdirectly comparable t o ,a n dt r e a t e d as, i n e r t i a lf o r c e s . The f o u r t h method recognized i s based on a f i n i t e e l e m e n t a p p r o a c h , t h e b a s i c f o r m u l a t i o n o f which has no r e f e r e n c e t o p r e s s u r e d i s t r i b u t i o n s o v e r t h e e n t i r e s u r f a c e .

The f i r s t t h r e e methods a r e u s u a l l y t h o u g h t of as stemmingfrom t h e kernelfunctionapproach of Reference 13. I n it t h ep r e s s u r ed i s t r i b u t i o n is assumed t o b e a l i n e a rc o m b i n a t i o no fp r e s s u r ed i s t r i b u t i o n modes p ( x , y ) J J J n The p r e s s u r e mode c o e f f i c i e n t s a are determined from a boundary j c o n d i t i o n r e q u i r i n g t h a t t h e n o r m a l i z e d i n d u c e d v e l o c i t y d i s t r i b u t i o n r e s u l t - i n g from t h e p r e s s u r e d i s t r i b u t i o n e q u a l s t h e a n g l e - o f - a t t a c k d i s t r i b u t i o n a t a s e t of downwash c o l l o c a t i o n p o i n t s : Combining e q u a t i o n s (5.3) and ( 5 . 4 ) l e a d st o where theelementsofmatrix [PKI] a r et h ei n t e g r a l s of theproductof p r e s s u r e d i s t r i b u t i o n mode and an aerodynamickernel.

The columns o f [p"] are l i n e a r l yi n d e p e n d e n tp r e s s u r ed i s t r i b u t i o n modes.

5.2.1 A n a l y t i c a lI n t e g r a t i o n - When p . ( x , y ) i s a l i n e a rc o m b i n a t i o n of J p r e s s u r ed i s t r i b u t i o n modes p n ( x , y ) ,a n a l y t i c a l modal functions ' f k ( x , y ) can be s e l e c t e d s u c h t h a t the i n t e g r a l s l J f k ( x , y ) p n ( x , y ) dx dy can be e v a l u a t e da n a l y t i c a l l y .G e n e r a l i z e da e r o d y n a m i cf o r c ec o e f f i c i e n t si n terms of modal coordinatescanthen be formed. The a n a l y t i c a l modes can be u s e d .

as arbitrary modes t o modalize the f l u t t e r e q u a t i o n , or a mode-dependent t r a n s f o r m a t i o n b e t w e e n t h e a n a l y t i c a l modes and t h e a c t u a l modes is used t o e x p r e s s t h e g e n e r a l i z e d a e r o d y n a m i c f o r c e . c o e f f i c i e n t s i n terms o f t h e a c t u a l modal c o o r d i n a t e s .

5.2.2 NumericalIntegrationof the ProductofDisplacementandPressure -

Reference 1 2 definesanapproachtoseparatingmode-independentoperations frommode-dependent o p e r a t i o n s i n which t h e s u r f a c e i n t e g r a l o f e q u a t i o n (5.1) i s evaluatednumerically. A Gaussianintegrationprocedure is s u g g e s t e dt o evaluate the i n t e g r a l . The p r e s s u r ep j( x , y ) and t h e d e f l e c t i o n h i ( x , y ) are e v a l u a t e d at i n t e g r a t i o n p o i n t s d e f i n e d b y t h e Gaussianprocedure.

W e i g h t i n gf a c t o r si nt h e form of a row m a t r i x , LwFJ, make it p o s s i b l et o write : The r i g h t h a n d s i d e o f e q u a t i o n ( 5 . 6 ) can be w r i t t e n as: The i n t e r c h a n g e o f t h e row a n d d i a g o n a l m a t r i x i n t h e l a t t e r p a r t o f t h e combined equation ( 5 . 7 ) makes it p o s s c b l e t o s e p a r a t e t h e mode-dependent operationsfromthemode-independentoperations.

' The column m a t r i x P F . ~ ~ } is a s e t of lumped aerodynamic f o r c e s . The

d e f l e c t i o n s h i c a n b e e x p r e s s e d i n terms of the d e f l e c t i o n s z a t t h e i s t r u c t u r a l nodes by t h e r e l a t i o n .

A v a r i a t i o n o f t h i s method i s o b t a i n e d i f i n s t e a d o f t h e p r e s s u r e d i s t r i b u t i o nt h ev e l o c i t yp o t e n t i a ld i s t r i b u t i o n 9. ( x s y )d u et oa n J a n g l e - o f - a t t a c kd i s t r i b u t i o n a. (x,y) is used. With t h e familiar l i n e a r i z e d r e l a t i o n b e t w e e n p r e s s u r e a n d v e l o c i t y p o t e n t i a l : p = - 2 ( g + ik.)

(5.9) equation (5.1) becomes: . . .I_ , I It canbe shown t h a t w i t h t h e h e l p o f n u m e r i c a l t e c h n i q u e s e q u a t i o n (5.10) c a n b e w r i t t e n as: where [WF] performsnumericallythe first i n t e g r a t i o ni ne q u a t i o n( 5 . 1 0 ) and [ W F D ] . p e r f o r m s t h e d i f f e r e n t i a t i o n and i n t e g r a t i o n i n t h e s e c o n d term o f t h a t equation.Equation (5.11) i s a t r i p l em a t r i xp r o d u c t , similar t o equa- t i o n ( 5 . 7 ) , i n which t h ec e n t e rm a t r i x is mode independent.Foradditional d e t a i l s see Reference 1.

5.2.3NumericalIntegrationofthePressures - When p . ( x , y ) is a l i n e a r

J c o m b i n a t i o no fp r e s s u r ed i s t r i b u t i o n modes p n ( x , y ) ,t h ei n t e g r a l $$pn(x,y) dx dycanbeevaluatedover small areas, o f t e n r e f e r r e d t o as aerodynamicboxes,intowhichthesurface is d i v i d e d . By e v a l u a t i n g n /$x p n ( x , y ) dx dy and $ $ y p ( x , y ) dx dy o v e rt h e same areas, lumped aerodynamicforcescanbedeterminedinmagnitudeandlocation. The modal displacement a t t h e l o c a t i o n o f e a c h lumped f o r c e ( i . e . , foreachaerodynamic boxandeach p"( x , y ) >c a n be e x p r e s s e di n terms o ft h es t r u c t u r a ld e g r e e s offreedom. Thus theproductofeach lumped f o r c ea n d its modal displacement can be formed. Summation overtheaerodynamicboxesandthepressuredis- t r i b u t i o n modes p a r t i c i p a t i n gi np .( x , y )l e a d st o A .

J i d

5.2.4FiniteElementApproach - I n a f i n i t ee l e m e n ta p p r o a c h , lumpedaerody-

(see equation ( 5 . 7 ) ) are expressed namic f o r c e s c o r r e s p o n d i n g t o [WF*pj] d i r e c t l yi nt e r m so fa n g l e - o f - a t t a c kd i s t r i b u t i o n s kj) u n d e ra p p r o p r i a t e s i m p l i m i n ga s s u m p t i o n s . The generalizedaerodynamicforcecoefficients are formed as i n S e c t i o n 5.2.2.

5.3 Basic Formulation Whatever theapproach, o r wh,ateveraerodynamictheory is chosen,the g e n e r a l i z e d a e r o d y n a m i c f o r c e c o e f f i c i e n t s i n terms o f modal c o o r d i n a t e s c a n b e ' e x p r e s s e d as t h e p r o d u c t o f five matricesofwhichonlythe first and last are mode-dependent: (5.12)

The m a t r i x [AIC] = [ A I C ( k ) ] , a f u n c t i o n of thereducedfrequency

k = - a' a n d t h e Mach number, i s the core of the aerodynamics and i s independent

v

o f mode shape. Its elementsarebasicaerodynamicinfluencecoefficients d e f i n i n g lumped aerodynamic f o r c e s {Za} at an aerodynamic force grid in terms of t h e a n g l e s o f a t t a c k a t downwash c o l l o c a t i o n p o i n t s : displacements ( z } . It i s independent of mode shape.

The m a t r i x [.3' i s independent of k and of mode shape, and distributes

lumpedaerodynamic forcesand moments o v e r t h e s t r u c t u r a l c o o r d i n a t e s .

I n t h ec a s et h a tt h ea p p r o a c ho fS e c t i o n 5.2.1 i s followed, [AIC] i s t h e m a t r i x of g e n e r a l i z e d a e r o d namic f o r c e c o e f f i c i e n t s i n t e r m s o f the a n a l y t i c a l modes; [HI and [Wj are e q u a t e d t o where columns o f [fe (x,y)] are t h ef i x e da n a l y t i c a l modes. The m a t r i x [ F ] is independentof k and of t h e a c t u a l mode s h a p e su s e dt o reduce t h eo r d e r of t h e f l u t t e r e q u a t i o n .

The operations performed by [W] and [HI' may be i n c l u d e di n [AIC].

Equation(5.12)thenreducestotheproductofthree matrices.

The matri? [Z] c o n t a i n st h e modal columns i n terms o ft h es t r u c t u r a l d e f l e c t i o n s (z}.

, . . .

The m a t r i c e s [AIC], [HIT and [ W ] are constant during an optimization procedure. They w i l l beused many times d u r i n gt h ed e s i g np r o c e s so fa n air- planewith a g i v e ne x t e r n a lc o n f i g u r a t i o n . It is t h e r e f o r ea d v a n t a g e o u st o form t h e s e m a t r i c e s i n a specialaerodynamicscomputerprogram.

Each time d u r i n g a n o p t i m i z a t i o n p r o c e d u r e t h a t a remodalization takes

p l a c e , Fi j ] must be recomputed.Depending on thedimensionsofthematrices

i n e q u a t i o n ( 5 . 1 2 ) , i t may b e more e f f i c i e n t t o compute t h e t r i p l e m a t r i x product

[HAW] = c . 3 ' [.IC] [I.3

(5.15) i n t h e aerodynamicsprogram, o r t o performone o r b o t h o f t h e m u l t i p l i c a t i o n s [E]T[H]T and [W][Z] i n t h e o p t i m i z a t i o n program.

I n t h e f o l l o w i n g s e c t i o n s f a c t o r s are d i s c u s s e d t h a t must beconsidered in determining which approach to numerically evaluating according to equation(5.12) i s most e f f i c i e n t .

5.4 F a c t o r sA f f e c t i n g The E f f i c i e n c yo f The NumericalEvaluationof The MatrixofGeneralized Aerodynamic F o r c e C o e f f i c i e n t s It i s b e l i e v e d t h a t t h e f o r m u l a t i o n

pij] = [ZIT p] [ E ]

whichfol .lows fromcombiningequations(5.12)and (5.15), is widely u s e d i n Detailed studyoftheformulationofequation(5.121,which i n d u s t r y .

d i r e c t l y followsfromReference12,however,indicatesthatthere are condi- tions under which it i s more e f f i c i e n tt o compute [Z]'[.3' and/or [W][E] i n t h e f l u t t e r o p t i m i z a t i o n module.Extensivecomparisonshavebeen made and are d i s c u s s e d i n d e t a i l i n R e f e r e n c e 1. I nt h ef o l l o w i n gt h ef a c t o r s a f f e c t i n g t h e s e c o m p a r i s o n s are discussed and major conclusions are presented.

5.4.1 'Matrix Population - When m a t r i c e s are sparsely populated, or populated , i n well d e f i n e d b l o c k s , p r o p e r programmingcan take advantage of t h i s .

I n e q u a t i o n ( 5 . 1 2 ) t h e m a t r i c e s [W] and [H] may be sparsely populated.

Thesematricesperformaninterpolationand [W] , i na d d i t i o n ,d e t e r m i n e s

streamwise s l o p e s at c o l l o c a t i o np o i n t s .I n t h e case o fs i m p l ei n t e r p o l a t i o n ( l i n e a r o r low degreepolynomial),each row i n [W] e x p r e s s e st h ea n g l e 0.f a t t a c k at a downwash c o l l o c a t i o n p o i n t i n terms of several surroundingstruc- t u r a lc o o r d i n a t e s .S i m i l a r l y ,e a c h row o f [H] e x p r e s s e st h ed e f l e c t i o n at a n i n t e g r a t i o n p o i n t i n terms o f s e v e r a l s u r r o u n d i n g s t r u c t u r a l c o o r d i n a t e s .

Thus each row i n [ W ] and [ H ] c o n t a i n s r e l a t i v e l y f e w , s a y < 20, nonzero e l e m e n t s ; f o r l i n e a r i n t e r p o l a t i o n e a c h row containsfournonzeroelements I n t h e c a s e o f i n t e r p o l a t i o n b y t h e s u r f a c e s p l i n e method, t h e m a t r i c e s [ W j and [H] are f u l l yp o p u l a t e d , a t least i nt h eb l o c k s that covertheaero- dynamic s u r f a c e s .

W i t h o u ts p e c i f i cs t i p u l a t i o n s , e q u a t i o n (5.16) i m p l i e st h a tt h eo r d e ro f t h em a t r i x [HAW] equals t h e number o fs t r u c t u r a lc o o r d i n a t e s . There may b e a c o n s i d e r a b l e number o f s t r u c t u r a l c o o r d i n a t e s that do notcarryanaero- dynamic l o a d . They c o r r e s p o n d t o z e r o e l e m e n t s i n [HAW]. It i s not expected t h a t t h e f r a c t i o n o f n o n z e r o e l e m e n t s w i l l be high enough t o j u s t i m t r e a t i n g [HAW] e s a s p a r s em a t r i x . However, b yp r o p e ro r d e r i n go ft h es t r u c t u r a l coordinates,thenonzeroelementsin [HAW] may b ec o n c e n t r a t e di n one or more blocks. Then t h e aerodynamics program may form [HAW] based on aerody- namic l o a dc a r r y i n gs t r u c t u r a lc o o r d i n a t e so n l y .C o r r e s p o n d i n g l y ,t h ef l u t t e r o p t i m i z a t i o n modulemust e l i m i n a t e s t r u c t u r a l c o o r d i n a t e s t h a t c a r r y no aero- dynamic l o a d from t h em o d a l i z i n gm a t r i x [Z] .

5 . 4 . 2I n t e r p o l a t i o nf o rA r b i t r a r y k Value - It i s g e n e r a l l ya c c e p t e dt h a t when t h eg e n e r a l i z e da e r o d y n a m i cf o r c ec o e f f i c i e n t s are d e t e r m i n e df o r a d i s c r e t e s e t o f v a l u e s , kt, of the reduced frequency, A..(k ) i n t e r - L J 1 1 p o l a t i o n i s adequate for approximating p i j ( k ) ] at a r b i t r a r yv a l u e so fk .

Two methodsofinterpolation are considered:cubicpolynomialandcubic s p l i n e .

c a nl e a dt o" h u n t i n g "( o s c i l l a t i o nb e t w e e n k v a l u e si na d j a c e n ti n t e r v a l s ) .

where [Aij(k,)] is t h e d e r i v a t i v e o f [ I A i j ( k ) ] e v a l u a t e d at k=k

e *

is a ni n p u tt ot h ei n t e r p o l a t i o ns u b r o u t i n e . Thus t h ed i f f e r e n c e betweenequations (5.19) and(5.20) i s t h a ti ne q u a t i o n (5.19) d i f f e r e n t i a t i o n occurs after thepolynomial f i t and i n e q u a t i o n ( 5 . 2 0 ) it o c c u r s b e f o r e t h e polynomial f i t . The formation of A. . ( k ) i nt h ef l u t t e ro p t i m i z a t i o n

L J Q I

module is based on equation ( 5.12) , equation (5.16) or any v a r i a n t t h a t i s

chosen as b e i n g most a p p r o p r i a t e . The d e r i v a t i v e [ A I C I (k)] or [HAW' ( k ) ] is n e e d e d a n d s h o u l d b e c a l c u l a t e d o u t s i d e t h e f l u t t e r o p t i m i z a t i o n module by anymethod that gives adequate accuracy.

To d e f i n e [Aij (k)] and ["lj ( k ) ] i n one k i n t e r v a l , f o u r m a t r i c e s

[AIC(kQ)] and four matrices AIC'(k ) , or four of each of the matrices

[ Q I

[HAW( k)] and [HAW' ( k ) ]m u s tb ei n p u ti n t ot h ef l u t t e ro p t i m i z a t i o n module.

If [AIC(k)] and [AIC'(k)] are input, [Aij(k)] follows from equation (5.12).

pij (E)] i s givenby:

[ A i j ( k ) ] = [Z]TIH]TIAIC'(ki] [.1[E] + i[Z]'[HIT[AIC(kj [ D ! Z ] [ Z ] . (5.21)

If k moves t o a na d j a c e n ti n t e r v a lo n l y two o ft h ei n p u tm a t r i c e s ,o n e f o r t h e a e r o d y n a m i c s c o e f f i c i e n t s a n d o n e f o r t h e i r d e r i v a t i v e s , n e e d be r e p l a c e d .

It should be n o t e dt h a t ,i ne f f e c t ,e a c h k i n t e r v a lh a s i t s own a s s o c i a t e dp o l y n o m i a l sf o rt h ev a l u eo f [Aij (k)] and its d e r i v a t i v e . ' The c u b i c s p l i n e method a l s o d e f i n e s d i f f e r e n t c u b i c p o l y n o m i a l s f o r each k i n t e r v a l . The c o e f f i c i e n t s f o r t h e p o l y n o m i a l , however, are derived

from'matrices defined for all k values ke, .f?= 1,2 . . . n. They follow

from the assumption of continuity of derivatives over the complete range of k values. The resultant expression, e .g., for [ A I C ( k ) ] is: I The matrices [MCeo] to [AICe3] should be formed outside the flutter optimization module.

Then :

bij ( k ) ] = [Axe0] +[Axel] (k-ke) + [ A X e 2 ] (k-kel2 + [ AXe3] (k-ke +

(5.23)

[AZeo] ik + [Ugl] ik(k-ke) + pe2] ik(k-ke) 2 + [A!Ze3] ik(k-ke) 3

where [ A X e o ] = [ E ] ' [ . ] ' [ A I C e o ] lp.3 [ Z ] etc., and [ A Z e o ] = [ E l T [ H ] '[ A I C e o ] F Z ] [ Z ] etc . , Because of the implied continuity of the derivatives, it is proper to differentiate equation (5.23) directly and thus no additional matrices for the derivative need to be formed.

To define [ I . i ( k ) ] and ["fj ( k ) ] in one k interval if the aerody- namics input is [ A I C ~ , ] to [ . I c e 3 ] requires eight coefficient matrices.

Switching k to any other interval requires replacing all eight matrices.

If the basic aerodynamics input is in the form of

[mb7eo] to [ u w e 3 ]

then only four coefficient matrices are needed for each k interval.

5.4.3 Number of k Intervals - Let the basic aerodynamics input into the

r -

flutter optimization module be HAW(k ) ; the number of k intervals to be

1 e l

considered is 1 .

Forthecubicpolynomia1,Lagrange’sinterpolationformula is considered t ob e most e f f i c i e n ts i n c e it expresses b i j ( k ) ]d i r e c t l yi n terms o f its value [Aij(k”)l at d i s c r e t e values k t , e = 1, 2, 3, 4: where L l ( k ) i s d e f i n e d by: a n dc y c l i cs u b s t i t u t i o nl e a d st o L2, x3 and Z4.

The i n t e r p o l a t i o nf o r m u l a ( 5 . 1 7 ) is u s e do n l yf o rt h ei n t e r v a l k2< k < k 3‘ For t h ei n t e r v a l k < k < k 4t h ei n d e x k? .must beincreasedbyoneandfor k < k < k2 t h e i n d e x must be lowered by one.

Since most methods of optimization require the computation of the deriv- a t i v eo ft h ea e r o d y n a m i c sm a t r i xw i t hr e s p e c tt ok ,t h ef o r m a t i o no ft h e d e r i v a t i v e , A ! . ( k ) , must be considered.

[ I 1 5 1

D i f f e r e n t i a t i n ge q u a t i o n (5.17) w i t hr e s p e c tt o k l e a d st oa n expression: t h a t i s based on t h e same aerodynamicmatrices as e q u a t i o n ( 5 . 1 7 ) . This approach, however, combined with the re-indexing of kl as k moves t o an a d j a c e n ti n t e r v a l ,l e a d s t o jumps i nt h ev a l u eo f [.;(k)l at a l l values k=ke.Apartfromconsiderations of a c c u r a c y ,t h i s is u n d e s i r a b l es i n c e it If cubic polynomial interpolation is used, ( 1 + 3 ) matrices and [HAW'( kt )] must be input and . pre- and postmultiplied by [ZIT and [Z] to form the 2(1+3) matrices

ki (kt ) ] and ( k1 )] needed in

Lagrange's interpolation formula.

If cubic spline interpolation is used, 41 matrices [mweO] to [ m ' 1 3 ]

must be'input and pre- and postmultiplied by [ Z I T and [ E ] to form 41

coefficient matrices

[Aeo] to [ . e 3 ] *

Under otherwise equal circumstances, polynomial interpolation is more efficient if This condition is valid for other sequences of operations to form [Aij(k)] according to equation (5.12). However, there are also sequences

for which 1 2 4 o r 1 2 5 is required for cubic polynomial interpolation

to be more efficient than cubic spline interpolation.

The number of intervals that should be used is difficult to predict.

If only one flutter constraint is active, k may stay within a rather smal range during the entire optimization process and that range may lie comp within one k interval. Obviously only aerodynamic matrices applicable to that one k interval need be computed. In general, however, several k intervals are required.

5.4.4 Sequence of Multiplications - Defining one computational operation as

one multiplication and one addition, the numbers of such operations requ inside the flutter optimization module for different sequences of multipli tions in equation (5.12) have been determined and compared.

The following options have been considered; the numerals indicate th sequence of multiplication.

H1:

[ A i j ] = [ E ] ' - [ . I T [.IC] - [.I [ z ]

H2 :

[ A i j ] = [ Z ] ' PIT [ . I C ] - b] [F]

H 3 : [ A I C ][ W ] = [ A W ] is computed outside the flutter optimization module

= [ Z I T - [ . 3 ' [ . . 3 [Z]

[ A i j ] 1 2 H 4 : [ H I T [ A I C ] = [ H A ] is computed outside the flutter module

[ A i j ] = [ Z I T - [ H A ] [ W ] [ z ]

" 1 2 H 5 : [ H I ' [ A I C ] [ W ] = [ H A W ] is computed outside the flutter module

Pij] = [ T I T[ H A W ] [ E ]

The number of computational operations is independent of sequence of multiplications in H 5 .

Formulas defining the number of numerical operations have been derived and a r er e p o r t e di nR e f e r e n c e 1. N o o p t i o ns t a n d so u t as c l e a r l y s u p e r i o r or i n f e r i o r t o a l l o t h e r s , b u t some comments are o f f e r e d i n S e c t i o n 5.5.

5.4.5 Form ofInputtingtheAngle-of-AttackGeneratingMatrix - I n t h e p r e -

v i o u ss e c t i o nt h eo p t i o n s are defined as i f one complex m a t r i x [tT(k)] was inputforeachvalue of k. Suboptionsoftheoptionsthatrequireinputting [ I ! ] c a nb eo b t a i n e db yc o n s i d e r i n gt h ed e f i n i t i o no f [ I T ] .

[W(k)] = [[DX] + i k [DZ]] Thus i n o p t i o n s H1, H2 and H 4 an "a" and a "b" version can be recognized.

I n t h e "a" v e r s i o n [W(k)] i s i n p u t f o r s e v e r a l k v a l u e s . I n t h e "b" v e r s i o n t h e real matrices [DX] and [DZ] are i n p u t .

Whether o p t i o n ?'a''. i s more e f f i c i e n t t h a n o p t i o n "b" dependson t h e o t h e r f a c t o r s . I n t h e c a s e o f o p t i o n H1 it seems t h a t Hla is favored i f i n t e r p o l a t i o n for d e f l e c t i o n s is heldsimple.Option Hlb i s favored i f s u r f a c e s p l i n e i n t e r p o l a t i o n is used. If t h e "a" option is usedandthe derivativeoftheaerodynamicsmatrix is needed,matrix [DZ] must be i n p u t anyway.

5.5 Summary of Comparisons Detailed comparison of the options H 1 through H 5 is r e p o r t e d i n R e f e r - ence 1. The following summarizes thecomparisons.

5.5.1 InputStorageRequirements - If t h e number of k i n t e r v a l s t o be used

is t h r e e or m o r e ,c u b i cp o l y n o m i a li n t e r p o l a t i o nf o ra r b i t r a r yv a l u e s of k r e q u i r e s less i n p u t s t o r a g e t h a n c u b i c s p l i n e i n t e r p o l a t i o n f o r a l l options H1 through H 5 .

5.5.2CoreSpace - Foroptions H 5 and H 3 , c u b i cp o l y n o m i a li n t e r p o l a t i o n f o r k r e q u i r e s two times as much corespaceascubicsplineinterpolation.

For all o t h e r o p t i o n s , b o t h m e t h o d s o f i n t e r p o l a t i o n r e q u i r e t h e same core space.

5.5.3 Read-In - C u b i cp o l y n o m i a li n t e r p o l a t i o nf o ra r b i t r a r y k r e q u i r e s l e s sr e a d - i nt h a nc u b i cs p l i n ei n t e r p o l a t i o n as t h ev a l u eo f k moves i n t o an a d j a c e n t i n t e r v a l .

5.5.4 Number ofComputationalOperations - Cubicpolynomialinterpolation

f o ra r b i t r a r yv a l u e s of k requiresfewercomputationaloperationsthancubic s p l i n e i n t e r p o l a t i o n u n d e r t h e f o l l o w i n g c o n d i t i o n s : o p t i o n s H3 and H5: i f t h e number of k i n t e r v a l s i s more t h a nt h r e e .

options H1 and H4: i f t h e number of k i n t e r v a l s is more t h a nf o u r .

o p t i o n H2: i f t h e number of k i n t e r v a l s is more t h a nf i v e .

Which o f t h e o p t i o n s H1 through H5 i s most e f f i c i e n t depends s t r o n g l y on thedimensionsofthematrices.These,inturn,depend on t h e d e s i r e d accu- racy, and t h e methods used for integrating or lumping the aerodynamic pressures a n d i n t e r p o l a t i n g and d i f f e r e n t i a t i n g t h e s t r u c t u r a l d i s p l a c e m e n t s .

Let M be t h e number of modes, N t h e number of s t r u c t u r a lc o o r d i n a t e s before modalizing, D t h e number of downwash c o l l o c a t i o np o i n t s and K t h e number o fi n t e g r a t i o np o i n t s . Then equation (5.12) canbeannotated as follows:

[Aij] = [ZIT [.I' P C ] p] [E]

(5.33) M,N N,K K,D D,N N,M From t h i se q u a t i o n it c a nb es e e nt h a t i f K and D a r e small compared w i t h N it becomes advantageous t o p e r f o r m t h e m u l t i p l i c a t i o n s [ZIT [ " I ' and [IT] [E] i n t h e f l u t t e r o p t i m i z a t i o n module. If K and D a r e e q u a l t o N o rl a r g e r ,t h e n it becomes advantageous t o form t h ep r o d u c t [HIT [AIC] [ I T ) o u t s i d et h ef l u t t e r module. The r e l a t i o n s h i p sd e f i n i n g when an o p t i o n i s b e t t e rt h a na n o t h e ra r ec o m p l i c a t e d . They a r e documented i n d e t a i l i n R e f e r e n c e 1, b u t t h e y h a v e n o t l e d t o s i m p l e c r i t e r i a .

5.6 Conclusions and Recommendations The p r e c e d i n g s e c t i o n s l e a d t o t h e f o l l o w i n g c o n c l u s i o n s : 1. Formulation of thegeneralizedaerodynamicforces i n t h e form i s p o s s i b l e a n d p r a c t i c a b l e f o r a l l approaches to determining unsteady aerodynamicforces.

2. S e v e r a l o p t i o n s o f i n p u t t i n g t h e m a t r i c e s [HI' , [AIC(k)] ' , and . I

[W( k)] = [DX] + ik[DZ] can be recognized, i .e. , s e p a r a t e l y , after m u l t i -

p l i c a t i o n , i n the form o f c u b i c s p l i n e m a t r i c e s , o r i n d e r i v a t i v e form.

Which o p t i o n i s most e f f i c i e n t dependsstrongly on s i z e s o f t h e m a t r i c e s , w h e t h e r t h e d e r i v a t i v e o f t h e aerodynamicsmatrix is needed, t h e method o fi n t e r p o l a t i o nf o ra r b i t r a r yk , the p o p u l a t i o no ft h em a t r i c e s [ H I and [W] a n d t h e number of k i n t e r v a l se x p e c t e dt o be a c t i v ed u r i n g t h e o p t i m i z a t i o n p r o c e s s .

3. I n a d d i t i o n t o d e p e n d i n g on t h e number o f c o m p u t a t i o n a l o p e r a t i o n s - I .

/ 1 . . , _ a i . ' . .

r e q u i r e dt o form . A. . ( k )t h ee f f i c i e n c yo ft h ef l u t t e ro p t i m i z a t i o n

[ l J 1

process may a l s o dependon t h e r e q u i r e d i n p u t s t o r a g e , t h e p o s s i b i l i t y o f s t o r i n g a l l m a t r i c e s r e q u i r e d f o r i n t e r p o l a t i o n o f t h e a e r o d y n a m i c s f o r k i n one i n t e r v a li nc o r e ,a n dt h er e a d - i nr e q u i r e d i f t h ev a l u e o f k moves t oa na d j a c e n ti n t e r v a l .

4 . It i s p o s s i b l et od e s i g n a f l u t t e ro p t i m i z a t i o n module t h a ti n c l u d e s a l l optionssuch that t h e u s e r c a n c h o o s e t h e o p t i o n t h a t i s most e f f i c i e n t , t h a t f i t s h i s a v a i l a b l e data or t h a t he p r e f e r s f o r some o t h e rr e a s o n .

I n viewof t h e aboveconclusionsthefollowingrecommendation is made: I n d e s i g n i n g a f l u t t e r o p t i m i z a t i o n module f o r a f a c i l i t y , the c a l c u l a t i o n o f t h e g e n e r a l i z e d a e r o d y n a m i c f o r c e c o e f f i c i e n t s a n d t h e i r derivatives should bebased on t h ef o r m u l a t i o n sp r e s e n t e di nt h i ss e c t i o n .T h a t i s , a mode- i n d e p e n d e n t p a r t s h o u l d b e g e n e r a t e d o u t s i d e t h e f l u t t e r o p t i m i z a t i o n module l e a v i n g an o f t e n t o be r e p e a t e d mode-dependent p a r t o f the c a l c u l a t i o n s t o be p e r f o r m e di n s i d et h ef l u t t e r module. The number ofoptions is largeand it may n o tb ep r a c t i c a b l et oi n c l u d e a l l o p t i o n s i n t h e f l u t t e r module. The choiceofoptions i s f a c i l i t yd e p e n d e n t .C e r t a i np r a c t i c e so fg e n e r a t i n g generalizedaerodynamicforcecoefficients may a l r e a d y e x i s t a n d t h e e x i s t i n g computer system may i n f l u e n c et h ec h o i c es i g n i f i c a n t l y . The module, however, shouldallow t h e u s e r c o n s i d e r a b l e freedom i n c h o o s i n g t h e o p t i o n t h a t i s most e f f i c i e n t f o r h i s problem. The number o fo p t i o n st o be includedshould be decided on t h e basis o f a s t a n d - a l o n ef l u t t e ro p t i m i z a t i o n module. It should not be r e s t r i c t e d b e c a u s e o f t h e module b e i n g p a r t o f a g e n e r a l a n a l y s i s s y s t e m which a t p r e s e n t has o n l y a r e s t r i c t e d c h o i c e o f o u t p u t t i n g a e r o d y n a m i c c o e f f i c i e n t s .

6. METHODS OF OPTIMIZATION FOR FLUTTER 6.1 General I n t h i s s e c t i o n f i v e a p p r o a c h e s t o s t r u c t u r a l o p t i m i z a t i o n w i t h f l u t t e r c o n s t r a i n t are reviewed. A l l five b e l o n g t o t h e c a t e g o r y . o f d i r e c t methods'; i.e. methods i n which t h e mathematica1,formulation d e f i n e s r e s i z i n g s t e p s aimed d i r e c t l y at determiningtheextremevalueoftheobjectivefunction, i nt h i sc a s em i n i m i z i n gt h es t r u c t u r a l mass. I n c o n t r a s t , i n t h e i n d i r e c t methods t h em a t h e m a t i c a lf o r m u l a t i o nd e f i n e sr e s i z i n gs t e p s aimed at satisf'y- i n g a c r i t e r i o n t h a t , when s a t i s f i e d , i n d i c a t e s t h a t t h e optimum c o n d i t i o n i s reached.

The f i v e methodsreviewedrepresentdistinctlydifferentmathematical formulations. They are, i nt h eo r d e r of review: the gradient methods of R u d i s i l l a n d Bhatia (Reference 14), t h e w e i g h t g r a d i e n t method o f Simodynes (Reference 151, a p e n a l t y f u n c t i o n method (Reference 1 6 ) , a method of f e a s i b l e d i r e c t i o n s ( R e f e r e n c e 17), and a method t h a t e v o l v e d *om t h e method

ofIncrementedFlutterAnalysis(References 4 and 18). A l l areformulated

undertheassumptionofmodalizationmatricesthat are independentofthe d e s i g n v a r i a b l e s , but can be updated at any r e s i z i n g s t e p .

I n a d d i t i o n t o t h e q u a l i t a t i v e e v a l u a t i o n o f t h e methodspresented i n t h i s s e c t i o n t h e r e i s a numericalevaluationandcomparisonin Appendix A .

I n t h a t a p p e n d i x t h e r e s u l t s are p r e s e n t e d o f a p p l y i n g t h e s e m e t h o d s t o a n o p t i m i z a t i o n t a s k i n which t h e f l u t t e r e q u a t i o n i s w r i t t e n i n terms of 49 discretedegreesoffreedomand i s notmodalized.

6.2 R u d i s i l l - B h a t i a Approach Reference 1 4 d e f i n e s f o u r d i f f e r e n t r e s i z i n g columns t h a t a l o n e or i n combination can be used to design a minimum weight, or near minimum weight, s t r u c t u r et h a ts a t i s f i e s a f l u t t e rs p e e dc o n s t r a i n t . The r e s i z i n g columns are d e f i n e di n terms o fi n c r e m e n t a lv a l u e s ,A P i ,o ft h ed e s i g nv a r i a b l e s 1 4 is followed.)Threeof them i n v o l v et h e Pi. (The notation of Reference g r a d i e n to ft h ef l u t t e rs p e e dw i t hr e s p e c tt ot h ed e s i g nv a r i a b l e s .I n con- n e c t i o nw i t ht h i s ,R e f e r e n c e 14 p r e s e n t s c l o s e d form a n a l y t i c a l e x p r e s s i o n s f o r t h e d e r i v a t i v e o f t h e f l u t t e r s p e e d w i t h r e s p e c t t o a d e s i g n v a r i a b l e .

Theseexpressionshavebeenusedsuccessfullyinnumerical t e s t casesper- formedduringthisstudy.Numericalvaluesofthederivatives are i n good agreementwithvaluesobtainedusingtheapproachofReference 15. I n t h e f o l l o w i n gt h er e s i z i n g columns d e f i n e di nR e f e r e n c e 1 4 are d i s c u s s e d . The terminologyofReference 14 i s followed.

6.2.1 VelocityGradientSearch - Equation (23) ofReference 1 4 d e f i n e s a

column ofdesign variable increments as: Equation (6.1) d e f i n e s a d i r e c t i o n b y means of of t h ei n c r e m e n t s i s d i r e c t l yr e l a t e dt ot h ev e l o c i t yi n c r e m e n t AV through AV , where A V = k P i ] i s the approximate t h e c o e f f i c i e n t i=l i n c r e a s e i n f l u t t e r s p e e d due t ot h ed e s i g nv a r i a b l ei n c r e m e n t sb p i ) .

The formulation i s known as t h e methodof s t e e p e s t a s c e n t o f t h e f u n c t i o n V(Pi). The d i r e c t i o ni m p l i e d by equation ( 6 . 1 ) d e f i n e s a column o fd i r e c t i o n a l cosines [z] for which [E] = [ Jw] dV is maximum.

and the Assuming a l i n e a r r e l a t i o n s h i p between the design variables pi a s s o c i a t e d mass, m = CiPi, equation ( 6 . 1 ) can be written as: i

av

Taking - as a r e f e r e n c e it c a nb es e e nt h a tt h ed i r e c t i o n of

a m i dependson t h e s c a l i n g betweenthedesignvariableandtheassociated mass.

S i n c e t h e q u a n t i t y o f d i r e c t i n t e r e s t is t o t a l mass, andnotdesignvariables r e l a t e d t o mass, it seems l o g i c a lt oc h o o s ee l e m e n t a r y mass as design variables and choose C i = l . I nt h a tc a s ee q u a t i o n ( 6 . 1 ) becomes: whichrepresents a d e s i g n c h a n g e i n t h e d i r e c t i o n o f maximum

J + +

The u s e f u l n e s s o f t h e v e l o c i t y g r a d i e n t s e a r c h is related t o its a b i l i t y t o raise t h e f l u t t e r speed more e f f i c i e n t l y , i.e., w i t h a smaller increase i n t o t a l mass m t h a n a s i m p l ei n c r e a s eo ft h eo v e r a l ls t i f f n e s s level. Simple i p h y s i c a l c o n s i d e r a t i o n s , h o w e v e r , l e a d t o t h e c o n c l u s i o n t h a t t h e most dV dV e f f i c i e n t move i s i n t h e d i r e c t i o n i n which - = - .is maximum.

dM zdm

i The n u m e r i c a l e v a l u a t i o n s i n Appendix A i n d i c a t e t h a t a r e s i z i n g column p r o p o r t i o n a lt o [EJ is a ne f f i c i e n t means o fr e s i z i n g a s t r u c t u r ei n one s t e p t o s a t i s f y a f l u t t e r s p e e d c o n s t r a i n t w i t h a moderate mass p e n a l t y , t h u s providing a good s t a r t i n g p o i n t f o r a p r o c e d u r e t h a t m i n i m i z e s t h e t o t a l mass at c o n s t a n t f l u t t e r s p e e d .

The r e l a t i v e e f f i c i e n c y o f a r e s i z i n g column p r o p o r t i o n a l t o

1E.J

f o l l o w s f r o m t h e f a c t t h a t it t e n d s t o a d d more material where it i s most e f f i c i e n t i n r a i s i n g t h e f l u t t e r s p e e d . Designvariablesforwhich - Bv i s ami negative are r e d u c e d i n v a l u e , which a l s o raises t h e f l u t t e r s p e e d .

6.2.2 Mass GradientSearch - Equation (30) ofReference 14 d e f i n e s a column

ofdesignvariableincrements: where M i s t h e t o t a l mass: M=Zm..

Equation ( 6 . 4 ) d e f i n e s a d i r e c t i o nb y means of [gj . The magnitudeof

- I theincrements i s d i r e c t l yr e l a t e dt ot h ev e l o c i t yi n c r e m e n t A V throughthe AV c o e f f i c i e n t The d i r e c t i o n d e f i n e d b y e q u a t i o n ( 6 . 4 ) c o r r e s p o n d s t o a maximum valueof Again assuming a l i n e a rr e l a t i o n between t h ed e s i g nv a r i a b l e s Pi and t h ea s s o c i a t e d mass m =C P it c a nb es e e nt h a t i i i

a M

and - - - ci .

api . . . I Thus equation ( 6 . 4 ) becomes: It i s a p p a r e n tt h a tt h ed i r e c t i o no f {APi} a g a i n depends on t h es c a l i n g between design variable and associated mass.

Choosing m as d e s i g n v a r i a b l e s e q u a t i o n (6.4) becomes: i which r e p r e s e n t s a uniformlydistributedweightincrement.

Reference 1 4 us'es equation ( 6 . 4 ) t o d e f i n e a d e c r e a s e i n t o t a l mass, thus a negative A V i s used. By following a p a t ho fs t e e p e s td e s c e n tf o r M(P.) the emphasis i s on d e c r e a s i n g t o t a l mass. The r e l a t i o n A M vs. A V i s notconsidered.

The mass g r a d i e n t s e a r c h c o u l d b e u s e d i n c o m b i n a t i o n w i t h t h e v e l o c i t y g r a d i e n ts e a r c ht of o r m u l a t ea no p t i m i z a t i o np r o c e d u r e .A l t e r n a t ea p p l i c a t i o n of t h e s e s e a r c h e s t e n d s t o l o w e r t h e d e s i g n v a r i a b l e w e i g h t r e q u i r e d f o r s a t i s f y i n g t h e f l u t t e r c o n s t r a i n t s i n c e t h e v e l o c i t y g r a d i e n t s e a r c h t e n d s t o i n c r e a s e t h e f l u t t e r s p e e d b y a d d i n g a r e l a t i v e l y e f f i c i e n t mass d i s t r i b u t i o n .

Although t h e mass g r a d i e n ts e a r c h removes mass i n d i s c r i m i n a t e l y ,r e p e a t e d a p p l i c a t i o n of b o t hs e a r c h e st e n d st o an optimum mass d i s t r i b u t i o n .R a t h e r t h a n remove mass i n d i s c r i m i n a t e l y , it seems l o g i c a l t o remove mass first where it reduces f l u t t e r s p e e d t h e least. That would be t h e c a s e i f most mass would

av

be removed where - i s smallest. Note t h a t i f t h e r e are d e s i g n variables

am, I a V - is n e g a t i v et h e i rr e d u c t i o ni n c r e a s e st h ef l u t t e rs p e e d , f o r which ami allowing more mass t o b e removedfrom design variables w i t h a small p o s i t i v e

v a l u eo f - . A n e f f i c i e n t method t o remove mass is t o remove it propor-

am,

1 av

t i o n a l l yt o - f o r d e s i g n v a r i a b l e s for which - is p o s i t i v e .

av/am, am, I I I n R e f e r e n c e 1 t h e s e q u e n t i a l a p p l i c a t i o n o f a v e l o c i t y g r a d i e n t s e a r c h and a mass g r a d i e n t s e a r c h i s r e p l a c e d b y t h e a p p l i c a t i o n of an e q u i v a l e n t two-component column ofdesignvariableincrements.Regroupingofthe ele- ments i n t h e twocomponentsshows c l e a r l y t h a t t h i s methoddoes result i n a

av

r e s i z i n g wheremore mass i s addedwhere -

i s l a r g e r andmore mass i s ami

av

removed where - i s smaller, r e g a r d l e s so ft h ea l g e b r a i cs i g n of - .

am, am, I I 6 . 2 . 3G r a d i e n tP r o j e c t i o nS e a r c h e s - The g r a d i e n tp r o j e c t i o ns e a r c hf o l l o w s a d i r e c t i o no fs t e e p e s ta s c e n tw h i l es a t i s f y i n g a c o n s t r a i n t . The g r a d i e n t p r o j e c t i o ns e a r c hu s e di nR e f e r e n c e 14 is aimed at f o l l o w i n g t h e s t e e p e s t a s c e n t o f t h e f l u t t e r s p e e d as a f u n c t i o n o f d e s i g n variables whilekeeping t h e t o t a l w e i g h t c o n s t a n t .

From equations(32)and(34)inReference 14 t h ef o l l o w i n g column o f

designvariableincrementscan be derived: n where AS2 = t h e s t e p s i z e i n terms of design variables.

x b p i ) , i=l Equation ( 6 . 8 ) c a n a l s o b e w r i t t e n as I n b o t h e q u a t i o n s

(6 . l o >

whichfollowsfromthecondition (6.11) Usually V(P.1 h a s a maximum f o r a g i v e nt o t a lw e i g h t . Although i n t h e a p p l i c a t i o n shown i n R e f e r e n c e 1 4 t h i s maximum is notsought, it may be ,reached. It would seem, t h e r e f o r e , that t h ef o r m u l a t i o n o f equation (6.8) with AS d e t e r m h i n g t h e s t e p s i z e , i s preferableovertheformulationofequa- t i o n (6.9) i n which A V d e t e r m i n e st h es t e p s i z e .

Reference 14 m e n t i o n s t h e p o s s i b i l i t y o f a g r a d i e n t p r o j e c t i o n s e a r c h i n which t h e f l u t t e r s p e e d i s heldconstantand t h e t o t a l w e i g h t i s reduced.

The p r e s e n t a u t h o r s c o n s i d e r t h i s a more s i g n i f i c a n t p r o c e d u r e from a prac- t i c a l p o i n t o f view. A s i n d i c a t e di nR e f e r e n c e 1 4 t h e f o r m u l a sf o rt h i s approachcan be obtained from the constant total weight approach by inter- changing the symbols V and M and changing the algebraic sign on AS.

S i n c et h et o t a l mass M has a minimum value,only t h e e q u i v a l e n to f equation (6.8) w i l l be given: (6.12) [APi} = - A S i=l where : L 1 (6.13)

A 1 = - -

[Ei

whichfollowsfromthecondition (6.14) Assuming a linear relat.ionship between the elementary masses, m and i' t h e d e s i g n v a r i a b l e s , Pi: m = CiPi (6.15) i equations(6.12)and(6.13). become:

- A S av

(6.16) + A 1 T i i=l

xl= -

= 1, i.e.,choosingtheelementary masses as L e t t i n g ,i na d d i t i o n , 'i d e s i g n v a r i a b l e s , t h e s e two equations become: (6.18) - A S i=l n

c:av

i=larni

A , = - , I i=l The s t e p s i z e i s determinedby n ( 6 . 2 0 )

!AS2 = x Ami

I i=l Equation (6.18) w i l l now be discussed under the assumption that i n e q u a l i t y i n d i c a t e s t h a t a uniformincrementof a l l t h e d e s i g n variables i n c r e a s e st h ef l u t t e rs p e e d .T h i s is n o t an unreasonable assumption. Addi- I I n view of t h e s e c o n s i d e r a t i o n s e q u a t i o n ,(.6.18) can be w r i t t e n as: (6.21) AS

- > 0 and Al am c a n b e l a r g e r o r smaller

where i i=l t h a n 1.

The r e s i z i n g column definedbyequation(6.21) is a l i n e a r c o m b i n a t i o n o f t h e columns a s s o c i a t e d w i t h t h e v e l o c i t y a n d mass g r a d i e n t s e a r c h d i s c u s s e d i n t h ep r e v i o u ss e c t i o n s . The a l g e b r a i c sum o ft h e two c o n t r i b u t i o n st e n d st o i n c r e a s et h ed e s i g n variables w i t h a l a r g e rv a l u eo f - a n d t o d e c r e a s e

ami

av

t h o s ew i t h a smaller v a l u eo f - i n c l u d i n g t h o s e w i t h a n e g a t i v e v a l u e o f

ami ’

av

. T h i sa g r e e sw i t hp h y s i c a lr e a s o n i n g . A s d e f i n e di ne q u a t i o n (6.18) t h e a m i two c o n t r i b u t i o n st o [Ami} i ne q u a t i o n( 6 . 2 1 ) are i n a f i x e dr a t i o . Due t o n o n l i n e a r i t i e s t h i s r e s u l t s i n a d r i f t i n f l u t t e r s p e e d due t o a r e s i z i n g s t e p bmi] .

By w r i t i n g (6.22) an e q u i v a l e n to fe q u a t i o n( 6 . 2 1 ) i s f o r m u l a t e d t h a t a l l o w s t h e u s e o f t h e

method ofIncrementedFlutterAnalysis(References 4 and 1) f o rd e t e r m i n i n g a

v a l u e o f s u c h t h a t t h e v e l o c i t y i n c r e m e n t , AV, a s s o c i a t e dw i t ht h ev e l o c i t y g r a d i e n t column i s e x a c t l yc a n c e l l e d .

I ne q u a t i o n (6.211, K l . d e t e r m i n e st h es t e ps i z e .I n a follow-on paper (Reference 19) R u d i s i l l a n d B h a t i a d e s c r i b e a d e t e r m i n i s t i c method of d e f i n i n g s t e p s i z e i n t h e case t h a t t h e g r a d i e n t p r o j e c t i o n s e a r c h i s used t o i n c r e a s e t h e f l u t t e r s p e e d a t c o n s t a n t t o t a l mass. The method i s basedon t h e a s s u m p t i o n t h a t t h e f l u t t e r s p e e d is a n e a r l y q u a d r a t i c f u n c t i o n o f t h e design variables.

S i n c e t h e t o t a l mass i s a l i n e a r f u n c t i o n o f t h e d e s i g n variables t h i s method o f d e t e r m i n i n g t h e s t e p s i z e is n o t a p p l i c a b l e t o t h e g r a d i e n t p r o - j e c t i o ns e a r c h at c o n s t a n tf l u t t e rs p e e d . Choosing a s t e p s i z e t h e n is a matterofexperienceandjudgment.

6.2.4Concluding Remarks - The p r e c e d i n gd i s c u s s i o nd e a l sw i t ht h eb a s i c r e s i z i n g columns as d e f i n e db yf o u rd i f f e r e n ta p p r o a c h e s .I n any p r a c t i c a l application,minimumsizeconstraintsmust be takenintoaccount.Conceptually t h i s i s a simple matter. Computer programming r e q u i r e sc l o s ea t t e n t i o nt o d e t a i l s i n c e a v a r i e t y o f p o t e n t i a l minimum s i z e c o n s t r a i n t s , some o f them e x p e c t e dt oo c c u ri n f r e q u e n t l y , must be foreseen.Reference 1 4 doesnotgo i n t o d e t a i l on t h i s .

The c a p a b i l i t y o f t h e approachespresentedinReference 1 4 f o r s t r u c t u r a l o p t i m i z a t i o n w i t h a f l u t t e r c o n s t r a i n t l i e s e x c l u s i v e l y i n t h e use o f t h e g r a d i e n t o f t h e f l u t t e r s p e e d w i t h r e s p e c t t o t h e d e s i g n variables. Repeated a d d i t i o n o f s t r u c t u r a l material a c c o r d i n g t o a d i s t r i b u t i o n p r o p o r t i o n a l t o 1 %] , followedeach time by a r a t h e ri n d i s c r i m i n a t er e m o v a lo f material t o m a i n t a i n t h e d e s i r e d f l u t t e r s p e e d , c o n v e r g e s t o a minimum mass design.

6 . 3 The WeightGradient Method of Simodynes I n March of 1973, E. E. Simodynes p r e s e n t e d a method f o r t h e o p t i m i z a - t i o no fs t r u c t u r a lw e i g h t at a s p e c i f i e df l u t t e rs p e e d( R e f e r e n c e 1 5 ) . The method employs t h eg r a d i e n to ft o t a lw e i g h tw i t hr e s p e c tt o n-2 design v a r i a b l e s i n a r e s i z i n g a l g o r i t h m t o m i n i m i z e s t r u c t u r a l w e i g h t w h i l e main- t a i n i n g a c o n s t a n tf l u t t e rs p e e d . Two o ft h ed e s i g nv a r i a b l e s are dependent v a r i a b l e s . The r e s i z i n g column i s such t h a t a c o n s t a n tf l u t t e rf r e q u e n c y i s a l s o m a i n t a i n e d , a c h a r a c t e r i s t i c which s i m p l i f i e s t h e f o r m a t i o n o f t h e r e q u i r e dd e r i v a t i v e s . The r e s t r i c t i o n on f l u t t e rf r e q u e n c y , however, repre- s e n t s a n a r b i t r a r y c o n s t r a i n t on theprocedurewhich may r e s u l t i n w e i g h t p e n a l t i e s w h i c h c a n n o t b e j u s t i f i e d b y t h e c o m p u t a t i o n a l s i m p l i f i c a t i o n s achieved. A m o d i f i c a t i o n o f t h e method i s p r o p o s e dw h e r e i nt h ef l u t t e r frequency i s p e r m i t t e d t o v a r y , t a k i n g t h e p l a c e o f o n e o f t h e d e p e n d e n t d e s i g nv a r i a b l e s .I nt h ef o l l o w i n g ,t h eo r i g i n a l method i s d i s c u s s e d f i r s t , followed by a discussion of the modified procedure.

6.3.1 The Method of Simodynes - The o p t i m i z a t i o n method p r e s e n t e di nR e f e r - ence 15 uses a r e s i z i n g a l g o r i t h m b a s e d on t h e g r a d i e n t o f t o t a l w e i g h t , s u b j e c t t o f l u t t e r speedand f l u t t e r f r e q u e n c yc o n s t r a i n t s . As w i t h most o p t i m i z a t i o n m e t h o d s , t h e s t i f f n e s s a n d i n e r t i a terms are e x p r e s s e d i n l i n e a r form: (6.23) The m a t r i c e s [KO] and Po] r e p r e s e n t t h e s t i f f n e s s a n d i n e r t i a o f t h e f i x e d s t r u c t u r e , a n d t h e m a t r i c e s b K i ] and [ . M i ] r e p r e s e n ts t i f f n e s s a n di n e r t i ai n c r e m e n t sp e ru n i tw e i g h to ft h e ith d e s i g nv a r i a b l e . The f l u t t e r e q u a t i o n f o r n e u t r a l s t a b i l i t y is (6.24) where [ Q ] i s a matrixofunsteadyaerodynamicforcecoefficients.

C o n s i d e r i n g f l u t t e r f r e q u e n c y a n d f l u t t e r v e l o c i t y t o be s p e c i f i e d , t h e f i x e d a n d v a r i a b l e c o e f f i c i e n t s o f t h e f l u t t e r e q u a t i o n may begrouped as i n e q u a t i o n s( 6 . 2 5 )a n d( 6 . 2 6 ) ,a n dt h ef l u t t e re q u a t i o ne x p r e s s e d as i n equation(6.27).

[.I = Po] - w 2 p o 1 - w 2 b ]

Fi] = [ . K i ] - w2[aMi] (6.26)

Two o f t h e d e s i g n v a r i a b l e s are now d e s i g n a t e d as t h e dependentdesign variables m and m a n d d e r i v a t i v e s o f t h e f l u t t e r e q u a t i o n w i t h r e s p e c t U V' t ot h ei n d e p e n d e n td e s i g nv a r i a b l e s rn formed as shown i ne q u a t i o n( 6 . 2 8 ) .

i (6.28) I nt h i se q u a t i o n , {q} is t h ef l u t t e re i g e n v e c t o r of equation(6.27), and LrJ i s t h e c o r r e s p o n d i n g row e i g e n v e c t o r . S i n c e t h e complex c o e f f i - cientsofequation(6.28) are known, t h ep a r t i a ld e r i v a t i v e s amu/ami and amv/ami are r e a d i l yo b t a i n e d . The w e i g h to ft h es t r u c t u r e is now elrpressed inequation(6.291,where Wo is t h ew e i g h to ft h ef i x e ds t r u c t u r e ;t h e d e r i v a t i v e o f t h e w e i g h t w i t h r e s p e c t t o e a c h i n d e p e n d e n t d e s i g n v a r i a b l e

is computed (equation (6.30) ) , and the gradient of total weight formed

( e q u a t i o n (6.31)).

n -2

W = W + m + m + Emi

(6.29) o u v i=l

aw am amv

-="- u + - + 1 (6.30) am, am, am, I I I A r e s i z i n g column ofincrements is thenformedfor a specificweightreduc- t i o n , -AW ( e q u a t i o n (6.3211, and new values of m and m are c a l c u l a t e d U v u s i n gt h i s column o fi n c r e m e n t sa n dt h ep a r t i a ld e r i v a t i v e s amu/ami and amv/ami.

(6.32) 6.3.2Discussion of t h e Method - A s i n d i c a t e d i n Reference 1 5 , t h e p r i n c i p a l f e a t u r e of t h e method o f Simodynes i s t h e w e i g h t g r a d i e n t , a n d t h e r e s i z i n g column d e r i v e d from it. The computationofthisweightgradient i s p a r t i c - u l a r l y s t r a i g h t f o r w a r d due t o t h e c o n s t r a i n t s on f l u t t e r s p e e d and f l u t t e r frequencyimposedontheresizingprocedure. I n p a r t i c u l a r ,t h eu n s t e a d y aerodynamicparameters are c o n s t a n tt h r o u g h o u tt h eo p t i m i z a t i o nc y c l e , so t h a t t h e f l u t t e r d e r i v a t i v e s do notinvolvederivativesoftheaerodynamic parameters.

T h i s s i m p l i f i c a t i o n , however, is n o to b t a i n e dw i t h o u ta na s s o c i a t e d disadvantage. The i m p o s i t i o no ft h ef r e q u e n c yc o n s t r a i n t on t h e optimizatfon p r o c e s s r e s u l t s i n a n a r t i f i c i a l c o n s t r a i n t on t h e d i s t r i b u t i o n o f t h e d e s i g n variable masses d u r i n gt h er e s i z i n gp r o c e d u r e . An exampleof t h i s may be s e e n i n t h e n u m e r i c a l e v a l u a t i o n s r e p o r t e d i n Appendix A . The e f f e c t o f t h i sc o n s t r a i n t is d i f f i c u l t t o assess, s i n c e it depends t o a g r e a td e g r e e : . .

on t h e p a r t i c u l a r s i t u a t i o n . For a c o n f i g u r a t i o n far from t h e f i n a l optimum d i s t r i b u t i o n , w i t h a f r e q u e n c y o f t h e c r i t i c a l mode s i g n i f i c a n t l y d i f f e r e n t .

from t h e f i n a l v a l u e , t h e e f f e c t o f the frequency constraint would be expected t o be s i g n i f i c a n t . For a c o n f i g u r a t i o nc l o s e rt o the optimum d i s t r i b u t i o n , w i t h a c r i t i c a l mode f r e q u e n c y a p p r o x i m a t e l y e q u a l t o t h a t o f t h e f i n a l con- f i g u r a t i o n ,t h ee f f e c tw o u l db ec o r r e s p o n d i n g l y less. Reference 1 5 states t h a t a f i x e d s e t of v i b r a t i o n modes i s used throughout a completeoptimization cycle,and it is c o n c l u d e dt h a tt h ef l u t t e rf r e q u e n c yc o n s t r a i n t i s maintained t h r o u g h o u tt h eo p t i m i z a t i o nc y c l ea l s o .I nt h a tc a s e , a t least o n ea d d i t i o n a l o p t i m i z a t i o n c y c l e m u s t b e p e r f o r m e d i n o r d e r t o assure that no e f f e c t s o f the c o n s t r a i n tr e m a i n . The choiceofdependentdesignvariablesalsoinfluences themagnitude of t h e c o n s t r a i n t e f f e c t , s i n c e d e s i g n v a r i a b l e s h a v i n g l i t t l e i n f l u e n c e o n f r e q u e n c y w o u l d p r o d u c e r e l a t i v e l y g r e a t e r d i s t o r t i o n s o f t h e u n c o n s t r a i n e d d i s t r i b u t i o n s .

A s w i t h o t h e r methodsemployingan a r b i t r a r y s t e p - s i z e ( e . g . Refer- ence 14), t h i s parameter must b e established on the basis of judgment, i n t u i t i o n ,e x p e r i e n c e or (probably) a combinationofthese. For t h ep r e s e n t method, s t e p - s i z e i s determined by t h e weightreduction, -AW, s p e c i f i e d by t h eu s e r . The c h o i c eo ft h i sp a r a m e t e ri n v o l v e s a compromise; t o ol a r g e a valueof A W c o u l dr e s u l ti nu n a c c e p t a b l yl a r g ee x c u r s i o n si nf l u t t e rs p e e d and f r e q u e n c y ;t o o small a v a l u e of AW would r e s u l ti na ne x c e s s i v e number o f r e s i z i n g s t e p s r e q u i r e d t o r e a c h optimum.

Minimum s i z e c o n s t r a i n t s a r e h a n d l e d i n a s t r a i g h t f o r w a r d manner; when a d e s i g nv a r i a b l er e a c h e s minimum s i z e , it i s temporarilyeliminated as a d e s i g nv a r i a b l e .I ns u b s e q u e n ts t e p s , t h e d e r i v a t i v e sa r ec a l c u l a t e df o r t h i s d e s i g n v a r i a b l e a n d it i s r e i n s t a t e d as a n a c t i v e d e s i g n variable i f t h e s e c a l c u l a t i o n s s o i n d i c a t e .A l t h o u g hn o ts p e c i f i c a l l ys t a t e di nt h er e f e r e n c e , t h i s is p r e s u m a b l yt r u eo ft h ed e p e n d e n td e s i g nv a r i a b l e s as w e l l as t h ei n d e - pendentdesign variables. Here a g a i n , a poorchoiceofdependentdesign v a r i a b l e s , r e q u i r i n g f r e q u e n t s h i f t i n g t o o t h e r d e s i g n v a r i a b l e s d u r i n g t h e o p t i m i z a t i o n c y c l e , w o u l d r e s u l t i n a n i n e f f i c i e n t r e s i z i n g p r o c e s s .

6.3.3 A M o d i f i c a t i o no ft h e Method ofSimodynes - A s d i s c u s s e d e a r l i e r , t h e c o n s t r a i n t on t h e f l u t t e r f r e q u e n c y p r o v i d e s a s i m p l i f i c a t i o n i n t h e computa- t i o n o f t h e r e q u i r e d d e r i v a t i v e s , b u t r e s u l t s i n a nonoptimum weightincre- ment implying a weightpenalty that is d i f f i c u l t t o p r e d i c t . A m o d i f i c a t i o n i s s u g g e s t e dw h i c he l i m i n a t e st h i sf r e q u e n c yc o n s t r a i n t .T h i s i s doneby a l l o w i n g t h e f l u t t e r frequency t o become a dependentvariable,taking the place of one of the dependent design variables. Following a procedure si-milarto thatof the original method, derivatives of the flutter equation are obtained after first expressing the flutter equation as a function of th reduced frequency, k. The result is equation (6.331, which is comparable to equation (6.28) of the original procedure. Equation (6.33) is solved for the' unknown parameters amu/ami and ak/ami; the total weight is expressed in equation (6.34), and the derivative of the weight with respect to each n-1

w = w + mu + E m i

i=l iables is shown in equation (6.35).

of the independent design var The gradient of the weight and the column of increments are formed as be- fore (equations (6.31) and (6.32)). Except for this modification in the weight gradient formation, the optimization cycle proceeds as in the original method.

6.3.4 Discussion of the Modified Method - The modification of Simodynes'

method suggested here unquestionably results in a less approximate procedure.

The frequency constraint, with the associated distortion of the resulting mass distribution, is removed. Not only does this eliminate an undesirable feature of the original method, but fewer optimization cycles should be required in order to reach a satisfactory approximation of the optimum dis- tribution. A s a result of the modification, however, some additional complica- tion of the computational procedure is required. In solving the flutter equation to obtain the eigenvectors, a fixed matrix of unsteady aerodynamic coefficients can no longer be used throughout the optimization cycle. In addition, the derivatives of the aerodynamic parameters with respect to the reduced frequency, k, must be obtained. How troublesome these complications are dependson t h e v e r s a t i l i t y o ft h ec o m p u t a t i o n a l system used. For t h e numerical e v a l u a t i o n o f t h e m o d i f i e d method p r e s e n t e d i n Appendix A, t h e Lockheed p-k method o fs o l v i n gt h ef l u t t e re q u a t i o n was u s e d ;t h i sp r o c e d u r e has a b u i l t - i n s u b r o u t i n e f o r i n t e r p o l a t i n g m a t r i c e s o f a e r o d y n a m i c c o e f f i - c i e n t s . A similar r o u t i n e was u s e dt oo b t a i nt h ea p p r o x i m a t e derivatives of the aerodynamicparametersusing a f i n i t e - d i f f e r e n c ep r o c e d u r e .S i n c et h e s e c o m p u t a t i o n a l t o o l s were r e a d i l y a v a i l a b l e , t h e m o d i f i c a t i o n s r e s u l t e d i n no p a r t i c u l a r c o m p u t a t i o n a l d i f f i c u l t y .

It should be n o t e d t h a t a c e r t a i n s i m i l a r i t y e x i s t s b e t w e e n t h e column of i n c r e m e n t s r e s u l t i n g from the present procedure and the,column of incre- ments r e s u l t i n g from t h e g r a d i e n t p r o j e c t i o n s e a r c h . w i t h , c g n s t a n t f l u t t e r s p e e do fR u d i s i l l - B h a t i a( S e c t i o n6 . 2 . 3 ) . The e x p r e s s i o nf o rt h e column o f incrementsofthepresentprocedure is shown inequation(6.36)andfrom e q u a t i o n ( A . 6 ) of Appendix A it i s seen that equation(6.37) is a ne q u i v a l e n t e x p r e s s i o n .

1 + -

" " " -

( 6 . 3 6 ) 1 -

av/am

U

" " _ " " " "

(6.37) R e c o g n i z i n g t h a t t h e s c a l a r p r e m u l t i p l i e r of t h e column i s a r b i t r a r i l y c h o s e n ,t h ei n c r e m e n t sf o r t h e independent design variables m can be made i i d e n t i c a l t o t h o s e o f e q u a t i o n ( 6 . 2 1 ) o f S e c t i o n 6 . 2 . 3 i f t h e normalization f a c t o r l/av/amu i s equal t o -A1 o f e q u a t i o n ( 6 . 2 1 ) . It w i l l b e r e c a l l e d t h a t AI i s chosen s o as t o r e s u l t i n a f l u t t e rv e l o c i t yi n c r e m e n t , on a l i n e a r basis, e q u a lt oz e r o .I ng e n e r a l , the n o r m a l i z a t i o nf a c t o rf o r the presentprocedure w i l l not be e q u a lt o -Al, s i n c e it r e s u l t s from t h e choice of t h ed e p e n d e n td e s i g nv a r i a b l e . The ( l i n e a r )f l u t t e rs p e e di n c r e m e n t i s t h e n h e l d t o z e r o b y t h e i n c r e m e n t i n t h e d e p e n d e n t d e s i g n v a r i a b l e d e f i n e d i n e q u a t i o n ( 6 . 3 7 ) .

6.3.5 Assessmentof t& Method - The o r i g i n a l Simodynesmethod is a simple, Y

straightforward procedure which involves a minimum o f c o m p u t a t i o n a l d i f f i c u l t y i n t h e g e n e r a t i o n o f t h e f l u t t e r derivatives. The f l u t t e rf r e q u e n c y con- s t r a i n t m i g h t be a . u s e f u l d e v i c e i n a d o p t i n g e x i s t i n g r o u t i n e s f o r t h e s o l u - t i o n o f t h e f l u t t e r e q u a t i o n a n d g e n e r a t i o n o f u n s t e a d y a e r o d y n a m i c p a r a m e t e r s f o r u s e i n a f l u t t e ro p t i m i z a t i o np r o c e d u r e .I nt h ed e v e l o p m e n to fa ni n t e - grateddesignprocedure,however,there wouldappear t o b e no c l e a r a d v a n t a g e i n r e t a i n i n g th'efrequencyconstraint.Instead, a m o d i f i c a t i o no ft h ep r o - ceduresuch as i n d i c a t e d h e r e would b e h l g h l y d e s i r a b l e .

6.4 An I n t e r i o rP e n a l t yF u n c t i o n Method I n s e e k i n g t o e v a l u a t e a p e n a l t y f u n c t i o n method, t h e i n i t i a l t a s k i s one o fd e f i n i n gb o t ht h e method i t s e l f and t h es c o p eo ft h ee v a l u a t i o n . A s u s e d h e r e , t h e term p e n a l t y f u n c t i o n method r e f e r s t o any s t r u c t u r a l o p t i m i z a - t i o n t e c h n i q u e i n whichpenaltyterms,which are r e l a t e d t o t h e c o n s t r a i n t e q u a t i o n s , are added t o a n o b j e c t i v e f u n c t i o n t o form a m o d i f i e d o b j e c t i v e f u n c t i o n , which i s thenminimized. For t h ep u r p o s e so ft h i se v a l u a t i o n , however, only a s i n g l e ,r e p r e s e n t a t i v ep r o c e d u r e w i l l beconsidered. The particularprocedurechosen i s a n i n t e r i o r p e n a l t y f i n c t i o n methodand follows c l o s e l y t h e method d e s c r i b e di nR e f e r e n c e 16, and i s e s s e n t i a l l y i d e n t i c a l t o t h a tu s e di nt h en u m e r i c &e v a l u a t i o n sp r e s e n t e di n Appendix A . A b r i e f d e s c r i p t i o n o f t h i s method i s p r e s e n t e d i n the n e x t s e c t i o n , p r e p a r a t o r y t o t h e d i s c u s s i o n and e v a l u a t i o n whichfollow.

t h e method considered 6 . 4 . 1 Description of Method - A s i n d i c a t e dp r e v i o u s l y , here is based on t h e method d e s c r i b e d i n R e f e r e n c e 1 6 . I nt h a tp r o c e d u r e , a m o d i f i e do b j e c t i v ef u n c t i o n ,P ( m i , r ) , i s formed as shown i ne q u a t i o n( 6 ; 3 8 ) .

The term W(mi) i s t h eq u a n t i t yt ob em i n i m i z e d , or o b j e c t i v ef u n c t i o n , a n dr e p r e s e n t st h ew e i g h ta s s o c i a t e d w i t h t h ed e s i g nv a r i a b l e s m . The i secondtermofequation(6.38)expressesthepenaltyterms as f u n c t i o n so ft h e d e s i g n c o n s t r a i n t s ; r i s a p e n a l t y f u n c t i o n w e i g h t i n g f a c t o r . By means o f thisformulation,theproblemofminimizing W ( m i ) , s u b j e c tt ot h ec o n s t r a i n t s g,(mi), i s t r a n s f o r m e dt oo n eo fa nu n c o n s t r a i n e dm i n i m i z a t i o no fP ( m i , r )

I

u s i n g a S W (Sequence of UnconstrainedMinimizationTechnique)approach (Reference 20 and 21). One suchminimization is c a r r i e do u tf o re a c hs u c c e s - sive r e d u c t i o no f t h e v a l u eo f the penaltyf'unctionweightingfactor, r, u n t i l t h e minimizedvalue of P ( m . , r ) is a p p r o x i m a t e l ye q u a lt ot h ec o r r e - spondingvalueof W ( mi .

The unconstrainedminimizationofP(m.,r) i s accomplished by first g e n e r a t i n g a move-vector d i r e c t i o n ,t h e nd e t e r m i n i n gt h e minimum of P(mi,r) i n t h i s d i r e c t i o n b y means o f a one-dimensionalsearch. For determining t h i s move d i r e c t i o n , several d i r e c t i o n - g e n e r a t i n ga l g o r i t h m s are available, t h e best known ofwhich i s t h e D F F algorithm(Reference 22) as u s e d i n t h e preliminarydesignprocedurereportedinReference 9 . Reference 16, how- e v e r , u s e s a v a r i a t i o n o f Newton'smethodwherein t h e s e c o n d derivatives of P(m.,r) are approximated,andthisprocedure w i l l be used here. The second d e r i v a t i v e s shown i n e q u a t i o n ( 6 . 3 9 ) may beapproximatedbyneglecting the second term o f t h e summation,under t h e a s s u m p t i o n s s t a t e d i n R e f e r e n c e 16. I na d d i t i o n , the o b j e c t i v ef u n c t i o n , W ( m i ) , i s assumed t o be a l i n e a r func;tionofthedesignvariables, so that the first term ofequation(6.39) i s e q u a lt oz e r o . The secondderivatives are thenapproximated as i n equa- t i o n ( 6 . 4 O ) , andan estimate f o r t h e column o fd e s i g nv a r i a b l e (6.40) increments which w i l l minimize P(mi,r) i s formed as shown i ne q u a t i o n (6.41).

(6.41) It should be n o t e d t h a t e q u a t i o n (6.41) would e x a c t l y d e f i n e t h e r e q u i r e d column o fd e s i g nv a r i a b l ei n c r e m e n t s i f P(m. ,r) were a q u a d r a t i cf u n c t i o n o ft h ed e s i g nv a r i a b l e sa n dt h ee x a c ts e c o n dd e r i v a t i v e su s e d .I nt h ep r e s e n t case,however,equation (6.41) i s u s e d o n l y t o d e f i n e t h e move d i r e c t i o n f o r theone-dimensionalsearch.

6.4.2Discussionofthe Method - I n e v a l u a t i n g t h e p e n a l t y f u n c t i o n method d e s c r i b e dh e r e ,f o u rp r i n c i p a le l e m e n t s or c h a r a c t e r i s t i c s c a n be discerned: t h e t r e a t m e n t o f t h e c o n s t r a i n t s , t h e p e n a l t y term w e i g h t i n g f a c t o r s , t h e directiongeneratingalgorithmandtheone-dimensionalsearch.Inthe f o l - lowingsections,eachof these c h a r a c t e r i s t i c s w i l l bediscussed,followed byanassessmentofthemethod as a completeprocedure.

The t r e a t m e n to fc o n s t r a i n t s i s t h e p r i n c i p a l d i s t i n g u i s h i n g feature of t h e p e n a l t y f u n c t i o n method,and t h i s t r e a t m e n t p r o v i d e s a number of advantages. A s u s e di nt h ep r e s e n tp r o c e d u r e ,t h ei n e q u a l i t yc o n s t r a i n t s s e r v e two i m p o r t a n tf u n c t i o n s : 1) conditioningof a move v e c t o rs u c ht h a t it t e n d s t o a v o i d v i o l a t i o n o f t h e c o n s t r a i n t s , and 2 ) l i m i t i n g o f t h e move v e c t o ra m p l i t u d es u c ht h a tt h ec o n s t r a i n t s are n o tv i o l a t e d . The first o f t h e s ef u n c t i o n s i s implementedthroughthedirectiongeneratingprocedure, and t h e secondofthese i s a p a r t o f the one-dimensionalsearch.

The t r e a t m e n to f t h e c o n s t r a i n t s as i n e q u a l i t y c o n s t r a i n t s , r e s u l t i n g i n t h e c h a r a c t e r i s t i c s d e s c r i b e d a b o v e , c a n b e a verypowerfulapproachin s t r u c t u r a lo p t i m i z a t i o nf o rf l u t t e r . One of t h e more obviousadvantages is t h e f a c t t h a t t h e i n c l u s i o n o f m u l t i p l e f l u t t e r s p e e d c o n s t r a i n t s c a u s e s l i t t l e c o n c e p t u a ld i f f i c u l t y . Whether t h i sa d v a n t a g e is o fp r a c t i c a lv a l u e i s d i f f i c u l t t o assess w i t h o u tf u r t h e r work. To i n c l u d e a l l f l u t t e rs p e e d c o n s t r a i n t s (for s e v e r a l Mach numbersand a i r p l a n e l o a d i n g c o n d i t i o n s ) i n t h ep e n a l t yt e r m would b e a largecomputationalburden. Thus it would seem t h a t o n l y a c t i v e c o n s t r a i n t s s h o u l d b e i n c l u d e d i n t h e p e n a l t y term and t h a t s e p a r a t e program logicshouldbeusedtodeterminewhichconstraints are a c t i v e .E x p e r i e n c e a t theLockheed-California Company, p a r t l yo b t a i n e dd u r i n g t h i s s t u d y , i n d i c a t e s t h a t m u l t i p l e a c t i v e f l u t t e r s p e e d c o n s t r a i n t s may occur r a r e l y .P r o b a b l yt h e mostimportantadvantageresulting from t h i s method of h a n d l i n gc o n s t r a i n t s i s t h e f a c t t h a t a l a r g e number and v a r i e t y o f c o n s t r a i n t s canreadilybeincludedinanautomatedprocedure. A disadvantage,however, i s t h e f a c t t h a t d e r i v a t i v e s o f t h e c o n s t r a i n t q u a n t i t i e s w i t h r e s p e c t t o t h e d e s i g n v a r i a b l e s must b e o b t a i n e d .

The h a n d l i n g o f t h e p e n a l t y term w e i g h t i n g f a c t o r s i n d i c a t e d i n equa- t i o n (6.38) e x e r t s a s i g n i f i c a n t i n f l u e n c e on t h e r e s u l t a n tp e r f o r m a n c eo f t h e method. The t r e a t m e n to ft h e s ef a c t o r s is a matter ofjudgmentand depends on experience.Experience a t NASA, LangleyResearchCentersuggests an i n i t i a l v a l u e f o r t h e s e w e i g h t i n g f a c t o r s which makes thepenaltyterms approximatelyequal. t o t h e v a l u e of t h e o b j e c t i v e f u n c t i o n (Refereme 2 3 ) .

The f i n a l v a l u e f o r these f a c t o r s c a n b e d e t e r m i n e d b y e s t a b l i s h i n g a c c e p t a b l e values of t h e r e s i d u a l c o n s t r a i n t i n e q u a l i t i e s a n d t h e n s p e c i f y i n g a n a l l o w a b l e percentage of the modified objective function contributed by the penalty terms.

Having t h u s e s t a b l i s h e d t h e r a n g e o f t h e w e i g h t i n g f a c t o r s , t h e a p p r o p r i a t e r e d u c t i o n f a c t o r t o b e a p p l i e d d u r i n g e a c h s t e p is determinedfromtheselec- t i o n o f t h e d e s i r e d number o fs t e p s .M o n i t o r i n go ft h ep r o g r e s so ft h ew e i g h t minimization,however, may l e a d t h e a n a l y s t t o i n t e r r u p t t h e o p t i m i z a t i o n p r o c e d u r ea n dm o d i mt h ep e n a l t yw e i g h t i n gf a c t o r s .S p e c i f i c a l l y ,o p t i m i z a - t i o n s t e p s may continue after t h e number o f s t e p s on which t h e r e d u c t i o n f a c t o r is based are completed i f t h e t o t a l w e i g h t p r o g r e s s i o n i n d i c a t e s t h a t a minimum w e i g h t h a s n o t b e e n a t t a i n e d .

A t e a c h s t e p i n t h e o p t i m i z a t i o n p r o c e s s , t h e w e i g h t i n g f a c t o r s d e t e r - mine t h ee x t e n tt ow h i c ht h ec o n s t r a i n t si n f l u e n c et h er e s u l t a n t move. If t h e w e i g h t i n g f a c t o r s are l a r g e r e l a t i v e t o t h e d i s t a n c e from t h e c o n s t r a i n t , t h e d e s i g n t e n d s t o move away from t h e c o n s t r a i n t , i n t o t h e f e a s i b l e d e s i g n space. If t h ew e i g h t i n gf a c t o r s are small, t h ec o n s t r a i n t se x e r t l i t t l e i n f l u e n c e on t h e move and t h e d e s i g n may move i n t h e d i r e c t i o n o f t h e con- s t r a i n t s . The behavior i s dependent on t h er e d u c t i o nf a c t o ra p p l i e dt ot h e w e i g h t i n gf a c t o r s a t eachsuccessivestep,which i s i n t u r n r e l a t e d t o t h e r a n g eo ft h ew e i g h t i n gf a c t o r sa n dt h e number o fs t e p ss e l e c t e d . A l a r g e number o fs t e p s (or a small r e d u c t i o n f a c t o r ) c a n r e s u l t i n e r r a t i c b e h a v i o r o f t h e p r o c e s s d u e t o t h e s t r o n g l y r e p e l l i n g i n f l u e n c e o f t h e c o n s t r a i n t s .

Too few s t e p s ,w i t ht h ec o r r e s p o n d i n gl a r g er e d u c t i o nf a c t o r s , may r e s u l t i n moves whichimpactone or more c o n s t r a i n t s b e f o r e s i g n i f i c a n t w e i g h t r e d u c - tions have been accomplished.

The u s eo ft h ea p p r o x i m a t es e c o n dd e r i v a t i v e si n an adaptationof Newton'smethod r e s u l t s i n a more efficient unconstrained minimization pro- cedurethandoestheuseofthe DFF' a l g o r i t h m( R e f e r e n c e2 2 ) .S i n c et h i s latter p r o c e d u r er e q u i r e s a number ofone-dimensionalsearchesapproximately e q u a l t o t h e number ofdesignvariables,andthe number ofsuchsearches r e q u i r e d w i t h Newton'smethod i s independentofthe number ofdesignvari- a b l e s ,t h ea d v a n t a g eo f Newton'smethod i n c r e a s e s as t h e number of design v a r i a b l e si n c r e a s e s . A s i n d i c a t e di ne q u a t i o n ( 6 . 4 1 ) however, t h e m a t r i x G must b ei n v e r t e d ;t h i sm a t r i x i s an n x n matrix where n i s t h e number of designvariables.Reference 1 6 s t a t e st h a tt h i sm a t r i x i s s i n g u l a r or i l l - c o n d i t i o n e d when t h e number o fa c t i v ec o n s t r a i n t s i s small. To preclude - t h eo b v i o u sd i f f i c u l t i e s whichwouldotherwiseresult, a m a t r i x G i s used i np l a c eo ft h em a t r i x G, theelementsofwhich are shown i ne q u a t i o n( 6 . 4 2 ) I n t h i s e q u a t i o n , i s t h e Kronecker d e l t a and a v a l u e 6i j (6.42) of E=O.O1 i s found t ob es a t i s f a c t o r yf o rt h es y s t e m se v a l u a t e dt h u sf a r .

For e a c h s t e p i n t h e o p t i m i z a t i o n p r o c e s s , a one-dimensionalsearch is conducted t od e t e r m i n et h e minimum o ft h em o d i f i e do b j e c t i v ef u n c t i o n . The d i r e c t i o n o f t h e move v e c t o r i s determined as previouslydescribed(equa- t i o n ~( 6 . 4 1 ) ) , and only the magnitude of the move v e c t o r i s v a r i e d d u r i n g t h es e a r c h . The derivativesofthemodifiedobjectivefunctionneedonly be evaluated once during each step, but the modified objective function i s e v a l u a t e d f o r a number of v a l u e s o f t h e move v e c t o r m a g n i t u d e s u f f i c i e n t t o d e f i n e t h e minimum o ft h ef u n c t i o n .T h e s ee v a l u a t i o n so ft h em o d i f i e do b j e c - t i v e f ' u n c t i o nr e q u i r ef l u t t e rs p e e ds o l u t i o n s , stress analysesand/orother p r o c e d u r e sa p p r o p r i a t et ot h ed e t e r m i n a t i o no ft h ep e n a l t y terms. Although t h e p r o c e s s o f e v a l u a t i n g t h e a c c e p t a b i l i t y o f a d e s i g n . r e l a t i v e t o t h e con- s t r a i n t s i s common t o a l l optimizationmethods,thepenaltyfunction method, employing a one-dimensionalsearch,requires at least t h r e es u c he v a l u a t i o n s perstep,whereasmethodswhich do not employ theone-dimensionalsearchneed onlyonesuchevaluation.Theseothermethods,however,normallyrequire a g r e a t e r number o f s t e p s t o a c h i e v e a n a c c e p t a b l e l e v e l o f o p t i m i z a t i o n , a n d t h e r e f o r e u s u a l l y r e q u i r e a l a r g e r number o f f l u t t e r s p e e d d e r i v a t i v e d e t e r - m i n a t i o n s .C o n s i d e r i n gt h a td e t e r m i n i n gt h ef l u t t e rs p e e dd e r i v a t i v e s r e q u i r e s d e t e r m i n a t i o n o f t h e f l u t t e r r o o t a n d two c h a r a c t e r i s t i c v e c t o r s , it is e s t i m a t e d t h a t i n terms o fc o m p u t a t i o n a lo p e r a t i o n s( a n dt h u sc o s t ) each s t e p i n t h e p e n a l t y f u n c t i o n p r o c e d u r e , r e q u i r i n g t h e c a l c u l a t i o n o f t h r e e f l u t t e r r o o t s , i s a p p r o x i m a t e l ye q u i v a l e n tt o three s t e p so f a pro- ceduresuch as t h e m o d i f i e d Simodynesmethod d e s c r i b e d i n S e c t i o n 6.3.

T h i s t r a d e - o f f m u s t b e t a k e n i n t o a c c o u n t i n anycomparativeevaluationof methods,andmustbedeterminedfor a r e a l i s t i cd e s i g np r o c e d u r e . The principaladvantageoftheone-dimensionalsearch i s that it provides a s p e c i f i c c r i t e r i o n f o r t h e d e t e r m i n a t i o n o f s t e p s i z e , r e s u l t i n g ( u s u a l l y ) i n g r e a t e r w e i g h t r e d u c t i o n s p e r s t e p t h a n o b t a i n e d w i t h methodsemploying arbitrary s t e p s i z e . The determinationofstepsizebythepresentprocedure canreadilybeimplemented as a f u l l y automatedcomputationalsubroutine.

6.4.3 Assessmentof t h e Method - Based on t h e d e s c r i p t i o n of t h e o p t i m i z a t i o n method p r e s e n t e d i n S e c t i o n 6 . 4 . 1 and t h e d i s c u s s i o n o f t h e c h a r a c t e r i s t i c s of t h e method i n S e c t i o n 6 . 4 . 2 , it i s c o n c l u d e dt h a tt h ep e n a l t yf u n c t i o n method (as t h a t t e r m i s u s e d h e r e ) i s a n e f f i c i e n t o p t i m i z a t i o n p r o c e d u r e p o s s e s s i n gc h a r a c t e r i s t i c sn o tf o u n di no t h e rm e t h o d s . The unique means o f h a n d l i n gd e s i g nc o n s t r a i n t s i s , of c o u r s e ,t h e mostnoteworthyofthese.

T h i s f e a t u r e , a l o n g w i t h t h e u s e o f t h e o n e - d i m e n s i o n a l s e a r c h t o d e t e r m i n e s t e p s i z e , r e s u l t s i n a n o p t i m i z a t i o n p r o c e d u r e w h i c h i s p a r t i c u l a r l y w e l l s u i t e d t o u s e i n a completelyautomatedroutine.Detailedspecifications f o rs u c h a p r o c e d u r es h o u l db er e l a t i v e l ye a s yt od e v e l o p . The p r i n c i p a l e l e m e n t r e q u i r e d f o r t h e u s e o f t h i s method,overandabovetherequirements o fo t h e r methods i n v e s t i g a t e d , i s t h ed e t e r m i n a t i o no ft h ed e r i v a t i v e so ft h e c o n s t r a i n t q u a n t i t i e s ( o t h e r t h a n f l u t t e r d e r i v a t i v e s ) w i t h r e s p e c t t o t h e d e s i g nv a r i a b l e s .S i n c e it s h o u l da l w a y sb ep o s s i b l et oe v a l u a t et h e con- s t r a i n t f u n c t i o n s f o r s p e c i f i e d v a l u e s o f t h e d e s i g n v a r i a b l e s , t h e s e d e r i v a - tives may b e o b t a i n e d b y f i n i t e d i f f e r e n c e t e c h n i q u e s i f no b e t t e r method i s a v a i l a b l e .I n terms of c o m p u t a t i o n a le f f i c i e n c y , it i s d i f f i c u l t t c compare t h e p e n a l t y f u n c t i o n method w i t h t h e o t h e r f o u r methodsconsideredhere.

From t h ef o r e g o i n g it i s c l e a r t h a t t h e p e n a l t y f u n c t i o n methoddoes, i n g e n e r a l , r e q u i r e more computationsperstepthan do t h e Simodynes o r R u d i s i l l - Bhatiaprocedures.Whether or n o tt h e more e f f i c i e n to p t . i m i z a t i o ns t e po ft h e p e n a l t y f u n c t i o n methodovercomes t h a t d i s a d v a n t a g e c a n o n l y b e a s s e s s e d i n terms of a r e a l i s t i c d e s i g n c a s e .

6.5 A Method o fF e a s i b l eD i r e c t i o n s The method o f f e a s i b l e d i r e c t i o n s is anapproach t o s t r u c t u r a l o p t i - m i z a t i o nu s i n gd i r e c tm i n i m i z a t i o no f a c o n s t r a i n e df u n c t i o n . This i s i n c o n t r a s t t o p e n a l t y f u n c t i o n methods, t r e a t e d i n S e c t i o n 6.4, whichconvert t h e c o n s t r a i n e d d e s i g n p r o b l e m i n t o a sequence of unconstrained minimizations of a modified objective function.

The method d i s c u s s e d h e r e i s based primarily on t h e method of Gwin and McIntosh(Reference 17), which is i n t u r n a g e n e r a l i z a t i o n o f a method developedbyZoutendijk(Reference24).Additionbackground material i s derivedfromVanderplaatsand Moses i nR e f e r e n c e 25. It should be n o t e dt h a t t h e d i s c u s s i o n p r e s e n t e d h e r e i s l i m i t e d t o t h o s e c h a r a c t e r i s t i c s i n h e r e n t t o t h e f e a s i b l e d i r e c t i o n s method i t s e l f ; o t h e r p a r t i c u l a r s o f t h e method p r e s e n t e d i n Reference 17 a r en o tt r e a t e d . Thus,suchelements as t h e method ofgeneratingaerodynamicparameters,solutionoftheflutterequationand c o m p u t a t i o n o f f l u t t e r s p e e d d e r i v a t i v e s are c o n s i d e r e d t o b e s e p a r a t e from t h e method o ff e a s i b l ed i r e c t i o n s . Theseandotherdetailsof a complete o p t i m i z a t i o np r o c e d u r ea r ec o n s i d e r e de l s e w h e r ei nt h i sr e p o r t .

6.5.1 Description of Method - The method of f e a s i b l ed i r e c t i o n sg e n e r a t e s a sequenceofdesignchanges,eachofwhich i s b o t h f e a s i b l e ( d o e s n o t v i o l a t e a c t i v ec o n s t r a i n t s ) a n du s a b l e( r e d u c e st o t a l mass). Each r e s i z i n gd i r e c t i o n i s f o l l o w e du n t i l a new c o n s t r a i n t i s v i o l a t e d , a n a c t i v e c o n s t r a i n t is re-encountered or t h e t o t a l mass i s minimized. The p r o c e s sc a nb ev i s u a l i z e d w i t ht h eh e l po fF i g u r e 6-1, which reproducesFigure 6 o f Reference 17. I n t h i s f i g u r e , t h e minimum s i z e c o n s t r a i n t s are r e p r e s e n t e d b y t h e h o r i z o n t a l and v e r t i c a l b o u n d a r i e s , t h e minimum f l u t t e r s p e e d c o n s t r a i n t b y t h e c u r v e d boundaryand theconstantweightcontours by t h e d i a g o n a l s t r a i g h t l i n e s .

The s t a r t i n g p o i n t f o r t h e p r o c e s s i s at p o i n t A, which i s on t h e f l u t t e r s p e e d c o n s t r a i n t b o u n d a r y ; t h e d e s i g n p r o c e e d s i n a d i r e c t i o n away from t h e f l u t t e r s p e e d c o n s t r a i n t a n d i n a d i r e c t i o n o f d e c r e a s i n g w e i g h t u n t i l a con- s t r a i n t boundary i s encountered. A t t h a tp o i n t , a new d i r e c t i o n i s generated and a new move executed.Thisprocess i s c o n t i n u e du n t i l a p o i n t B i s reached which approximates an optimum design.Figure 6-2, a l s o fromReference 17, i l l u s t r a t e s t h e r a n g e o f a c c e p t a b l e d e s i g n d i r e c t i o n s s t a r t i n g from a p o i n t B on a g e n e r a l n o n l i n e a r c o n s t r a i n t b o u n d a r y .

For a c o n s t r a i n t e q u a t i o n of t h e formgiven i n e q u a t i o n (6.43), thecondi- t i o n t h a t t h e d i r e c t i o n i s f e a s i b l e i s givenbyequation ( 6 . 4 4 ) , where (Vh} is t h e g r a d i e n t v e c t o r 'of t h e c o n s t r a i n t w i t h r e s p e c t t o t h e d e s i g n variables, m and {S} i s t h e d i r e c t i o n v e c t o r o f t h e d e s i g n variables.

i' h I O (6.43) LS) (Vh} 5 0 (6.44) W = Constant

"2

Figure 6-1: HypotheticalDesignSpace c ? ?

F i g u r e 6-2: Direction-Finding Problem a t a ConstraintBoundary I n t h e p r e s e n t c o n t e x t , a f e a s i b l e d i r e c t i o n is onewhichdoesnot v i o l a t e a n a c t i v e c o n s t r a i n t , as defined by a l i n e a r a p p r o x i m a t i o n o f t h a t c o n s t r a i n t . The l i n e segment BC i nF i g u r e 6-2 l i e s on theboundaryofthe feasible r e g i o n . The c o n d i t i o nt h a t the d i r e c t i o n is u s a b l e is givenby equation ( 6 . 4 5 ) , where {VW} is t h eg r a d i e n tv e c t o ro ft h eo b j e c t i v ef u n c - t i o n , i . e . , t h e t o t a l s t r u c t u r a l w e i g h t . Any d i r e c t i o nb e t w e e nl i n e s BD and BC i n F i g u r e 6-2 is t h e n i n t h e u s a b l e - f e a s i b l e s e c t o r .

The p a r t i c u l a r d i r e c t z o n v e c t o r u s e d i n the. p r e s e n t method i s fbundsuch t h a t a s c a l a r p i s maximized s u b j e c tt ot h ec o n d i t i o n se x p r e s s e di n equa- t i o n( 6 . 4 6 ) .I nt h i s set o fe q u a t i o n s , 8 i s anadjustmentfactorwhich (6.46) ( c ) The l e n g t ho f L S J i s bounded.

c o n t r o l st h ed i r e c t i o no f L S ) w i t h i nt h eu s a b l e - f e a s i b l er e g i o n ;l a r g e values of 8 f o r c et h ed i r e c t i o n away from the constraintandtoward t h e usableboundary BD, w h i l e a valueof 8 = 0 r e s u l t s i n a move d i r e c t i o na l o n g BC. F o rn o n l i n e a rc o n s t r a i n t s ,s u c h as a minimum f l u t t e rs p e e d con- l i n e s t r a i n t ,a ni n t e r m e d i a t ev a l u eo f 8 i s usedwhichapproximatelybisectsthe usable f e a s i b l es e c t o r .R e f e r e n c e 17 suggests a value of 8 = 1 . 0 . It w i l l be shown, however, that t h e e f f e c to f any p a r t i c u l a rv a l u eo f 8 depends on t h en o r m a l i z a t i o no ft h ec o n s t r a i n tg r a d i e n t s and t h ew e i g h tg r a d i e n t . For l i n e a rc o n s t r a i n t s ,s u c h as minimum s i z e c o n s t r a i n t s , a valueof 8 = 0 i s u s e d i n o r d e r t o p r o d u c e a move d i r e c t i o n p a r a l l e l t o t h e c o n s t r a i n t b o u n d a r y .

The c o n d i t i o nt h a tt h el e n g t h of L S ] is bounded i s usuallyaccomplishedby l i m i t i n gt h ee l e m e n t s as shown i n e q u a t i o n ( 6 . 4 7 ) .

Once t h ev a l u eo f 8 i s s e l e c t e d ,t h es u b o p t i m i z a t i o np r o b l e mi n d i c a t e d by equations (6.46) i s t r a n s f o r m e d t o a s t a n d a r d formand solvedby means of theSimplexalgorithm. Some o ft h e details of t h i s procedure are g i v e ni n Reference 17, and a more comprehensivedescription may be found i n R e f e r e n c e 26. The d e t a i l so ft h eS i m p l e xa l g o r i t h m w i l l n o tb er e p e a t e d here, b u t some a s p e c t so ft h ep r o c e d u r e are d i s c u s s e di nt h ef o l l o w i n gs e c t i o n . It should be n o t e d t h a t t h e S i m p l e x a l g o r i t h m w a s u s e d t o g e n e r a t e t h e d i r e c t i o n v e c t o r s for t h e n u m e r i c a l e v a l u a t i o n s p r e s e n t e d i n Appendix A o f t h i s r e p o r t .

I n a p p l y i n g t h e c o n s t r a i n t c o n d i t i o n s i n d i c a t e d b y e q u a t i o n ( 6 . 4 6 ( a ) ) , o n l y t h o s e c o n s t r a i n t s w h i c h are c o n s i d e r e d t o be a c t i v e at a p a r t i c u l a r d e s i g np o i n t are i n c l u d e di nt h ed i r e c t i o n - f i n d i n gp r o c e s s . A c o n s t r a i n t i s c o n s i d e r e d t o be a c t i v e i f t h e d e s i g n i n q u e s t i o n falls w i t h i n a s p e c i f i e d tolerance band c o f t h e c o n s t r a i n t b o u n d a r y . T h i s c o n s t r a i n t t o l e r a n c e band is a r b i t r a r i l y s p e c i f i e d and may v a r yw i t ht y p eo fc o n s t r a i n t . A s s t a t e d earlier, a n y p a r t i c u l a r d e s i g n s t e p c o n t i n u e s i n t h e s p e c i f i e d d i r e c - t i o n u n t i l a c o n s t r a i n t i s v i o l a t e d ; at t h a t p o i n t , a c o r r e c t i o n s t e p i s t a k e n b a c k i n t o t h e c o n s t r a i n t t o l e r a n c e b a n d a n d a new d i r e c t i o n g e n e r a t e d .

6.5.2 P r i n c i p a lC h a r a c t e r i s t i c so ft h e Method - The p r i n c i p a ld i s t i n g u i s h i n g feature o ft h e method o f f e a s i b l e d i r e c t i o n s i s , as t h e name i m p l i e s ,t h e d i r e c t i o ng e n e r a t i n gp r o c e d u r e . Some aspectsofthisprocedure,andofthe methodof e s t a b l i s h i n g t h e s t e p s i z e , are d i s c u s s e d i n t h e f o l l o w i n g .

Some u n c e r t a i n t y e x i s t s as t o t h e h a n d l i n g o f t h e c o n s t r a i n t t o l e r a n c e band, E . Reference 1 7 i n d i c a t e st h a t an a p p r o p r i a t ec o n s t r a i n tt o l e r a n c e is e s t a b l i s h e d f o r e a c h c o n s t r a i n t or c o n s t r a i n t t y p e , a n d t h a t when a con- s t r a i n t i s v i o l a t e d , t h e s t e p s i z e i s c o r r e c t e ds u c ht h a tt h ee n d - p o i n to f t h e s t e p i s midway i nt h et o l e r a n c eb a n d .R e f e r e n c e 25, however, recommends t h e u s e of a l a r g e t o l e r a n c e b a n d f o r t h e i n i t i a l p h a s e s o f t h e o p t i m i z a t i o n s o t h a t t h e c o n s t r a i n t s w i l l become a c t i v e e a r l y i n t h e o p t i m i z a t i o n p r o - cedureand will r e m a i na c t i v e . Combining these two approacheswouldnot a p p e a r t o r e s u l t i n a n e f f i c i e n t r e s i z i n g p r o c e d u r e , s i n c e a l a r g e c o n s t r a i n t tolerancebandwouldprevent a closeapproachtoanyconstraint.Forthe i d e a l i z e d t e s t c a s e o f Appendix A, no p a r t i c u l a r a d v a n t a g e c a n b e d i s c e r n e d i n m a i n t a i n i n g a l a r g e "pad"on t h e c o n s t r a i n t s ; t h i s wouldalmostcertainly r e s u l t i n a n i n c r e a s e d number o f r e s i z i n g s t e p s f o r t h e w e i g h t r e d u c t i o n s shown i n S e c t i o n A . 6 . It may b et h a ts u c h a procedurewouldbeusefulin t h e a p p l i c a t i o n o f t h e method t o more complex designproblems as a means of avoidingconvergence t o l o c a l i z e d minima, b u t it i s f e l t t h a ts u c hu s e f u l n e s s would b e rare i n p r a c t i c a l s i t u a t i o n s . Based on t h e i d e a l i z e d t e s t c a s e , it w o u l d a p p e a r t h a t t h e c o n s t r a i n t t o l e r a n c e f o r minimum s i z e c o n s t r a i n t s s h o u l d b ee q u a lt oz e r o . A minimum s i z e c o n s t r a i n t s h o u l d n o t b e a c t i v e u n t i l t h e minimum s i z e i s r e a c h e d ,s i n c eo t h e r w i s e no f u r t h e r r e d u c t i o n i n t h a t d e s i g n v a r i a b l e would t a k ep l a c ew h i l et h ec o n s t r a i n tr e m a i n e da c t i v e . For minimum f l u t t e r s p e e d c o n s t r a i n t s , it would seem t h a t t h e c o n s t r a i n t t o l e r a n c e b a n d f o rd e f i n i n ga na c t i v ec o n s t r a i n ts h o u l di n d e e db ef a i r l yl a r g e . The f l u t t e r s p e e dc o n s t r a i n t st h e n become a c t i v e e a r l y i n t h e d e s i g n p r o c e s s , a n d are t h e r e f o r ee f f e c t i v ei nt h ee f f i c i e n tr e s i z i n go ft h ed e s i g nv a r i a b l e s . The c o n s t r a i n t t o l e r a n c e b a n d f o r d e t e r m i n i n g t h e e n d - p o i n t o f a p a r t i c u l a r d e s i g n s t e p , h o w e v e r ,s h o u l db ee s s e n t i a l l ye q u a lt oz e r o .

Assuming a r e a s o n a b l ev a l u eo ft h ea d j u s t m e n tf a c t o r , 8 , t h en e x t d e s i g n s t e p w i l l b e d i r e c t e d w e l l i n t o t h e f e a s i b l e r e g i o n , s o t h a t no u s e f u l purpose i s s e r v e d b y o r i g i n a t i n g t h e s t e p anyappreciabledistancefromthe f l u t t e r c o n s t r a i n t b o u n d a r y .

For o t h e rt y p e so fc o n s t r a i n t ,t h et r e a t m e n to ft h ec o n s t r a i n tt o l e r a n c e bandcanbebased on similar reasoning.

As i n d i c a t e d earlier, t h e recommended a d j u s t m e n t f a c t o r f o r a minimum s i z ec o n s t r a i n t or o t h e rl i n e a rc o n s t r a i n t is 8 = 0. Thischoiceofadjust- ment f a c t o r results i n s u b s e q u e n t d e s i g n s t e p s p r o c e e d i n g p a r a l l e l t o t h e constraint boundary i f the d e s i g n s t e p s w o u l d o t h e r w i s e r e s u l t i n c o n t i n u e d r e d u c t i o n so ft h ec o n s t r a i n e dd e s i g nv a r i a b l e . For n o n l i n e a rc o n s t r a i n t s such as f l u t t e rs p e e dc o n s t r a i n t s , a nonzerovalueof 8 must be used i n o r d e r t o f o r c e t h e d e s i g n d i r e c t i o n away from t h e c o n s t r a i n t b o u n d a r y .

O t h e r w i s e , a n y f i n i t e move a m p l i t u d e w o u l d v i o l a t e t h e f l u t t e r s p e e d con- "his c h a r a c t e r i s t i co f the parameter 8 r e s u l t si nt h e term s t r a i n t .

"push-off" f a c t o rb e i n ga p p l i e dt o it i n some r e f e r e n c e s . The most e f f i c i e n t value of t h e "push-off"factordependsontheparticularstructuralopti- mizationunderconsideration,but a valueof 8='1.0 i s usuallychosen.

Reference 17 states t h a t t h i s v a l u e p r o d u c e s a v e c t o r d i r e c t i o n w h i c h a p p r o x i m a t e l yb i s e c t st h eu s a b l e - f e a s i b l es e c t o r .E x a m i n a t i o no fe q u a t i o n s (6.46) demonstrates, however, that the v a l u e 8 which accomplishes t h i s is dependentonthenormalizationof (Vh} and {VW}. A l a r g e rv a l u eo ft h e n o r m a l i z e dc o n s t r a i n tg r a d i e n tc o r r e s p o n d st o a smaller v a l u eo f f 3 , w h i l e a l a r g e r v a l u e o f t h e n o r m a l i z e d w e i g h t g r a d i e n t c o r r e s p o n d s t o a l a r g e r valueof 8 . I nt h en u m e r i c a le v a l u a t i o n sd e s c r i b e di n Appendix A, t h e weightgradient w a s a u n i t column as a consequence of t h e c h o i c e o f d e s i g n v a r i a b l e s , a n d t h e f l u t t e r s p e e d c o n s t r a i n t g r a d i e n t w a s normalizedsuchthat t h ev a l u eo f t h e average element i s u n i t y . A valueof 8 = 1.0, i n conjunc- t i o n w i t h t h a t normalization,producedapproximately the d e s i r e d result. The more usualprocedureofnormalizingthegradients on t h e l a r g e s t e l e m e n t would haveproducedapproximatelytwice as much "push-off".

It is notedthatVanderplaatsand Moses, Reference 2 5 , recommend a variable"push-off"factorwhich is a functionof the distancefromthe con- s t r a i n tb o u n d a r y .I nt h ep r e s e n tn o t a t i o n ,t h i s i s e x p r e s s e di ne q u a t i o n t h (6.48) , where E i s t h ec o n s t r a i n tt o l e r a n c e , hk i s t h ev a l u eo ft h e j j c o n s t r a i n t f u n c t i o n at t h e kth d e s i g n s t e p , and eo i s chosen as u n i t y .

(6.48) S i n c e t h e c o n s t r a i n t f u n c t i o n s h a v e n e g a t i v e v a l u e s anywhere w i t h i n t h e feasible r e g i o n , t h e e f f e c t o f t h i s treatmentofthe"push-off"factorswould be t o d r i v e t h e d e s i g n t o t h e c o n s t r a i n t t o l e r a n c e b o u n d a r y , w h e r e t h e v a l u e of eJ i s e q u a lt oz e r o .A l t h o u g ht h i sa p p r o a c hh a st h ea d v a n t a g eo fp r o - v i d i n g a uniformtreatment of t h e "push-off"factors, it would a p p e a r t h a t t h e r e s u l t a n t e x c u r s i o n s o f t h e d e s i g n i n a n d o u t o f t h e c o n s t r a i n t t o l e r a n c e bandmight w e l l o f f s e t any advantage derived from this approach.

The d i r e c t i o n v e c t o r f o r feasible d i r e c t i o n sp r o c e d u r e s is u s u a l l y o b t a i n e d by theSimplexmethod. A s n o t e di nR e f e r e n c e 25, " t h e r e s u l t i n g d i r e c t i o n t e n d s t o p o i n t t o w a r d s t h e c o r n e r s of a hypercube i n t h e d e s i g n space (Si = 1 or -l)." The d i r e c t i o nv e c t o r so b t a i n e di nt h en u m e r i c a l evaluationsofAppendix A c e r t a i n l ye x h i b i tt h ei n d i c a t e dt e n d e n c i e s . Refer- ence 25 goeson t o p r o p o s e a n a l t e r n a t e d i r e c t i o n v e c t o r f o r m u l a t i o n , b a s e d onimposingboundson t h e t o t a l v e c t o r r a t h e r t h a n t h e i n d i v i d u a l e l e m e n t s .

Although t h i s f o r m u l a t i o n h a s n o t b e e n e v a l u a t e d i n d e t a i l , it is presumed t h a t t h e r e s u l t i n g v e c t o r m i g h t b e somewhat more e f f i c i e n t s i n c e it i s s u b j e c t t o fewer c o n s t r a i n t s .

R e t u r n i n g t o t h e o r i g i n a l f o r m u l a t i o n , some elementaryconsiderations ofequations (6.46) andequation (6.47) w i l l i n d i c a t e t h e r e a s o n s f o r t h e r e s u l t i n g d i r e c t i o n v e c t o r c h a r a c t e r i s t i c s , a n d w i l l suggest a method o f g e n e r a t i n g t h e d i r e c t i o n v e c t o r w i t h o u t t h e use of the Simplex algorithm.

R e f e r r i n g t o e q u a t i o n s (6.46) , assume for t h e moment t h a t o n l y t h e f l u t t e r c o n s t r a i n t i s a c t i v e . It w i l l b es e e nt h a t maximum p canonlyoccur when bothequation ( 6 . 4 6 ( a ) ) a n de q u a t i o n( 6 . 4 6 ( b ) ) are e q u a lt oz e r o . If now t h e v a l u e o f 0 i s t a k e n as u n i t y , e q u a t i o n ( 6 . 4 9 ) r e s u l t s when p i s maximum.

R e t u r n i n gt oe q u a t i o n s (6.46) andobservingthattheelementsof {Vh) are g e n e r a l l yn e g a t i v e , it i s n o t d i f f i c u l t t o see t h a t t h e a d d i t i o n o f w e i g h t t o t h e d e s i g n v a r i a b l e h a v i n g t h e g r e a t e s t i n c r e a s e i n f l u t t e r s p e e d p e r pound,andremovingweightfromthedesignvariablehavingthe least i n c r e a s e i nf l u t t e rs p e e dp e r pound, t e n d st oi n c r e a s e p . The r e s t r i c t i o n on t h e s i z e o f t h e e l e m e n t s e x p r e s s e d i n e q u a t i o n (6.47) limits these elements t o +1 and -1 , r e s p e c t i v e l y , w h i l e t h e o b j e c t i v e of maximizing p i n s u r e s t h a tt h e s e limits w i l l be reached.Continuingthe same l i n eo fr e a s o n i n g , it w i l l b es e e nt h a tt h er e s u l t i n gd i r e c t i o nv e c t o r must have +1 elementsfor t h ed e s i g n variables w i t ht h eh i g h e rf l u t t e rs p e e dd e r i v a t i v e s , -1 elements f o r t h e d e s i g n v a r i a b l e s w i t h t h e l o w e r f l u t t e r s p e e d d e r i v a t i v e s , a n d o n e e l e m e n to fi n t e r m e d i a t ev a l u ei no r d e rt os a t i s f ' ye q u a t i o n ( 6 . 4 9 ) . If one or more d e s i g n v a r i a b l e s i n t h e n e g a t i v e g r o u p is l i m i t e d b y a n a c t i v e minimum s i z i n g c o n s t r a i n t , t h e c o r r e s p o n d i n g e l e m e n t s i n t h e d i r e c t i o n v e c t o r are s e t t oz e r oa n de q u a t i o n (6.49) balanced as b e f o r e . It should be n o t e dt h a tt h e ranking of t h e f l u t t e r d e r i v a t i v e s mustbe on t h e b a s i s o f r a t e ofchangeof f l u t t e r s p e e d p e r pound o f d e s i g n v a r i a b l e , which is t h e r a t i o o f t h e e l e m e n t s

of {Vh} t ot h ee l e m e n t so f {VW} . During the course of the numerical

e v a l u a t i o n s p r e s e n t e d i n Appendix A, it was f o u n d t h a t a move v e c t o r d i r e c t i o n i d e n t i c a l t o t h a t g e n e r a t e d b y t h e Simplexmethodcould be d e r i v e d b y t h e proceduresdescribed. When t h e r e are two a c t i v ef l u t t e rc o n s t r a i n t s ,

av, av, v1

and V , , t h ev a l u e so f -+ - determine which elements of I S ] are

a m , am, +1 and -1 ; two equations similar t o (6.49) w i l l determine two elements of LSJ t h a t h a v e i n t e r m e d i a t e v a l u e s . This w a s also demonstrated, numer- i c a l l y , t o l e a d t o t h e same I S ] as theonegeneratedbytheSimplexmethod.

It i s expectedthisapproachcan be expanded t o more a c t i v e f l u t t e r con- s t r a i n t s . It i s s u g g e s t e dt h a t it is more s t r a i g h t f o r w a r da n d may b e a more economical means o f g e n e r a t i n g t h e d i r e c t i o n v e c t o r .

Several methods of step-size selection are proposed i n R e f e r e n c e 17, thesimplestofwhich i s t o c o n t i n u e t o i n c r e a s e s t e p s i z e i n t h e p r e s c r i b e d d i r e c t i o n u n t i l a c o n s t r a i n tv i o l a t i o no c c u r s . A t t h i s p o i n t , a c o r r e c t e d d i s t a n c e t o a p o i n t w i t h i n t h e c o n s t r a i n t t o l e r a n c e b a n d is f o u n d b y l i n e a r i n t e r p o l a t i o n o f t h e a p p r o p r i a t e c o n s t r a i n t f ’ u n c t i o n v a l u e s .

A s mentioned earlier, it should be r e l a t i v e l y s i m p l e t o d e t e r m i n e t h e s t e p s i z e which would terminate exactly on the boundary of the nearest linear c o n s t r a i n t , a n d i n v i e w o f t h e f a c t t h a t no f u r t h e r r e d u c t i o n o f t h a t design v a r i a b l e w i l l t a k e p l a c e w h i l e t h e c o n s t r a i n t is a c t i v e , no c o r r e c t i o n i n t o thetolerancebandwouldappeartoberequired. A more directproceduremight t h e n be t o d e t e r m i n e t h e s t e p - s i z e t o t h e n e a r e s t l i n e a r c o n s t r a i n t a n d t h e n check f o r v i o l a t i o n of o t h e rc o n s t r a i n t su s i n g that s t e p - s i z e . If c o n s t r a i n t v i o l a t i o n s r e s u l t e d , t h e s t e p - s i z e w o u l d b e r e d u c e d u n t i l t h e c r i t i c a l con- s t r a i n t was j u s t s a t i s f i e d . For minimum f l u t t e rs p e e dc o n s t r a i n t s ,t h eu s e o f some f o r m o f I n c r e m e n t e d F l u t t e r A n a l y s i s t o s o l v e f o r the s t e p - s i z e n e c e s s a r y t o s a t i s f y t h e f l u t t e r c o n s t r a i n t s h o u l d r e s u l t i n a s u b s t a n t i a l improvement i n t h e s t e p - s i z e s e a r c h p r o c e d u r e .

6.5.3 Assessmentofthe Method - The methodof f e a s i b l ed i r e c t i o n s i s similar

t o t h e p e n a l t y f u n c t i o n methods i n t h a t t h e c o n s t r a i n tf u n c t i o n si n f l u e n c e b o t ht h ed i r e c t i o no ft h ed e s i g ns t e pa n dt h es t e p - s i z e .F u l la u t o m a t i o no f t h e method i s documented i nR e f e r e n c e 27. A wide r a n g eo fc o n s t r a i n tt y p e s canbe accommodated, the only requirement being t h a t it must b e p o s s i b l e t o e v a l u a t e b o t h t h e c o n s t r a i n t f u n c t i o n s a n d c o n s t r a i n t g r a d i e n t s f o r e a c h d e s i g ns t e p .I nc o n c e p t ,m u l t i p l ef l u t t e rc o n s t r a i n t sc a n be included, a l t h o u g h t h e r e may b e p r a c t i c a l d i f f i c u l t i e s t o b e overcome o u t s i d e the o p t i m i z a t i o n p r o p e r .

The move d i r e c t i o n v e c t o r r e s u l t i n g *om t h e u s u a l formoftheequations a p p e a r s t o be r a t h e rc r u d e ,a n d it is f e l t t h a ta na l t e r n a t ep r o c e d u r e ,s u c h as s u g g e s t e d i n R e f e r e n c e 25, might r e s u l t i n a more e f f i c i e n t move v e c t o r .

The t r e a t m e n t o f c o n s t r a i n t t o l e r a n c e s a n d d e t e r m i n a t i o n o f s t e p - s i z e seem overlycomplicated,andsignificantimprovementsinthese areas s h o u l d b e p o s s i b l e .

Overall, t h e method seems t o b e q u i t e r e p r e s e n t a t i v e o f t h e d i r e c t methods of f l u t t e r o p t i m i z a t i o n a n d c o m p e t i t i v e w i t h o t h e r methodsevaluated.

The n u m e r i c a l e v a l u a t i o n s r e p o r t e d i n Appendix A show t h e method t o be much better behaved than would be i n d i c a t e d b y c o n s i d e r a t i o n o f t h e move v e c t o r , a n d t h e rate of convergence demonstrated on the simplified test c a s e is q u i t e s a t i s f a c t o r y . A s a r e s u l t , t h e method of feasible d i r e c t i o n s must be con- s i d e r e d a s t r o n g c a n d i d a t e f o r f u r t h e r e v a l u a t i o n i n a r e a l i s t i c d e s i g n environment.

6.6 An Optimization Method UsirgIncrementedFlutterAnalysis IncrementedFlutterAnalysis w a s conceived as a method f o r e f f i c i e n t l y d e t e r m i n i n g d e s i g n p a r a m e t e r s f o r e x t e r n a l s t o r e s s a t i s f y i n g a predetermined f l u t t e rs p e e dr e q u i r e m e n t( R e f e r e n c e 4). Its c a p a b i l i t yo fd e t e r m i n i n gt h e v a l u e o f g e n e r a l d e s i g n v a r i a b l e s s u c h t h a t t h e f l u t t e r s p e e d h a s a given v a l u e makes it attractive as a t o o l i n o p t i n i z a t i o n w i t h f l u t t e r c o n s t r a i n t s .

A s t u d y was t h e r e f o r e i n i t i a t e d t o assess theusefulnessofIncremented F l u t t e r A n a l y s i s as a t o o li na no p t i m i z a t i o np r o c e d u r e . It was u s e d i n a h e a v i l y i n t e r a c t i v e a p p r o ' a c h t o o p t i m i z a t i o n f o r f l u t t e r , u s i n g t h e Computer Graphicssystem. The r e s u l t i n g method o fo p t i m i z a t i o n w a s demonstratedon a simulated design problem based on a subsonic transport wing (Appendix A, S e c t i o n A . 7 ) and was a l s o u s e d on anactualdesignproblem(arrowwingsuper- s o n i c t r a n s p o r t ) .

During t h i s d e v e l o p m e n t , t h e method o f IncrementedFlutterAnalysis was g e n e r a l i z e d t o be c o n s i s t e n t w i t h t h e n e e d s i n a complex optimizationprogram (Reference 1).

A breadboardprototypeofanautomatedcomputerprogram was developed and demonstrated on the same simulateddesignproblem.

I n t h e f o l l o w i n g , t h e main features oftheprogramand i t s presentform are presented based on d a t a i n R e f e r e n c e 18.

6.6.1 Main Features - The o p t i m i z a t i o n method i s a r e s i z i n g r o u t i n e ' t h a t m i n i m i z e s t o t a l mass w h i l e m a i n t a i n i n g t h e f l u t t e r s p e e d e x a c t l y at a r e q u i r e d value.

Key features o f t h e method a r e t h a t t h e r e s i z i n g column i s a l l o w e d t o c h a n g e d i r e c t i o n , w i t h o u t t h e n e e d t o r e c a l c u l a t e f l u t t e r s p e e d d e r i v a t i v e s , during a one-dimensional minimization process in which the value of a s c a l a r a ! i s d e t e r m i n e dt h a tm i n i m i z e st h et o t a l mass.

I n t h e methods of o p t i m i z a t i o n d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n s , a column of designvariableincrements(resizingcolumn) k m ! ] is d e f i n e d as :

(Am:} = ak [ dk]

k where [dk] d e f i n e s a d i r e c t i o ni nd e s i g nv a r i a b l es p a c ea n dt h es c a l a r (y a magnitude. The r e s i z e dd e s i g n is related t o t h e o r i g i n a l d e s i g n b y I n t h i s method t h e r e s i z i n g column is (Am:} = ( d k ( a k ) } i .e. , t h ed i r e c t i o n of k m : ] i s a functionof t h e s c a l a r CY. The s c a l a r c \ !

is d i s c u s s e di nS e c t i o n 6.6.2. Duringoneresizingcycle, i . e . , f o ro n e v a l u eo ft h es u p e r s c r i p t k , one set o fd e r i v a t i v e so ft h ef l u t t e rs p e e d

av

w i t hr e s p e c tt ot h ed e s i g nv a r i a b l e s , - i s used. The column m a t r i x

a m i ’

av

{dk(ak)] i s a f u n c t i o n o f - as w e l l as s i z i n gc o n s t r a i n t s .D u r i n g the ami one-dimensionalminimization o f the t o t a l mass w i t h CY as a v a r i a b l e , the method ofIncrementedFlutterAnalysis is u s e d t o m a i n t a i n t h e f l u t t e r s p e e d e x a c t l y at t h e desired v a l u e .

6.6.2Present Form of Program - The r e s i z i n g column P m i } is d e f i n e d as

t h e sum o f a b a s i cr e s i z i n g column I C i ] andanadjustment column 6 I The elements o f (Ci} s a t i s f yt h ee q u a t i o n andthus would c o r r e s p o n dt o a zerochange i n f l u t t e r s p e e d i f V were a l i n e a r f u n c t i o n o f t h e m i ' s . The s c a l a r 6 of the adjustment column 6@ij i s determined such that . .

[Ami} = {Ci} + (Ai} r e s u l t s e x a c t l y i n a zerochange i n f l u t t e r s p e e d .

av

I nt h ef o l l o w i n gt h en o t a t i o n - = i s used..

ami 'mi

The d e s i g nv a r i a b l e s are d i v i d e di n an R group and a Q group, s u c h t h a t The division between the R and Q group l i e sw i t h i nt h ep o s i t i v er a n g e of Vmi . Thus a l l d e s i g nv a r i a b l e sf o r which V . < 0 are i nt h e R group.

m l The l a r g e s t Vmi i s i d e n t i f i e db y i = m and the smallest p o s i t i v e 'mi by i = s p . The d i v i s i o n b e t w e e n t h e R and Q groups i s d e f i n e d by t h e i n e q u a l i t y

v /Vmi - 1

mm 2 VR vm/vmsp - 1 where VR i s a n e m p i r i c a l v a l u e . It was found, by numerical experimenta- t i o n ,t h a t VR = 0.3 i s a na c c e p t a b l ev a l u e for t h e t e s t c a s er e p o r t e di n Appendix A. For i = m t h e l e f t hand s i d ee q u a l sz e r o ; f o r i = s p - it equals 1. Thus t h ed e s i g nv a r i a b l e sf o r which t h ei n e q u a l i t y (6.56) is satisfied b e l o n gt ot h e R group.

The a l g o r i t h m i n R e f e r e n c e 18 is based on removing mass fromeachdesign v a r i a b l ei nt h e R g r o u pt h a t i s not at minimum s i z e . Each mass removal is individuallycoupledwith a changeof a l l d e s i g nv a r i a b l e si nt h e Q group suchthatequation ( 6 . 5 3 ) i s s a t i s f i e d . Mass removalfromadesignvariable i n t h e R group for which VmiER > 0 r e q u i r e sa d d i t i o no f mass i nt h e Q groupe If < 0 mass removal i n t h e R group is compensated by mass removal i n t h e Q group i no r d e rt os a t i s f ye q u a t i o n( 6 . 5 3 ) .

The column m a t r i x { C r ) } is t h e change i nt h ed e s i g nv a r i a b l e si n t h e Q group due t o removal of mass fromdesignvariable r .in t h e . R

. group. In Reference 18 t h ed i s t r i b u t i o n

> 0 ) and is used i f mass i s t ob e added t ot h e Q group ( C m x 0 and 'miER

v

cr

mm

A=

V ' m mq i f mass is t ob es u b t r a c t e d from t h e Q group ( C m C 0 and VmiER < 0 ) .

Equation ( 6 . 5 7 ) e x p r e s s e st h a t more is added t ot h ed e s i g nv a r i a b l e sw i t h t h eh i g h e rv a l u e so f Vmi. Equation (6.58) e x p r e s s e st h a t more is s u b t r a c t e d from t h ed e s i g nv a r i a b l e s w i t h thelowervalues of Vmi.

The d i s t r i b u t i o nf o r removal of mass i n t h e R group used i n Reference 18 is : \

cr - -

The f o r e g o i n g l e a d s t o a b a s i c r e s i z i n g column t h a t i s t h e s u m of two columns t h a t do not"overlap" and t h u s c a n b e w r i t t e n a s onecolumn (6.60) The column m a t r i x given by equation ( 6.59 ; {cq) by:

[cr } i s

where n i s t h e number of design var!.ables i nt h e Q group.

Equations ( 6 . 5 9 ) and ( 6 . 6 1 ) d e f i n et h eb a s i cr e s i z i n g column I C i } w i t h t h e s c a l a r C* d e f i n i n g a magnitude.

If t h ev a l u eo f C* is s u c ht h a te q u a t i o n (6.59) l e a d st ot h ev i o l a t i o n of minimum s i z e c o n s t r a i n t s , t h a t e q u a t i o n i s onlyusedfortheelements that do n o tv i o l a t et h es i z ec o n s t r a i n t s . The otherelements are g i v e nv a l u e s c o r r e s p o n d i n g t o t h e minimum s i z e c o n s t r a i n t s .

V i o l a t i o no f a minimum s i z ec o n s t r a i n tb yd e s i g nv a r i a b l e si nt h e Q group i s e x p e c t e dt ob ei n f r e q u e n t . The program,however,hasprovisions t o g u a r d a g a i n s t s u c h v i o l a t i o n .

The d i s t r i b u t i o n of the adjustment column 6 A i s d i s c u s s e d i n

I il

Reference 18. I nt h en u m e r i c a l example i n Appendix A , A = 0 except a t i aV i = m (maximum - ) . It i s suggested, however, that Ai = Ci f o r iEQ a m .

1 .

and Ai = 0 f o r i E R i s e x p e c t e dt ob e a b e t t e rc h o i c e .

The v a l u eo ft h es c a l a r 6 i s determinedby means o f IncrementedFlutter Analysis. The d e t e r m i n a n t a lf l u t t e re q u a t i o n( a c c o r d i n gt oe q u a t i o n ( 3 . 4 ) ) i s w r i t t e n as: k

D 1 ( 7 + i )k,g,V,p,mi+Ci,6Ai = 0 ( 6 . 6 2 )

I

k I n e q u a t i o n ( 6 . 6 2 ) , Y = 0 , g = 0, V and p have given values; m i are t h ev a l u e so ft h ed e s i g nv a r i a b l e s a t t h eb e g i n n i n go ft h ec u r r e n t r e s i z i n g s t e p ; s a t i s f i e s e q u a t i o n ( 6 . 5 3 ) . E q u a t i o n ( 6 . 6 2 ) is solved by

two-dimensional Regula F a l s if o r k and 6 . The complete resizing column i s

thendeterminedbyequation ( 6 . 5 4 ) .

If no s i z e l i m i t a t i o n s become a c t i v e , s i m p l e s u b s t i t u t i o n o f equa-' t i o n (6.59) i n t oe q u a t i o n (6.61) shows t h a t a l l e l e m e n t so ft h er e s i z i n g

column [ Ci] are p r o p o r t i o n a l t o C*. D i f f e r e n t values of C* can be

assumed; pm:} = [Ci} + 6 [ A d can be computedand M = b ] [ m t + A m : ] canbe

computed as a functionof C*. This i s , i np r i n c i p l e ,t h e one dimensional m i n i m i z a t i o n t h a t d e t e r m i n e s t h e d i r e c t i o n as w e l l as t h e m a g n i t u d e o f t h e r e s i z i n g column. The t o t a l mass M does have a minimum due t on o n l i n e a r e f f e c t s , which are t a k e n i n t o a c c o u n t b y means of Incremented Flutter Analysis.

I ~ . J . .

When s i z el i m i t a t i o n s are a c t i v en o t a l l elements of [ci] are propor-

* t i o n a lt o C* and nonalgebraic operations are needed t oo b t a i n M as a functionof C*. The l o g i cf o rt h i s is p r e s e n t e di nR e f e r e n c e 18.

To f a c i l i t a t e i n i t i a t i o n o f t h e onedimensionalminimization, a v a r i a b l e a = - g cr i s used. It i s t h e t o t a l mass removed from d e s i g nv a r i a b l e si n t h e R group and has a v e r y s i m p l e r e l a t i o n t o C*. The q u a n t i t y , a ! h a s a simplephysicalmeaning,independent of t h e number o fd e s i g nv a r i a b l e s . An i n i t i a l v a l u e c a n b e c h o s e n as a f r a c t i o n o f t h e t o t a l mass representedby t h e d e s i g n v a r i a b l e s .

The numericalexampleofReference 18 i s p r e s e n t e d i n t h e Appendix as Table A - 1 1 . It s u g g e s t st h a tt h e method i s v e r yp o w e r f u li nr e d u c i n gt h e t o t a l mass by a l a r g ef r a c t i o n ( > 8 0 % ) ofthedifferencebetweenthecurrent t o t a l mass andthe minimum mass i n a s i n g l e s t e p . * Although it i s n o t e s s e n t i a l t o t h e o v e r a l l method, it should be noted ,?* t h a t t h e program as d e s c r i b e d i n R e f e r e n c e 18 u s e s t h e method of Incremented F l u t t e rA n a l y s i st od e t e r m i n et h ev a l u e so f aV/ami by means of a f i n i t e - .

differenceapproach(Reference 1).

6.6.3 Concluding Remarks - The a p p r o a c ht a k e ni nt h i s method is d i s t i n c t l y d i f f e r e n t fromeachoftheothermethodsdiscussed,although it c o n t a i n s elementsofseveralofthesemethods. One d i s t i n g u i s h i n gf e a t u r e is t h a t t h e f l u t t e rs p e e d i s h e l de x a c t l y a t t h ec o n s t r a i n tv a l u e .I nf a c t ,t h i s feature is used t o i n c l u d e t h e n o n l i n e a r c h a r a c t e r o f t h e f l u t t e r s p e e d a s . a f u n c t i o n o f t h e d e s i g n v a r i a b l e s a n d makes it p o s s i b l e t o do a one-dimensional minimi- z a t i o n o f t h e o b j e c t i v e f u n c t i o n i t s e l f , r a t h e r t h a n o f a m o d i f i e d o b j e c t i v e f u n c t i o n as i n t h e p e n a l t y f u n c t i o n method.

A n o t h e r d i s t i n g u i s h i n g f e a t u r e i s t h e d e p a r t u r e from t h e t r a d i t i o n a l d i s - t r i b u t i o n s i n t h e r e s i z i n g column,which are l a r g e l y b a s e d o n t h e g r a d i e n t of t h e v e l o c i t y a n d t h e t o t a l mass. It hasbeendemonstratedthatempirically g e n e r a t e d d i s t r i b u t i o n s c a n l e a d t o r a p i d l y c o n v e r g i n g o p t i m i z a t i o n p r o c e d u r e s .

A t h i r d feature i s t h eu s eo fI n c r e m e n t e d ' F l u t t e rA n a l y s i s . It i s not c o n s i d e r e d a d v a n t a g e o u s t o use I n c r e m e n t e d F l u t t e r A n a l y s i s t o d e t e r m i n e t h e d e r i v a t i v e s o f t h e f l u t t e r s p e e d b y t h e f i n i t e d i f f e r e n c e method. However, I n c r e m e n t e d F l u t t e r A n a l y s i s , as executed with the help of the two-dimensional Regula Falsi method f o r s o l v i n g two nonlinear equations with twounknowns, is a n e f f i c i e n t method f o rk e e p i n gt h ef l u t t e rs p e e de x a c t l yc o n s t a n t .T h i s u s e o f I n c r e m e n t e d F l u t t e r A n a l y s i s c o u l d be u s e d a d v a n t a g e o u s l y i n some o f t h e o t h e r m e t h o d s o f o p t i m i z a t i o n : k e e p i n g t h e f l u t t e r s p e e d e x a c t l y c o n s t a n t f a c i l i t a t e s t h e o b s e r v a n c e o f a convergence crTterion for minimum t o t a l mass s i n c e s i d e e f f e c t s d u e t o d r i f t o f t h e f l u t t e r s p e e d are avoided.

6.7 Comparison of Optimization Methods

6.7.1 General - I n comparing t h eo p t i m i z a t i o n methodsdiscussed i n

S e c t i o n s6 . 2 - 6.6, many d i f f e r e n c e s i n p r o c e d u r a l d e t a i l a r e a p p a r e n t .

S p e c i f i c a l l y , d i f f e r e n c e s i n g e n e r a t i n g t h e d i s t r i b u t i o n a n d m a g n i t u d e o f t h er e s i z i n g column canberecognized. O f these two, however, t h e more b a s i c d i f f e r e n c e r e l a t e s t o t h e d e t e r m i n a t i o n o f t h e m a g n i t u d e o f t h e r e s i z i n g column, or s t e ps i z e .S e v e r a lo ft h e methods - t h ew e i g h tg r a d i e n to p t i m i z a - t i o n o f Simodynes ( S e c t i o n6 . 3 )a n dt h ev e l o c i t yg r a d i e n t , mass g r a d i e n t , a n d g r a d i e n t p r o j e c t i o n methodsofRudisill-Bhatia(Section6.2) - employ a r b i t r a r y s t e ps i z e s .I nc o n t r a s t ,t h ep e n a l t yf u n c t i o n method ( S e c t i o n 6 . 4 ) , t h e method o f f e a s i b l e d i r e c t i o n s ( S e c t i o n 6 . 5 ) a n dt h e method i n c o r p o r a t i n g IncrementedFlutterAnalysis(Section 6.6) a l l make useof a s t e p s i z e d e t e r - mined by w e l l d e f i n e dc r i t e r i a .I nt h ef o l l o w i n g ,t h e s e two groups are d i s - cussedseparatelyandthen a c a n d i d a t e r e s i z i n g p r o c e d u r e is s y n t h e s i z e d , based on theanalyticalandnumericalevaluationsconductedthus far.

6.7.2ArbitraryStep-SizeProcedures - The resizingproceduresemployingan

-~ a r b i t r a r y s t e p s i z e are c h a r a c t e r i z e d b y a well-defined resizing cycle which i s u s u a l l ys i m p l ea n ds t r a i g h t f o r w a r d .I ng e n e r a l ,o n ef l u t t e rs o l u t i o n ( w i t h two c h a r a c t e r i s t i c v e c t o r s ) a n d one set o f f l u t t e r d e r i v a t i v e s are r e q u i r e df o re a c hs t e p . The proceduresbased on c o n s t a n tf l u t t e rs p e e d ( S i m o d y n e ' s w e i g h t g r a d i e n t a n d t h e g r a d i e n t p r o j e c t i o n s e a r c h o f R u d i s i l l - B h a t i a ) r e l y on a l i n e a r i z a t i o n , b a s e d on t h e f l u t t e r v e l o c i t y d e r i v a t i v e s , t oh o l dt h ef l u t t e rs p e e dc o n s t a n t . A s a result, t h ea c t u a lf l u t t e rs p e e d t e n d s t o d r i f t (downward, i n most p r a c t i c a l r e s i z i n g e x e r c i s e s ) a n d must p e r i o d i c a l l yb ec o r r e c t e d . It is t h i st e n d e n c yw h i c he f f e c t i v e l y limits s t e p - s i z e , s i n c e l a r g e e x c u r s i o n s from t h e r e q u i r e d f l u t t e r s p e e d m a r e unde- sirable. R a t h e rt h a na t t e m p t i n gt o maximize step-size,however, a moderate s t e p s i z e i s chosen,basedonexperienceandengineeringjudgment,andthe a t t e n d a n tp e n a l t yo fa ni n c r e a s e d number o fs t e p s i s accepted.

Intermsofspecificprocedures,theRudisill-Bhatiamethodsshouldbe somewhat more e f f i c i e n tt h a nt h ew e i g h tg r a d i e n t method ofSimodynes. A s i n d i c a t e d i n S e c t i o n 6 . 3 , t h e f r e q u e n c y c o n s t r a i n t imposedby t h i s l a t t e r p r o c e d u r e r e s u l t s i n a degreeofapproximationwhichprobablycannot be j u s t i f i e d on t h e b a s i s o f t h e r e s u l t i n g s i m p l i f i c a t i o n s . I n t h e m o d i f i e d procedure(Section 6.3.3) , t h i s f r e q u e n c y c o n s t r a i n t i s removedand t h e r e s u l t i n g p r o c e d u r e i s shown t o b e similar t o t h e g r a d i e n t p r o j e c t i o n searchofRudisill-Bhatia.Incomparingthese two p r o c e d u r e s ,t h eR u d i s i l l - Bhatiaapproachhas a s i g n i f i c a n t a d v a n t a g e i n t h a t it d o e s n o t r e q u i r e t h e s e l e c t i o n o f a dependentdesignvariable,andthuseliminatestheresulting i n f l u e n c eo ft h i sc h o i c eo nt h ep e r f o r m a n c eo ft h ep r o c e d u r e . The u s eo f t h e f l u t t e r v e l o c i t y d e r i v a t i v e s as developedbyRudisill-Bhatia,ratherthan thenormalizedderivativesofSimodynes,has some advantage i n a procedure whichincorporates a n o n z e r of l u t t e rv e l o c i t yi n c r e m e n t .T h i s would b e t h e c a s e when u s i n g t h e f l u t t e r v e l o c i t y g r a d i e n t t o d e f i n e a design variable d i s t r i b u t i o n t o i n c r e a s e t h e f l u t t e r speedof an i n i t i a l l y d e f i c i e n t s y s t e m , or t o make small f l u t t e r v e l o c i t y c o r r e c t i o n s d u r i n g t h e r e s i z i n g c y c l e s .

For use i n c o n s t a n t f l u t t e r velocityprocedures,however, it i s shown i n S e c t i o n A . 3 . 3 , Appendix A, t h a t t h e two f o r m s o f t h e d e r i v a t i v e may be used interchangeably.

Thetwo most useful procedures employing arbitrary step size would then a p p e a r t o b e t h e v e l o c i t y g r a d i e n t s e a r c h and t h e g r a d i e n t p r o j e c t i o n s e a r c h ofRudisill-Bhatia. A s e n v i s i o n e di nS e c t i o n 6.7.4, t h i sf o r m e rp r o c e d u r e would notbeimplementedusinganarbitrarystep-size. The most useful a p p l i c a t i o n o f t h e v e l o c i t y g r a d i e n t s e a r c h p r o c e d u r e i s i n i n c r e a s i n g t h e f l u t t e r s p e e d of a f l u t t e r - d e f i c i e n t s y s t e m t o t h e r e q u i r e d f l u t t e r s p e e d i n a n e a r l y optimum manner. It i s shown i nS e c t i o n A . 4 . 1 , however, t h a t t h e u s e o f a s i n g l e s t e p t o a c c o m p l i s h t h i s i s p r o b a b l y t h e most e f f e c t i v e p r o c e d u r e .

A s a consequence, it seems r e a s o n a b l et ou s et h eI n c r e m e n t e dF l u t t e rA n a l y s i s t e c h n i q u e ( c f . p a g e 4 3 ) ' t o d e t e r m i n e t h e s t e p - s i z e n e c e s s a r y t o s a t i s f y t h e f l u t t e rs p e e dc o n s t r a i n te x a c t l y . The g r a d i e n tp r o j e c t i o ns e a r c hc o u l da l s o be improvedby a m o d i f i c a t i o n o f t h e r e s i z i n g columnsuch t h a t t h e mass g r a d i e n t component i s r e p l a c e d by t h e r e c i p r o c a l o f t h e f l u t t e r s p e e d d e r i v a t i v e s , r e s u l t i n g i n a r e s i z i n g column definedbyequations(A.3), ( A . 4 ) and ( A . 5 ) o f Appendix A. A comparisonofTables A-5 and A-8 of Appendix A i n d i c a t e s t h a t , a t l e a s t for t h ei d e a l i z e d t e s t c a s ee v a l u a t e dt h e r e , a s i g n i f i c a n t i n c r e a s e i n e f f i c i e n c y r e s u l t s from t h i s m o d i f i c a t i o n .

6.7.3 DefinedStep-SizeProcedures - Inresizingproceduresemploying a

d e f i n e ds t e p - s i z e ,a na t t e m p t i s made t o maximize t h es t e p - s i z e so as t o d e r i v e t h e maximum b e n e f i t from a s i n g l er e s i z i n gs t e p .T h i s maximum step- s i z e i s determinedby a well-defined set o f c r i t e r i a , u s u a l l y i n v o l v i n g t h e c o n d i t i o n o f t h e c u r r e n t d e s i g n w i t h r e s p e c t t o t h e d e s i g n c o n s t r a i n t s .

E v a l u a t i n g t h e s e c r i t e r i a u s u a l l y i n v o l v e s t h e d e t e r m i n a t i o n of t h e f l u t t e r s p e e d ,a l o n gw i t ho t h e rc o n s t r a i n tc o n d i t i o n s , at s e v e r a lp o i n t sa l o n gt h e move p a t hd u r i n go n es t e p .I nc o n t r a s tt ot h ea r b i t r a r ys t e p - s i z ep r o c e d u r e s , t h e n , t h e d e f i n e d s t e p - s i z e p r o c e d u r e s n o r m a l l y r e s u l t i n a g r e a t e r mass r e d u c t i o n f o r e a c h set o f f l u t t e r d e r i v a t i v e s c a l c u l a t e d , b u t a t t h ee x p e n s e o f a g r e a t e r number o f r e q u i r e d f l u t t e r s o l u t i o n s .

Each o f t h e t h r e e d e f i n e d s t e p - s i z e p r o c e d u r e s d i s c u s s e d i n t h e p r e c e d - i n g s e c t i o n s h a s d i s t i n c t c h a r a c t e r i s t i c s , andmeaningfulcomparisons are d i f f i c u l t t o make. Some g e n e r a lo b s e r v a t i o n s are p o s s i b l e , however, a l l o w i n g some t e n t a t 7 V F ! Ponclusions to be drawn.

The p e n a l t y f u n c t i o n p r o c e d u r e ( S e c t i o n 6.4) i s perhapsthemost versatile o ft h et h r e ep r o c e d u r e sc o n s i d e r e d . The t r e a t m e n to fc o n s t r a i n t s i s s t r a i g h t - forward,makingautomationoftheprocedureparticularlysimple. A step is t e r m i n a t e d b e f o r e a c o n s t r a i n t v i o l a t i o n o c c u r s , b u t t h e c o n s t r a i n t s are c o n t i n u o u s 3 y a c t i v e a n d e x e r t some i n f l u e n c e o n t h e d i r e c t i o n o f e a c h r e s i z i n g step. The e f f i c i e n c y o f t h e method is t o a s i g n i f i c a n td e g r e ed e p e n d e n to n t h e h a n d l i n g o f t h e p e n a l t y term w e i g h t i n g f a c t o r s a s s o c i a t e d w i t h t h e con- straints ( e q u a t i o n ( 6 . 3 8 ) ) , so t h a t some judgementandexperience are b o t h r e q u i r e d i n t h e s e l e c t i o n of t h e s e f a c t o r s . A moretroublesomedifficulty might be e n c o u n t e r e d i n t h e u s e o f t h e d i r e c t i o n g e n e r a t i n g a l g o r i t h m b a s e d on Newton's method. A s shown i ne q u a t i o n (6.41), t h em a t r i xo fc o e f f i c i e n t s o f t h e d e s i g n variable secondderivativesmust be inverted,and it i s i n d i c a t e d i n R e f e r e n c e 16 t h a t t h i s m a t r i x may b e s i n g u l a r o r . i l 1 - c o n d i t i o n e d .

Although means t o a v o i d t h i s problem are s u g g e s t e d , c o m p u t a t i o n a l d i f f i c u l t i e s may s t i l l arise i n p r a c t i c a l d e s i g n t a s k s i n v o l v i n g l a r g e numbers ofdesign v a r i a b l e s .

The method o f feasible d i r e c t i o n s ( S e c t i o n 6 . 5 ) provides a t r e a t m e n t o f c o n s t r a i n t s which i s d i f f e r e n t from t h a t o f e i t h e r o f t h e o t h e r two methods c o n s i d e r e dh e r e . For t h ef l u t t e rs p e e dc o n s t r a i n t( a n dp r e s u m a b l yo t h e r n o n l i n e a rc o n s t r a i n t s )t h ea p p r o a c ha p p e a r st o be q u i t es a t i s f a c t o r y . It i s similar t o t h a t o f t h e p e n a l t y f u n c t i o n method, i n t h a t t h e "push-off" f a c t o r can be considered as a n a l o g o u s t o t h e p e n a l t y w e i g h t i n g f a c t o r s o f that p r o c e d u r e .I nt h ef e a s i b l ed i r e c t i o n s method, however, t h e move d i r e c t i o n i s r e s t r i c t e d t o t h e u s a b l e as w e l l as f e a s i b l er e g i o n . A s a consequence,each move must r e d u c e t h e o b j e c t i v e f u n c t i o n ( t o t a l mass) as w e l l as avoid a v i o l a t i o no ft h ed e s i g nc o n s t r a i n t s .I nc o n t r a s t ,t h ep e n a l t yf u n c t i o n move must r e d u c e t h e m o d i f i e d o b j e c t i v e f u n c t i o n , b u t n o t n e c e s s a r i l y t h e o b j e c t i v e f'unction i t s e l f . For l i n e a rc o n s t r a i n t s ,s u c h as minimum s i z i n gc o n s t r a i n t s , t h e f e a s i b l e d i r e c t i o n s method i s f o r m u l a t e d s u c h t h a t t h e c o n s t r a i n t s are not a c t i v e u n t i l a c o n s t r a i n t v i o l a t i o n o c c u r s , and t h e r e s i z i n g s t e p is terminated a t t h a tp o i n t .P r i o rt ot h ec o n s t r a i n tv i o l a t i o n ,t h ec o n s t r a i n t exerts no i n f l u e n c e o n t h e move d i r e c t i o n and t h e r e f o r e s u c h c o n s t r a i n t v i o l a t i o n s are normaloccurrences. For t h ei d e a l i z e d t e s t caseevaluated i n Appendix A, t h i s c h a r a c t e r i s t i c d i d n o t a p p r e c i a b l y d e g r a d e t h e e f f i c i e n c y of t h e p r o c e d u r e ; o n l y f o u r s i z i n g c o n s t r a i n t s became a c t i v e d u r i n g t h e optimization,and a s i g n i f i c a n tw e i g h tr e d u c t i o nr e s u l t e df r o me a c hs t e p .I n a more r e a l i s t i c d e s i g n c a s e , w i t h a l a r g e number of minimum s i z e c o n s t r a i n t s , it is a n t i c i p a t e d t h a t a s i g n i f i c a n t number o f s h o r t , i n e f f e c t i v e moves would r e s u l t from s i z i n gc o n s t r a i n te n c o u n t e r s . Once a l i n e a rc o n s t r a i n t becomes a c t i v e , however, t h e c o n s t r a i n t i s i n c o r p o r a t e d i n t h e move d i r e c t i o n so thatsubsequent moves t a k ep l a c ea l o n gt h ec o n s t r a i n tb o u n d a r y . The c o n d i t i o n s imposedon t h e d i r e c t i o n o f t h e move v e c t o r r e s u l t i n d e s i g n v a r i a b l e i n c r e - ments t h a t ,i ng e n e r a l ,a r e , e q u a li nm a g n i t u d e , are p o s i t i v ef o rt h ed e s i g n variables w i t ht h eh i g h e r values o f a V / a m and are n e g a t i v ef o rt h ed e s i g n

variables w i t ht h el o w e rv a l u e so f aV/arn. Thus there appears a l a c ko f

d i f f e r e n t i a t i o n between t h e d e s i g n v a r i a b l e i n c r e m e n t s w i t h e q u a l a l g e b r a i c s i g n . It should be noted,however,thattheresultsofthenumericalevalua- t i o n s d e s c r i b e d i n Appendix A do n o t s u b s t a n t i a t e t h i s l a c k o f e f f i c i e n c y .

I The p r o c e d u r e i n c o r p o r a t i n g t h e u s e o f I n c r e m e n t e d F l u t t e r A n a l y s i s i n a formalizedresizingprocedure(Section 6.6) u t i l i z e s a c o n c e p t t o d e f i n e step-sizewhich i s s i g n i f i c a n t l y d i f f e r e n t from t h a t o f e i t h e r o f t h e o t h e r two methodsdiscussedhere. I n p r i n c i p l e , a d i s t r i b u t i o no fd e s i g n variable increments is defined which reduces total mass andwhich,on a l i n e a r b a s i s , i s not a l i n e a r .

produces no change i n f l u t t e r s p e e d . S i n c e f l u t t e r s p e e d functionofthedesignvariables,however,any f i n i t e v a l u e o f t h i s i n c r e m e n t a l d i s t r i b u t i o n w i l l produce ( i n a l l p r a c t i c a l c a s e s ) a d e c r e a s e i n f l u t t e r s p e e d .F o rs e v e r a lv a l u e so ft h ei n c r e m e n t a ld i s t r i b u t i o n ,t h e value o fa n adjustmentincrement i s determinedbytheIncrementedFlutterAnalysistech- nique which b r i n g s t h e f l u t t e r s p e e d e x a c t l y b a c k t o t h e r e q u i r e d v a l u e . For some v a l u e o f t h e i n c r e m e n t a l d i s t r i b u t i o n t h e t o t a l mass w i l l be a minimum; t h i s p o i n t d e f i n e s , t h e e n do ft h es t e p . By t h i s m e a n s , , t h e n o n l i n e a r i t i e s i n t h e f l u t t e r c o n s t r a i n t are e x p l i c i t l y a c c o u n t e d f o r , and t h e maximum mass r e d u c t i o n for a g i v e n s e t o f v a l u e s o f t h e f l u t t e r s p e e d d e r i v a t i v e s i s achieved.Intheprocessofdeterminingthestep-sizeassociatedwiththe mass, minimum s i z ec o n s t r a i n t s are enforced as necessary. To t h a t minimum e x t e n t , t h e d i r e c t i o n v e c t o r o f t h e d e s i g n v a r i a b l e s is a f u n c t i o n o f t h e move a m p l i t u d e , t h e d i r e c t i o n c o n f o r m i n g t o t h e s i z e c o n s t r a i n t s .

A s p r e s e n t e d i n R e f e r e n c e 18 and as used i n t h e n u m e r i c a l e v a l u a t i o n s o f Appendix A , t h e method d e s c r i b e d i n S e c t i o n 6.6 d i f f e r s i n two o t h e r respectsfromtheothermethodsevaluated. The f l u t t e rd e r i v a t i v e s are o b t a i n e d t h r o u g h t h e u s e o f I n c r e m e n t e d F l u t t e r A n a l y s i s i n t h e form o f i n c r e m e n t si ne a c hi n d i v i d u a ld e s i g nv a r i a b l er e q u i r e df o r a referencechange i nf l u t t e rs p e e d . The r e s u l t so ft h en u m e r i c a le v a l u a t i o ni n d i c a t et h a tt h i s f i n i t e d i f f e r e n c e f o r m o f t h e f l u t t e r d e r i v a t i v e s r e s u l t s i n v a l u e s compa- r a b l e t o t h o s e o b t a i n e d from t h e a n a l y t i c form,and t h a t t h e two forms may b e usedinterchangeably. It is recognized, however, t h a tt h eu s eo ft h i sp r o - c e d u r e f o r o b t a i n i n g t h e f l u t t e r d e r i v a t i v e s - which r e q u i r e s t h e e q u i v a l e n t of a f l u t t e r s o l u t i o n f o r eachdesignvariable - is n o t e f f i c i e n t f o r a p r a c t i c a ld e s i g nt a s ki n v o l v i n g a l a r g e number ofdesignvariables.Insuch a c a s e , t h e u s e o f t h e a n a l y t i c f o r m o f t h e d e r i v a t i v e would b e more econom- i c a l . The o t h e r a r e a i n which t h i s method d i f f e r s from t h o s ep r e v i o u s l y discussed i s i nt h ef o r m a t i o n o f t h er e s i z i n g move v e c t o r . The s e p a r a t i o n o ft h ed e s i g nv a r i a b l e si n t o two groups,thosewithhighervalues .of a V / a m and thosewithlowervaluesof a V / a m , is empirical,and it i s n o tc l e a r t h a tt h ec r i t e r i o nu s e di nS e c t i o n 6.6 would b e e f f i c i e n t i n a l l c a s e s . The . .

d i s t r i b u t i o n o f t h e i n c r e m e n t s f o r t h e d e s i g n v a r i a b l e s w i t h t h e h i g h e r v a l u e s ' o f N / a m i s p r o p o r t i o n a lt ot h ev e l o c i t yg r a d i e n t ,b u tt h ed i s t r i b u t i o nf o r t h ed e s i g nv a r i a b l e sw i t ht h el o w e rv a l u e so f a V / a m is a secondorder f u n c t i o no ft h er e c i p r o c a l so ft h ev e l o c i t yd e r i v a t i v e s . The r e s u l t s o f a n u m e r i c a l e v a l u a t i o n u s i n g t h e move v e c t o r o f S e c t i o n A . 3 . 3 , Appendix A, i n p l a c e o f t h e move v e c t o r d e s c r i b e d i n S e c t i o n 6.6 i n d i c a t e t h a t t h e e f f i - c i e n c y o f t h e two move v e c t o r s i s a p p r o x i m a t e l y e q u a l , a t , l e a s t i n terms o f t h e i d e a l i z e d t e s t c a s e o f Appendix A. I n view o f t h i s , it i s considered t h a t t h e morecomplex move v e c t o r p r e s e n t e d i n S e c t i o n 6.6 i s n o t j u s t i f i e d on t h e basis o f p r e s e n t r e s u l t s .

6.7.4 Formulationof a ResizingProcedure - Basedon t h e e v a l u a t i o n s

p r e s e n t e d i n t h i s s e c t i o n and the results o f t h e n u m e r i c a l e v a l u a t i o n s - p r e s e n t e d i n Appendix A, a r e s i z i n g p r o c e d u r e c a n be formulatedwhich wili' result i n a n improvedperformanceoverthatofanyofthespecificmethods discussed:'. It i s r e c o g n i z e d t h a t t h e e v a l u a t i o n s o f t h e r e s i z i n g p r o c e d u r e s are notcomplete;anypromisingproceduremust be f u r t h e r e v a l u a t e d i n terms of a r e a l i s t i c d e s i g n t a s k i n o r d e r t o a r r i v e a t f i r m conclusions. I n p a r t i c u l a r , f u r t h e r e v i d e n c e must b e o b t a i n e d t o d e t e r m i n e t h e relative e f f i c i e n c y o f t h e a r b i t r a r y s t e p - s i z e a n d d e f i n e d s t e p - s i z e p r o c e d u r e s .

For t h e time being,however, it w i l l bepremisedthattheproblemsassociated with a p r a c t i c a l d e s i g n t a s k w i l l d i c t a t e t h e u s e o f a d e f i n e d s t e p - s i z e procedure.Theseproblems, some ofwhich are d i s c u s s e di nS e c t i o n 7, would seem t o i n d i c a t e t h a t t h e g r e a t e s t p o s s i b l e mass reduction should be obtained f o r e a c h s t e p , s i n c e t h e s t r u c t u r a l r e a n a l y s i s r e q u i r e d p e r r e s i z i n g s t e p may be much more e x t e n s i v e t h a n i s generally recognized.

A s q u a l i f i e d b y t h e p r e c e d i n g p a r a g r a p h , t h e p r e f e r r e d r e s i z i n g p r o - cedure may b e d e s c r i b e d i n t e r m s o f t h e f o l l o w i n g c h a r a c t e r i s t i c s : F l u t t e r s p e e d d e r i v a t i v e s are of t h e a n a l y t i c t y p e , c a l c u l a t e d b y t h e method of R u d i s i l l - B h a t i a ( S e c t i o n 6.21, p o s s i b l y g e n e r a l i z e d by t a k i n g i n t o a c c o u n t t h e d e r i v a t i v e s o f t h e v i b r a t i o n modes w i t h r e s p e c t t o t h e d e s i g n v a r i a b l e s .

The i n i t i a l r e s i z i n g s t e p is o n ew h i c hi n c r e a s e st h ef l u t t e rs p e e do f t h ef l u t t e r - d e f i c i e n td e s i g nt ot h er e q u i r e dv a l u e .T h i s i s done i n a nearly-optimummannerby t h e a d d i t i o n o f d e s i g n v a r i a b l e i n c r e m e n t s d i s t r i b u t e d a c c o r d i n gt ot h ev e l o c i t yg r a d i e n t . The t o t a lr e q u i r e df l u t t e rs p e e di n c r e - ment i s o b t a i n e d i n a s i n g l e s t e p , a n d a form ofIncrementedFlutterAnalysis i s u s e d t o d e t e r m i n e t h e m a g n i t u d e o f t h e s t e p r e q u i r e d t o s a t i s f y t h e f l u t t e r c o n s t r a i n te x a c t l y .

Subsequentresizingstepsareperformed a t c o n s t a n t f l u t t e r s p e e d , u s i n g thetechniqueofminimizationof t h e o b j e c t i v e f u n c t i o n (mass) d e s c r i b e d i n S e c t i o n 6.6 i no r d e rt od e t e r m i n et h es t e p - s i z e . A s d i s c u s s e di nt h a ts e c t i o n , t h e p r i m a r y d i s t r i b u t i o n o f d e s i g n v a r i a b l e i n c r e m e n t s i s such as t o produce z e r of l u t t e rv e l o c i t yc h a n g eo n a l i n e a r i z e d b a s i s . I n c r e m e n t e d F l u t t e r Analysis i s t h e n u s e d t o d e t e r m i n e t h e m a g n i t u d e o f a n a d j u s t m e n t column o f i n c r e m e n t sr e q u i r e dt om a i n t a i nt h ea c t u a lf l u t t e rs p e e dc o n s t a n t . The t o t a l mass o f t h e d e s i g n v a r i a b l e s , i n c l u d i n g b o t h t h e p r i m a r y a n d a d j u s t m e n t d i s t r i b u t i o n s , i s determined as a f u n c t i o no fs t e p - s i z e ,a n dt h es t e p - s i z e c o r r e s p o n d i n gt o minimum mass chosen.Using t h i s c o n f i g u r a t i o n as a s t a r t i n g p o i n t , t h e r e s i z i n g c y c l e i s r e p e a t e d .

The s i z i n g c o n s t r a i n t s a r e s a t i s f i e d i n t h e manner o f S e c t i o n 6.6, w i t h t h e d i r e c t i o n o f t h e move vector being modified as c o n s t r a i n t s are encountered d u r i n gt h em i n i m i z a t i o no ft h eo b j e c t i v ef u n c t i o n .N o t et h a tt h ef l u t t e r c o n s t r a i n t i s s a t i s f i e d a t eachsubstepoftheminimization.

The move v e c t o r f o r d e s i g n variables c o r r e s p o n d i n g t o positive values of av/am i s b a s e do nt h ec o m b i n a t i o no ft h ev e l o c i t yg r a d i e n ta n dt h e 7 2 n e g a t i v e r e c i p r o c a l s o f t h e f l u t t e r d e r i v a t i v e s shown i n S e c t i o n A . 3 . 3 .

For design variables c o r r e s p o n d i n gt on e g a t i v ev a l u e so f aV/am, modified d i s t r i b u t i o n s will be used, some optionsofwhich are d i s c u s s e d i n R e f e r e n c e 1.

A summary discussion 'of this procedure and an example of nymerical results are p r e s e n t e d i n R e f e r e n c e 28.

7. CONSIDERATIONS RELATED TO A REALISTIC D E S I G N ENVIRONMENT . , I n t h e literature o n o p t i m i z a t i o n w i t h f l u t t e r c o n s t r a i n t s , t h e methods p r e s e n t e d are i l l u s t r a t e d w i t h examples ofvaryingcomplexity. One example i n R e f e r e n c e 16 i s basedon 156 structuraldegreesoffreedomand23design variables. The example i nR e f e r e n c e 29 i s basedon 150 degreesoffreedom and 100 design variables. A s much as t h e s e numbers surpassthecorresponding numbers i n earlier examples i n t h e l i t e r a t u r e , t h e y f a l l s h o r t o f what may b e e n c o u n t e r e di n a r e a l i s t i cd e s i g ne n v i r o n m e n t . Thus,problems t h a t may r e s u l t fromsuchanenvironmentremainunexposed.Duringthepresentwork, several a s p e c t s o f d e a l i n gw i t ha na c t u a ld e s i g n havebeenexamined. They are d i s - c u s s e d i n t h e f o l l o w i n g s e c t i o n s , t o g e t h e r w i t h o t h e r a s p e c t s t o which l i t t l e or no a t t e n t i o n c o u l d be given.

7.1 S t r u c t u r a l Model The mathematical model r e p r e s e n t i n g t h e s t r u c t u r e o b v i o u k l y is a n i m p o r t a n te l e m e n to fs t r u c t u r a lo p t i m i z a t i o nw i t hf l u t t e rc o n s t r a i n t s .F i n i t e elementstructuralmodelswiththousandsofelementsand a corresponding number ofnodaldisplacements as degreesoffreedom are u s e d f o r stress and s t i f f n e s s a n a l y s i s o f a g i v e ns t r u c t u r e . A d u p l i c a t i o no fe f f o r tc a nb e avoided i f t h e same s t r u c t u r a l model c a n b e u s e d f o r f l u t t e r o p t i m i z a t i o n .

It should be n o t e d t h a t f o r a f l u t t e r a n a l y s i s , o r a l o a d s a n a l y s i s i n c l u d i n g a e r o e l a s t i c e f f e c t s , t h e r e f i n e m e n t o f a multi-thousandelement s t r u c t u r a l model is notrequired. For t h ec u r r e n tt y p eo fs u b s o n i ct r a n s p o r t s , a.relatively simple beam model s u f f i c e s f o r f l u t t e r . For supersonictrans- p o r t s , however, as are i n e x i s t e n c e a n d p r o j e c t e d f o r t h e f u t u r e , a simple beam model i s inadequateand a f i n i t e elementmodelmust be used.This i m p l i e s t h a t m e t h o d s o f o p t i m i z a t i o n ' f o r f l u t t e r must b e a b l e t o h a n d l e f i n i t e e l e m e n t t y p e s t r u c t u r a l r e p r e s e n t a t i o n .

The t y p i c a l s t r u c t u r a l model f o r stress a n a l y s i s h a s a number of degrees o f freedom t h a t e x c e e d s what a t p r e s e n t seems p r a c t i c a b l e f o r t h e r e p e t i t i v e v i b r a t i o na n a l y s i se x p e c t e di n a f l u t t e ro p t i m i z a t i o n program. A reduced number of degrees of freedom can be obtained by using a s t i f f n e s s m a t r i x o f 7 3 r e d u c e d s i z e i n t h e v i b r a t i o n a n a l y s i s or by generating a c o a r s e r f i n i t e elementmodel f o r a e r o e l a s t i c a n a l y s e s , w h i c h may or may n o t r e q u i r e f i r t h e r s i z e r e d u c t i o n .

The common approach t o c o o r d i n a t e r e d u c t i o n i s onei'nwhichcoordinates t ob ee l i m i n a t e d are assumed t o havezeroloads.This i s o f t e n c a l l e d s t a t i c r e d u c t i o n i n c o n t r a s t w i t h t h e a p p r o a c h o f R e f e r e n c e 30; which can be called dynamic r e d u c t i o n ,s i n c et h er e d u c t i o n is a f u n c t i o n of t h ef r e q u e n c y . If t h eb a s i cs t i f f n e s sm a t r i x [\] i s p a r t i t i o n e d as i n d i c a t e di ne q u a t i o n ( 7 . 1 ) , r - .

the reduced matrix, Fr]., i s given by equation (7.2).

If t h ei n c r e m e n t a ls t i f f n e s s due t oi n c r e a s i n gt h ed e s i g nv a r i a b l e p, I a u n i t amount i s [AKi], t h eb a s i cs t i f f n e s sm a t r i x , as a f u n c t i o n o f t h e pi's i s : where pi i s d e f i n e dr e l a t i v et o a r e f e r e n c ev a l u e .

~ due t o t h e I n g e n e r a l , t h e r e f o r e , [Kr] i s a n o n l i n e a r f u n c t i o n o f t r i p l ep r o d u c ta n di n v e r s i o ni ne q u a t i o n( 7 . 2 ) . Thus, t o com2ute t h e c o o r d i n a t e r e d u c t i o n r e p r e s e n t e d b y e q u a t i o n ( 7 . 2 ) must b e r e p e a t e d for

eachcombinationofvaluesof Pi *

I n most f l u t t e r o p t i m i z a t i o n p r o c e d u r e s , t h e d e r i v a t i v e o f t h e s t i f f n e s s , w i t h r e s p e c t t o many d e s i g n v a r i a b l e s pi is r e q u i r e d .

i s as follows: The procedure Equation(7.2) i s e q u i v a l e n tt o (7.4)

[..I = [GRIT [%] [ G R ]

where The d e r i v a t i v e o f t h e r e d u c e d s t i f f n e s s m a t r i x w i t h r e s p e c t t o any

design variable pi, evaluated at a given combination of values of pi, is

then defined by: where AKi is c o n s i s t e n tw i t he q u a t i o n( 7 . 3 ) .

The a p p l i c a t i o no fe q u a t i o n ( 7 . 6 ) is as follows. The incremental stiff- ness matrices [ . I C i ] are i n v a r i a n t d u r i n g t h e o p t i m i z a t i o n p r o c e s s . A s a new set o fd e s i g nv a r i a b l e s , Pi, i s d e f i n e dd u r i n g a s t e pi nt h eo p t i m i z a -

t i o n p r o c e s s , [%] i s computed according to equation (7.3) and kr] accord-

i n gt oe q u a t i o n ( 7 . 2 ) . The r e d u c e ds t i f f n e s sm a t r i x [..3 can then be used i n a v i b r a t i o na n a l y s i s . The a s s o c i a t e dc o o r d i n a t er e d u c t i o nm a t r i x [GR] is formedandused i n t h e t r i p l e p r o d u c t o f e q u a t i o n ( 7 . 6 ) t o compute t h e d e r i v a t i v e so f kr] w i t hr e s p e c tt o a l l d e s i g nv a r i a b l e s .

When t h e number o fs t r u c t u r a lc o o r d i n a t e s is n o tt o o high, t h e c o o r d i n a t e s t o b e e l i m i n a t e d c a n b e r e s t r i c t e d t o t h o s e t h a t are of no i n t e r e s t t o a e r o - e l a s t i ca n a l y s e sa n d ,i nf a c t ,c a n be consideredunloaded. However, when t h e s t r u c t u r a l model i s d e s i g n e d f o r stress a n a l y s i s it may b e d e s i r a b l e t o e l i m i n a t ec o o r d i n a t e st h a t are o f i n t e r e s t t o a e r o e l a s t i c a n a l y s e s , s u c h as d e f l e c t i o n sp e r p e n d i c u l a rt ol i f t i n gs u r f a c e s .T h i su s u a l l y means t h a t coordinatesmust be e l i m i n a t e dt h a th a v ea s s o c i a t e di n e r t i a . If t h a t is t h e case,equations (7.4) and ( 7 . 6 ) must be a p p l i e d t o the mass m a t r i x as w e l l (Reference 31).

I n u s i n g a c o a r s e g r i d f i n i t e e l e m e n t model f o r a e r o e l a s t i c a n a l y s i s , t h e aim is t o r e d u c e t h e number o f c o o r d i n a t e s t o be e l i m i n a t e d t o a minimum. I n a f i n i t e d i f f e r e n c e a p p r o a c h , as d e s c r i b e di nR e f e r e n c e 32, t h e o n l y s t r u c - tural degrees of freedom are d e f l e c t i o n s p e r p e n d i c u l a r t o t h e l i f t i n g s u r f a c e .

Which approach will be favored in future optimization work is hard to foresee. There seem to be three areasof investigation that could lead to significant devebpment.

It seems most advisable, because of the directness of the approach, to

br( p i ) ] ' for arbitrary sets of pi in the basic

speed up the computation of finite element analysis system. Possibly approximate methods can be developed, which are valid for a few resizing steps,. after which an exact updating place. It seems self-evident that the last updating in an optimization should be exact.

A second area of investigation could be based on an approach used with some success at the Lockheed-California Company. In it the reduced stiff- ness matrix is approximated by a polynomial function of the design variables: where the summation is over i = 1- n and j = 1- n.

Such a polynomial can be an acceptable approximation over limited ranges of the values of the design variables. Since the stiffness is represented as an explicit functionof the design variables, it can readily be evaluated for

any arbitrary combination of values of p . The derivative of the stiffness

i matrix is:

One element of [ K r ( pi 4 is approximated by

To determine the values of the coefficients Ai and B i j ' Kr(Pi)

must be computed for n+n2 values of p . Thus, to define the polynomial

i

expression in equation ( 7 . 7 ) it is necessary to compute [K~ ( pi ) I for

l+n+n2 linearly independent columns , with the help of equations (7.3)

and (7.2).

On an arrow wing, where design variables were torsional and bending s t i f f n e s s o v e r c e r t a i n areas o f t h e o u t e r wing, it w a s foundthere w a s l i t t l e couplingbetweenthedesign variables. When t h a t is t h ec a s ee q u a t i o n (7.7) c a n b e w r i t t e n as: Only 1+2n evaluations of- K r ( P i ) a r e n e c e s s a r y . t o compute

[ I

and a l l t h ec o e f f i c i e n tm a t r i c e s Fi] and Fi] i n e q u a t i o n ( 7 .lo).

A t h i r d area o f i n v e s t i g a t i o n is t h e u s e o f t h e a e r o e l a s t i c model. I n t h a t c a s e it seems mandatory t h a t a d i r e c t two-way r e l a t i o n s h i p be developed between t h e s i z i n g i n t h e stress modeland t h e s i z i n g i n t h e a e r o e l a s t i c model.

7.2 M u l t i p l eF l u t t e rS p e e dC o n s t r a i n t s Although t h e p r o b l e m o f m u l t i p l e f l u t t e r s p e e d c o n s t r a i n t s i s addressed i n t h e l i t e r a t u r e ( e . g . , R e f e r e n c e 3 3 ) , e x a m p l e s i n t h e l i t e r a t u r e , u s e d t o i l l u s t r a t e methods of o p t i m i z a t i o n f o r f l u t t e r , are a l l r e s t r i c t e d t o one f l u t t e rs p e e d .O f t e n it i s i n d e e dp o s s i b l et oe l i m i n a t e a l l f l u t t e r con- s t r a i n tv i o l a t i o n sb ye l i m i n a t i n gt h e most c r i t i c a l f l u t t e r s p e e d . I n gen- eral, however, t h e p o s s i b i l i t y of more t h a n one a c t i v e f l u t t e r s p e e d c o n s t r a i n t must be a n t i c i p a t e d .

F o r m a l l y , t h e p e n a l t y f u n c t i o n method ( S e c t i o n 6.4) a n d t h e method o f f e a s i b l e d i r e c t i o n s ( S e c t i o n 6.5) h a v e b u i l t - i n c a p a b i l i t y t o h a n d l e m u l t i p l e f l u t t e rs p e e dc o n s t r a i n t s . The o t h e r m e t h o d sd i s c u s s e di nS e c t i o n 6 r e q u i r e added l o g i c t o h a n d l e m u l t i p l e f l u t t e r s p e e d c o n s t r a i n t s .

Reference 34 makes u s e o f t h e m u l t i - c o n s t r a i n t c a p a b i l i t y of t h e p e n a l t y f u n c t i o n methodby r e q u i r i n g t h a t t h e f l u t t e r r o o t s f o r s e l e c t e d v a l u e s o f thereducedfrequency, k, correspondtocombinationsofspeedand damping t h a tp r o v i d ea d e q u a t e damping w i t h i n t h e f l i g h t e n v e l o p e (see a l s o S e c t i o n 7 . 3 ) .

The o p t i m a l i t y c r i t e r i o n f o r o n e f l u t t e r s p e e d c o n s t r a i n t is: where i and j refer t o free design variables, i.e., d e s i g n v a r i a b l e s t h a t axe n o t at a s i z i n gc o n s t r a i n t( R e f e r e n c e s 29 and 1).

Fortwo f l u t t e r s p e e d c o n s t r a i n t s t h e o p t i m a l i t y c r i t e r i o n is (Reference 1): 1 1 1 av, avl av,

-"

= 0 ('7.12) a m am, a m 1 3 av, av, av,

"-

am, am, a m

Equation(7.12)must be s a t i s f i e d f o r anycombinationofthree free design variables. It i s s a t i s f i e d i f : E x t e n s i o n o f t h i s c r i t e r i o n t o more t h a n two f l u t t e r s p e e d c o n s t r a i n t s is s t r a i g h t f o r w a r d .

av, av2

It was f o u n dt h a tt h ev a l u e -+ - d e t e r m i n e st h er e s i z i n g column

ami ami

t h a t i s g e n e r a t e d b y t h e f e a s i b l e d i r e c t i o n method ofReference 17 (Sec- t i o n 6.5) w i t h a push-offfactor 8 = 1. It i s b e l i e v e dt h a tt h i sv a l u ec a n a l s o b e u s e d i n d e f i n i n g a r e s i z i n g column i f t h e o p t i m i z a t i o n is based on t h e methodsdiscussedinSections6.2,6.3and 6.6. This i s f u r t h e rd i s c u s s e d i n R e f e r e n c e 1.

To d e m o n s t r a t em u l t i p l ef l u t t e rs p e e dc o n s t r a i n tc a p a b i l i t y ,t h en u m e r i c a l examplesmust relate t o a r e a l i s t i c d e s i g n e n v i r o n m e n t i n w h i c h two or more i n - f l i g h t modes l e a d t o f l u t t e r s p e e d s below t h e minimum r e q u i r e d f l u t t e r speed.Thesein-flight modes may b e u n r e l a t e d f l u t t e r modes f o r a p a r t i c u l a r w e i g h t c o n f i g u r a t i o n o f t h e a i r p l a n e a n d o n e p a r t i c u l a r Mach number, or t h e y may b e r e l a t e d o r u n r e l a t e d f l u t t e r modes for more thanoneweightconfigura- t i o n and Mach number.

7 . 3 Damping C o n s t r a i n t s I n t h e d i s c u s s i o n s i n S e c t i o n 6 , t h e emphasis is on f l u t t e r s p e e d con- s t r a i n t s . This i s i nr e c o g n i t i o no ft h ef l u t t e rs p e e dm a r g i n s as definedby t h eF e d e r a lA i r w o r t h i n e s sR e g u l a t i o n sa n dm i l i t a r ys p e c i f i c a t i o n s . The requirementsoftheFederalAirworthinessRegulations are i l l u s t r a t e d i n Figure 7-1. The a i r p l a n e s h a l l b e d e s i g n e d t o be f l u t t e r f r e e - w i t h i n t h e 1 . 2 I$,, 1 . 2 VD and h = -3100 m altitude-speedenvelopedefinedby M 30 ' (-10,200 f t ) . I n a d d i t i o n t o t h i s f l u t t e r s p e e d r e q u i r e m e n t t h e r e is a n impliedrequirementforadequate modal damping withinthisenvelope.This is i l l u s t r a t e d i n F i g u r e 7-2: f o r a l l f l i g h t c o n d i t i o n s w i t h i n t h e f l i g h t envelopetheremustbe a c e r t a i n amount o f p o s i t i v e damping. I n k-method terminology this means g 5 gmax, and i n p-k-method terminology y 5 y I max' Both gmax and Ymax a r en e g a t i v eq u a n t i t i e s . From t h ed e f i n i t i o n so f g and Y it f o l l o w s t h a t f o r small values g E 2V.

To s a t i s f y t h e g e n e r a l damping c o n s t r a i n t , t h e i n e q u a l i t y c o n s t r a i n t ) must be invoked at severalspeedsbelow 1.2 VD, ( o r Y' Ymax gmax f o r all i n - f l i g h t modes o f i n t e r e s t , for s e v e r a l Mach numbersand f o r t h e a i r p l a n ew e i g h tc o n f i g u r a t i o n st o be c o n s i d e r e d .I n a r e a l i s t i c d e s i g n environment t h i s may lead t o hundreds o f i n e q u a l i t yc o n s t r a i n t s . It is o b v i o u s t h e r e a r e p r a c t i c a l d i f f i c u l t i e s a s s o c i a t e d w i t h that many c o n s t r a i n t s .

Experience shows t h a t u s u a l l y o n l y v e r y f e w damping c o n s t r a i n t s are a c t i v e and, t h e ya r ea s s o c i a t e dw i t h hump modes.Sometimesdamping c o n s t r a i n t v i o l a t i o n s b y hump modes disappear as t h e s t r u c t u r e i s r e s i z e d t o e l i m i n a t e t h e most c r i t i c a l f l u t t e r s p e e d v i o l a t i o n ( s ) . I n a n t i c i p a t i o n o f t h e n e e d for invoking a minimum hump mode damping c o n s t r a i n t , S e c t i o n 3 . 4 p r e s e n t s a p r o c e d u r e t o d i r e c t l y d e t e r m i n e t h e minimum damping of a hump mode.

The o p t i m a l i t y c r i t e r i o n f o r a n a c t i v e damping c o n s t r a i n t is: where i and j r e f e r t o f r e e d e s i g n v a r i a b l e s . It c o r r e s p o n d s t o t h e o p t i r n a l i t y c r i t e r i o n f o r a f l u t t e r s p e e d c o n s t r a i n t , t h e d e r i v a t i o n o f which (given in Reference 1) c a n b e g e n e r a l i z e d t o a r b i t r a r y c o n s t r a i n t s .

For a combined f l u t t e r s p e e d a n d damping c o n s t r a i n t , t h e o p t i m a l i t y c r i t e r i o n is: = o velocity, IiEAs Figure 7-1: Example of Flight Envelope Figure 7-2: Min.lmrtm D a r q e i n g Requirement Equation (7.15) must be s a t i s f i e d f o r any combinationof three f r e e d e s i g nv a r i a b l e s . It is s a t i s f i e d i f Here C i s a n arbitrary c o n s t a n tt h a tc a nb eu s e dt oc r e a t ec o m p a t i b l e u n i t s o r t o a s s i g n a d i f f e r e n t w e i g h t i n g t o t h e two c o n s t r a i n t s .

I n several methods of o p t i m i z a t i o n t h i s c r i t e r i o n c a n p r o v i d e a guide towards generating a r e s i z i n g column.

. Further development of a p r a c t i c a lm e t h o d . o fi n c l u d i n g damping con- s t r a i n t s i n f l u t t e r o p t i m i z a t i o n s h o u l d be based onnumericalexamples i n a r e a l i s t i c d e s i g n e n v i r o n m e n t . .

7.4 Mass Ballast The literature pays l i t t l e o r no e x p l i c i t a t t e n t i o n t o ballast (dead weight) as a d e s i g nv a r i a b l e . The r e a s o nf o r t h i s omission i s understandable: any method o f o p t i m i z a t i o n that c a n h a n d l e d e s i g n v a r i a b l e s r e p r e s e n t i n g related s t i f f n e s s a n d mass changescanhandle a d e s i g n v a r i a b l e r e p r e s e n t i n g a mass changeonly.

mass ballast may b e a more e f f i c i e n t way o f r a i s i n g t h e f l u t t e r Adding speed t o i t s r e q u i r e dv a l u et h a ns t r u c t u r a ls t i f f e n i n g . That a p p e a r e dt ob e t h e c a s e o n o n e o f t h e U n i t e d S t a t e s s u p e r s o n i c t r a n s p o r t d e s i g n s a n d i n t h e example t r e a t e di nR e f e r e n c e 29. I nt h el a t t e rc a s e , however, it is n o tc l e a r whetherthemodifiedstrengthrequirements due t o t h e a d d i t i o n o f ballast a r e accountedfor.Reference 29 demonstrates a p o t e n t i a lc o m p l i c a t i o na s s o c i a t e d w i t h mass b a l l a s t as a d e s i g nv a r i a b l e : as ballast i s added i n a p a r t i c u l a r r e g i o n , t h e f l u t t e r s p e e d d e r i v a t i v e w i t h r e s p e c t t o t h e mass ballast changes f r o mn e g a t i v et op o s i t i v e . A s s t a t e di nR e f e r e n c e 29, i f t h i s phenomenon occurs,anautomatedresizingprocedure may f a i l t o r e c o g n i z e t h e b e n e f i c i a l e f f e c t o f a l a r g e r amount o f b a l l a s t s i n c e an i n f i n i t e s i m a l amount o f b a l l a s t proved t o l o w e rt h ef l u t t e rs p e e d .U n t i l t h i s a s p e ' c to ff l u t t e ro p t i m i z a t i o n hasreceived more attention,considerableengineeringjudgmentshouldbeused i n h a n d l i n g mass ballast as d e s i g n v a r i a b l e s .

I n view of theprecedingparagraph, it may b e c o n v e n i e n t t o first consider 1 s t i f f n e s sd e s i g n ' v a r i a b l e sa n dt h e i ra s s o c i a t e dm a s s e so n l yf o rr a i s i n gt h e most c r i t i c a l f l u t t e r s p e e d t o t h e d e s i r e d v a l u e andthesubsequentoptimiza- t i o n at c o n s t a n tf l u t t e rs p e e d .S t a r t i n gw i t h the o r i g i n a ld e f i c i e n t con- f i g u r a t i o n , it is thendeterminedwhetherany mass changewithoutstiffness change is more e f f i c i e n t t h a n . t h e most e f f i c i e n t s t i f f n e s s change i n r a i s i n g t h e f l u t t e r s p e e dt ot h ed e s i r e dv a l u e .I n c r e m e n t e dF l u t t e rA n a l y s i sc a nb e u s e d t o d i r e c t l y d e t e r m i n e t h e amountof ballast needed t o meet t h e f l u t t e r c o n s t r a i n t , r e g a r d l e s s o f a n y c h a n g e s i n s i g n o f t h e f l u t t e r s p e e d d e r i v a t i v e s as a f u n c t i o n o f t h e amount of mass ballast. If mass b a l l a s t i s more e f f e c - t i v e t h a n optimum s t i P f e n i n g , t h e mass b a l l a s t d e s i g n v a r i a b l e s o f i n t e r e s t can be added as d e s i g n v a r i a b l e s f o r a f i n a l o p t i m i z a t i o n p r o c e s s .

7.5 I n t e r f a c e With StrengthOptimization Combined o p t i m i z a t i o n f o r f l u t t e r a n d stress hasbeendemonstratedwith s i m p l es t r u c t u r a lm o d e l sa n d / o ru n d e rs i m p l i e i n ga s s u m p t i o n s( R e f e r e n c e s 9 , 16 and 2 9 ) .

InReferences 9 and 1 6 the p e n a l t y f u n c t i o n method i s u s e d a n d , i n o r d e r t o r e d u c e t h e number o f s t r e s s c o n s t r a i n t d e r i v a t i v e s t o b e e v a l u a t e d , t h e s t r e s sc o n s t r a i n t i s r e d u c e dt o one c o n s t r a i n tp e rl o a d i n gc o n d i t i o n . The e f f e c t o f t h i s c a n b e e a s i l y s e e n i n t h e f o l l o w i n g f o r m u l a t i o n o f t h e p e n a l t y functionapproach.

The m o d i f i e d o b j e c t i v e f u n c t i o n may be represented by: where : V ( m i ) = f l u t t e r s p e e d VR = minimum d e s i r e df l u t t e r speed a . ( m i ) = s t r e s s i n j t h element j = 1 ... n .

J - c r . = maximum a l l o w a b l e s t r e s s i n j t h e l e m e n t J m = d e s i g n v a r i a b l e ; mass a s s o c i a t e d w i t h i t h d e s i g n e l e m e n t ; i i = 1 ... n.

-

m = minimum allowable value of m i i rV, rcr,rm = p e n a l t y w e i g h t i n g f a c t o r s

terms as t h e r e a r e f l u t t e r s p e e d c o n s t r a i n t s , a n d as many 2 -

j=l crj = c r . (mi 1

J terms as t h e r ea r ed e s i g nl o a dc o n d i t i o n s . For t h i sd i s c u s s i o n it i s suffi- c i e n t t o assumeone f l u t t e r s p e e d c o n s t r a i n t a n d one design load condition.

A s a p a r t o f t h e d e t e r m i n a t i o n of t h e r e s i z i n g column, the p a r t i a l d e r i v a t i v e s a@(mi) /ami are used.

For each design variable, one derivative - and n derivatives

n am, I I Q

a

-

- must b e e v a l u a t e d . Thus a t o t a l o f n f l u t t e r s p e e d

ami Q. - (T. (mi)

J J d e r i v a t i v e s and n s t r e s s d e r i v a t i v e s must be evaluated. In References 9 and 16 t h e number of stress d e r i v a t i v e s is r e d u c e d ' t o n b ye v a l u a t i n gt h e d e r i v a t i v e s by means of f i n i t e d i f f e r e n c e s b u t p e r f o r m i n g t h e d i f f e r e n t i a - t i o n a f t e r t h e summation; i.e., t h e f o l l o w i n g i d e n t i t y which r e q u i r e st h ee v a l u a t i o n of n d e r i v a t i v e sp e rs i n g l ed e s i g nv a r i a b l e , is r e p l a c e d b y t h e f i n i t e d i f f e r e n c e r e p r e s e n t a t i o n : r e q u i r i n g t h e e v a l u a t i o n o f o n l y o n e d e r i v a t i v e p e r s i n g l e d e s i g n v a r i a b l e .

This substitution does not affect the one-dimensional minimization that is p a r t o f t h e methodused i n Reference 1 4 , b u t it does a f f e c t t h e d i r e c t i o n o ft h er e s i z i n gv e c t o r .I n s t e a d ofeachelementalstresscontributing

a@

i n d i v i d u a l l yt o - , o n e c o n t r i b u t i o n r e p r e s e n t i n g a n a v e r a g e s t r e s s ami penaltyterm i s used. N o s t u d i e si n v e s t i g a t i n gt h ee f f e c to ft h i ss u b s t i t u t i o n have been reported.

In Reference 28, the o p t i m a l i t yc r i t e r i o n am = c o n s t a n t for a l l i is i u s e d i n t h e flutter o p t i m i z a t i o n a n d t h e f u l l y - s t r e s s e d - d e s i g n c r i t e r i o n f o r s t r e n g t ho p t i m i z a t i o n . The two o p t i m i z a t i o n s are p e r f o r m e da l t e r n a t e l yu n t i l a convergeddesign is obtained.

The f u l l y - s t r e s s e d - d e s i g n c r i t e r i o n d o e s n o t , i n g e n e r a l , l e a d t o a minimum w e i g h t s t r u c t u r e , a t l e a s t n o t f o r a redundantstructure(Refer- ence 35). F u r t h e r m o r e ,w i t h o u tf u r t h e ri n v e s t i g a t i o nt h e r e is l i t t l e ground f o r e x p e c t i n g t h a t a l t e r n a t i n g f l u t t e r a n d s t r e n g t h o p t i m i z a t i o n w i l l l e a d t o a convergeddesign i f b o t h f l u t t e r c o n s t r a i n t s a n d stress c o n s t r a i n t s a r ea c t i v e .T h i s leads t o t h e c o n c l u s i o n t h a t i d e a l l y f l u t t e r and s t r e n g t h o p t i m i z a t i o n s h o u l d t a k e p l a c e s i m u l t a n e o u s l y as i s done i n R e f e r e n c e s 9 and 16.

It would seem t h a t t h e adequacyofthemethodsofReferences 9 , 16 and 29 has not been demonstrated when a p p l i e d t o a practical design problem.

Conceptually the methodsofReferences 9 and 16 allow as many stress con- s t r a i n t s p e r l o a d c o n d i t i o n as t h e r e are independent stress c o n s t r a i n t s (which may be l a r g e r t h a n t h e number o fe l e m e n t s ) . Thus t h e number ofinde- p e n d e n ts t r e s sc o n s t r a i n t sc a nb ev e r yl a r g e .T h i s has l e d t o t h e d e f i n i t i o n ofone stress c o n s t r a i n t p e r l o a d c o n d i t i o n , w h i c h , however,removes t h e p o s s i b i l i t y o f i n d e p e n d e n t stress c o n s t r a i n t s c o n t r i b u t i n g i n d i v i d u a l l y t o t h e move v e c t o rd i r e c t i o n .P o s s i b l yo t h e rc o m p o s i t e stress c o n s t r a i n t sc a n be d e f i n e d s u c h t h a t t h e number o f d e r i v a t i v e s t o b e e v a l u a t e d i s reduced w h i l e r e t a i n i n g t h e c o n t r i b u t i o n o f e a c h c o n s t r a i n t t o the move v e c t o r d i r e c t i o n .

A d d i t i o n a l i n v e s t i g a t i o n s i n t h e areas ofstructuralmodelingand combined o p t i m i z a t i o n f o r f l u t t e r a n d s t r e n g t h are needed before conclusions regarding t h e best approachcan be formulated.Suchconclusionsshouldbebasedonthe results o f a n a l y s e s i n a r e a l i s t i c d e s i g n e n v i r o n m e n t .

8. COMPUTATIONAL ASPECTS OF THE FLUTTER TASK The purposeof t h i s s e c t i o n is t o d e l i n e a t e t h e c o m p u t a t i o n a l a s p e c t s o f t h e c o m p l e t e f l u t t e r t a s k , w h i c h i n c l u d e s f l u t t e r a n a l y s i s as w e l l as s t r u c - tural s y n t h e s i s , i . e . t h e d e s i g n o f a s t r u c t u r e t h a t satisfies t h e f l u t t e r r e q u i r e m e n t s .A l t h o u g ht h es u b j e c to ft h i sr e p o r t is f l u t t e ro p t i m i z a t i o n , i . e . s t r u c t u r a l s y n t h e s i s aimed at a minimum w e i g h t s t r u c t u r e t h a t s a t i s f i e s t h e f l u t t e r r e q u i r e m e n t s , it is u s e f u l t o i n c l u d e f l u t t e r a n a l y s i s , or f l u t - t e r s u r v e y , i n t h i s d i s c u s s i o n , s i n c e t h e r e is a l a r g e common data baseand many common a n a l y t i c a l t o o l s .

S t r u c t u r a l d e s i g n aimed at s a t i s v i n g f l u t t e r r e q u i r e m e n t s m u s t , t o r e s u l t i n a viable a i r p l a n e , a l s o t a k e i n t o a c c o u n t s t r e n g t h r e q u i r e m e n t s a n d r e q u i r e - ments relatedtomanufacturingcost.Examinationof a m e r i t f u n c t i o n t h a t combines s t r u c t u r a l w e i g h t a n d m a n u f a c t u r i n g c o s t falls o u t s i d e t h e s c o p e of t h i ss t u d y . The i n t e r f a c eb e t w e e ns t r u c t u r a ls y n t h e s i sw i t hs t r e n g t h con- s t r a i n t s a n d s y n t h e s i s w i t h f l u t t e r c o n s t r a i n t s is d i s c u s s e d i n S e c t i o n 7.5.

There it i s c o n c l u d e d t h a t a d d i t i o n a l i n v e s t i g a t i o n s i n t h e a r e a s o f s t r u c - tural m o d e l i n g a n d s t r u c t u r a l o p t i m i z a t i o n w i t h combined f l u t t e r a n d s t r e n g t h c o n s t r a i n t s are needed before the best a p p r o a c h t o t o t a l s t r u c t u r a l s y n t h e s i s canbeformulated. The organizationofcomputerprogramsandmodulesdis- c u s s e d i n this s e c t i o n as a p o s s i b l e b a s i s f o r d e f i n i t i o n o f s p e c i f i c a t i o n s f o r computer software envisions combining flutter optimization with satis- f a c t i o n o f stress c o n s t r a i n t s .

8.1 The Complete F l u t t e r Task The c o m p l e t e f l u t t e r task can be considered as b e i n g composed o f three s u b t a s k s , s u b s e q u e n t l y t o be discussed: 1. F l u t t e r s u r v e y o f t h e o r i g i n a l d e s i g n .

2. I n i t i a ls t r u c t u r a lr e s i z i n gt os a t i s f y a l l f l u t t e rr e q u i r e m e n t s .

3. Flutter optimization: weight minimization w h i l e e x p l i c i t l y s a t i s f y i n g f l u t t e r r e q u i r e m e n t s a n d n o t v i o l a t i n g s t r e n g t h r e q u i r e m e n t s .

8.1.1 F l u t t e rS u r v e y - The o r i g i n a l d e s i g n i s definedby t h e e x t e r n a l geometry,by a s t r u c t u r a l mass d i s t r i b u t i o n d e r i v e d b y s a t i s f y i n g s t r e n g t h requirements,byanadditional mass d i s t r i b u t i o n r e p r e s e n t i n g f i x e d , non- s t r u c t u r a l airframe masses(e.g.powerplants,controlsystem,furnishings) and mass d i s t r i b u t i o n sr e p r e s e n t i n gu s e f u ll o a d s( e . g .f u e l ,p a y l o a d ) .

The f l u t t e r s u r v e y is a series o f f l u t t e r a n a l y s e s s u f f i c i e n t t o i d e n t i f y any f l u t t e r d e f i c i e n c y t h a t may e x i s t i n t h e o r i g i n a l d e s i g n over a range of operatingconditions.Although details of the a c t u a le x e c u t i o no ft h ef l u t t e r survey may b e d i f f e r e n t f o r d i f f e r e n t e n g i n e e r i n g f a c i l i t i e s , it is b e l i e v e d t h a tt h ef l o wd i a g r a mi nF i g u r e 8-1 is g e n e r a l l ya p p l i c a b l e . The g e n e r a l procedure i s not new; it i s d i s c u s s e d h e r e i n o r d e r t o r e l a t e it t o t h e o v e r a l l d e s i g np r o c e s s .I nF i g u r e 8-1 o v a lb o x e sd e f i n ee n g i n e e ra c t i o np o i n t s , althoughnotnecessarilymanualoperations;rectangularboxesrepresent com- putingmodules. The computingprocesscanproceedfromonemodule t o t h e n e x t withoutengineeraction,althoughanengineer'sreview may be i n s e r t e d a t any point.

S t a r t i n g w i t h t h e d e s i g n d e f i n i t i o n box i n F i g u r e 8-1, t h e engineerpro- c e e d st op r e p a r e ,n o tn e c e s s a r i l ym a n u a l l y ,s t r u c t u r a l model data, i n e r t i a dataandaerodynamics data. The s t r u c t u r e s module forms t h e s t i f f n e s s and i n e r t i a m a t r i c e s t h a t are u s e di nt h ev i b r a t i o na n a l y s i s . It performs, i f n e c e s s a r y , t h e s t a t i c c o o r d i n a t e r e d u c t i o n t o k e e p t h e number ofdegreesof a d d i t i o n a l m i g h t , s t i f f n e s s configuration( e) """""""""""""" Y I I

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I I I I I I improve I I I I d e f i n i t i o n 1 L"".-.d I h e p a r a t i o n of resulta I I I I I additional Mach numbers I ~ . " - - - - - - - - - - - - - - - " . - , - . - - - - , , J Figure 8-1: Flutter Survey Task freedom f o r t h e v i b r a t i o n a n a l y s i s w i t h i n a p r a c t i c a l limit. The s t r u c t u r e s module may a l s o f o r m t h e i n e r t i a m a t r i x a s s o c i a t e d w i t h t h e masses o f t h e If t h i s is t h ec a s e ,t h ei n e r t i ad a t ap r e p a r e db yt h e s t r u c t u r a le l e m e n t s .

engineer refer t o n o n s t r u c t u r a l a n d u s e f u l l o a d masses o n l y . I n e r t i a m a t r i c e s f o r a number o f u s e f u l l o a d c o n f i g u r a t i o n s , c h o s e n on t h e b a s i s o f e x p e r i e n c e , and t h e o u t p u t o f t h e s t r u c t u r e s module are i n p u t i n t o t h e v i b r a t i o n a n a l y s i s moduleand v i b r a t i o n a n a l y s e s , l e a d i n g t o n a t u r a l f r e q u e n c i e s a n d v i b r a t i o n modes , are performed.

The n a t u r a l f r e q u e n c i e s a n d v i b r a t i o n modes can be reviewed by the engineerforcheckingpurposesand, after t h e f l u t t e r a n a l y s i s , f o r o b t a i n - i n g a b e t t e ru n d e r s t a n d i n go ft h ef l u t t e rb e h a v i o ro ft h ea i r p l a n e .I nt h e caseof a f i n a l d e s i g n , t h e r e s u l t s o f t h e v i b r a t i o n a n a l y s i s c a n be com- p a r e d w i t h t h e r e s u l t s fromgroundvibration tests. From a n a n a l y t i c a l p o i n t of view, however, t h e v i b r a t i o n a n a l y s i s is onlynecessary i f t h e o r i g i n a l number ofdegrees-of-freedomexceeds a p r a c t i c a l l i m i t f o r t h e f l u t t e r a n a l y s i s . I n t h a t c a s e , t h e v i b r a t i o n modes a s s o c i a t e dw i t ht h el o w e rv i b r a - t i o nf r e q u e n c i e s are, i ng e n e r a l ,u s e d as g e n e r a l i z e dc o o r d i n a t e sf o rt h e f l u t t e r a n a l y s i s . The o u t p u to ft h ev i b r a t i o na n a l y s i sc a nb ef o r m u l a t e dt o i n c l u d eg e n e r a l i z e ds t i f f n e s s and i n e r t i am a t r i c e s .I nt h a tc a s e ,t h e func- t i o no ft h eg e n e r a l i z e d - m a t r i c e s module is t o form onlythegeneralizedaero- dynamics m a t r i c e s . However, t o b e more g e n e r a l l yu s e f u l ,t h i s module should a l s o i n c l u d e t h e c a p a b i l i t y o f g e n e r a t i n g g e n e r a l i z e d mass and s t i f f n e s s matrices bypre-andpostmultiplicationby modal m a t r i c e s .T h i sc a p a b i l i t y are notupdated after e a c h r e s i z i n g may beused i f t h e g e n e r a l i z e d c o o r d i n a t e s s t e p .I nv i e wo ft h eo p t i o n sa v a i l a b l ef o rf o r m i n gt h eg e n e r a l i z e da e r o - dynamics m a t r i c e s( S e c t i o n5 . 3 ) ,t h eg e n e r a l i z e d - m a t r i c e s module may do more than a pre- and postmultiplication by t h e modal m a t r i c e s o b t a i n e d f r o m t h e v i b r a t i o na n a l y s i s .C o n s e q u e n t l y ,t h eo u t p u t of the aerodynamics module may b e a s e to f [HAW] matrices(Equation ( 5 . 1 6 ) ) f o rd i s c r e t ev a l u e so ft h e reducedfrequency k , or may b e a s e t of b a s i c aerodynamicinfluencecoeffi- c i e n t m a t r i c e s [ A I C ( k ) ] and matrices [ H I , [DX] and [ D Z ] (Equations ( 5 . 8 ) and ( 5 . 3 2 ) ) , or anycombinationbetweentheseextremes as d i s c u s s e d i n S e c t i o n 5 . 4 . 4 . The aerodynamics data p r e p a r a t i o nc o n s i s t so fd e f i n i n ga e r o - dynamics g r i ds y s t e m sf o rt h e downwash c o l l o c a t i o np o i n t s and t h e aerodynamics l o a d s p o i n t s , and t h e s e l e c t i o n o f Mach numbers f o r which t h e f l u t t e r a n a l y s i s w i l l b ep e r f o r m e d .S t r u c t u r a lg r i dd a t a are inputintotheaerodynamics module t o form t h e g r i d t r a n s f o r m a t i o n m a t r i c e s [HI, [DX] and [DZ].

S t a t i c r e d u c t i o n o f t h e s t i f f n e s s m a t r i x i s a g e n e r a l l y a c c e p t e d method ofreducingthe number o f d e g r e e s o f f r e e d o m i n t h e v i b r a t i o n a n a l y s i s .

Reference 36 p r e s e n t s an a l t e r n a t i v et h a t i s w o r t h . c o n a i d e r a t i o n .I nt h e approach of Reference36, no r e d u c t i o n o f t h e s t i f f n e s s m a t r i x t a k e s p l a c e ; i nf a c t ,t h ec o m p l e t es t i f f n e s sm a t r i x i s notassembled.Instead,theoutput o f t h e s t r u c t u r e s module i s a c o l l e c t i o n o f s u b m a t r i c e s t h a t are u s e d d i r e c t l y i n t h e v i b r a t i o n a n a l y s i s module t o compute g e n e r a l i z e d s t i f f n e s s a n d i n e r t i a d a t a , a n d v i b r a t i o n modes.

From t h e g e n e r a l i z e d - m a t r i c e s module t h ef l u t t e rs u r v e yp r o c e d u r e I e n t e r st h ef l u t t e ra n a l y s i s module. The f l u t t e ra n a l y s i s module should contain an interpolation routine for computing the generalized aerodynamics matrix f o ra r b i t r a r y k v a l u e s . The f l u t t e r e q u a t i o n is solvedbyany s u i t a b l e methodand t h e o u t p u t is a series o f f-g-V diagrams(Figure 3-1) f o r d i f f e r e n t Mach numbers, d i f f e r e n t d i s t r i b u t i o n s o f u s e f u l l o a d a n d ' f o r symmetric,anti-symmetricandpossibly asymmetric modes.

Figure 8-1 i n d i c a t e st h r e ep o t e n t i a lr e a n a l y s i sl o o p s . Any realistic p r o c e d u r e m u s t a c c o u n t f o r t h e p o s s i b i l i t y t h a t t h e i n i t i a l c h o i c e o f i n p u t parameters d o e s n o t p r o v i d e s u f f i c i e n t d e f i n i t i o n o f t h e f l u t t e r c h a r a c t e r . - i s t i c s o f t h e o r i g i n a l d e s i g n . It is p o s s i b l e t h a t t h e i n i t i a l c h o i c e o f k-values or s p e e d s f o r w h i c h t h e f l u t t e r a n a l y s i s is performed is i n s u f f i - c i e n t t o d e t e r m i n e f l u t t e r s p e e d s or minimum damping i n hump modes. Thus, review o f t h e f l u t t e r a n a l y s i s r e s u l t s may r e q u i r e r e t u r n t o t h e f l u t t e r a n a l y s i s module f o r improved d e f i n i t i o no ft h e f-g-V diagrams.Review,of t h e r e s u l t s may a l s o i n d i c a t e the need f o r i n c l u d i n g more Mach numbers, i n e r t i a c o n f i g u r a t i o n s a s s o c i a t e d w i t h d i f f e r e n t u s e f u l l o a d d i s t r i b u t i o n s , or s t i f f n e s s m a t r i c e s i n t h e f l u t t e r s u r v e y . The p o s s i b i l i t yo fi n c l u d i n g more t h a n o n e s t i f f n e s s m a t r i x f o l l o w s from t h e f a c t t h a t f a i l e d c o n d i t i o n s must be considered.

After s u f f i c i e n t f l u t t e r d a t a havebeengenerated, it is determined whether any f l u t t e rd e f i c i e n c i e se x i s t . If t h e r e are no d e f i c i e n c i e s ,t h e f l u t t e r t a s k i s completed,unlessdesignchangesoccurwhich m a k e it neces- s a r yt or e p e a tt h ef l u t t e rs u r v e y .S i n c et h e first s u r v e yr e s u l t e di n e n g i n e e r i n g f a m i l a r i t y w i t h t h e f l u t t e r c h a r a c t e r i s t i c s o f t h e d e s i g n , a d d i t i o n a l s u r v e y s u s u a l l y c a n b e r e s t r i c t e d t o fewer c o m b i n a t i o n s o f i n e r t i a c o n f i g u r a t i o n s , Mach numbers a n d f a i l e d c o n d i t i o n s t h a n were i n v e s t i g a t e d d u r i n g t h e f i r s t survey.

If t h e r e a r e f l u t t e r d e f i c i e n c i e s , a s t r u c t u r a l r e s i z i n g is i n i t i a t e d which i s aimed a t s a t i s f y i n g a l l f l u t t e r r e q u i r e m e n t s .

8.1.2 I n i t i a l S t r u c t u r a l R e s i z i n g - If t h e f l u t t e r s u r v e y o f t h e o r i g i n a l d e s i g ni n d i c a t e st h ep r e s e n c eo ff l u t t e rd e f i c i e n c i e s , it i s d e s i r e d t o remove t h e s ed e f i c i e n c i e sw i t h a minimum weightpenalty.Reference 1 i n d i c a t e st h a t it i s t h e o r e t i c a l l y p o s s i b l e t o a t t a i n a minimum w e i g h t d e s i g n b y j u d i c i o u s l y a d d i n g s t r u c t u r a l mass i n small q u a n t i t i e s t o t h o s e s t r u c t u r a l e l e m e n t s t h a t , at e a c hs t e p ,a r e most e f f i c i e n t i n removing t h ed e f i c i e n c i e s .F o r a s t r u c - t u r e w i t h a l a r g e number o f d e s i g n v a r i a b l e s a n d s e v e r a l f l u t t e r d e f i c i e n c i e s , t h i s i s , however,animpracticableapproach. It i s more e f f i c i e n t t o first g e n e r a t e , i n o n e or very f e w r e s i z i n g s t e p s , a s t r u c t u r e w i t h o u t f l u t t e r d e f i c i e n c i e s , b u t n o t n e c e s s a r i l y w i t h minimum weight, and then minimize the w e i g h tw h i l ea v o i d i n gf l u t t e rd e f i c i e n c i e s . Most methods of f l u t t e ro p t i m i z a - t i o n d i s c u s s e d i n t h i s r e p o r t a r e b a s e d on t h i s a p p r o a c h .

Two t y p e so ff l u t t e rd e f i c i e n c i e s are recognized: 1) t o o low a f l u t t e r speed (Vf < V R ) and 2) i n s u f f i c i e n t damping a t t h et o po f a hump mode Althoughbothdeficiencies are u n d e s i r a b l e , t h e Y . ) .

"hump t o p > max allowed f l u t t e r speeddeficiencyoccurs more frequentlyand i s emphasizedthroughout thisreport.Optimizationtechniquesaimed at s a t i s f y i n gf l u t t e rs p e e d requirementscan be g e n e r a l i z e dt oi n c l u d e damping r e q u i r e m e n t s .I nt h i s discussion,re.ference w i l l be made, f o r c o n v e n i e n c e , t o f l u t t e r s p e e d r e q u i r e - ments, or c o n s t r a i n t s ,o n l y .

Experience at theLockheed-California Company, r e l a t e d t o a . r e a l i s t i c

designenvironment , s u g g e s t s t h a t on thebasisofengineeringjudgment,one

f l u t t e r d e f i c i e n c y o f t e n c a n b e i d e n t i f i e d as being most c r i t i c a l . T h a t i s , f o r a p a r t i c u l a r Mach number a n d u s e f u l l o a d c o n f i g u r a t i o n t h e r e e x i s t s a deficiency,'theremovalofwhich i s e x p e c t e d t o r e s u l t i n a l l f l u t t e r defi- c i e n c i e sb e i n g removed. If t h i s i s not the c a s e ,t h e n one or more a d d i t i o n a l a p p l i c a t i o n s of the following approach w i l l l e a d t o a d e s i g n w i t h o u t f l u t t e r d e f i c i e n c i e s . Neither c a s e ,i ng e n e r a l ,l e a d st o an optimum d e s i g n .

Reference (1) i n d i c a t e s that a r e s i z i n g column i s the most c r i t i c a lf l u t t e rs p e e d , and C d e f i n e s a magnitude where 'rnc s u c h t h a t V = VR, i s a n e f f i c i e n t i n i t i a l r e s i z i n g . The column [*} mc is recognized as t h e g r a d i e n to ft h ef l u t t e rs p e e d .O t h e rd i s t r i b u t i o n s of Ami, however, may be considered, e.g.,

[Ami] = Cspositive elements of (I:'] - - lavmc:ami J)

If it i s d i f f i c u l t t o d e f i n e onemost c r i t i c a l f l u t t e r mode, a r e s i z i n g column based on a weighted sum of two or more f l u t t e r s p e e d g r a d i e n t s may be a b e t t e r approach.Inanycase , i n i t i a l r e s i z i n g s h o u l d t a k e i n t o a c c o u n t t h e e f f i c i e n c y w i t h w h i c h d e s i g n v a r i a b l e s c a n i n c r e a s e t h e f l u t t e r s p e e d and,thus , it i s n e c e s s a r y , at t h i s s t a g e , t o d e f i n e d e s i g n v a r i a b l e s a n d t o d e t e r m i n e p a r t i a l d e r i v a t i v e s o f t h e f l u t t e r s p e e d s w i t h r e s p e c t t o t h e design variables. I n g e n e r a l , t h e i n i t i a l r e s i z i n g column canbedefined as: where t h es u b s c r i p t j refers t ot h ef l u t t e rs p e e d s that are less t h a nt h e required speed.

The i n i t i a l r e s i z i n g p r o c e d u r e , a g a i n , may b e d i f f e r e n t f o r d i f f e r e n t e n g i n e e r i n g f a c i l i t i e s a n d it probablydepends,in i t s d e t a i l s , on t h e p r e - ceding f l u t t e r surveyprocedure as well.as on thesubsequentoptimization p r o c e d u r e st ob eu s e d . With t h i s i n mind t h e e s s e n t i a l features of t h e pro- cedure are shown i nt h ef l o wd i a g r a m of Figure 8-2. Note t h a t i n F i g u r e 8-2 ovalboxes s t i l l i n d i c a t e e n g i n e e r a c t i o n p o i n t s , b u t t h e r e c t a n g u l a r b o x e s no longer define computational modules but computing activity in general, and no attempt i s made t o d e f i n e s p e c i f i c modules.

The endpointofFigure 8-1 i s t h es t a r t i n gp o i n tf o rF i g u r e 8-2: review of t h ef l u t t e rs u r v e yr e s u l t s . If t h e r e are f l u t t e rd e f i c i e n c i e s ,d e s i g n v a r i a b l e s may bedefinedand most c r i t i c a l f l u t t e r c o n d i t i o n s s e l e c t e d . To

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g e n e r a t et h ef l u t t e rs p e e dd e r i v a t i v e s A, t h ec h a r a c t e r i s t i cr o o t s and ami v e c t o r s c o r r e s p o n d i n g t o t h e f l u t t e r p o i n t s must bedetermined(e.g.,bythe two-dimensionalRegula F a l s i f o l l o w e d by a subroutinefordeterminingchar- a c t e r i s t i cv e c t o r s ) .I na d d i t i o n ,d e r i v a t i v e so f t h e mass, s t i f f n e s s and aerodynamicsmatrices are r e q u i r e d . If t h e r e i s a s t a t i cr e d u c t i o no ft h e s t i f f n e s s m a t r i x , t h e s t a t i c r e d u c t i o n m a t r i x must b e e x p l i c i t l y g e n e r a t e d i n o r d e r t o compute t h ed e r i v a t i v e so ft h er e d u c e ds t i f f n e s sm a t r i x . The

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f l u t t e rs p e e dd e r i v a t i v e s - j canbe computed f o l l o w i n gt h ef o r m u l a t i o no f ami Reference 14, a compact versionofwhich i s i n c l u d e di nR e f e r e n c e 1. The

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a n a l y s t may want t o r e v i e w t h e v a l u e s of - j before deciding on t h e d i r e c - am,

t i o no ft h er e s i z i n g column, l(z)] , or it may be formed automatica1,ly

by theprogram.IncrementedFlutterAnalysis(References 1 and 4) , is a con-

venient method of determining each C t h a t r e s u l t si n If j [ f ( - ) ] c o n t a i n so n l yp o s i t i v ee l e m e n t s , t h el a r g e s tv a l u eo f C deter- j mines t h er e s i z i n g column b m i ] a c c o r d i n gt oe q u a t i o n ( 8 . 2 ) which r e s u l t s i n a l l V j 2 VRj. If t h e r e i s uncertaintyabout t h i s r e s u l t ,t h es t r u c t u r e f - l i s incrementedby {Ami} and a new v i b r a t i o na n df l u t t e ra n a l y s i s i s per- formed f o rs e l e c t e dc o m b i n a t i o n so f Mach number andusefulload. If necessary, more c r i t i c a lf l u t t e rs p e e d s V a r e s e l e c t e d a n d t h e p r o c e s s i s repeated.

j The proceduresdescribed, and i l l u s t r a t e d i n F i g u r e 8-2, are aimed at a o n e - s t e p r e s i z i n g t o r e a c h t h e g o a l of s a t i s f y i n g a l l f l u t t e r r e q u i r e m e n t s .

V a r i a t i o n s o f t h i s p r o c e d u r e a r e p o s s i b l e b u t would i n c l u d e t h e same b a s i c computationalmodules.

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I I

t

I Compute Flutter Static Reduction Review cl Analysis Matrix [GR] Exit if a l l flutter requirewntr are ratirfied Figure 8-2: Initial StructuralResizing Task 8.1.3 F l u t t e rO p t i m i z a t i o n - O f t h e t h r e e s u b t a s k s d i s c u s s e d i n t h i s s e c t i o n , t h e f l u t t e r o p t i m i z a t i o n t a s k i s mostdependenton thecomputationalmethods a d o p t e db ya ne n g i n e e r i n gf a c i l i t y . However, t h i s method dependency is m a i n l y c o n c e n t r a t e d i n t h e a c t u a l r e s i z i n g p r o c e d u r e w i t h a c o n s t a n t s e t of generalizedcoordinates(.invariantvibrationmodes). The s t r u c t u r a la n a l y s i s a n d v i b r a t i o n a n a l y s i s , t o be r e p e a t e d several times d u r i n g t h e o p t i m i z a t i o n t a s k , c a n b e d e f i n e d i n g e n e r a l terms, r e l a t i v e l y i n d e p e n d e n t o f t h e r e s i z i n g procedureused.

The methods of optimization discussed in Section 6 d e a l w i t h o n l y a small a s p e c t o f t h e o p t i m i z a t i o n t a s k , namely, t h e r e s i z i n g when given a con- s t a n t s e t o fg e n e r a l i z e dc o o r d i n a t e s l ( i . e . , i n v a r i a n tv i b r a t i o nm o d e s ) . It is most l i k e l y t h a t modal updating i s necessary(Section 4 ) . If a s t a t i cr e d u c - t i o n o f t h e s t i f f n e s s m a t r i x i s u s e d t o r e d u c e t h e number ofdegreesof freedom i n t h e v i b r a t i o n a n a l y s i s ( S e c t i o n 7 . 1 ) , t h e r e d u c e d s t i f f n e s s m a t r i x may b e a n o n l i n e a rf u n c t i o n of t h ed e s i g nv a r i a b l e s .T h i s results i n addi- t i o n a lc o m p u t a t i o n st od e t e r m i n ed e r i v a t i v e s of the s t i f f n e s sm a t r i xa n d p o s s i b l y t h e i n e r t i a m a t r i x .

I n t h i s s e c t i o n t h e o v e r a l l f l u t t e r o p t i m i z a t i o n t a s k , e x c e p t f o r t h e r e s i z i n g f o r a c o n s t a n t set o f g e n e r a l i z e d c o o r d i n a t e s , i s d e l i n e a t e d u n d e r - t h e followingassumptions: 1. There w i l l b e modal updating.

2. S t a t i c r e d u c t i o n of t h e s t i f f n e s s m a t r i x i s r e q u i r e d .

For e a c h r e s i z i n g s t e p t h e d e r i v a t i v e o f t h e r e d u c e d s t i f f n e s s m a t r i x i s 3.

determinedexactly(Equation ( 7 . 6 ) ) .

4. No s t a t i c r e d u c t i o n o f t h e mass m a t r i x i s r e q u i r e d .

There i s o n e a c t i v e f l u t t e r s p e e d c o n s t r a i n t .

5.

6. S t r e n g t hr e q u i r e m e n t sa r e satisfied.

Keeping i n mind these assumptions, a flowdiagram i s formulated(Fig- ure 8-3) t h a td e l i n e a t e st h o s ec o m p u t a t i o n a ls t e p st h a t are considered independentofthe methodchosen f o r d e t e r m i n i n g t h e r e s i z i n g s t e p s , g i v e n t h ef l u t t e rs p e e dd e r i v a t i v e s .S e v e r a lo ft h ec o m p u t a t i o n si n d i c a t e di n Figure 8-3 are i d e n t i c a l t o t h o s e i n F i g u r e 8-2.

The i n p u t si n t ot h ef l u t t e ro p t i m i z a t i o nt a s ka r e :t h es t a r t i n g stiff- nessand mass m a t r i c e s t h a t are o u t p u t f r o m t h e i n i t i a l r e s i z i n g t a s k ; t h e i n c r e m e n t a ls t i f f n e s sa n d mass m a t r i c e s ;t h eb a s i ca e r o d y n a m i c si n p u t ;a n d t h e aerodynamicsderivativesinput.

The s t a r t i n g (first c u r r e n t ) s t i f f n e s s and mass m a t r i c e s are u s e d t o d e f i n eg e n e r a l i z e dc o o r d i n a t e s via a v i b r a t i o na n a l y s i s .T h e s eg e n e r a l i z e d c o o r d i n a t e s a l o n g w i t h t h e a s s o c i a t e d g e n e r a l i z e d s t i f f n e s s a n d mass m a t r i c e s ,

i

Current Aerodynamics

Stiffness 1

Mass Input Hatrix [K] MatrFx [M] Define neu ah- b u m sizing where stress constraint

I ' P

. 1

L

is violated Current Reduced Generalized Determine Vibration

I

Stiffness - Stiffness

-

Coordinates "c rlutter roots

bktrh [KJ M t r k IRK] Analysis

[.I and modes

I I Resizing step(s) with consttrnt generalized coordinates. Minimum sizing defined by ori- ginal stiffness matrix (beforeinitialresizing)

+l Lbrrent Total

Figure 8-3: The Flutter Optimization Task and t h e b a s i c a e r o d y n a m i c s i n p u t are u s e d t o o b t a i n a p o i n t s o l u t i o n of t h e f l u t t e re q u a t i o na n dt h ea s s o c i a t e dc h a r a c t e r i s t i cv e c t o r s . The c h a r a c t e r - i s t i c v e c t o r s are combined i n t u r n w i t h t h e i n c r e m e n t a l s t i f f n e s s m a t r i x p r e -a n dp o s t m u l t i p l i e db yt h es t a t i cr e d u c t i o nm a t r i x [GR], w i t ht h e incremental mass matrix,andwiththeaerodynamicsderivativesinput,toform t h ed e r i v a t i v es c a l a r s( S e eR e f e r e n c e 1). The f l u t t e rs p e e dd e r i v a t i v e s are formedand i n p u t i n t o t h e r e s i z i n g m o d u l e .O t h e ri n p u t si n t ot h er e s i z i n g moduledependon t h e methodof optimization used, but include minimum s i z e c o n s t r a i n t s e q u a l t o t h e o r i g i n a l d e s i g n , i . e . , b e f o r e t h e i n i t i a l r e s i z i n g , andprogramcontrolparameters t o b e c h o s e n by t h e a n a l y s t .

The r e s i z i n g module g e n e r a t e s a column ofdesignvariableincrements d e f i n i n g one s t e p i n a r e s i z i n g p r o c e s s t h a t u s u a l l y c o m p r i s e s - several s t e p s .

The t o t a l mass a s s o c i a t e d w i t h t h e new v a l u e s o f t h e d e s i g n v a r i a b l e s i s compared w i t h t h e p r e v i o u s t o t a l mass. If t h e t o t a l mass hasnotconverged t o a minimum, t h e d e s i g n v a r i a b l e i n c r e m e n t s are used t o g e n e r a t e new c u r r e n t s t i f f n e s s and mass matricesandtheprocess i s repeated.

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

One approach,whichdoesnotinvolvetheresizingmodule, is shown i n Figure 8-3. I n it, when a minimum t o t a l mass i s reached,theloadsand stress a n a l y s i s i s redone. Due t os t i f f e n i n g of t h e o r i g i n a ld e s i g n ,t h e r e may be stress v i o l a t i o n s due t o r e d i s t r i b u t i o n o f i n t e r n a l l o a d s or changes i nt h ee x t e r n a ll o a d s . Where s u c hv i o l a t i o n so c c u r ,t h ee l e m e n ts i z e s are i n c r e a s e d t o satisfy t h e stress c o n s t r a i n t s and new c u r r e n t s t i f f n e s s and mass m a t r i c e s are formedand t h e f l u t t e r o p t i m i z a t i o n i s r e p e a t e du s i n g updated minimum s i z e s . After a new convergence on a minimum t o t a l mass, t h e stresses are checkedagainand, if n e c e s s a r y ,t h ee n t i r ep r o c e s s is repeated.

The increasedelement s i z e w i l l , i n g e n e r a l , c a u s e a change i n f l u t t e r s p e e d .

This i s expected t o b e small, s i n c e t h e s t r e s s v i o l a t i o n s are most l i k e l y t o b e i n e l e m e n t s t h a t h a v e n o t i n c r e a s e d i n s i z e during t h e f l u t t e r o p t i m i z a - t i o n a n d , t h u s , are i n e f f e c t i v e i n c h a n g i n g t h e f l u t t e r s p e e d .

I n a n o t h e r a p p r o a c h t h e r e i s a s t r o n g i n t e r a c t i o n b e t w e e n a stress a n a l y s i s moduleand t h e r e s i z i n g module s u c h t h a t e a c h r e s i z i n g s t e p i s con- s t r a i n e ds u c ht h a ts t r e s sc o n s t r a i n t sa r en o tv i o l a t e d . Two p o s s i b i l i t i e s c a nb ed i s t i n g u i s h e d : t h e l o a d s are assumed c o n s t a n t or t h el o a d sa r er e c a l - c u l a t e d at eachstep.Intheformercasetheloadsand stress a n a l y s i s shown i n F i g u r e 8-3 must f o l l o w t h e f l u t t e r o p t i m i z a t i o n as d e s c r i b e d i n t h e p r e v i o u s paragraph.

It i s n o t e dt h a tt h ea e r o d y n a m i c si n p u t ,i nF i g u r e 8-3, i s n o td e f i n e d i n terms o fs p e c i f i cm a t r i c e s . The d i s c u s s i o n si nS e c t i o n 5 andReference 1 i n d i c a t e t h a t t h e r e are many o p t i o n s f o r f o r m u l a t i n g t h e m a t r i c e s o f g e n e r a l - izedaerodynamicforcecoefficients,and it i s consideredoutsidethescope o ft h i sd i s c u s s i o nt op r e s e n t a d e f i n i t ec h o i c e . It i s worthwhilenoting, however, t h a t i f c u b i c s p l i n e i n t e r p o l a t i o n i s usedfortheaerodynamics, thebasicaerodynamicsinputandtheaerodynamicsderivativesinput are i d e n t i c a l m a t r i c e s .

Since norecommendations f o r a s p e c i f i c methodof o p t i m i z a t i o n are made as a r e s u l t o f t h i s s t u d y , a f u r t h e r d e f i n i t i o n o f t h e r e s i z i n g module, i n Figure 8-3, i s c o n s i d e r e d t o f a l l o u t s i d e t h e s c o p e o f t h i s r e p o r t .

8.2 Aspectsofthe Computing System G e n e r a l a s p e c t s o f t h e c o m p u t i n g s y s t e m r e q u i r e d t o p e r f o r m t h e f l u t t e r t a s k are d i s c u s s e d w i t h o u t a t t e m p t i n g t o d e f i n e d e t a i l e d s p e c i f i c a t i o n s f o r such a system.

well d e f i n e db u i l d i n gb l o c k st h ec o m p l e t e , , When p r o p e r l y .d i v i d e di n t o f l u t t e r t a s k , i n c l u d i n g t h e o p t i m i z a t i o n , i s r e l a t i v e l y s t r a i g h t f o r w a r d , seems e s p e c i a l l y i n v i t i n ge x t e n s i v ea u t o m a t i o n . The f l u t t e ro p t i m i z a t i o nt a s k w e l l s u i t e df o rc o m p l e t ea u t o m a t i o n . It is questionable,however,whether t h ea c t u a ld e s i g ne x p e r i e n c ea v a i l a b l e i s s u f f i c i e n t t o d e c i d e on a l l a s p e c t s o f t h e o p t i m i z a t i o n t a s k a n d t o embark on t h e d e s i g n o f a computingsystem t h a t c a n h a n d l e e f f i c i e n t l y a s t r u c t u r a l d e s i g n t h a t i s d e f i n e d by s e v e r a l t h o u s a n d so ff i n i t ee l e m e n t s .I n view of t h i st h eL o c k h e e d - C a l i f o r n i a Company has f i r s t developed a semi-automaticsystem,based on i t s Computer Graphics system. It hasbeenused on t h ec o n f i g u r a t i o n upon which thenumerical examplesofAppendix A are based, onan arrowwingsupersonictransportstudy (Reference 1) and on an ac.tualhardwareproblem. The a u t h o r sb e l i e v e , how- e v e r , t h a t a batchprocesssystemwith maximum automationoptions i s a d e s i r a b l ed e s i g n asset, even i f i n i t s i n i t i a l v e r s i o n it i s somewhat r e s t r i c t e d i n t h e number o f s t r u c t u r a l d e g r e e s o f f r e e d o m , d e s i g n v a r i a b l e s and f l u t t e rc o n s t r a i n t s it c a nh a n d l e .I nt h i ss e c t i o n some a s p e c t so f a batchprocesscomputingsystem are discussed.

A n e n g i n e e r i n g f a c i l i t y t h a t a l r e a d y h a s a c o m p u t i n g s y s t e m f o r f l u t t e r a n a l y s i s ( f l u t t e r s u r v e y t a s k ) may want t o r e s t r i c t i t s batchprocesssystem t o t h e i n i t i a l r e s i z i n g t a s k and t h e f l u t t e r o p t i m i z a t i o n t a s k . T h i s , how- e v e r , seems o n l y j u s t i f i e d i f t h e d a t a i n p u t andoutput of t h e e x i s t i n g f l u t - t e r a n a l y s i s s y s t e m c a n e a s i l y b e made compatiblewiththeinputrequirements f o rt h eo t h e rt a s k s .S i n c et h er e p e t i t i v ea n a l y s e sa s s o c i a t e dw i t hf l u t t e r optimizationputextraemphasis on c o m p u t a t i o n a l e f f i c i e n c y , a f a c i l i t y may d e c i d e t o u p d a t e i t s f l u t t e r a n a l y s i s s y s t e m as p a r t of t h e i n t r o d u c t i o n o f a f l u t t e ro p t i m i z a t i o nc a p a b i l i t y .I nt h ef o l l o w i n g it i s assumed t h a t a b a t c h p r o c e s s s y s t e m f o r t h e c o m p l e t e f l u t t e r t a s k i s t o be designed.

When comparing theflowdiagrams i n t h e F i g u r e s 8-1 through 8-3 it i s c l e a r t h a t t h e t h r e e t a s k s r e p r e s e n t e d i n t h e s e t h r e e f i g u r e s h a v e a l a r g e number ofcomputationalfunctions i n common. Thus, t h e f i r s t p o i n to f con- s i d e r a t i o n i s whether three independent programs should be developed within an existing computing system or whether oneprogram, or a new system should b ed e v e l o p e df o rt h ec o m p l e t ef l u t t e rt a s k . The choicedependsonwhat computingsystem i s a v a i l a b l e at a f a c i l i t y a n d , t o a c e r t a i n e x t e n t , p e r s o n a l p r e f e r e n c e so ft h ee n g i n e e r s . From a p r a c t i c a le n g i n e e r i n gp o i n to f v i e w it seems s e l f - e v i d e n tt h a t ,w h a t e v e rc o u r s e i s chosen,dataformatcompatibility i s mandatoryand t h a t a d a t a management system i s r e q u i r e d .

9 5 I n s e l e c t i n g a computing system f o r t h e c o m p l e t e f l u t t e r t a s k , con- s i d e r a t i o n s h o u l d be given t o a s y s t e m t h a t h a s a c c e s s t o a n e x i s t i n g matrix algebracomputingsystem. If t h i sa c c e s s is n o ta v a i l a b l e ,t h ec o m p l e t e f l u t - ' t e r t a s k s y s t e m s h o u l d i n c l u d e some g e n e r a l i z e d matrix a l g e b r a c a p a b i l i t y t o e n a b l e t h e u s e r t o d e p a r t f r o m a r i g i d f o r m a t .

I n d e s i g n i n g a computersystem f o r t h e c o m p l e t e f l u t t e r t a s k , two approachescanbedistinguished:Self-containedProgramsand Variable Job Stepping.

I n t h e f i r s t approach,which i s t h e more common o f t h e t w o , t h e r e may b e a self-contained program for each of the subtasks comprising the complete f l u t t e r t a s k , or some or a l l s u b t a s k s may b e combined i n t o oneself-contained program.Eachprogramhas i t s own e x e c u t i v e module which c o n t r o l sc a l l i n g i n t oc o r et h ev a r i o u sc o m p u t i n g modules(sub-programs) as t h e y are needed d u r i n gt h ee n t i r ec o m p u t e ro p e r a t i o nf o rt h a tp r o g r a m .O r g a n i z i n gt h e com- p l e t e f l u t t e r task i n oneprogramwithoneexecutive modulewould r e s u l t i n a verylargeprogramwith some complex input/outputinterfaceproblems as w e l l as coreoverlayproblems. If e x i s t i n gb a t c h programs are t o b e i n t e g r a t e d intosuchanoverallcomputingprogram,extensivemodificationmightbeneeded i n o r d e r t o r e s o l v e some oftheseproblems. If t h e c o m p l e t e f l u t t e r t a s k i s coveredby more thanoneprogram,automatictransferfromoneprogram t o anothermightproveimpracticable,thuslimitingtheoverallautomation a t t a i n a b l e .S i n c ee n g i n e e rr e v i e w s are r e q u i r e dd u r i n gt h ec o m p u t a t i o n a l e f f o r t , however, t h i s may n o tb e a ni m p o r t a n tl i m i t a t i o n .C o n s i d e r a t i o n must be g i v e n t o t h e d e g r e e o f commonalitybetweenmodules i n t h e s e p a r a t e pro- grams t h a tp e r f o r mt h e same function.Failingtoachievecompletecommonality, e.g., due t od i f f e r e n to v e r l a yr e q u i r e m e n t s ,i n c r e a s e st h ee f f o r tn e e d e dt o update a f u n c t i o n a l module.

V a r i a b l e J o b S t e p p i n g c o n s i s t s o f a number ofseparatecomputerprograms eachrepresenting a computingmodule,such as t h o s e d e f i n e d i n F i g u r e 8-1, which a r ec o n t r o l l e db ya n o t h e r program c a l l e dt h eE x e c u t i v e .V a r i a b l eJ o b S t e p p i n g ,t h e r e f o r e , i s a sequenceofseparatecomputerjobstepsinwhich e a c hj o bs t e p i s a f u n c t i o n o f t h e p r e c e d i n g j o b s t e p ( s ) as determinedby theExecutive Module. The Executive Module ( a separateprogram)monitors t h e t a s k c o m p l e t i o n c o d e s o f t h e p r e c e d i n g s t e p s a n d , b a s e d o n t h e i n s t r u c t i o n codesuppliedbytheengineer,determineswhichcomputing module i s next r e q u i r e d . P r i o r t o t r a n s f e r r i n g t h e c o n t r o l t o t h e computingmodule, t h e Executive Module p r e p a r e s t h e i n p u t d a t a for t h a t computingmodule i n accor- dancewith i t s dataformatrequirements. The VariableJobSteppingsystem i s i n h e r e n t l y modular i n approach. T o addanothermodule,onlytheExecutive Module (program)needs t o b e m o d i f i e d andreloaded as a new executablepro- gram i n a d d i t i o n t o t h e new computingmodule. Existingbatchprocessprograms c a n b e i n c l u d e d i n a Variable Job Stepping system with l i t t l e m o d i f i c a t i o n t o thebatchprograms. If a new method f o r computing,say,aerodynamicmatrices , becomes a v a i l a b l e ,a g a i no n l yt h eE x e c u t i v e Module needs t ob em o d i f i e d . It i s a l s o w o r t h n o t i n g t h a t e a c h program w i t h i n t h e Variable Job Stepping system could be executed as anindependentprogramoutsidetheExecutive Module Control.Figure 8-4 illustrates t h ep r i n c i p l e . The Executive i s loadedand Core System Data Disc or DM External Storage EXEC i s loaded.

The user supplied coC.ing i s interpreted.

Themodule t o beloculed into core i s , established (e.g., k d u l e C).

Writ Executive Input data for Module: C i s w r i t t e n on Disc i n the fbrrat required by Module C.

The data required by E E C when reloaded i s written on Disc.

""""-b""""" Module Module C i s loadedoverthe same space occupied by EXEC.

The required data i s read from Disc by Module C.

Data output fromModule C i s written on Disc.

fibdule C completion code and s t a t u s f o r use by EXEC i s written on Disc.

Desirable Features 1 " " " " " " " " " .

o KO system overhead except EXEC i s loaded. EXEC: data i s read data preparation by EXEC.

and the next module t o be loaded i s o The EXEC and all computing determined.

modules have access t o " " 1 1 " " " " " " " the fill coreallocated .

s t o the computer run.

s

Figure 6-4: Computing System Using Variable Job Stepping

c a l l s i n t h e n e x t module needed,say C y which is l o a d e d o v e r t h e same space as w a s occupiedbytheExecutive Module. When module C hascompleted i t s t a s k it c a l l s b a c k t h e E x e c u t i v e Module, a t t h e same time g i v i n g it i n s t r u c - t i o n s t h a t depend on t h e o u t p u t of module C . The Executive Module c a l l s i n the next computational module, again while annihilating i t s e l f fromcore.

Which o f t h e twoapproachesdiscussed i s preferreddepends on many f a c t o r s a n d , w i t h o u t a more in-depthlook at t h e d e t a i l s o f t h e computing system,cannot be determined at t h i s time.

A s s t a t e da b o v e ,t h ec o m p l e t ef l u t t e r task i s r e l a t i v e l y s t r a i g h t f o r - ward, when properly defined, and w i l l , i n p r i n c i p l e a t least , n o t g i v e rise' t o g r e a t programming d i f f i c u l t i e s . The g r e a tc h a l l e n g ei nd e s i g n i n gt h e system l i e s i n maximizing i t s c a p a c i t y , i n terms o f number o f s t r u c t u r a l c o o r d i n a t e s ,d e s i g nv a r i a b l e sa n df l u t t e rc o n s t r a i n t s ,w h i l ek e e p i n g computing c o s tr e a s o n a b l e . This doesrequire a v e r ye f f i c i e n tu s eo f a l l computer resources : maximum u t i l i z a t i o n o f a v a i l a b l e c o r e , e f f i c i e n t i n p u t / o u t p u t r o u t i n e s , s y s t e m a t i c l a b e l l i n g and e x t e r n a l s t o r a g e o f data blocks , e f f i c i e n t compactingofdatablocks ( e . g .s p a r s em a t r i c e s ) , e t c .O b v i o u s l y ,c l o s e c o o p e r a t i o n b e t w e e n e x p e r t s i n f l u t t e r a n a l y s i s , optimization procedures and computerprogramming i s r e q u i r e d i n o r d e r t o o b t a i n an e f f i c i e n t computing system.

9 . CONCLUSIONS AND RFlCOMMENDATIONS Based on t h e i n v e s t i g a t i o n s and evaluationsperformedduringthisstudy andaugmentedby t h e a d d i t i o n a l s u p p o r t i n g a c t i v i t i e s c o n d u c t e d c o n c u r r e n t l y at theLockheed-California Company, it i s concluded t h a t a n e f f i c i e n t , p r a c t i c a b l e f l u t t e r o p t i m i z a t i o n module canbeformulatedandimplemented with a reasonable amount offurtherdevelopment. Some of t h e p a r t i c u l a r con- c l u s i o n ss u p p o r t i n gt h i sg e n e r a lc o n c l u s i o na r ep r e s e n t e di nt h ef o l l o w i n g : Aerodynamics parameters may b e e f f i c i e n t l y r e p r e s e n t e d by t h e f i v e - matrixproduct shown i ne q u a t i o n ( 5 . 1 2 ) . The most e f f i c i e n t method of imple- m e n t a t i o n o f t h i s e q u a t i o n depends upon a complex r e l a t i o n s h i p i n v o l v i n g t h e numbers of aerodynamic integration points, downwash p o i n t s , g e n e r a l i z e d modal c o o r d i n a t e s ,d i s c r e t es t r u c t u r a lc o o r d i n a t e s andreducedfrequenciesused, as w e l l as t h ei n t e r p o l a t i o np r o c e d u r e s employed. To becompletelygeneral, s e v e r a l o p t i o n s s h o u l d b e a v a i l a b l e i n o r d e r t o p r o v i d e a l t e r n a t e p r o c e d u r e s f o rt h eg e n e r a t i o no ft h e s ep a r a m e t e r s .I nr e l a t i o nt ot h ec o m p l e t ef l u t t e r optimizationtask,however, it i s concluded t h a t s e v e r e l y l i m i t i n g t h e number of suchoptionswouldnotsignificantlyincreasetherequiredcomputing r e s o u r c e s .

R e p e t i t i v e f l u t t e r s o l u t i o n s of t h e t y p e r e q u i r e d i n s t r u c t u r a l r e s i z i n g procedures may b e e f f i c i e n t l y o b t a i n e d u s i n g t h e 2-D Regula F a l s i method.

A l t h o u g h o t h e r f l u t t e r s o l u t i o n p r o c e d u r e s m i g h t b e d e v e l o p e d t o t h e same degree of efficiency and reliability demonstrated by the Regula Falsi in o b t a i n i n g r e p e t i t i v e f l u t t e r s o l u t i o n s , it i s c o n c l u d e d t h a t t h i s p r o c e d u r e a widerrangeofapplicationthanotherproceduresconsidered. The 2-D has Regula F a l s i methodcan b e u s e d i n t h e s o l u t i o n o f a general form of the f l u t t e r e q u a t i o n i n o r d e r t o o b t a i n t h e v a l u e s o f anytwodependent variables r e q u i r e d t o s a t i s f y t h e e q u a t i o n . It can be u s e d i n a d i r e c t formofIncre- mented F l u t t e r A n a l y s i s t o s o l v e f o r t h e m a g n i t u d e o f t h e i n c r e m e n t o f a s p e c i f i e d d e s i g n v a r i a b l e , a l o n g w i t h a n a d d i t i o n a l v a r i a b l e , n e c e s s a x y t o satisfy a f l u t t e rc o n s t r a i n t .O t h e ra p p l i c a t i o n s , such as t h ed e t e r m i n a t i o n o f t h e minimum damping o f a hump mode, are a l s o p o s s i b l e .

R e s i z i n g p r o c e d u r e s v a r y w i d e l y i n p a r t i c u l a r d e t a i l , w i t h e a c h p r o c e - d u r ec o n s i d e r e de x h i b i t i n g one or more advantageous features. I nt e r m so f a realistie; d e s i g n . e f f o r t , however, it i s c o n c l u d e d t h a t t h e g e n e r a l d i f f e r - e n c e s b e t w e e n t h e a r b i t r a r y s t e p - s i z e p r o c e d u r e s a n d t h e d e f i n e d s t e p - s i z e procedures are more s i g n i f i c a n t t h a n t h e d i f f e r e n c e s b e t w e e n t h e i n d i v i d u a l procedures i ne a c hc a t e g o r y . Based on t h en u m e r i c a le v a l u a t i o n so ft h e i d e a l i z e d t e s t c a s e o f Appendix A , it would appear that the arbitrary s t e p - s i z e proceduresproducethe same mass r e d u c t i o n as t h e d e f i n e d s t e p s i z e procedures w i t h less t o t a l computing c o s t . It is n o tc l e a rt h a tt h i s same r e s u l t would b e o b t a i n e d f o r a more complex d e s i g n e f f o r t . It i s concluded, be i n c l u d e d i n a f l u t t e r o p t i m i z a - however, t h a t e i t h e r t y p e o f p r o c e d u r e c a n t i o n module w i t h noundue d i f f i c u l t y .

P r a c t i c a l c o n s i d e r a t i o n s i n t h e implementationof a f l u t t e r o p t i m i z a t i o n procedurecanhave a t least as profound an e f f e c t on theperformanceofthe f l u t t e r module as does t h e s e l e c t i o n of t h e three majorelementsconsidered t h u s far. These c o n s i d e r a t i o n si n c l u d e the c h o i c eo fs t r u c t u r a lm o d e l , num- b e r o f s t r u c t u r a l d e g r e e s o f freedom r e t a i n e d , method of g e n e r a t i o n o f i n c r e m e n t a l s t i f f n e s s m a t r i c e s , number ofmodalcoordinatesused,frequency of updating vibration modes andfrequency of u p d a t i n g f l u t t e r d e r i v a t i v e s .

No d e f i n i t ec o n c l u s i o n sa r ea v a i l a b l er e g a r d i n gt h e s ec o n s i d e r a t i o n s ,s i n c e it w a s n o t p o s s i b l e t o c o n d u c t t h e r e q u i r e d i n v e s t i g a t i o n s w i t h i n t h e s c o p e o f t h e p r e s e n t s t u d y .

The conclusionsreachedin the courseof t h e p r e s e n t s t u d y , a n d p r e s e n t e d above, lead t o t h e followingrecommendations f o r development of a f l u t t e r o p t i m i z a t i o n module: 1. Computer coding for the aerodynamics submodule should be based on the _ .

five-matrixproductofequation ( 5 . 1 2 ) . The number o f o p t i o n s i n form- i n g t h e f i v e - m a t r i x p r o d u c t s h o u l d be limited t o t h o s e forms that are most g e n e r a l l y u s e f u l .

2. The f l u t t e rs o l u t i o n submodule f o rt h er e p e t i t i v ef l u t t e rs o l u t i o np r o - cedure should be based on t h e two-dimensionalRegulaFalsiapproach.

A global solution procedure, such as t h e p-k method ofReference 3 or t h e Desmarais-Bennettmethod(Reference 7) should be i n c l u d e d f o r t h e i n i t i a l and f i n a l f l u t t e r s u r v e y .

3. A t least twocandidate'resizingproceduresshould be e v a l u a t e di n a r e a l i s t f c d e s i g n e f f o r t s u c h as t h e a r r o w w i n g f l u t t e r o p t i m i z a t i o n t a s k performedunderContract NAS-1-12288. One r e s i z i n gp r o c e d u r es h o u l d be o f t h e arbitrary s t e p - s i z e t y p e a n d t h e o t h e r a d e f i n e d s t e p s i z e t y p e .

To f a c i l i t a t e t h e c o m p a r i s o n , it would b e u s e f u l t o m a i n t a i n as much ' commonalitybetween t h e two methods as p o s s i b l e . As anexample,the r e s i z i n g p r o c e d u r e f o r m u l a t e d i n S e c t i o n 6.7.4 could be used as t h e d e f i n e d s t e p - s i z e p r o c e d u r e , a n d t h e g r a d i e n t p r o j e c t i o n s e a r c h o f R u d i s i l l - B h a t i a ( S e c t i o n 6 . 2 ) , m o d i f i e d t o i n c o r p o r a t e t h e move v e c t o r of Section A . 3 . 3 , could be used as t h e arbitrary step-size procedure.

4. Depending on t h e outcome o ft h e s ee v a l u a t i o n s ,s p e c i f i c a t i o n ss h o u l d bedeveloped for t h e s e l e c t e d p r o c e d u r e a n d t h e r e q u i r e d c o m p u t e r c o d i n g accomplished. It s h o u l db er e c o g n i z e dt h a t it may be d e s i r a b l e t o retain t h e o p t i o n o f u s i n g e i t h e r t y p e o f r e s i z i n g p r o c e d u r e i n t h e f i n a l module .

5 . I n v e s t i g a t i o no ft h ef l u t t e ro p t i m i z a t i o nt a s ks h o u l db ee x t e n d e dt o i n c l u d e t h o s e p r o b l e m s e n c o u n t e r e d i n a r e a l i s t i c d e s i g n 'environment whichhave a d i r e c t i n f l u e n c e on the performance o f an optimization procedure.TheseproblemsarediscussedinSection 7 andmentioned b r i e f l yi nt h ec o n c l u s i o n sa b o v e . To beof most u s e , it i s f e l t t h a t s u c h i n v e s t i g a t i o n s mustbe made u s i n g a s t r u c t u r a l d e s i g n t a s k o f t h e complexity of the arrow wing study performed by Lockheed f o r NASA ( DTAS-1-12288) .

6. The e x t e n tt o which it i s f e a s i b l ea n dd e s i r a b l et oi n t e g r a t et h ef l u t t e r o p t i m i z a t i o n t a s k w i t h t h e s t r e n g t h o p t i m i z a t i o n t a s k s h o u l d be i n v e s t i - gatedwithemphasison means o f s i m p l i f y i n g t h e f o r m u l a t i o n o f t h e s t r e n g t h c o n s t r a i n t s .

APPENDIX A NUMERICAL EXAMPLES OF R E S I Z I N G PROCEDURES A. 1 INTRODUCTION To p r o v i d e t h e b a s i s f o r a d i r e c t comparisonofthe several candidate r e s i z i n g p r o c e d u r e s ( S e c t i o n 6 ) i n performing a s i m p l i f i e d f l u t t e r o p t i m i z a - t i o n t a s k , an i d e a l i z e d t e s t case was formulated and numerical evaluations were conducted. Not a l l candidateprocedures were e v a l u a t e dt ot h e same d e g r e e .I n most c a s e st h ep r o c e s s was d i s c o n t i n u e d as soon as t h er e s u l t so f i n t e r e s t were obtained.

The s t r u c t u r a l modelonwhich t h e test c a s e w a s based i s a simple E I , GJ beam r e p r e s e n t a t i o no f a subsonictransportairplane.Althoughnoattempt was made t o simulate a realistic d e s i g n p r o c e s s i n d e t a i l , t h e r e s u l t i n g optimization might be regarded as t y p i c a l of t h e p r e l i m i n a r y d e s i g n p h a s e o f t h e developmentofsuchanairplane. The test case was d e s i g n e d t o a v o i d t h e d i f f i c u l t i e s a s s o c i a t e d w i t h m o d a l i z a t i o n ( S e c t i o n 4) and t h e n o n l i n e a r s t i f f n e s s e f f e c t s ( S e c t i o n 7.1) e n c o u n t e r e d i n more p r a c t i c a l o p t i m i z a t i o n e f f o r t s .

I n i m p l e m e n t i n g t h e v a r i o u s r e s i z i n g p r o c e d u r e s , no specialized computer programs were f o r m e d .I n s t e a d ,e x i s t i n gb a t c h ,g r a p h i c sa n dr e m o t et e r m i n a l systems andprograms were employed,augmented by handcomputationswhere

i necessary. As a result , no d i r e c t comparison of the computer resources

.i; . r e q u i r e d b y t h e v a r i o u s r e s i z i n g p r o c e d u r e s i s a v a i l a b l e ; s u c h i n f o r m a t i o n f relative t o t h i s i d e a l i z e d test c a s e is of l i t t l e p r a c t i c a l s i g n i f i c a n c e i n any e v e n t .

A. 2 STRUCTURAL MODEL The s t r u c t u r a l model used for t h e test c a s e i s an E I , G J beam r e p r e s e n t a - t i o n o f a s u b s o n i c t r a n s p o r t a i r p l a n e .

There are 9 grid-points on t h e wingsemi-span, 1 2 g r i d - p o i n t s a l o n g t h e f u s e l a g e c e n t e r l i n e ( F i g u r e A-1) andothermiscellaneousgrid-pointsusedto d e f i n e a r i g i d empennageand c o n t r o l s u r f a c e , f o r a t o t a l o f 67 e l a s t i c degrees of freedom. Symmetric boundary conditions are imposed. A l l i n e r t i a and aerodynamicscoordinates are r e t a i n e d .E i g h t e e ne l a s t i cd e g r e e s of freedom,notassociatedwithdesignvariables, are e l i m i n a t e d b y s t a t i c s t i f f - nesscondensation,reducing t h e number ofdegreesoffreedomto 49.

The design variables are t h e t o r s i o n a l s t i f f n e s s e s o f t h e e i g h t s t r u c - tural e l e m e n t si n d i c a t e di nF i g u r e A-1. The b e n d i n gs t i f f n e s s of t h e wing i s n o tv a r i e d . The d e s i g nv a r i a b l e sa r ed e f i n e d as incrementsovervalues d e f i n i n g a s i m u l a t e d s t r e n g t h d e s i g n a n d t h e a s s o c i a t e d i n e r t i a a n d s t i f f n e s s m a t r i c e s are e x p r e s s e d i n t e r m s o f a u n i t mass, s o that t h e t o t a l i n e r t i a and s t i f f n e s s m a t r i c e s are as e x p r e s s e di ne q u a t i o n s ( A . l ) and ( A . 2 ) .

a

[K] = [KO] + E mi+i]

i=l

{MI = [Mol + E m. 1 bi]

i=l 10 1 " rigid element elastic element F i g u r e A-1: S t r u c t u r a l Model where ko] and Po] are t h e m a t r i c e s o f t h e f i x e d s t i f f n e s s a n d i n e r t i a and pi]- and F M J are t h e s t i f f n e s s and i n e r t i a m a t r i c e s a s s o c i a t e d w i t h a u n i t mass o ft h ed e s i g n variable m For t h ep r e s e n tp u r p o s e s ,t h e i' cr s t r e n g t h d e s i g n e d p o r t i o n s o f t h e d e s i g n v a r i a b l e s are minimum-size i n c l u d e di nt h e K m a t r i c e s , s o t h a tt h e AKi, AMi matrices include

0' Mo

o n l y t h e d e s i g n variable i n c r e m e n t s r e l a t i v e t o t h e s t r e n g t h d e s i g n .

The s t r e n g t h d e s i g n e d c o n f i g u r a t i o n i s a f i c t i t i o u s c o n f i g u r a t i o n d e f i n e d s u c h t h a t a f l u t t e r p r o b l e m i s a s s u r e d w i t h i n t h e d e s i g n e n v e l o p e o f t h e air- p l a n e . The c h a r a c t e r i s t i c s o f t h i s c r i t i c a l f l u t t e r r o o t a r e shown i n Fig- ure A-2, i n d i c a t i n g a f l u t t e r speed of approximately 226.4 m / s EAS (440 KEAS).

Using t h i s as a s t a r t i n g p o i n t , t h e t e s t c a s e r e q u i r e d t h a t t h e f l u t t e r s p e e d be i n c r e a s e d t o 270.1 m / s EAS (525 KEAS) with a minimum o f w e i g h t i n c r e a s e i n t h e t o r s i o n a l s t i f f n e s s d e s i g n v a r i a b l e s .

10 2 4 . 0 Requency Hz 2.0 1 . 0

. 0 2

. 0 1 Logarithnic Increment 0

- -02

Figure A-2: Frequencyand Damping of C r i t i c a l F l u t t e r Root I n a d d i t i o n t o t h e f l u t t e r - d e f i c i e n t ( 2 2 6 . 4 m / s E M ) c o n f i g u r a t i o n , two a u x i l i a r yc o n f i g u r a t i o n s are r e q u i r e d .F o rt h ec o n s t a n tf l u t t e rs p e e dp r o - cedures, a c o n f i g u r a t i o n h a v i n g t h e r e q u i r e d f l u t t e r s p e e d (270.1 m / s E M ) b u t a non-optimum d i s t r i b u t i o no ft h ed e s i g nv a r i a b l e s i s needed.This was ‘ o b t a i n e d b y i n c r e a s i n g t h e d e s i g n v a r i a b l e s i n a manner e q u i v a l e n t t o r a i s i n g a u n i f o r m f a c t o r u n t i l t h e r e q u i r e d t h e t o r s i o n a l s t i f f n e s s o f t h e wingby f l u t t e r s p e e d w a s reached.Forthepenaltyfunctionprocedure(Reference 1 6 ) , an i n i t i a l c o n f i g u r a t i o n h a v i n g a f l u t t e r s p e e d i n e x c e s s o f t h e f i n a l f l u t t e r speed is r e q u i r e d .T h i sc o n f i g u r a t i o n w a s o b t a i n e d i n t h e same manner as t h e p r e v i o u s o n e , e x c e p t t h a t t h e f l u t t e r s p e e d w a s i n c r e a s e d t o 280.4 m/s EAS (545 K E A S ) . The d e s i g nv a r i a b l ed i s t r i b u t i o n sa n dt o t a ld e s i g nv a r i a b l e mass f o r t h e s e c o n f i g u r a t i o n s a r e shown i n T a b l e A-1. A man-in-the-loop r e s i z i n g p r o c e d u r e , d i f f e r e n t from any o f t h e methods d i s c u s s e d i n S e c t i o n 6 and u s i n g Incremented Flutter Analysis as t h e p r i n c i p a l t o o l , i n d i c a t e d t h a t the mini- . mum mass f o r t h e r e q u i r e d f l u t t e r speed i s approximately 249.6 kg (550.2 lbs) .

Thisvalue i s used as a bench mark forfurthercomparisons.

10 3

I F l u t t e r I Design Variable Mass, kg* I

Speed

m / s EAS Tot a1 a 6 4 3 2 1

7 5 226.4 0 0 0 0 0 0 0 0 0 21.8 31.3 63.6 113.5 506.6 15.3 65.5 90.4 105.2 270.1 280.4 630.9 . 81.6 131.0 141.3 19.1 27.1 . 38.9 79.2 112.5 * I? a c c o r d a n c e w i t h t h e d e f i n i t i o n o f d e s i g n variable t h i s i s a mass i n c r e a s e above t h e s i m u l a t e d s t r e n g t h level design.

Table A-1: I n i t i a lC o n f i g u r a t i o n ; Non-Optimum D i s t r i b u t i o n A. 3 METHOD OF SIMODYNES The method o f Simodynes i s r e p o r t e d i n R e f e r e n c e 15 a n d d i s c u s s e d i n S e c t i o n 6.3 o f t h i s r e p o r t . The n u m e r i c a le v a l u a t i o n sr e p o r t e dh e r e are i n f a c te v a l u a t i o n s , or p a r t i a le v a l u a t i o n s ,o ft h r e ed i s t i n c tm e t h o d s . The Simodynesmethod i t s e l f was e v a l u a t e d o n l y t o t h e e x t e n t o f t h e first r e s i z i n g c y c l e . A t t h a t p o i n t , twoundesirable features o f t h e method were i d e n t i f i e d and a m o d i f i c a t i o n o f t h e method was implemented.Thenumerical e v a l u a t i o n was t h e nc o n t i n u e d ,u s i n gt h i sm o d i f i e d method. I n t h em o d i f i e d methodone o f t h e o b j e c t i o n a b l e features o f t h e o r i g i n a l method, t h e f r e q u e n c y c o n s t r a i n t , i s avoided, as is d i s c u s s e di nS e c t i o n 6.3. F i n a l l y , a second modification was i n c o r p o r a t e d b y w h i c h t h e i n f l u e n c e o f t h e c h o i c e o f t h e dependentdesignvariable on t h e r e s i z i n g s t e p is eliminated,andanaddi- t i o n a l n u m e r i c a l e v a l u a t i o n was performed using this procedure.

A.3.1 O r i g i n a l Method of Simodynes - As d i s c u s s e d i n S e c t i o n 6.3, t h e method of Simodynes i s a procedure for m i n i m i z i n g t h e t o t a l w e i g h t o f a set o f d e s i g nv a r i a b l e sw h i l em a i n t a i n i n g a f i x e df l u t t e rs p e e da n df r e q u e n c y .I n o r d e r t o s a t i s f y t h e s e c o n s t r a i n t s , two o f t h e d e s i g n variable masses are considered to be dependent functions of the remaining design variable masses.

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

A n i n i t i a l r e s i z i n g s t e p was g e n e r a t e d f o r a n a r b i t r a r i l y c h o s e n t o t a l mass r e d u c t i o n , W1, o f 45.4 kg (100 l b s )f o re a c ho ft h r e ep a i r so f depen- d e n td e s i g nv a r i a b l e s : 1 and 2, 4 and 5, and 7 and 8. The r e s u l t i n g values o f t h e d e s i g n v a r i a b l e i n c r e m e n t s are shown i n Table A-2.

I Dependent .sign Variable Incrc nent s , 2 6 ~~ -198.6 -0 .o -0.3 -1 .o -1.4 -45.4 155 - 9 " -0.2 -0.5 -45.4 -10 .o -4.3 20.6 -15.3 -45.4 Table A-2: I n i t i a l Design Variable Increments (first r e s i z i n gs t e p ) f o r T h r e e P a i r s o f DependentDesign Variables The d u a l c o n s t r a i n t of f l u t t e r s p e e d andfrequencyleads t o relatively l a r g e p o s i t i v e a n d n e g a t i v e i n c r e m e n t s i n t h e d e p e n d e n t d e s i g n v a r i a b l e s .

When design variable p a i r s 1, 2 and 4, 5 are chosen as dependentdesign variables the magnitude of the negative increment i s l a r g e r t h a n t h e avail-

able amount ( c f . Table A-1) . The q u e s t i o n arises whether t o i n v o k et h e

minimum s i z e c o n s t r a i n t a n d a c c e p t t h e r e s u l t i n g l a r g e r d r i f t i n f l u t t e r speed, or r e c o g n i z e t h e v i o l a t i o n o f t h e minimum s i z e c o n s t r a i n t as a t e m - p o r a r y s i t u a t i o n t h a t w i l l b e c o r r e c t e d i n s u b s e q u e n t s t e p s , or reduce w1 It shouldbenoted,however,that as a r e s u l t o f t h e l a r g e i n c r e m e n t s o f t h e dependentdesign variables, t h e amount o f r e s i z i n g t o w a r d s t h e g o a l o f minimum t o t a l mass o c c u r r i n g i n t h e i n d e p e n d e n t v a r i a b l e s i s small when t h e p a i r s 1, 2 and 4 , 5 are t h ed e p e n d e n td e s i g nv a r i a b l e s . How s e r i o u s a drawback is i m p l i e d b y t h e s e c o n s i d e r a t i o n s h a s n o t b e e n p u r s u e d i n d e t a i l , s i n c e it i s b e l i e v e d t h a t t h e f r e q u e n c y c o n s t r a i n t b y i t s e l f p u t s Simodynesmethod at a d i s t i n c t d i s a d v a n t a g e , c e r t a i n l y i n viewof t h e r a t h e r s t r a i g h t f o r w a r d manner i n which t h i s c o n s t r a i n t c a n be removed, as i s d e m o n s t r a t e d i n t h e modified Simodynesmethodwhich i s t h e s u b j e c t o f t h e n e x t s e c t i o n .

A.3.2 ModifiedSimodynes Method - The method of Simodynes w a s m o d i f i e d t o

e l i m i n a t e t h e f l u t t e r f r e q u e n c y c o n s t r a i n t , s u b s t i t u t i n g f l u t t e r f r e q u e n c y as a d e p e n d e n t v a r i a b l e i n p l a c e o f one of t h e two dependent design variables u s e d i n t h e o r i g i n a l method ( S e c t i o n 6.3). - . . - . , ., r .

As i n t h e c a s e o f t h e o r i g i n a l method,an i n i t i a l r e s i z i n g s t e p w a s g e n e r a t e d f o r a t o t a l mass r e d u c t i o no f W1 = 45.4 kg (100 l b s ) f o r e a c h o ft h r e ec h o i c e so ft h ed e p e n d e n td e s i g n variable. The r e s u l t s , shown i n Table A-3, i n d i c a t e more r e a s o n a b l e d i s t r i b u t i o n s o f t h e r e s i z i n g i n c r e m e n t s due t o t h e e l i m i n a t i o n o f t h e f r e q u e n c y c o n s t r a i n t , a l t h o u g h t h e s e n s i t i v i t y t ot h ec h o i c eo fd e p e n d e n td e s i g n variable r e m a i n s .S p e c i f i c a l l y ,t h e result

ofchoosing as a dependentdesign variable oneforwhich - i s small i n

a m Table A-3: I n i t i a l DesignVariableIncrement (first r e s i z i n gs t e p ) f o r T h r e e Dependent Design Variables; Modified I Simodynes Method magnitude is demonstrated i n T a b l e A-3. With number 1 as thedependent v a r i a b l e , a largenegativeincrement of t h a t v a r i a b l e i s r e q u i r e d t o b a l a n c e r a t h e r small i n c r e m e n t s i n t h e i n d e p e n d e n t d e s i g n v a r i a b l e s i n k e e p i n g t h e f l u t t e r speedconstant on a l i n e a r basis. For a givenweightdecrement w1 , t h e amount o f r e s i z i n g t o w a r d s t h e g o a l o f minimum t o t a l mass o c c u r r i n g i n theindependent variables i s small. It would seem, t h e r e f o r e ,t h a t it i s important t o choose as a dependentdesignvariableone t h a t corresponds t o a l a r g ev a l u e - With t h e p r o p e r c h o i c e o f t h i s v a r i a b l e , it wouldappear a m ' t h a t a r e a s o n a b l ye f f i c i e n tp r o c e d u r em i g h tr e s u l t .I no r d e rt oa s s e s st h i s , a d d i t i o n a l r e s i z i n g s t e p s were e x e c u t e d ,u s i n gd e s i g nv a r i a b l e 5 as t h e independentvariable. The f l u t t e rs p e e d sa n dd e s i g nv a r i a b l ed i s t r i b u t i o n s f o r s i x r e s i z i n g s t e p s are p r e s e n t e d i n Table A-4. One otherminordeparture from t h e o r i g i n a l method i s t h a t t h e f l u t t e r s p e e d w a s a d j u s t e d t o a p p r o x i m a t e l y 270.1 m / s E M at t h e b e g i n n i n g o f e a c h r e s i z i n g c y c l e b y a uniformpercentage i n c r e a s e i n t h e c u r r e n t d e s i g n v a r i a b l e d i s t r i b u t i o n .

It i s b e l i e v e d t h a t t h e e f f i c i e n t p e r f o r m a n c e i n d i c a t e d b y t h e results shown i n T a b l e A-4 may b e somewhat b e t t e r t h a n might typically be achieved by t h i s method. These e v a l u a t i o n s were p e r f o r m e d r e l a t i v e l y l a t e i n t h e con- t r a c t e f f o r t , s o t h a t much experiencewith t h i s t e s t casehadbeenaccumulated.

Thisallowed a c h o i c e o f t o t a l mass r e d u c t i o n s t e p s whichminimized t h e r e q u i r e d number o fs t e p st oa c h i e v et h e mass r e d u c t i o n s shown. S p e c i f i c a l l y , t h e s u c c e s s i v e v a l u e s o f W chosen ( 9 0 . 7 , 90.7, 90.7, 4 5 . 4 , 9 . 1 ,9 . 1 kg) a n t i c i p a t e t h e minimum weight.

A . 3 . 3 Improved Move Vector - To e l i m i n a t e t h e s e n s i t i v i t y o f t h e m o d i f i e d method t o t h e c h o i c e ofdependentdesignvariable, a modified move v e c t o r was f o r m u l a t e d a n d t e s t e d i n a f u r t h e r e v a l u a t i o n .

F l u t t e r SpeeC Design Variable Mass. Kg_ 4 6 T o t a l . m / s EAS Step 1 7

2 1 3

L " 270 . I 21.8 15.3 506.6 113.5 31.3 65.5 267.2 88 .o 48.3 39.3 53.4 1.7 415.9 0 .o 268.3 48.9 35.4 354.7 60.7 38.5 .o 48.1 267.8 22.2 63 15 -9 24.6 17.2 30.5 278 - 9 6 . 1 249.1 268.6 6 . 1 26.3 64.5 47.2 3.6 5.5 68.1 49.8 248.8 0.6 0.4 1 . 0 7.6 269.7 27 * 5 51.2 8.6 270 .O 0 .o 0.0 0.0 28.2 249.2 69.9

~~~~ 1 L

. . . .~ Table A-4: Mass Reductions with Modified Simodynes Procedure

The new move v e c t o r i s shown i n equation ( A . 3 ) , where - i s t h e

am.

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

The f i r s t , p o s i t i v e , componentof t h i s move v e c t o r w i l l berecognized as t h e v e l o c i t yg r a d i e n t . The s c a l a r "a" is determinedfromequation ( A . 4 ) , where W2 i s t h e t o t a l mass s p e c i f i e d for t h ep o s i t i v e component of t h ev e c t o r

b m i ) . The s c a l a r "b" definesthemagnitude of t h en e g a t i v e component of

t h ev e c t o r e m i ) s u c ht h a tt h ec h a n g ei nf l u t t e rs p e e d , on a l i n e a r i z e d b a s i s , i s zero.

The f l u t t e r v e l o c i t y d e r i v a t i v e s , n e e d e d f o r t h i s move v e c t o r , are not computed d i r e c t l y i n e i t h e r t h e Simodynes or modified Simodynes procedure.

I n s t e a d , p a r t i a l d e r i v a t i v e s o f t h e d e p e n d e n t v a r i a b l e s w i t h r e s p e c t t o i n d e p e n d e n tv a r i a b l e sa r eg e n e r a t e d . For t h em o d i f i e d Simodynes procedure, c o n s i d e r i n g t h e d e p e n d e n t d e s i g n v a r i a l b e , m and one independent design U, m keeping a l l other independent variables constant, the condition v a r i a b l e , i' V(mu, m . ) = c o n s t a n tl e a d st o :

-

where - i s a n o r m a l i z e df l u t t e rs p e e dd e r i v a t i v e . The negative of t h e ami complete set o f s u c h d e r i v a t i v e s i s t h e n e q u i v a l e n t t o t h e f i u t t e r v e l o c i t y d e r i v a t i v e s n o r m a l i z e d t o t h e f l u t t e r v e l o c i t y d e r i v a t i v e o f t h e d e p e n d e n t

designvariable.Examinationofequations (A. 3) , ( A . 4 ) and ( A . 5 ) shows t h a t

t h e column o fr e s i z i n gi n c r e m e n t s , Ami , remainsinvariantwithnormalization o ft h ef l u t t e rv e l o c i t yd e r i v a t i v e s .A c c o r d i n g l y ,t h en o r m a l i z e dd e r i v a t i v e s i n d i c a t e d i n e q u a t i o n ( A . 6 ) a r e u s e d t o e v a l u a t e t h e r e s i z i n g i n c r e m e n t s shown i n equation(A.3).

Using t h e move v e c t o rd e s c r i b e dh e r e ,w i t h a valueof W = 45.4 kg (100 l b s ) , t h e r e s u l t s of e i g h t r e s i z i n g s t e p s a r e shown i n Table A-5. I n c o m p a r i n g t h e s e r e s u l t s w i t h t h e r e s u l t s i n Table A-4 a n d n o t i n g t h a t t h e "improved" move v e c t o r r e q u i r e s more s t e p s t o r e a c h t h e m i n i m u m t o t a l mass , r e f e r e n c e i s again made t o t h e f a c t t h a t t h e r e s u l t s o f T a b l e A-4 weregen- e r a t e d a f t e r c o n s i d e r a b l e e x p e r i e n c e h a d b e e n g a i n e d w i t h t h i s i d e a l i z e d t e s t c a s e .S p e c i f i c a l l y , it w a s f o u n d t h a t t h e v a l u e o f W c h o s e n a f f e c t s t h e W2 i nt h ep r e s e n t convergenceoftheprocedure.Notethatthequantity p r o c e d u r eh a sn od i r e c tr e l a t i o n s h i pt ot h eq u a n t i t y W1 used i nt h em o d i f i e d Simodynes procedure.

To d e t e r m i n et h ee f f e c t of varying W2, s t e p s 5 , 6 and 7 wererepeated

with values of W2 of 90.7, 90.7 and136kg,respectively. The r e s u l t s , p r e s e n t e d i n Table A-6, demonstrate that improved performance of the method couldbeexpectedfor a more f a v o r a b l ec h o i c eo ft h ev a l u eo f W 2' 10 8 " ..

r F l u t t e r S p e e l Desig: n Variable Mass, 8 m / s EAS 4 6 7 T o t a l ltep 1 2 3 5 ~~ ~~ ~~ ~~ "~ ___- 21.8 506.6 270.1 63.6 31.3 L5.3 0 105.2 65.5 L13 -5 30.1 269 .2 66.3 58.6 64.8 39.4 6.9 392.3 1 67.5 58.7 36.6 6 . 8 269.6 43.2 68 .O 46.8 332.8 2 25 .O 50.7 55.4 41.7 269.8 53 -1 7.9 290.7 . 3 0.3 24.6 38.5 53 - 2 271.1 26.9 50.8 74 .O 58 .O 45.3 8.5 269 -9 4 0 .o 7.5

76.2 8.8 260. o 270 .O

0 .o 16.5 48.6 61.9 47.9 0 .o

49.6 254.8 270. o

0 .o 46.4 78.1 64.8 8.9 6 0 .o 7.0 251.2 270.0 0 .o 0 .o 0 .o 44.5 67.2 50 -9 9 -1 8.6 250.8 270.1 0 .o 0 .o 42.1 80.4 68.5 51.3 8 0 .o ~~ . ... ~ " . " - Table A-5: Mass Reductions w i t h Improved Move Vector Design Variable Mass, kg F l u t t e r Spee m / s U S T o t a l 8 6 . 4 3 7

I ' -~

0 . 0 0 . 0 269.8 252.0 8.7 49.8 65.1 77.7 5 90 - 7 45.6 5.0 270 .o 0 . 0 6 250.0 8 . 3 51.5 80.2 0 . 0 0 . 0 90 - 7 69.0 40.9 0 . 0 270 .o 249.6 8.4 52.0 82.2 0.0 0.0 7 71.9 35.1 -. . . . ~ ." " Table A-6: Mass ReductionswithIncreasedValuesof W A. 4 ' GRADIENT METHODS OF RUDISILL AND BHATIA In Reference 1 4 , R u d i s i l l and B h a t i a p r e s e n t a methodofgenerating f l u t t e r v e l o c i t y d e r i v a t i v e s andtheysuggestseveralresizingprocedures u s i n gt h e s ed e r i v a t i v e s .T h e s ep r o c e d u r e sa r ed i s c u s s e di nS e c t i o n 6.2. Two o f t h e s e p r o c e d u r e s , t h e v e l o c i t y g r a d i e n t s e a r c h and t h e g r a d i e n t p r o j e c t i o n s e a r c h , werechosen for n u m e r i c a l e v a l u a t i o n a n d t h e r e s u l t s a r e p r e s e n t e d i n t h e f o l l o w i n g s e c t i o n s .

A . 4 . 1 VelocityGradientSearch - "he v e l o c i t yg r a d i e n ts e a r c h i s u s e d t o i n c r e a s e f l u t t e r s p e e d b y a series o f r e s i z i n g s t e p s i n which the increments i n t h e d e s i g n v a r i a b l e s are p r o p o r t i o n a l t o t h e c o r r e s p o n d i n g e l e m e n t s of t h e v e l o c i t yg r a d i e n t . ' The r e s u l t i n g d i s t r i b u t i o n i s not optimum, b u t is a good i n i t i a l d i s t r i b u t i o n f o r p r o c e d u r e s i n which t h e t o t a l mass i s minimized at c o n s t a n tf l u t t e rs p e e d . One obviousapplication of t h i sp r o c e d u r e is i n i n c r e a s i n g ' t h e f l u t t e r s p e e d o f a f l u t t e r - d e f i c i e n t d e s i g n t o t h e r e q u i r e d f l u t t e r s p e e d . S t a r t i n g w i t h t h e 226.4'm/s EAS c o n f i g u r a t i o n of t h e test case s t r u c t u r a l model, t h e f l u t t e r s p e e d w a s i n c r e a s e d t o a p p r o x i m a t e l y 270.1 m / s EAS i nf i v es t e p s( T a b l e A-7). The d e s i g nv a r i a b l ei n c r e m e n t s were formed a c c o r d i n gt oe q u a t i o n (A.71, where t h e nominalvelocityincrements, AV, were 12.9,10.3,10.3, 5.1 and 3.1 m / s E M . The a c t u a lv e l o c i t yi n c r e m e n t s do 1) the nonlinear n o t c o r r e s p o n d e x a c t l y t o t h e n o m i n a l i n c r e m e n t s b e c a u s e o f r e l a t i o n s h i p b e t w e e n t h e i n c r e m e n t i n f l u t t e r s p e e d a n d i n c r e m e n t s i n the designvariables,and 2 ) t h e f a c t t h a t t h e v e l o c i t y d e r i v a t i v e s , as d e f i n e d i n R e f e r e n c e 1 4 , are notbased on matchedatmosphericconditionsof Mach number, s p e e d a n d a l t i t u d e , w h e r e a s t h e f l u t t e r e q u a t i o n was s o l v e d d i r e c t l y f o r matchedconditions.

To e v a l u a t e t h e e f f e c t o f t h e number o f r e s i z i n g s t e p s u s e d t o p r o d u c e a givenvelocityincrement,themagnitude of t h e o r i g i n a l i n c r e m e n t was d e t e r m i n e d w h i c h r e s u l t e d i n t h e same f l u t t e r speed of 270.5 m / s EAS i n one r e s ' i z i n gs t e p . The r e q u i r e d mass, 292.3 kg,doesnotdiffergreatlyfrom the m u l t i - s t e p r e s u l t o f 286.7 kg, when compared w i t h the optimum t o t a l d e s i g nv a r i a b l e mass 0 f ~ 2 4 9 . 6 kgpounds ( f o r Vf = 270.1 m / s EAS).

DesignVariable Mass, kg F l u t t e r Speed 4 T o t a l m / s EAS 3 5

" 1

- 2 . 4 1 0 . 1 8.1 5.7 239 * 0 6.0 13.6 23.2

15.7 130 - 9 249.9

1 1 . 2 24.6 40.3 23.6 211.0 261.1 14.7 50.4 31.5 27.5 260.9 250.8 17.2 36.2 29.8 286.7 57 -1 270 * 5

I

T 7.9 11.6 '

Table A-7: DesignVariable Mass a n dF l u t t e r Speed;VelocityGradientSearch A . 4 . 2 GradientProjectionSearch - The g r a d i e n tp r o j e c t i o ns e a r c h is a p r o c e d u r e f o r r e d u c i n g t h e t o t a l mass o f t h e d e s i g n v a r i a b l e s w h i l e a t t e m p t i n g t o m a i n t a i n a c o n s t a n tf l u t t e rs p e e d . The column ofresizingincrements is derived from the velocity gradient and the mass g r a d i e n t as i n d i c a t e d by equation ( A . 8 ) where t h es c a l a r hl i s determined as i ne q u a t i o n (A.9): Comparison w i t h equations ( A . 3) , ( A . 4 ) and ( A . 5) shows that t h i s r e s i z i n g column i s similar t o t h a t o f S e c t i o n A.3.3 except t h e mass g r a d i e n t is u s e d i n p l a c e o f t h e r e c i p r o c a l so f t h e f l u t t e r d e r i v a t i v e s . For comparison w i t h t h i s previous procedure, t h e product AoA1, t h ec o e f f i c i e n to f the v e l o c i t yg r a d i e n t , was chosen t o g i v e t h e same 45.4 kg p o s i t i v ei n c r e m e n t as b e f o r e . The results of t h e first three r e s i z i n g s t e p s , s t a r t i n g w i t h t h e 270.1 m / s EAS configura- t i o n , are g i v e ni n Table A-8. Comparison w i t h Table A-5 i n d i c a t e s that t h e first t h r e e s t e p s o f t h e g r a d i e n t p r o j e c t i o n s e a r c h are less e f f e c t i v e t h a n t h e i n i t i a l s t e p o f t h e p r o c e d u r e o f S e c t i o n A . 3 . 3 . The e v a l u a t i o no f t h e g r a d i e n t p r o j e c t i o n s e a r c h was terminated at t h i s p o i n t .

Design Variable Mass, kg F l u t t e r Speed 8 4 2 1 T o t a l 6 3 m / s EAS S t e p 7 5 0 21.8 1 5 . 3 31.3 63.6 105.2 113.5 270.1 506.6 65.5 90.4 61.0 1 270.0 479.4 10.3 25.2 34.5 62.3 83.3 97.4 105.4 28.2 2 458.3 7.2 37.8 61.9 77.1 90.6 98.1 270.0 57.4 3 84.1 439.5 5.4 30.8 40.7 61.8 54.2 71.3 91.2 270.0 L ~ ~ . . _. " . " .

Table A-8: Mass ReductionswithGradientProjectionSearch A.5. INTERIOR PENALTY FUNCTION METHOD The i n t e r i o r p e n a l t y f u n c t i o n method,described i n R e f e r e n c e 16 and d i s c u s s e d i n S e c t t o n 6.4, employs a series of unconstrained minimizations of a m o d i f i e d o b j e c t i v e f u n c t i o n i n o r d e r t o m i n i m i z e t h e o b j e c t i v e f u n c t i o n of interest ( u s u a l l yt o t a l mass). The m o d i f i e do b j e c t i v ef u n c t i o n , @ ( m i ) , i s formedbyaddingpenalty terms, r e f l e c t i n g t h e c o n t r a i n t . e q u a t i o n s , t o t h e o b j e c t i v ef u n c t i o n W(mi) ( e q u a t i o n (A.10)). The minimizationofthemodified o b j e c t i v e n (A. 10) f u n c t i o n is c a r r i e d o u t f o r r e p e a t e d r e d u c t i o n s o f t h e p e n a l t y w e i g h t i n g f a c - t o r , r, u n t i l t h e minimum ofthemodifiedobjectivefunctionapproximates t h e minimum of t h e o b j e c t i v e f u n c t i o n . I n t h e p r e s e n t c a s e , t h e p e n a l t y terms r e p r e s e n t e d minimum s i z e (mass) c o n s t r a i n t s f o r e a c h o f t h e e i g h t d e s i g n v a r i a b l e s , i n a d d i t i o n t o t h e f l u t t e r s p e e d c o n s t r a i n t . S t a r t i n g w i t h t h e 280.4 m / s EAS (545 KEAS) c o n f i g u r a t i o n , t h e i n i t i a l p e n a l t y w e i g h t i n g f a c t o r s werechosen t o produce a t o t a l o f t h e p e n a l t y terms approximately equal to t h eo b j e c t i v ef u n c t i o n .T h e s ep e n a l t yw e i g h t i n gf a c t o r sw e r er e d u c e di nf i v e s t e p s t o a valuewhichresultedinpenaltytermsapproximatelyequaltoone p e r c e n to ft h e minimum v a l u eo ft h eo b j e c t i v ef u n c t i o n( t o t a l mass). The move d i r e c t i o n was generated using approximate second derivatives with Newton’s method ( S e c t i o n 6.4 andReference 1 6 ) . The r e s u l t s o f t h e five r e s i z i n g s t e p s are given i n Table A-9.

Table A-9: Mass ReductionswithPenaltyFunctionProcedure It is r e c o g n i z e d t h a t a d d i t i o n a l r e s i z i n g s t e p s i n which t h e p e n a l t y weightingfactor i s furtherreduced would l e a d t o a lower t o t a l mass. How- ever, it hasbeenobserved(Section 6.4.2) t h a t e a c h s t e p i n t h e p e n a l t y functionprocedurerequiresapproximatelythe same number ofnumericalevalua- t i o n s as t h r e e s t e p s i n a r b i t r a r y s t e p s i z e p r o c e d u r e s , so t h a t t h e number of s t e p s i n this numerical evaluation of the penalty function method i s s u f f i - s i z e procedures c i e n t t o p r o v i d e t h e c o m p a r i s o n w i t h t h e a r b i t r a r y s t e p (Tables A-4 and A-5).

A.6 METHOD OF FEASIBLE DIRJXTIONS The method of feasible directions (Reference 17 and Section 6.5) gen- erates a s e r i e s o f r e s i z i n g move vectors,eachofwhich i s b o t h f e a s i b l e ( d o e sn o tv i o l a t ea c t i v ec o n s t r a i n t s ) andusable(reducestotal mass). Each r e s i z i n g d i r e c t i o n i s f o l l o w e d u n t i l a new c o n s t r a i n t i s v i o l a t e d , an a c t i v e c o n s t r a i n t i s r e - e n c o u n t e r e do rt h et o t a l mass is minimized. The r e s i z i n g d i r e c t i o n i s found using the Simplex procedure to determine an optimum move d i r e c t i o n .

I n t h e p r e s e n t c a s e , t h e 270.1 m / s EAS configuration was taken as t h e s t a r t i n g p o i n t , and minimum s i z e (mass) c o n s t r a i n t s wereimposed on t h e eightdesignvariables i n a d d i t i o nt ot h ef l u t t e rs p e e dc o n s t r a i n t . The r e s u l t s o f t h e first e i g h t r e s i z i n g s t e p s a r e p r e s e n t e d i n T a b l e A-10.

-

Design Variable Mass, kg I F l u t t e r Speed

' t e p 1 4 8 I Total I m / s EAS

-

1135 63.6 15.3 ' 506.6 270.1

65.5

Y 31.3 21.8

1 98.6 46.1 -7 69.3 0.5 273 - 0 50 36.7 467.6 2 28.1 80.4 44.1 54.9 89.8 270.1 7 . 0 382.7 3 23.6 38.3 t10.6 12.8 59.4 49.0 309.2 273 -0 61.4 4 11.2 71.1 0.5 47.7 98.0 293.8 273.7 61.4 82.5 11.2 274.0 5 0.5 36.9 87.2 283.0 30.8 81.2 270.1 5 A 0.5 76.5 55.3 5.3 253.0 6 270 . I 0.5 31.9 '79.0 78.7 52.8 7.8 251.6 6 A 31.8 270.1 0.5 78 - 9 250.8 7.6

- ~~

Table A-10: Mass ReductionswithFeasibleDirectionsProcedure It should be n o t e d t h a t s t e p s 5A and 6 A are so d e s i g n a t e d b e c a u s e t h e f l u t t e r s p e e d c o n s t r a i n t was n o t active f o r t h o s e s t e p s , a n d t h e r e f o r e no f l u t t e r s p e e d d e r i v a t i v e s were r e q u i r e d . The r e s u l t i n g move d i r e c t i o n w a s t h e n oneof"steepestdescent ,I' determined by t h e mass gradient.These two s t e p s r e q u i r e d somewhat less computingresourcesthandidtheremainder of t h e s t e p s , a n d it t h e r e f o r e seemed r e a s o n a b l e n o t t o i d e n t i m . t h e m as f u l l s t e p s . It s h o u l da l s o be n o t e d t h a t 0.5 kg (onepound) was e s t a b l i s h e d as t h e minimum mass ( s i z e ) a l l o w e d f o r anydesign variable, r a t h e r t h a n z e r o as i no t h e rp r o c e d u r e s .T h i s was later d e t e r m i n e dt o be unnecessary as i s d i s c u s s e di nS e c t i o n6 . 5 . 2 . Had t h i s n o t b e e n t h e c a s e , t h e t o t a l mass i n each of t h e l a s t s e v e r a l s t e p s wouldhavebeen s l i g h t l y less.

A . 7 AN OPTIMIZATION METHOD USING INCREMENTED FLUTTER ANALYSIS Reference 18 d e s c r i b e s a resizingproceduredeveloped at Lockheedand u s e dp r i m a r i l yi n an i n t e r a c t i v e mode employinggraphicsdisplays.This procedure i s d i s c u s s e d i n S e c t i o n 6.6.

Although t h i s p r o c e d u r e i s p r e s e n t l y i n an incomplete state ofdevelop- ment, it i s o f i n t e r e s t i n t h a t it employs a uniquemethod of determining r e s i z i n gs t e ps i z e .T h i s i s done by a d i r e c tm i n i m i z a t i o n of t h eo b j e c t i v e function(weight)using a d i r e c t i o nd e t e r m i n e d by v e l o c i t y d e r i v a t i v e s and t h ec o n s t r a i n t s . This minimizationresultsfrom,and takes i n t oa c c o u n t , t h e n o n l i n e a r r e l a t i o n s h i p b e t w e e n d e s i g n v a r i a b l e i n c r e m e n t s a n d f l u t t e r speed. A new d i r e c t i o n i s generated arter eachminimization,andtheprocess i s r e p e a t e d until anacceptableapproximationofthe minimum t o t a l mass i s obtained.

The results o f 4 r e s i z i n g s t e p s a r e p r e s e n t e d i n T a b l e A-11.

DesignVariable Mass, kg F l u t t e r Speed

1 2 3 8 1 T o t a l m / s EAS

6 1 7 T

~ " 105.2 90.4 113 * 5 270.1 0 .o 0 . 0 0.0 270 .I 0.0 0.0 0.0 270.1 0.0 0 . 0 0.0 36.0 82.7 270.1 0 .o 0.0 0.0 32.7 84.9 270.1

i I

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