A method for analyzing the aeroelastic stability of a helicopter rotor in forward flight

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A METHOD FOR .ANALYZING

THE AEROELASTIC STABILITY

OF A HELICOPTER ROTOR

I N FORWARD FLIGHT

by Peter Crimi

Prepared by ROCHESTER APPLIED SCIENCE ASSOCIATES, INC.

Rochester, N . Y .

for Langley Research Center N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N W A S H I N G T O N , D . C. A U G U S T 1 9 6 9 ~ TECH LIBRARY KAFB, NM 0060505 NASA CR-1332 A METHOD FOR ANALYZING THE AEROELASTIC STABILITY OF A HELICOPTER ROTOR IN FORWARD FLIGHT By Peter Crimi Distribution of this report is provided in the interest of informationexchange.Responsibility for thecontents resides in the author or organization that prepared it.

Issued by Originator as RASA Report No. 68-10 Prepared under Contract No. NAS 1-7411 by ROCHESTER APPLIED SCIENCE ASSOCIATES, INC.

Rochester, N.Y.

for Langley Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For s a l e by the Cleoringhouse for Federal Scientific and Technical Information Springfield,Virginio 22151 - CFSTl price $3.00 CONTENTS Page

S U M M A R Y . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I N T R O D U C T I O N . . . . . . . . . . . . . . . . . . . . . . . 2

SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . 3

DEVELOPMENT O F T H E E Q U A T I O N S O F M O T I O N O F A ROTOR I N F O R W A R D F L I G H T . . . . . . . . . . . . . . . . . . . . . .

1 7 DEVELOPMENT O FT H EB A S I C METHOD . . . . . . . . . . . . . .

O U T L I N E O F B A S I C C O M P U T E R PROGRAM . . . . . . . . . . . . . 26

3 1 A P P L I C A T I O N O F T H E METHOD . . . . . . . . . . . . . . . . .

4 2 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . .

A P P E N D I X A : R e l a t i o n s h i p Among t h e S o l u t i o n s of O r i g i n a l and D i f f e r e n t i a t e d Systems . . . . .

4 5 A P P E N D I X B : L i s t i n g of B a s i c C o m p u t e r Program . . . . . .

A P P E N D I X C : Method for E x t r a c t i o n of t h e C o m m o n Polynomial Factor From t h e T w o H i g h e r - D e g r e e

Polynomials . . . . . . . . . . . . . . . . . 1 0 1

R E F E R E N C E S . . . . . . . . . . . . . . . . . . . . . . . . 1 0 6

iii A METHOD FOR ANALYZING THE AEROELASTIC STABILITY OF A HELICOPTER ROTOR I N FORWARD FLIGHT By Peter C r i m i ROCHESTER APPLIED SCIENCE ASSOCIATES, I N C .

SUMMARY A methodhasbeendevelopedwhichprovidestheexactsolution t o t h e p e r t u r b a t i o n e q u a t i o n s o f m o t i o n o f a r o t o r i n forward f l i g h t .E f f e c t so fc o m p r e s s i b i l i t y ,s t a l l and reversed flow are t a k e ni n t oa c c o u n t ,a n dt h e r e is no r e s t r i c t i o n on t h e number of degreesoffreedomwhichcan be analyzed.

The equationsofmotionof a r o t o r b l a d e i n f o r w a r d f l i g h t were d e r i v e d i n terms ofthenormal modes of f r e e v i b r a t i o n o f t h e blade.Aerodynamicforces were e x p r e s s e d on t h eb a s i s .o fq u a s i - s t e a d yf l o w ,u s i n gs t r i pt h e o r y .P e r t u r b a t i o ne q u a t i o n s were gen- e r a t e d byexpanding a l l a e r o d y n a m i cf o r c e si nt h ep e r t u r b a t i o n are of v a r i a b l e sa b o u tt h e i rs t e a d y - s t a t ev a l u e s . The e q u a t i o n s t h e formof a coupled set o fl i n e a r ,s e c o n d - o r d e rd i f f e r e n t i a l e q u a t i o n s w i t h p e r i o d i c a l l y v a r y i n g c o e f f i c i e n t s .

A method was t h e n d e v e l o p e d f o r a n a l y z i n g t h e s t a b i l i t y o f l i n e a r dynamicsystemswithperiodicallyvaryingparameters.

S t a b i l i t y i s determined by c a l c u l a t i o n of a l l t h e c h a r a c t e r i s t i c values ofthesystem.Thus, a s o l u t i o np r o v i d e sr a t e s of growth o r decayofthemotionfollowing a d i s t u r b a n c e , i n a d d i t i o n t o a d e t e r m i n a t i o no fw h e t h e ro rn o tt h es y s t e m is stable.

A d i g i t a l computerprogram was preparedtoimplementthegen- eral method f o r t h e c a s e o f a r o t o rb l a d ew i t ho n e , two o r t h r e e d e g r e e so ff r e e d o m .S t a b i l i t yc a l c u l a t i o n s were performedfor c o m p a r i s o nw i t hr e s u l t so b t a i n e d by direct t i m e i n t e g r a t i o n o f t h e n o n l i n e a re q u a t i o n s of motionof a r i g i db l a d ew i t hf l a p p i n ga n d lead-laghinges. Limited c a l c u l a t i o n s were a l s o made o ft h ea e r o - e l a s t i c s t a b i l i t y of a model r o t o r b l a d e f o r whichexperimental f l u t t e r d a t a was a v a i l a b l e .C o m p a r i s o n so ft h er e s u l t si n d i c a t e q u a l i t a t i v e a g r e e m e n t i n b o t h cases.

INTRODUCTION The a e r o e l a s t i c s t a b i l i t y o f h e l i c o p t e r r o t o r s is ofconcern f o r t w or e a s o n s .F i r s t , a t t h eh i g ha d v a n c er a t i o sa s s o c i a t e d w i t h compound a n ds t o w a b l e - r o t o ro p e r a t i o n ,t h eh o s t i l ea e r o d y n a m i c e n v i r o n m e n tc o u l dl e a dt od a n g e r o u si n s t a b i l i t i e s .S e c o n d l y ,i n c o n v e n t i o n a lo p e r a t i o n ,i n s t a b i l i t i e sc a no c c u rw h i c h ,w h i l eg e n - e r a l l y n o t of a c a t a s t r o p h i c n a t u r e b e c a u s e o f n o n l i n e a r e f f e c t s , are n o n e t h e l e s ss e r i o u s from a f a t i g u ea n dc o n t r o ls t a n d p o i n t .

The a n a l y s i s o f r o t o r f l u t t e r f o r h o v e r i n g f l i g h t i s a r e l a - t i v e l ys t r a i g h t f o r w a r dp r o b l e m ,t h ef o r m u l a t i o n sb e i n ge s s e n t i a l l y t h e same a s t h o s e f o r c l a s s i c a l f l u t t e r o f c o n v e n t i o n a l a i r c r a f t .

C o n s i d e r a b l er e s e a r c h ,b o t ht h e o r e t i c a la n de x p e r i m e n t a l ,h a sd e a l t w i t ht h ef l u t t e ro fh o v e r i n gr o t o r s (see, forexample,References 1 through 4 ) . Agreementbetweentheoryandexperimenthasgenerally beengood.

The problem f o r a r o t o r i n f o r w a r d f l i g h t i s fundamentally d i f f e r e n t from t h a to ft h eh o v e r i n g case, due t o t h e n a t u r e o f t h e e q u a t i o n so fm o t i o n .B e c a u s et h er e l a t i v ef l o wi m p o s e d on a g i v e n b l a d e s e c t i o n v a r i e s p e r i o d i c a l l y as t h e b l a d e r o t a t e s , t h e e f f e c - t i v e dynamic p r e s s u r e , a n d h e n c e t h e c o n s t a n t o f p r o p o r t i o n a l i t y o f t h ea e r o d y n a m i cf o r c e s ,a l s ov a r i e sp e r i o d i c a l l y . A s a consequence, t h e e q u a t i o n s o f m o t i o n o f t h e r o t o r h a v e p e r i o d i c a l l y v a r y i n g co- e f f i c i e n t s .S y s t e m sd e s c r i b e d by e q u a t i o n s o ft h i st y p e ,e v e nt h o u g h l i n e a r , d i s p l a y many u n u s u a lp r o p e r t i e s and i n some r e s p e c t s resem- b l en o n l i n e a rs y s t e m s (see Reference 5 ) .

L i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h p e r i o d i c a l l y v a r y i n g c o e f f i - c i e n t s havebeentheconcernofappliedmathematiciansforover a century. The d i f f e r e n t i a le q u a t i o no fs e c o n do r d e rb e a r i n g h i s name w a s d i s c u s s e d byMathieu i n 1868 i n r e f e r e n c e t o t h e problem of a v i b r a t i n g e l l i p t i c membrane. The more generalsecond-order e q u a t i o n d e r i v e d by H i l l f o r d e t e r m i n i n g t h e m o t i o n o f t h e l u n a r p e r i g e e w a s presentedby him i n 1877. In1883,Floquetdetermined t h e f o r m o f t h e s o l u t i o n f o r any l i n e a r d i f f e r e n t i a l e q u a t i o n w i t h

p e r i o d i c c o e f f i c i e n t s . *

More r e c e n ta n a l y s e sr e l a t e dt od y n a m i cs y s t e m sw i t hp e r i o d i c p a r a m e t e r sa r ep r e s e n t e di nR e f e r e n c e s 7 , 8 and 9. InReference t h 7 , t h e g e n e r a l N - orderproblem i s t r e a t e d ,w h i l eR e f e r e n c e 8 i s c o n c e r n e dw i t hs p i n - s t a b i l i z e ds a t e l l i t e sa n dR e f e r e n c e 9 d e a l s ~.

* A h e e a r l yh i s t o r yo fM a t h i e ua n dr e l a t e d -

f u n c t i o n s i s g i v e ni nR e f e r e n c e 5. Many p a p e r so fh i s t o r i c a l i n t e r e s t are a l s on o t e di nR e f e r e n c e 6.

withmechanical instabilities o fh e l i c o p t e rr o t o r s .T h e s ea n a l y s e s and similar o n e s t r e a t i n g r e l a t e d p r o b l e m s g e n e r a l l y e i t h e r r e l y on e x p a n s i o n i n a s m a l l p a r a m e t e r t o o b t a i n a s o l u t i o n o r are re- s t r i c t e d t o s p e c i a l f o r m s o f t h e d i f f e r e n t i a l e q u a t i o n s .

S i n c e a g e n e r a l method f o r a n a l y z i n g l i n e a r s y s t e m s w i t h p e r i - o d i cp a r a m e t e r sh a sn o tb e e na v a i l a b l e , it h a sb e e nn e c e s s a r yi nt h e p a s t , i n i n v e s t i g a t i n g r o t o r f l u t t e r , t o r e l y p r i m a r i l y on d i r e c t i n t e g r a t i o n o f t h e e q u a t i o n s o f m o t i o n u s i n g an a n a l o g o r d i g i t a l computer. The r e s u l t so fa n a l o gs t u d i e so fr o t o rs t a b i l i t y are re- p o r t e di nR e f e r e n c e s 1 0 and 11. D i g i t a lc o m p u t e r s were u s e dt o o b t a i nr e s u l t sd i s c u s s e di nR e f e r e n c e s 1 2 and 13. Computer solu- t i o n s are u s e f u l p a r t i c u l a r l y i n t h a t n o n l i n e a r e f f e c t s c a n r e a d i l y b e i n c o r p o r a t e d t o g i v e an a c c u r a t e i n d i c a t i o n of systemdynamics f o r a s p e c i f i cr o t o r. a n df l i g h tc o n d i t i o n s . However, t h er e s u l t s are o f t e n d i f f i c u l t t o i n t e r p r e t a n d e v a l u a t e i n terms of system s t a b i l i t y , a n d , e s p e c i a l l y when a d i g i t a lc o m p u t e r is u s e d ,t h e i r c o s tp r e c l u d e sc o n d u c t i n g a thoroughparametricstudy.

I t h a s b e e n p o s s i b l e t o g a i n some i n s i g h t i n t o r o t o r a e r o - e l a s t i c c h a r a c t e r i s t i c s by t r e a t i n g a singledegreeoffreedom a n a l y t i c a l l y( R e f e r e n c e s 1 4 and 15). A s i n g l es e c o n d - o r d e rd i f f e r - e n t i a le q u a t i o nr e p r e s e n t st h es y s t e m . T h i s equationcan be reduced t o t h e form of H i l l ' s equation(Reference 61, for w h i c ht h e r ea r e t e c h n i q u e sa v a i l a b l et oo b t a i ns o l u t i o n s . The analyses have been c o n f i n e dt ot h ef l a p p i n gd e g r e eo ff r e e d o m ; it was f o u n dt h a tt h i s simplesystem i s q u i t e s t a b l e , b u t t h a t f l u t t e r c a n o c c u r a t very h i g ha d v a n c er a t i o s( o ft h eo r d e ro fu n i t y ) .

The s t u d yr e p o r t e dh e r e was d i r e c t e d t o o b t a i n i n g t h e e x a c t a n a l y t i c a l s o l u t i o n s of t h e p e r t u r b a t i o n e q u a t i o n s of motionof a r o t o ri nf o r w a r df l i g h t . The v a l u e so fs y s t e mp a r a m e t e r so rf l i g h t c o n d i t i o n s were n o t r e s t r i c t e d , n o r were l i m i t a t i o n s p l a c e d on t h e number o rt y p e s 0.f degreesoffreedom.Becausethegeneralproblem was a t t a c k e d ,t h es o l u t i o no b t a i n e dh a sa p p l i c a t i o n in a r e a so t h e r t h a nr o t o ra e r o e l a s t i cs t a b i l i t y . The methodpresentedcan be used t o d . e t e r m i n e t h e s t a b i l i t y o f any l i n e a r dynamicsystemwithperi- o d i c a l l yv a r y i n gp a r a m e t e r s . Also, the computer program developed t o implement t h e methodhasbeensegmented i n s u c h a way t h a t it can be a p p l i e d t o t h e s t a b i l i t y a n a l y s i s ofany l i n e a r dynamic s y s t e mw i t ht h r e ed e g r e e s of freedom.

SYMBOLS d i m e n s i o n l e s s ,p e r i o d i cc o e f f i c i e n t si nt h eb a s i c amnf bmn set o f d i f f e r e n t i a l e q u a t i o n s d i m e n s i o n l e s s , p e r i o d i c c o e f f i c i e n t s i n t h e cmnf d e r i v e d set of d i f f e r e n t i a l e q u a t i o n s b l a d ec h o r d , m C b l a d e s e c t i o n l i f t c o e f f i c i e n t b l a d es e c t i o nd r a gC o e f f i c i e n t ‘ d b l a d e s e c t i o n moment c o e f f i c i e n t f o r moment about ‘ m m i d - c h o r d ,p o s i t i v et oi n c r e a s ei n c i d e n c e d d r a g p e r u n i t s p a n o f b l a d e , N/m aerodynamicforcecomponentperunitbladespan FV i nt h ey - d i r e c t i o n , N/m aerodynamicforcecomponentperunitbladespan i nt h ez - d i r e c t i o n , N/m c o e f f i c i e n t s i n t h e f i n i t e s u m o ft r i g o n o m e t r i c r j f u n c t i o n s r e l a t e d t o A IO l i f t p e r u n i t s p a n o f b l a d e , N/m h o r i z o n t a ld i s t a n c ef o r w a r do ft h eX o a x i so ft h e

, m

e l a s t i c a x i s ( x - a x i s ) aerodynamic moment perunitspanaboutmid-chord, Mc/;L p o s i t i v e t o increase i n c i d e n c e , N aerodynamic moment p e r u n i t s p a n a b o u t t h e e l a s t i c a x i s ,p o s i t i v et oi n c r e a s ei n c i d e n c e , N m b l a d e mass p e r u n i t s p a n , kg/m N number o fd e g r e e so ff r e e d o mo ft h ee l a s t o - mechanicalsystem r o t o r r a d i u s , m R m a g n i t u d e o f r e s u l t a n t f l u i d v e l o c i t y r e l a t i v e V t o a b l a d e s e c t i o n , m / s component o f f l u i d v e l o c i t y , r e l a t i v e t o a b l a d e s e c t i o n ,n o r m a lt ot h e X O - Z O p l a n e , m / s componentof f l u i d v e l o c i t y , r e l a t i v e t o a b l a d e Vt s e c t i o n , t a n g e n t t o t h e Xo-Zo p l a n e , m / s magnitudeof free-stream v e l o c i t y ( a i r c r a f t vf f o r w a r ds p e e d ) , m / s V displacement of b l a d e s e c t i o n i n t h e y - d i r e c t i o n , m W d i s p l a c e m e n to fb l a d es e c t i o ni nt h ez - d i r e c t i o n , m wake-inducedinflow, m / s c o o r d i n a t e s r o t a t i n g w i t h t h e b l a d e , t h e Y O a x i s c o i n c i d e n t w i t h t h e s h a f t c o o r d i n a t e s f i x e d w i t h r e s p e c t t o a l o c a l b l a d e s e c t i o n , t h e x - a x i s c o i n c i d i n g w i t h t h e e l a s t i c a x i s and t h e z - a x i s p a r a l l e l t o t h e c h o r d d i s t a n c e o f t h e e l a s t i c axisaheadofmid-chord,m 'a u b l a d es e c t i o na n g l eo fa t t a c k , rad c1 s h a f t tilt w i t h r e s p e c t t o n o r m a l t o f r e e stream, S p o s i t i v e f o r a f t tilt ofthenegativeYo-axis,rad A i n f i n i t e d e t e r m i n a n t E d i s t a n c ef o r w a r do ft h e e l a s t i c a x i so fb l a d e s e c t i o n mass center, m t h d i s p l a c e m e n to ft h e n- coupled mode o f f r e e 'n v i b r a t i o no ft h ee l a s t o - m e c h a n i c a ls y s t e m f l a p w i s eb e n d i n gs l o p e r e a l p a r t o fs y s t e mc h a r a c t e r i s t i cv a l u e i w i m a g i n a r yp a r to fs y s t e mc h a r a c t e r i s t i cv a l u e i w r o o t sd e f i n i n gp o i n t sa tw h i c h A is s i n g u l a r mass d e n s i t yo ff r e e stream, kg/m3 P anglebetweenlocal blade chordandthe Xo-Yo p l a n e ,r a d t o r s i o n a ld e f l e c t i o na b o u tt h e e l a s t i c a x i s ,p o s i - t i v e t o i n c r e a s e i n c i d e n c e , r a d chordwisebendingslope r o t o rr o t a t i o n a ls p e e d , rad/s i w c h a r a c t e r i s t i c v a l u e of t h e b a s i c o r d e r i v e d systems, i w = X + i X , R t h n a t u r a l f r e q u e n c y o f t h e k - coupled mode o f f r e e v i b r a t i o n o f t h e r o t a t i n g s y s t e m , r a d / s DEVELOPMENT O F THE EQUATIONS OF MOTION O F A ROTOR I N FORWARD FLIGHT The p e r t u r b a t i o n e q u a t i o n s o f m o t i o n f o r a r o t o r b l a d e i n f o r w a r df l i g h t are d e r i v e di nt h i ss e c t i o n . The s e t ofcoupled, l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h p e r i o d i c c o e f f i c i e n t s w h i c h a r e o b t a i n e d form t h e s u b j e c t f o r a n a l y s i s i n t h e n e x t s e c t i o n .

The e l a s t o - m e c h a n i c a ls e g m e n to ft h ea e r o e l a s t i cs y s t e m , t o have N degreesoffreedom, i s c o n v e n i e n t l yr e p r e s e n t e d assumed i n terms of t h e N normal modes o ff r e ev i b r a t i o n . Normal modes a n d f r e q u e n c i e s f o r a r o t a t i n g beam c a n b e o b t a i n e d e i t h e r from a c o n t i n u o u sr e p r e s e n t a t i o n( R e f e r e n c e 1 6 ) o r from a lumped-mass model (Reference 1 7 ) .

Aerodynamic f o r c e s are d e r i v e dh e r ei na c c o r d a n c ew i t ht h e usual strip-theory assumptions. Quasi-steady flow is assumed as w e l l . A p e r t u r b a t i o na n a l y s i s is thenperformed,withexpansion o fe x p e r i m e n t a l l yo ro t h e r w i s ed e t e r m i n e da e r o d y n a m i cc o e f f i c i e n t s i n a T a y l o rs e r i e s , a b o u tt h en o m i n a lf l o wc o n d i t i o n s , i nt h eq u a n - tities d e f i n i n g t h e p e r t u r b a t i o n s i n t h eb l a d em o t i o n s .

Consider a r o t o rb l a d e ,t h e n ,w i t ha n g u l a rs p e e d R s u b j e c t e d t o a u n i f o r l nf r e e stream ofmagnitude Vf and d e n s i t y p ,a s shown i n Figure 1. The Xo-axis i s t a k e nc o i n c i d e n t w i t h t h ep i t c h ,o r f e a t h e r i n g ,a x i s , when t h eb l a d e i s i n i t s undeformed p o s i t i o n ,a s shown i n th.e f i g u r e . The c o o r d i n a t e s( x , y , z )a l s or o t a t ew i t ht h e b l a d e a n d a r e f i x e d w i t h r e s p e c t t o a l o c a l b l a d e s e c t i o n o f t h e undeformed blade, w i t ho r i g i n a t t h e e l a s t i c a x i s andwiththe z - a x i sp a r a l l e lt ot h ec h o r d l i n e . The h o r i z o n t a lo f f s e to ft h e e l a s t i c a x i sw i t hr e s p e c tt ot h eX o - a x i s i s denoted Cz. The a n g l e which a s e c t i o n of t h e undeformedblademakeswiththeXo-Zoplane, combiningbuilt-in t w i s t a n d p i t c h c o n t r o l s e t t i n g s , i s denoted @.

Blac?e motions are d e f i n e d by t h e f i v e v a r i a b l e s v , w , ( 9 , e and

I#, The d e f l e c t i o n s of t h e e l a s t i c axis i n t h e y and z d i r e c t i o n s are v and w , r e s p e c t i v e l y , t h e t o r s i o n a l d e f l e c t i o n i s (9, w h i l e e and $ a r e b e n d i n g s l o p e s : A - A Figure 1. Coordinate systems f o r a r o t o r i n forward flight.

These f i v e v a r i a b l e s are e x p r e s s e d i n terms o f t h e N coupled modes of free v i b r a t i o n s e l e c t e d t o r e p r e s e n t t h e r o t a t i n g m e c h a n i c a l s y s t e m .S p e c i f i c a l l y ,t h e s ev a r i a b l e s are e x p r e s s i b l ei nt h e form: ’ (1) where ( n ) (x) , Aw ( n ) (x), e t c . , d e n o t et h ev a l u e s of v , w, e t c . , AV t h o b t a i n e d o n e x c i t a t i o n o f t h e n- coupled mode.

I fL a g r a n g e ’ se q u a t i o n sa r ea p p l i e dt ot h eS y s t e m ,w i t ht h e f u n c t i o n s 5 1 , 5 2 , ... , cN a sg e n e r a l i z e dc o o r d i n a t e s ,t h es y s t e m e q u a t i o n s of motion are givenby t h where w’ i s t h en a t u r a lf r e q u e n c yo f t h e k- mode and Fk is t h e k g e n e r a l i z e df o r c ea p p l i e dt ot h a t mode. I t is assumed t h a t t h e mode shapeshavebeennormalized s o as t o y i e l d a g e n e r a l i z e d mass o fu n i t y .

The g e n e r a l i z e d f o r c e s are e x p r e s s e d by where R i s r o t o r r a d i u s .

O m i t t i n g t h e s t e a d y - s t a t e f o r c i n g terms, which are n o t o f c o n c e r n t o a s t a b i l i t y a n a l y s i s , t h e f o r c e and moment d i s t r i b u t i o n s Q,, Q w r etc. a r eg i v e n by

~ , ( x , t ) = - 2 m ~ E s i n cp$ + ~ ~ ( x , t )

Q $ ( x , ~ ) = - 21,n; + 2 m n s

where Fv and Fw are aerodynam i c f o r c e s i n t h e y and z d i r e c t i o n s , r e s p e c t i v e l y , M i s theaerodynamic moment about t h e e l a s t i c a x i s ,

+

E i s t h e d i s t a n c e o f t h e s e c t i o n mass c e n t e r f o r w a r d o f t h e e l a s t i c a x i s , m i s b l a d e mass p e ru n i ts p a na n d Io i s mass moment o f - i n e r t j a a b o u tt h e e l a s t i c a x i sp e ru n i ts p a n . The terms c o n t a i n i n g Q and + r e p r e s e n tg y r o s c o p i cc o u p l i n g terms. They are p l a c e d on t h e r i g h t - hand s i d e b e c a u s e t h e i r i n c l u s i o n i n t h e v i b r a t i o n a n a l y s i s t o o b t a i n mode shapesandfrequencies would d i s r u p t t h e o r t h o g o n a l i t y o f t h e modes. N o q u a n t i t y Q similar t o t h e q u a n t i t y Q a p p e a r s e ’ Q r i n E q s . ( 3 ) o r ( 4 ) b e c a u s et h e mass moment o fi n e r t i ao f a b l a d e s e c t i o na b o u t i t s c h o r d l i n eh a sb e e na s s u m e dt ob en e g l i g i b l e .

The n e x t s t e p is t o o b t a i n e x p r e s s i o n s f o r t h e aerodynamic f o r c e sa n d a p p r o p r i a t e l i n e a r i z a t i o n s . To accom- t o perform the d e f i n e ,f i r s t , a , an e f f e c t i v es e c t i o ni n c i d e n c e p l i s h t h i s where Vn and Vt are c o m p o n e n t s o f f l u i d v e l o c i t y r e l a t i v e t o t h e b l a d e , a t m i d c h o r d ,n o r m a la n dt a n g e n t i a lt ot h er o t o r . p l a n e , r e s p e c t i v e l y .I f as d e n o t e ss h a f ta n g l e( p o s i t i v ef o r a f t tilt of t h e Q - v e c t o r ) , Z a i s d i s t a n c e o f t h e e l a s t i c a x i s forwardof mid- chordand Qt i s t h eb l a d ea z i m u t hr e l a t i v et ot h ed o w n s t r e a md i r e c - t i o n ,t h e nt h e s ev e l o c i t yc o m p o n e n t s are g i v e n by

= [ ; + z a i - n ( z a - L ~ ) ( 8 cos cp + Q s i n 0 ) I cos cp

vn

- w s i n cp + Vf cos c1 cos n t ( e cos @ + I ) s i n cp)

e S

+ Vf s i n 0 : - w

S i ’ ( 5 )

Vt = Ox + Vf cos c1 s i n a t + w cos cp

S

+ 1 ; + z a i - R ( Z , - L,) ( e cos cp + Q s i n @ ) I s i n cp

i where w i s t h e wake-inducedinflow,assumedconstantover t h e i r o t o rp l a n e . The f a c t o r Q ( 8 cos cp + Q s i n 0 ) a p p e a r si n E q s . ( 5 ) b e c a u s et h ea n g u l a rv e l o c i t yd i r e c t e da l o n gt h e Y o ( s h a f t )a x i s , ofmagnitude R , h a s a componentwhich i s t a n g e n t t o t h e b l a d e when t h eb l a d e i s i n c l i n e d t o t h e X 0 - Z O p l a n e ,d u et ob e n d i n g , t h ea n g l e of i n c l i n a t i o nb e i n g 8 c o s @ + $ s i n 0.

The aerodynamic l i f t 1, a c t i n gn o r m a lt ot h er e l a t i v ef l u i d v e l o c i t y ,a n dc o n t a i n i n g a term t oa c c o u n tf o rt h ea p p a r e n t camber due t o r o t a t i o n o f t h e s e c t i o n a b o u t t h e x - a x i s , t h e d r a g d a c t i n g P a r a l l e l t o t h e r e l a t i v e v e l o c i t y , and t h e moment M ~ , ? a c t i n g about - I - mid-chord,aregiven by ( 6 ) where v2 = v i + v : .

The b l a d ec h o r d i s C and CL, Cd and Cm are e x p e r i m e n t a l l y deter- mined two-dimensional force coefficients. The apparent-camber term arises due t o t h e l i n e a r v a r i a t i o n o f q u a s i - s t e a d y downwash a l o n gt h eb l a d ec h o r d when t h e b l a d e s e c t i o n i s r o t a t e d a b o u t mid- chord. The aerodynamic forces appearing in Eqs. ( 4 ) are related t ot h o s ei n Eqs. ( 6 ) by t h ef o l l o w i n ge x p r e s s i o n s :

= - L cos a - d s i n a

FV

Fw = 1 s i n a - d c o s a

The n e x ts t e p i s t op e r f o r m a c o n s i s t e n t l i n e a r i z a t i o n o f Eqs. ( 7 ) by expandingthevariousZunctionsappearing i n t h o s e e q u a t i o n si nT a y l o r series a n dd i s c a r d i n gs e c o n d - o r d e rq u a n t i t i e s

i nv , w , 4 , e o r I). The expansion i s made about the nominal i n c i -

dence c10 andnominalrelativespeed V o , where I " 1 S i 5 = t a n I

R X + vf cos c1 s i n R - t

S as o b t a i n e d when a l l d e p e n d e n tv a r i a b l e sv a n i s h i n t h ee x p r e s s i o n s f o r V and a. S p e c i f i c a l l y ,t h ee x p a n s i o n s of a

and v y i e l d

a = a0 + Aa + ...

= a0 I V = Vo + AV + ...

= V O + - [vf s i n a - wi] { [G + z a i

VO S

- R ( Z a - L z ) ( e c o s @ + $ s i n @ ) ] c o s @

- G s i n @ + vf c o s as c o s n t ( e c o s @ + 9 s i n I

- R ( Z , - l Z ) ( e cos @ + $ s i n a ) I s i n @ } + . . .

Each t e r m i n Eqs. ( 7 ) i s thenexpanded,usingtheseexpressions f o r c1 and V. Forexample,the f i r s t t e r m i n FV is expanded by w r i t i n g L cos a as f o l l o w s :

- s i n a0Aa + ... I

Thus, on d i s c a r d i n gh i g h e r - o r d e r terms and t h e s t e a d y s t a t e term ~ p V ~ C C e ( a o ) c o s a o ,w h i c h i s n o to fc o n c e r ni n a s t a b i l i t y a n a l y s i s , L c o s a i s g i v e n by

+ I ) s i n @ ) ] c o s @

- G s i n + vf c o s a s c o s a t ( e cos CP + I ) s i n 0 1

+ $ s i n Q ) l c o s @

- G s i n + vf c o s a s c o s n t ( e c o s o + q, s i n @ ) I

+ q, s i n @ ) ] s i n @ I

I

I t can beseen a t t h i s p o i n t t h a t , by s y s t e m a t i c a l l y andcon- s i s t e n t l y e x p a n d i n g a l l terms i n t h e manner o u t l i n e da b o v e ,t h e effects o fc o m p r e s s i b i l i t y , s t a l l and reversedflowhave been r e t a i n e di nt h ef o r m u l a t i o n . The c o e f f i c i e n t s CL, Cd and Cm and t h e i 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 i n c i d e n c e , a l l e v a l u a t e d a t t h e n o m i n a li n c i d e n c dc c ~ ( x , t ) ,a p p e a ri nt h ef o r m u l a t i o n s as p e r i - o d i c a l l yv a r y i n gf a c t o r si nt h ee x p r e s s i o n sf o rt h ec o e f f i c i e n t s i nt h ee q u a t i o n so fm o t i o n .T h u s ,p r o v i d e dt h ec o r r e c ta n d c o m p l e t ev a r i a t i o n s of t h o s ec o e f f i c i e n t sw i t hn o m i n a li n c i d e n c e andnominal Mach number a r e r e t a i n e d , t h e e f f e c t s o f c o m p r e s s i - b i l i t y , s t a l l andreversedflow on t h e l i n e a r s t a b i l i t y o f t h e system are p r o p e r l y t a k e n i n t o a c c o u n t .

Once E q s . (7) have been expanded as o u t l i n e da b o v e ,t h o s e re- l a t i o n s are s u b s t i t u t e di n Eqs. (4), which are i nt u r ns u b s t i t u t e d i n Eqs. ( 2 ) . With t h eb l a d em o t i o n se x p r e s s e di n terms o ft h e g e n e r a l i z e dc o o r d i n a t e s ,t h r o u g h E q s . (11, t h ep e r t u r b a t i o n equa- t i o n s of motion a r eo b t a i n e d . Specifically, t h o s ee q u a t i o n s are:

k = 1,2, ..., N

!

i n which r 7

v cos u - cos 5 cosnt

S i Y L t

I -

i n which I !

i n which

A ( x , t ) = - 2 1 0 Q A ( n ) + 2 m R ~ A;")sin cp + Aw (n) cos cp J

9 1 Jln

DEVELOPMENT OF THE BASIC METHOD The s p e c i f i c p r o b l e m a t hand here i s t o establish t h e means f o r d e t e r m i n i n g t h e s t a b i l i t y o f a set o f l i n e a r e q u a t i o n s w i t h p e r i o d i c c o e f f i c i e n t s , s u c h as t h o s e r e p r e s e n t i n g a h e l i c o p t e r

r o t o ri nf o r w a r df l i g h t , Eqs. ( 8 ) . Thoseequations are p u t i n a

somewhatmore convenientformbydefininganindependentvariable z, a n dc h a n g i n gt h en o t a t i o n of t h e c o e f f i c i e n t s , a c c o r d i n g t o m,n = 1 , 2 , ..., N .

t

The e q u a t i o n s t o be analyzedthen become m = 1 , 2 , . . . , N ; where amn and bmn are p e r i o d i c f u n c t i o n s :

a ( z + IT) = amn (.z)

m n As w a s n o t e d i n t h e i n t r o d u c t i o n , e q u a t i o n s of t h e formof Eqs. ( 9 ) h a v er e c e i v e dc o n s i d e r a b l ea t t e n t i o no v e r the pastcen- tury.Althoughattempts a t g e n e r a ls o l u t i o n sh a v e m e t w i t h little s u c c e s s , t h e f o r m o f t h e s o l u t i o n i n t h e g e n e r a l case has been I .

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

developed. The theory of Floquet (see Reference 6 ) y i e l d st h a t where $,(z) i s p e r i o d i c ,w i t h a p e r i o d 71, and i w i s a complex con- s t a n t .S i n c et h ed i f f e r e n t i a ls y s t e m( E q s . ( 9 ) ) is o fo r d e r 2 N , t h e r e a r e 2 N v a l u e so f i w d e f i n i n g t h e s o l u t i o n f o r a given case.

I f anyone of t h e s e c h a r a c t e r i s t i c v a l u e s h a s a p o s i t i v e real p a r t , t h es y s t e m is u n s t a b l e .

I n o r d e r t o s e c u r e t h e t h e o r e t i c a l b a s i s f o r t h e s o l u t i o n and t oa v o i dn u m e r i c a ld i f f i c u l t i e s , it i s n e c e s s a r y t o f i r s t o p e r a t e on Eqs. ( 9 ) t oo b t a i n two r e l a t e d sets o fe q u a t i o n s . I t i s convenient f o rt h i sp u r p o s et ou s em a t r i xn o t a t i o n . A column m - t r i x X and s q u a r e matrices A and B canbedefined as 5 1 5 2

x = A = B =

5N I so t h a t Eqs. ( 9 ) canbe w r i t t e n d2 X

dX + BX = 0

- + A d z dz w h e r e t h e d e r i v a t i v e o f a m a t r i x i s t h em a t r i xf o r m e d by r e p l a c i n g eachelement by i t s d e r i v a t i v e .

i n o b t a i n i n g the two r e l a t e d sets o fe q u a t i o n s The f i r s t s t e p

is t o d i f f e r e n t i a t e Eq. ( 1 0 ) , y i e l d i n g

d2X +

- d 3 X + A -

dz dz If Eq. ( 1 0 ) i s s o l v e df o r d2X/dz2and t h e r e s u l t s u b s t i t u t e d i n E q . (ll), it i s f o u n dt h a t dx

- d 3 X + c - + DX = 0

dz d~ 3 where

c = - dA + B-A*

d z I f , now, Eq. ( 1 2 ) is d i f f e r e n t i a t e d and t h es e c o n dd e r i v a t i v e o f X e l i m i n a t e d as b e f o r e , it i s f o u n dt h a t d X

- d 4 X + E - + F X = 0

dz d~ 4 where E = -

dC + D - CA

d z F = - - dD CB .

d z I t can be shown t h a t a set o ff u n c t i o n s is a s o l u t i o n of t h e o r i g i n a le q u a t i o n (Eq. (lo)), i f a n d o n l y i f it i s a s o l u t i o n t o b o t h of t h ed e r i v e de q u a t i o n s , Eq. (12) and Eq. (13). As a r e s u l t , t h es o l u t i o n so f Eq. ( 1 0 ) can be found by solvingEqs. (12) and (13) a n d i d e n t i f y i n g t h o s e s o l u t i o n s common t o t h e l a t t e r two e q u a t i o n s .

Theproof i s s t r a i g h t f o r w a r d , and i s g i v e ni n Appendix A.

C o n s i d e r ,f i r s t ,t h es o l u t i o nt o Eq. (12). I t i s convenient a t t h i s p o i n t t o a b a n d o nt h em a t r i xn o t a t i o n .T h u s ,i f cm and d e n o t et h ee l e m e n t so f matrices C and D , r e s p e c t i v e l y , Eq. (12) dmn g i v e st h a t m = 1 , 2 , ..., N .

From t h e t h e o r y of F l o q u e t ,t h es o l u t i o n of Eqs. (14) must be of t h e form m = l f 2 , . . . , N ; upon expansion of +m i n a complex F o u r i e r series. Also, t h ep e r i - o d i cc o e f f i c i e n t s i n Eqs. (14) canbeexpanded i nF o u r i e r series: I ft h eF o u r i e rr e p r e s e n t a t i o n sf o rt h es o l u t i o na n df o rt h e c o e f f i c i e n t sa r es u b s t i t u t e di n Eqs. ( 1 4 ) a n dt h ec o e f f i c i e n t so f 2ikz e aregroupedand s e t e q u a l t o z e r o , a s e t o f l i n e a r r e c u r s i o n r e l a t i o n si nt h e unknown c o e f f i c i e n t s pmk i s o b t a i n e d .S p e c i f i - c a l l y , it i s f o u n dt h a t = . . . f - 2 , - l , 0 , 1 , 2 f . . . ; n

r = 1 , 2 , ..., N

l i n e a r a l g e b r a i c T h e s e r e l a t i o n s c o n s t i t u t e a n i n f i n i t e set of s e t e q u a t i o n s i n t h e unknown F o u r i e r c o e f f i c i e n t s prnk. F o r t h i s o fe q u a t i o n s t o have a n o n t r i v i a l s o l u t i o n , t h e a s s o c i a t e d i n f i n i t e d e t e r m i n a n t A ( w ) mustvanish ( a d i s c u s s i o n of i n f i n i t e d e t e r m i n a n t s is g i v e ni nR e f e r e n c e 6 ) . The r e q u i r e m e n tt h a t A v a n i s h is t h e con- d i t i o n whichdetermines w, a n dh e n c et h es t a b i l i t y of thesystem.

I n o r d e r t h a t A b em e a n i n g f u l l yd e f i n e d , it i s n e c e s s a r y t o divideeachofEqs. (15) through by t h ec o e f f i c i e n to f prn. With t h e unknowns t h e na p p r o p r i a t e l yo r d e r e d ,t h ed i a g o n a le l e m e n t so f A are a l l u n i t y a n dt h eo f f - d i a g o n a le l e m e n t s a l l v a n i s h i n t h e l i m i t a s a row o r column i n d e xo ft h ed e t e r m i n a n tt e n d st o either p o s i t i v eo rn e g a t i v ei n f i n i t y .S p e c i f i c a l l y ,t h ee l e m e n t s u U v of

A ( w ) ( p , v = 0 , *I, i 2 ...) are g i v e n by

- a Nn+r, Nk+s - a (n,k;w) rs n,k = ... , - 2 , - 1 , 0 , 1 , 2 ,... ; r,s = 1 , 2 ,..., N , where i n which = l , i = j .

Note t h a t t h e diagonal elements u a r e a l l u n i t y . The con- u p s t r u c t i o n o f t h e d e t e r m i n a n tc a np e r h a p sb eb e s tv i s u a l i z e da s made upof a c o l l e c t i o no fs u b a r r a y s which a r e NxN i n s i z e . The l o c a t i o no f any e l e m e n tw i t h i n a s u b a r r a y is determinedfrom i n d i c e s r and s. The l o c a t i o no fe a c hs u b a r r a y is s p e c i f i e d throughindices n and k. The g e n e r a la r r a n g e m e n to ft h e a's i n A i s i l l u s t r a t e d i n F i g u r e 2 f o r t h e case N = 2 .

The v a l u eo f A i s o b t a i n e d , f o r a g i v e n w , by e v a l u a t i n g t h e f i n i t ed e t e r m i n a n tf o r m e d by ranging n and k from,say, -L t o L.

i . 0 .

. ...

...

Figure 2 . The arrangement of thedeterminant elements for t h e case N = 2 S u c c e s s i v e l y l a r g e r v a l u e s f o r L a r e t h e n t a k e n u n t i l t h e limit be- !

comes apparent '(see Reference 6 ) .

The expression by i t s e l f i s n o t a p a r t i c u l a r l y u s e f u l r e l a t i o n f o r d e t e r m i n i n g w , b e c a u s e o f t h e l i m i t i n g p r o c e s s i n v o l v e d i n e v a l u a t i n g i n f i n i t e determinants. I t w i l l be shown, however, t h a t A ( w ) i n f a c t con- s t i t u t e s acombinedseries-productexpansionin w of a f i n i t e sum of t r i g o n o m e t r i cf u n c t i o n s . With A ( w ) e x p r e s s e di nt h e latter form, Eq. (16) c a nb es o l v e de x p l i c i t l yf o r w .

I t s h o u l d b e n o t e d h e r e t h a t H i l l obtained such a r e l a t i o n f o r t h e s e c o n d - o r d e r d i f f e r e n t i a l e q u a t i o n w h i c h bears h i s name (see Reference 6 ) . Thedevelopmentwhichfollows i s a g e n e r a l i z a - t i o n o f t h a t r e s u l t .

Two p r o p e r t i e s o f t h e f u n c t i o n A(w) must f i r s t be e s t a b l i s h e d , S p e c i f i c a l l y , it i s a s s e r t e d t h a t !

1. A ( w ) is an a n a l y t i cf u n c t i o n of w , e x c e p tf o rs i m p l e , i d e n t i f i a b l e p o l e s ; 2. A ( w ) i s p e r i o d i c i n w w i t h a p e r i o do f 2.

That A ( w ) i s a n a l y t i cc a nb ec o n c l u d e d by n o t i n g t h a t A is an a b s o l u t e l yc o n v e r g e n td e t e r m i n a n t - - i . e . ,t h ep r o d u c to ft h ed i a g - o n a le l e m e n t sc o n v e r g e sa b s o l u t e l y( i nt h i s case t ou n i t y )a n dt h e double sum oftheoff-diagonalelemengsconvergesabsolutely.

Uniform convergenceandanalyticitycanthen be e s t a b l i s h e d (see Reference 6 ) .

N o t e t h a t i f t h e o r i g i n a l e q u a t i o n s (Eqs. (9)), r a t h e rt h a n d e r i v e de q u a t i o n s ( E q s . ( 1 4 ) ) , h a db e e nu s e d t o g e n e r a t e A , t h a t determinantwouldnothavebeenabsolutelyconvergentsincethe double sum o f o f f - d i a g o n a l e l e m e n t s g e n e r a t e d f r o m t h e o r i g i n a l e q u a t i o n s is onlyconditionallyconvergent. What f o l l o w sh i n g e s an the a n a l y t i c i t y o f A , h e n c et h en e c e s s i t yf o rw o r k i n gw i t ht h e d i f f e r e n t i a t e d sets o f e q u a t i o n s , I w h e r et h e X's are t h e 3N r o o t s of t h e N cubic e q u a t i o n s + r = 1 , 2 , . .. , N .

X 3 + X C r r 0 n = o , r r 0

The p e r i o d i c i t y o f A i s shown by s u b s t i t u t i n g w + 2 f o r w i n

t h ee x p r e s s i o nf o ra r S , whereupon it is found t h a t

a (n,k;w + 2 ) = a r s ( n + 1 , k + 1 ; w )

rs Thus, i n t h e l i m i t , t h ev a l u eo f A is unchanged i f w i s r e p l a c e d by w + 2 : A ( w + 2 ) = A ( w ) .

Now, c o n s i d e rt h ef u n c t i o n D ( w ) , d e f i n e d by N 3 where the f ' s are c o n s t a n t s . O b s e r v e t h a t i s an a n a l y t i c r j f u n c t i o na n dt h a t F u r t h e r ,n o t et h a tt h e term

h a ss i m p l ep o l e sa t w = - 2n - i h f o r n = 0 , * 1 , * 2 , .. . , andhas

r j no o t h e rs i n g u l a r i t i e s .T h u s ,i ft h ev a l u ef o re a c hc o n s t a n t f r j i s properly chosen, D ( w ) w i l l have no poles. L e t t h e f ' s be s o r j chosen,making D ( w ) a n a l y t i ct h r o u g h o u tt h ef i n i t ep a r to ft h e w- plane.But D i s c l e a r l y bounded a t i n f i n i t y ; A ( w ) has a l i m i t of one and t h ec o t a n g e n t sh a v e limits of *i f o r I m ( w ) - + * m . T h e r e f o r e , by L i o u v i l l e ' st h e o r e m , D ( u ) is simply some c o n s t a n t ,s a y c: It o n l yr e m a i n st od e t e r m i n et h ev a l u e so ft h e f ' s and of r j c. T h i s is f a c i l i t a t e db y first c o n s i d e r i n gt h e limits of D ( w ) w i t h i n f i n i t e o: C l e a r l y ,t h e n , c=l and Replacing c by u n i t ya n ds o l v i n gf o r A i n Eq. (18) g i v e st h a t Now, l e t 3N - 1 a r b i t r a r y ( b u t f i n i t e )v a l u e so f w , s a y w l , U2,... b ea s s i g n e di nt h e aboveequation.Theresulting 'w3N-1' 3N - 1 e q u a t i o n s ,t o g e t h e rw i t ht h eo n eo b t a i n e df o ri n f i n i t e w , p r o v i d es u f f i c i e n tr e l a t i o n s t o s o l v ef o rt h e f I s . More s p e c i - r j f i c a l l y ,t h o s ec o n s t a n t sa r et h es o l u t i o no f N 3 1 1 f cot[:(wk + i A ) 3 = 1 - A(wk) , k = 1,2 ,..., 3N - 1; r=1 j = 1 r j r j With t h e f 's known, t h ed e t e r m i n a n t a le q u a t i o n , A ( w ) = 0 , r j canbereplacedby N 3 irw A polynomialofdegree 3 N i n e canbereadilyderivedfrom Eq. ( 2 0 ) . The 3 N r o o t so ft h a tp o l y n o m i a ld e t e r m i n et h ec h a r a c - t e r i s t i c v a l u e sf o rE q s . ( 1 2 ) .

OUTLINE O F B A S I C COMPUTER PROGRAM I n o r d e r t o implement t h e m e t h o dd e r i v e df o ra n a l y z i n g sta- b i l i t y o f dynamicsystemswithperiodicparameters, a b a s i c d i g i t a l w a s p r e p a r e d f o r t h e a n a l y s i s of systemswith computerprogram threedegreesoffreedom. The p r o g r a ma c c e p t st h ep e r t i n e n td a t a f o r a g i v e n s y s t e m i n t h e form o f F o u r i e r c o e f f i c i e n t s of t h e p e r i o d i cc o e f f i c i e n t si nt h ee q u a t i o n so fm o t i o n( E q s . ( 9 1 , w i t h N = 3 ) , a n d t h e n p r o c e e d s t o c a l c u l a t e t h e s i x c h a r a c t e r i s t i c valuesofthesystem by t h e m e t h o d d e r i v e d i n t h e p r e v i o u s s e c t i o n .

F o rs p e c i f i ca p p l i c a t i o n s , a s m a l ls u b r o u t i n ec a nb ep r e p a r e d ,i f n e c e s s a r y , t o g e n e r a t e t h e F o u r i e r c o e f f i c i e n t s u s e d a s i n p u t b y thebasiccomputerprogram.

The o v e r a l lf l o wo fi n f o r m a t i o ng u i d e db yt h ef o r m u l a t i o n s of t h eb a s i cp r o g r a m are outlinedbelow. The numbers i nt h eo u t - l i n ec o r r e s p o n dt ot h o s ei nt h eb l o c k so nt h ef l o wd i a g r a mo f Figure 3 . The o u t l i n ei n c l u d e sr e f e r e n c et ot h ep e r t i n e n te q u a - t i o n so ft h ep r e v i o u ss e c t i o n . The r e l a t i o n sg u i d i n gt h e more r o u t i n ep r o c e d u r e s ,s u c h as e x t r a c t i o no ft h er o o t so fp o l y n o m i a l s a n d s o l v i n g o f l i n e a r a l g e b r a i c e q u a t i o n s , h a v e b e e n o m i t t e d , b u t are embodied i n t h e computerprogram, a l i s t i n g o f which i s g i v e n i n Appendix B.

1. F o u r i e rE x p a n s i o no fC o e f f i c i e n t s Given t h e c o e f f i c i e n t s of t h e e q u a t i o n s of m o t i o nf o r a l i n e a r dynamicsystemwiththreedegreesoffreedomandperiodically varyingparameterswithnormal modes u s e d a s g e n e r a l i z e d co- o r d i n a t e s ,s u c ha s E q s . ( 9 1 , t h ec o e f f i c i e n t s are expanded 1 1 1 I1 I II I1 I. 1 1 1 -111 I I I and d e t e n m i n a n t e v a l u a t i o n

I , ( 4 a 1 J

b d e t e a m i n a n t + v a l u e 4 S o l u t i o n o b L i n e a n eq u a t i o nn

t

S y b t e m c h a n a c t e k i b t i c V a l u e d F i g u r e 3. Procedure f o r c a l c u l a t i n gc h a r a c t e r i s t i cv a l u e so f a dynamicsystemwiththreedegrees of freedomand p e r i o d i c a l l y v a r y i n g p a r a m e t e r s a set of F o u r i e r i nF o u r i e r series. The s p e c i f i co u t p u t is am and bmn appear- c o e f f i c i e n t s f o r t h e p e r i o d i c c o e f f i c i e n t s (9) w i t h N = 3 ; m,n=1,2,3. This subroutine i s i n gi n ' E q s .

a s s o c i a t e d w i t h t h e p a r t i c u l a r dynamicsystembeinganalyzed.

andcanbedescribed as p r e p a r i n g t h e i n p u t t o t h e m a i n s t a b i l i t y - a n a l y s i s program.

2 . Matrix M u l t i p l i c a t i o n and FourierExpansionof C o e f f i c i e n t s Given t h e F o u r i e r c o e f f i c i e n t s of t h ee l e m e n t s of t h e 3x3 m a t r i c e s A and B a p p e a r i n gi n E q . (101, t h eF o u r i e rc o e f f i c i - e n t so ft h ee l e m e n t so f matrices C and D , a p p e a r i n g i n Eq.

( 1 2 ) ,andof E and F, a p p e a r i n gi n Eq. (13), are c a l c u l a t e d .

I n t h i s way, t h ec o e f f i c i e n t so ft h eh i g h e r - o r d e r , or d i f f e r - e n t i a t e d ,s y s t e m so fe q u a t i o n s are r e l a t e d t o t h e o r i g i n a l systemofequations,Eqs. ( 9 ) .

3. Polynomial Factorization The t h r e e cubic p o l y n o m i a l sa s s o c i a t e dw i t ht h ed e t e r m i n a n t f o r t h e n i n t h - o r d e r s y s t e m a r e f a c t o r e d t o o b t a i n t h e n i n e s i n g u l a r i t i e s of t h a td e t e r m i n a n t( s u b r o u t i n e3 a ) . The t h r e e q u a r t i c p o l y n o m i a l s a s s o c i a t e d w i t h t h e t w e l f t h - o r d e r s y s t e m are f a c t o r e d t o o b t a i n t h e t w e l v e s i n g u l a r i t i e s o f t h a t d e t e r -

minant(subroutine3b) . The p o i n t s uk a t which t h e i n f i n i t e

d e t e r m i n a n t s are t o b ee v a l u a t e d are a l s o s e l e c t e d , by s p e c i - fyingeachofthem t o be a set d i s t a n c e 6 fromoneofthe s i n g u l a rp o i n t so ft h ed e t e r m i n a n t . The accuracyofthe s o l u - t i o n was found t o b eq u i t es e n s i t i v et ot h ev a l u e of 6 , it g e n e r a l l y b e i n g n e c e s s a r y t o make 6 a s small as p o s s i b l e , w i t h o u ts a c r i f i c i n gn u m e r i c a la c c u r a c ye l s e w h e r e i n t h ep r o - gram, i no r d e rt og e ts a t i s f a c t o r yr e s u l t s . The r e a s o nf o r t h i sb e h a v i o r i s n o t known, b u t i s presumablyconnectedin some way t o t h e a c c u r a c yw i t hw h i c ht h ei n f i n i t ed e t e r m i n a n t s are c a l c u l a t e d ,t h i sb e i n gt h em o s td i f f i c u l t ,a n dh e n c e least a c c u r a t e ,o ft h et a s k sp e r f o r m e d by t h e program. A v a l u e f o r 6 o f was u s e d f o r m o s t o f t h e c a l c u l a t i o n s r e p o r t e d h e r e .

4 . Determinant Element and Determinant Evaluation For each value of w the determinant elements are computed k ' and the i n f i n i t ed e t e r m i n a n t i s e v a l u a t e d . It was a n t i c i p a t e d that this s t e p wouldbethemosttime-consumingoneinthe program,and so c o n s i d e r a b l ec a r e was t a k e n t o employ the mosteconomical means f o rd e t e r m i n a n te v a l u a t i o n . A p r o c e s s o f t r i a n g u l a r i z a t i o n was s e l e c t e d f o r e v a l u a t i n g f i n i t e determinants.

The l i m i t i n gv a l u eo f a d e t e r m i n a n t ,f o r a g i v e n uk, as t h e number ofelementsincreaseswithoutbound, is e s t i m a t e d as f o l l o w s .T h r e ed e t e r m i n a n tv a l u e s ,f o rt h r e es u c c e s s i v e l y l a r g e r numbers of rows and columns, are first obtained. The determinant i s t h e n assumed t o v a r y i n v e r s e l y as some unknown power o f t h e number o f rows and columns. The assumed :form h a s t h r e e unknown c o n s t a n t s , so t h e t h r e e d e t e r m i n a n t v a l u e s , c o r r e s p o n d i n gt o three d i f f e r e n t d e t e r m i n a n t s i z e s , p r o v i d e s u f f i c i e n t i n f o r m a t i o n t o c a l c u l a t e t h e c o n s t a n t s andhence a ne s t i m a t eo ft h e l i m i t as t h e number ofelements becomes i n f i n i t e . It has been found t h a tt h r e ed e t e r m i n a n t s , re- s p e c t i v e l y 33x33, 39x39 and 45x45 i n s i z e , g e n e r a l l y are s u f f i c i e n tt op r o v i d et h r e e - p l a c ea c c u r a c yi n t h e r e s u l t .

5. S e t - u pa n dS o l u t i o no ft h eL i n e a rA l g e b r a i cE q u a t i o n s For t h e n i n t h - o r d e rs y s t e m , t h e e i g h t v a l u e s o f u k andthe e i g h tv a l u e so f A ( W ) are u s e dt o compute t h e c o e f f i c i e n t s k and inhomogeneous terms of Eqs. ( 1 9 ) . The n i n ee q u a t i o n s a r e t h e n s o l v e d € o r t h e n i n e c o n s t a n t s f A s i m i l a r p r o - r j ' cedure i s f o l l o w e dt oo b t a i n t h e t w e l v ec o n s t a n t sf o rt h e twelfth-ordersystem. The e q u a t i o n s are s o l v e d by t r i a n g u - l a r i z a t i o n .

6. Determinationofthe Characteristic Polynomialsof t h e Higher-OrderSystems For t h e ninth-o.rdersystem, the c o e f f i c i e n t so ft h ep o l y - nomialof the n i n t h ategree i n e i'w a r e d e r i v e d from t h e deter- m i n a n t a l e q u a t i o n , Eq. ( 2 0 ) ( s u b r o u t i n e 6 a ) . S i m i l a r l y t h e c o e f f i c i e n t s of t h e c h a r a c t e r i s t i c p o l y n o m i a l o f t h e t w e l f t h - ordersystem are o b t a i n e d from t h e a p p r o p r i a t ed e t e r m i n a n t a l e q u a t i o n( s u b r o u t i n e6 b ) .

7. E v a l u a t i o no f t h e Common Polynomial Factor At t h i sp o i n t ,o n ec o u l dp r e s u m a b l y compute t h e n i n e r o o t s o f t h en i n t h - d e g r e ep o l y n o m i a l ,t h et w e l v er o o t so ft h et w e l f t h - degreepolynomialandidentify t h e r o o t s common t o the two polynomials as b e i n gt h ec h a r a c t e r i s t i cv a l u e so ft h eo r i g i n a l system. However, t h i sp r o c e d u r er e q u i r e se v a l u a t i o no ff i f t e e n e x t r a n e o u sr o o t s ,f o l l o w e d by p o s s i b l e d i f f i c u l t i e s i n i d e n t i - fyingwhichroots are indeed common ones.

It w a s found,though, t h a t t h e c o e f f i c i e n t so ft h ec h a r a c t e r - i s t i c e q u a t i o n f o r t h e o r i g i n a l s y s t e m , which is a c t u a l l y a p o l y n o m i a lf a c t o r common t o the two higher-degreepolynomials, c a nb eo b t a i n e di n terms o f t h e c o e f f i c i e n t s o f t h e s e h i g h e r - degreepolynomials. The s t e p st a k e nt od e r i v et h en e c e s s a r y e x p r e s s i o n sa r eo u t l i n e di n Appendix C. Subroutine 7 imple- m e n t s t h o s ee x p r e s s i o n s ,y i e l d i n gt h ec o e f f i c i e n t so ft h e s i x t h - d e g r e ep o l y n o m i a lc h a r a c t e r i z i n gt h eo r i g i n a ls y s t e m .

8. Polynomial Factorization A s t a n d a r d l i b r a r y s u b r o u t i n e is u s e d t o o b t a i n t h e s i x r o o t s o ft h ec h a r a c t e r i s t i cp o l y n o m i a l . The l o g a r i t h m of e a c hr o o t is t h e ne v a l u a t e d ( r e c a l l t h a tt h ep o l y n o m i a l i s formedof irw powers of e ) t oo b t a i nt h es i xc h a r a c t e r i s t i cv a l u e s , and h e n c et h es t a b i l i t y ,o ft h es y s t e m .

Duringthecheck-outofthecomputerprogram, it was found necessary t o e x t r a c t t h e r o o t s o f t h e h i g h e r - d e g r e e p o l y - nomials. Since l i t t l e a d d i t i o n a lr u n n i n g t i m e i s consumed by t h o s ec a l c u l a t i o n s ,d e t e r m i n a t i o no ft h er o o t sa n dc h a r a c - t e r i s t i c v a l u e sf o rt h en i n t h and t w e l f t h - o r d e rs y s t e m sh a s b e e nr e t a i n e di nt h ep r o g r a mf o rp u r p o s e s of comparison.

I

APPLICATION O F THE METHOD The computerprogramimplementingthegeneralmethod w a s first checkedout bymeans of a test case, t h e characteristic valuesof whichcouldbederivedinadvance.Theprogram w a s u s e dn e x tt o analyze a r o t o r s y s t e m w i t h two degrees of freedomforwhichsolu- t i o n s by d i r e c t time i n t e g r a t i o n h a d b e e n o b t a i n e d p r e v i o u s l y .

L a s t l y , c a l c u l a t i o n s were conducted f o r a model r o t o rs y s t e mh a v i n g t h r e ed e g r e e so ff r e e d o m ,f o rw h i c hf l u t t e rb o u n d a r i e sh a db e e n o b t a i n e de x p e r i m e n t a l l y .T h e s ea p p l i c a t i o n s are d i s c u s s e di nd e t a i l below.

Test CaseCalculations It was foundnecessary i n t h ec o u r s eo ft h ec h e c k - o u t of t h e computerprogram t o h a v ea v a i l a b l e a test c a s e f o r which t h ec h a r - a c t e r i s t i cv a l u e so ft h eo r i g i n a ls i x t h - o r d e rs y s t e m as w e l l as of t h ed e r i v e dn i n t h - o r d e rs y s t e m were known i n advance. Such a c a s e w a s g e n e r a t e d by a p p r o p r i a t e l y t r a n s f o r m i n g M a t h i e u ' s e q u a t i o n .

The s i x t h - o r d e rs y s t e m was t h e n made up from t h r e eo ft h e s ee q u a - t i o n s , t r e a t e d as a coupledsystem,theprogramnotbeing able t o d i s t i n g u i s hw h e t h e rt h ee q u a t i o n s are c o u p l e do rn o t .

The d e r i v a t i o n o f t h e test case was c a r r i e d o u t as f o l l o w s .

ConsiderMathieu'sequation,which i s g e n e r a l l y w r i t t e n i n t h e form d2 u

- + ( a - 2q cos 2 z ) u = 0

dz2 where a and q a r es p e c i f i e dc o n s t a n t s . The c h a r a c t e r i s t i cv a l u e s * p o fM a t h i e u ' se q u a t i o n are a f u n c t i o no f a and q (see Reference 5 ) .

Now, l e t a new d e p e n d e n tv a r i a b l e y be d e f i n e d by

- z f ( z )

y = ue where f (2) is p e r i o d i c w i t h a p e r i o d 71, b u t o t h e r w i s e may be re- g a r d e da sa r b i t r a r y .I f u i s s u b s t i t u t e d i n Eq. ( 2 1 ) i n terms of y and f t h e f o l l o w i n g d i f f e r e n t i a l e q u a t i o n f o r y i s obtained: * + A * + B y = O dz dz where

A = A f ( z )

~ = a - 2 q c o s 2 z + - f ~ + - - 1 1 df ti dz Suppose, now, f i s g i v e n b y f ( 2 ) = 1 ( A ~ cos 2nz + B~ s i n2 n z ) .

o n= I t t h e nf o l l o w st h a tt h ec h a r a c t e r i s t i cv a l u e so fy ,a s a s o l u t i o n

o f E q . ( 2 2 ) , mustbe p - A 0 / 6 and - p - A o / 6 . Thus, w e havederived

a s e c o n d - o r d e r d i f f e r e n t i a l e q u a t i o n w i t h p e r i o d i c c o e f f i c i e n t s , o f a q u i t eg e n e r a lf o r m ,f o r which t h e c h a r a c t e r i s t i c v a l u e s are known.

a t h i r d - o r d e re q u a t i o n is d e r i v e d by t h e F u r t h e r ,s u p p o s e methoddescribedpreviouslywhich eliminates t h e s e c o n d d e r i v a t i v e , namely * + C * + D y = O dz 3 dz where

C = B + - dA - A2

dz D = - - dB AB dz It i s p o s s i b l e , t h r o u g h f u r t h e r t r a n s f o r m a t i o n a n d u t i l i z a t i o n o f the r e l a t i o n sg i v e n on p. 134 ofReference 5 , t o show t h a t t h e I a d d i t i o n a lc h a r a c t e r i s t i cv a l u e of Eq. (23) i s t h en e g a t i v eo ft h e sum o f t h e o t h e r t w o v a l u e s , i-e., A0/3.

The t h r e e e q u a t i o n s o f t h e test case were a l l d e r i v e df r o mt h e Mathieu'sequationhaving a = 1, q = 0.32.Forthosevaluesof a a n dq , it f o l l o w st h a t 1-1 = -15813 + i (see p.105ofReference 5 ) .

1 Only t h ef i r s tt h r e e terms o ft h e series f o rf ( z ) were r e t a i n e d

f o re a c he q u a t i o n of t h e test case. The v a l u e so f A o , A1 and A, c h o s e n a n d t h e a s s o c i a t e d c h a r a c t e r i s t i c v a l u e s f o r t h e t h r e e de- greesoffreedomare as follows : A0 A0 A0 / 3 1-1 - 7 A2 A1 A0 - v - 7

- 1 . 5 0 . 5 9 1 8 7 - i

- 4.5 0 . 9 0 8 1 3 + i 0 . 6 2 . 0

- 0 . 6 - 0 . 2 - 0 . 0 5 8 1 3 - i 0 . 2 5 8 1 3 + i - 0 . 6 2.0

1.0 - 0 . 6 5 8 1 3 - i - 0 . 3 4 1 8 7 + i 0 . 6 - 2 . 0

3.0 i The F o u r i e rc o e f f i c i e n t so ft h ef u n c t i o n s A and B appearing i n E q . ( 2 2 ) w e r e c a l c u l a t e df o rt h et h r e ed e g r e e s of freedom and were s u p p l i e da si n p u t st ot h ec o m p u t e rp r o g r a m . The c h a r a c t e r - i s t i c valueswhich were c a l c u l a t e d f o r t h e n i n t h a n dt w e l f t ho r d e r s y s t e m sa r et a b u l a t e db e l o w ,t o g e t h e rw i t ht h ei n d e p e n d e n t l y d e r i v e de x a c t( a n t i c i p a t e d )v a l u e s . The q u a n t i t i e s X R and A I given are r e l a t e d t o w by i w = X R + i X , .

CHARACTERISTIC VALUES OF THE 9TH ORDER SYSTEM Calculated

l R l R 5

0 . 9 0 8 1 3 1 . 0 0 . 9 0 8 1 3 1 . 0

0 . 5 9 1 8 5 - 1 . 0 0 . 5 9 1 8 7

- 1 . 0

0 . 2 5 8 1 4 1 . 0 0 . 2 5 8 1 3 1 . 0

- 0 , 0 5 8 1 6 - 1 . 0 - 0 . 0 5 8 1 3 - 1 . 0

- 0 . 3 4 1 8 4

1 . 0 - 0 . 3 4 1 8 7 1 . 0

- 0 . 6 5 8 1 3 - 1 . 0 - 0 . 6 5 8 1 3 - 1 . 0

- 1 . 5 0 0 0 0

- 1 . 5 0 0 0 0 0.0 0.0

- 0 . 1 9 9 9 8 - 0 . 2 0 0 0 0

0 . 0 0.0 1 . 0 0 0 0 0 0 . 0 1 . 0 0 0 0 0 0 . 0 CHARACTERISTIC VALUES O F THE 12TH ORDER SYSTEM Calculated Anticipated

r 1

~ - ~ _ _ _ - -~ x R x R - ~ ~~ 0 . 9 0 8 0 3 1 . 0 0 . 9 0 8 1 3

- 1 . 0 0 . 5 9 1 8 7

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

- 0 . 0 5 9 6 0 - 0 . 0 5 8 1 3

- 1 . 0

- 0 . 3 4 1 8 8 1 . 0 - 0 . 3 4 1 8 7

- 0 . 6 5 8 1 3 - 1 . 0 - 0 . 6 5 8 1 3

-

0 . 9 7 6 9 9 1 . 0

-

0 . 0 2 3 6 9 - 1 . 0

-

- 0 . 1 0 9 7 0 1 . 0

-

- 0 . 4 1 4 2 5 0 . 0

-

0 . 2 1 4 3 1 0 . 0

-

- 1 . 3 8 9 3 5 - 1 . 0

- -

I

A s c a nb es e e nf r o mt h et a b u l a t e dr e s u l t s ,t h ec h a r a c t e r i s t i c v a l u e s common t o t h e n i n t h a n dt w e l f t ho r d e rs y s t e m s are t h o s eo f t h es i x t h - o r d e rs y s t e m .N o t e ,t o o ,t h a tt h ep r o g r a mf o rt h em o s t p a r t p r e d i c t s t h e c h a r a c t e r i s t i c v a l u e s f o r t h i s c a s e t o f o u r s i g n i f i c a n t f i g u r e s .

The c h a r a c t e r i s t i c v a l u e s were a l s o c a l c u l a t e d from t h e s i x t h - degreepolynomialwhich was derivedfromthe two higher-degree polynomials. The r e s u l k i n gr o o t s were u n a c c e p t a b l yd i f f e r e n t from t h ec o r r e c tv a l u e s ,a p p a r e n t l yb e c a u s et h ep o l y n o m i a la s so d e r i v e d is s e n s i t i v e t o s l i g h t i n a c c u r a c i e s i n t h e c o e f f i c i e n t s o f t h e h i g h e r - d e g r e ep o l y n o m i a l sf o rt h i s case. Thus, it may benecessary t o r e l y on directcomparisonofninth-orderandtwelfth-order s o l u - t i o n st od e t e r m i n et h ec o r r e c ts i x t h - o r d e rs o l u t i o n si n some i n - s t a n c e s .

ComparisonWith Direct T i m e I n t e g r a t i o n The a n a l y s i s o f s t a b i l i t y by d i r e c t t i m e i n t e g r a t i o n on a d i g i - t a l computerof a r i g i d r o t o r blade w i t hf l a p p i n ga n dl e a d - l a g h i n g e s is r e p o r t e d i n Reference 1 2 . The n o n l i n e a rr e p r e s e n t a t i o n s ofboth i n e r t i a l a n da e r o d y n a m i cf o r c e sa r eu t i l i z e d i n theequa- t i o n so fm o t i o n . The basic b l a d ef o rw h i c hn u m e r i c a lr e s u l t s were o b t a i n e d had z e r oo f f s e to ft h ef l a p p i n qh i n g ea n d 0 . 0 5 R o f f s e t o f t h el e a d - l a gh i n g e . The bladehad a c o n s t a n tc h o r de x c e p tf o r a c u t - o u tf r o mt h ea x i so fr o t a t i o nt o 0 . 2 R. I nt h ec a l c u l a t i o n s r e p o r t e di nR e f e r e n c e 1 2 , t h e r o t o r was unloadedand a t z e r o s h a f t a n g l e .F u r t h e rd e t a i l sc a n be found i n Reference 1 2 .

For two of t h e casesanalyzedinReference 1 2 , t h e v a r i a t i o n s offlappingandlead-lagangleswith t i m e are presented.These caseshadvaluesofadvanceratio u of 0 . 6 and 1 . 4 , r e s p e c t i v e l y .

The mass c o n s t a n t y l f o rb o t hc a s e s was 1 . 6 ( y ' = o c R 4 / I h , where Ih i s t h e mass moment o fi n e r t i aa b o u tt h el e a d - l a gh i n g e ) . The c a s e for t h el o w e ra d v a n c er a t i o is r e p o r t e d t o be v e r ys t a b l ea n d t h e case w i t h 1 ~ . = 1 . 4 i s i n d i c a t e d t o b e s t a b l e b u t n e a r a boundary o fn e u t r a ls t a b i l i t y . The t i m e h i s t o r i e so ft h e blade displacements, fromReference 1 2 , are r e p r o d u c e di nF i g u r e s 4 and 5.

The a p p r o p r i a t ep a r a m e t e rv a l u e sf o rt h e s e two c a s e s were i n - s e r t e d i n t h e l i n e a r i z e d e q u a t i o n s of motionof a r o t o r b l a d e d e v e l o p e dp r e v i o u s l yi nt h i sr e p o r t . Series r e p r e s e n t a t i o n so ft h e c o e f f i c i e n t s i n t h e e q u a t i o n s , i n c l u d i n g t h e f i r s t e l e v e n h a r m o n i c s o f r o t o r r o t a t i o n a l s p e e d , w e r e g e n e r a t e d a n d s u p p l i e d t o t h e main s t a b i l i t y - a n a l y s i s program. The c h a r a c t e r i s t i cv a l u e sc a l c u l a t e d are as f o l l o w s : 3 5 CHARACTERISTIC VALUES DEGREE OF p = 0.6 p = 1.4 FREEDOM XI X R XI l R

- 1 . 0

- 1.13400 - 0.73379 0.60206

Flapping

1 . 0 - 2,56010

- 0 . 6 0 2 0 6 - 1.13400

0.56206

- 0.00534

- 0.00353 0.56195

Lead-Lag

- 0.56206

- 0.00534 - 0.56195

- 0.00353

E x a m i n a t i o n o f t h e s e r e s u l t s i n d i c a t e s q u a l i t a t i v e a g r e e m e n t w i t h t h e r e s u l t s of t h e a n a l y s i s by d i r e c t t i m e i n t e g r a t i o n b u t t h e r e is e v i d e n c eo f some d i f f e r e n c e si nq u a n t i t y . A t 1 . 1 = 0 . 6 , t h e flappingmotionshould damp by a f a c t o r e-’ = 0.135 i n 4 / ( - X R ) = 3.527 radiansofazimuthchange, by t h e r e s u l t o b t a i n e d h e r e , whereasFigure 4 i n d i c a t e s a much more r a p i dd e c r e a s e . On t h e o t h e rh a n d , a t 1 . 1 = 1 . 4 , t h e f a c t o r e-* s h o u l da p p l yt ot h ef l a p p i n g m o t i o nf o r a changeof 4 / ( - X R ) = 5.452 r a d i a n s ,b u tF i g u r e 5 i n d i - cates t h a tt h ef l a p p i n gm o t i o n i s c o n s i d e r a b l y less s t a b l et h a n t h a t .

From t h e q u a l i t a t i v e v i e w p o i n t , t h e c o m p a r i s o n i s morefavor- a b l e . The d e c r e a s ei ns t a b i l i t y of t h ef l a p p i n gd e g r e eo ff r e e d o m w i t hi n c r e a s i n g 1 . 1 i s i n e v i d e n c e i n t h e r e s u l t o b t a i n e d h e r e , s i n c e X R i s less n e g a t i v e a t 1 . 1 = 1 . 4 t h a n a t I . I = 0 . 6 . The sta- b i l i t y o f a givensystem i s d e t e r m i n e d ,o fc o u r s e , by t h e l e a s t n e g a t i v e ,o rm o s tp o s i t i v e ,v a l u eo f X R . Also,inagreementwith t h ei n d i c a t i o n so fF i g u r e s 4 and5,thelead-lagmotion i s o n l y s l i g h t l y damped, w i t h a f a c t o r e-’ d e c r e a s eo c c u r r i n gi n1 , 1 3 0 r a d i a n s a t 1 . 1 = 0.6 and i n 7 5 0 r a d i a n s a t 1 . 1 = 1 . 4 . .

The q u a n t i t a t i v e d i f f e r e n c e s i n t h e p r e d i c t i o n s o f t h e f l a p - p i n g m o t i o n c a n b e a t t r i b u t e d t o t h e n o n l i n e a r e f f e c t s which are i n c l u d e d i n t h e d i r e c t - i n t e g r a t i o n s o l u t i o n , b u t which are a b s e n t from t h ef o r m u l a t i o n sa n a l y z e dh e r e .T h i sc a nb es e e na sf o l l o w s .

With t h er o t o ru n l o a d e d ,a s i s t h e case h e r e ,t h ee q u a t i o n sf o r rigid-bodyflappingandlead-lagmotion become decoupled when linearized.Thus,anycouplingofthemotionwhich i s d e t e c t e d c a nb ea t t r i b u t e d , i n t h i sc a s e ,t on o n l i n e a rd y n a m i c - c o u p l i n g effects. S i n c et h em o t i o n sp l o t t e di nF i g u r e s 4 and 5 were i n i t i a t e d by a d i s t u r b a n c ei nf l a p p i n g ,t h ec o n s i d e r a b l el e a d - l a g motionmustthen a l l be due t on o n l i n e a re f f e c t s .F u r t h e r m o r e , . 2 a Id .1 k -.2 W a Id k .1 r c Q) ?I tn c Id tn c t n I I I I I 0 211 4 T 6l1 87i 10 a Azimuth angle,rad I . J = 0.6 Figure 4 . Blademotionsforadvanceratio (fromReference 1 2 ) .

a id k .

a, rl rn c id rn c a id k .

a, F c 0 2 a 4TI 6 a 8 a 10 TI Azimuth a n g l e ,r a d Figure 5. Blade m o t i o n sf o ra d v a n c er a t i o p = 1 . 4 (fromReference 1 2 ) .

t h ef l a p p i n gm o t i o nf o r LI = 1.4 a t t h eh i g h e ra z i m u t ha n g l e s a p p e a r s t o h a v e b e e n e x c i t e d by t h e p e r s i s t i n g l e a d - l a g m o t i o n , r e s u l t i n g i n anapparent dampingof themotionwhich i s much less thanwouldotherwisebethe case.

ComparisonWithExperimental Data Reference 1 0 r e p o r t st h er e s u l t so fa ne x p e r i m e n t a li n v e s t i - g a t i o n o f t h e f l u t t e r o f a model h e l i c o p t e r r o t o r i n f o r w a r d f l i g h t . The model r o t o r had a s i n g l eb l a d ew i t h a r a d i u so ff o u r f e e t andwith a f l a p p i n gh i n g et h r o u g ht h ea x i so fr o t a t i o n . The bladehad a c o n s t a n tc h o r do f 3.5 inchesand a r o o t c u t o u t o f 6 i n c h e s .I n e r t i a l and e l a s t i c p r o p e r t i e so ft h eb l a d e are g i v e ni n Reference 1 0 . The b l a d e w a s r e l a t i v e l y s t i f f i n t o r s i o n ,b u tt h e c o n t r o ls y s t e m w a s made f l e x i b l e , so t h e p r i m a r y c o n t r i b u t i o n s t o blademotionsderivedfromrigid-bodypitch,flappingmotionsand d e f l e c t i o n si nt h ef i r s tf l a p w i s eb e n d i n g mode. The main parame-' ters o ft h es t u d y were a d v a n c er a t i o ,c o n t r o ls y s t e ms t i f f n e s s andchordwisemasscenterlocation.

The c a s e selected forcomparisonhad a r a t i o of f i r s t f l a p - w i s e bending frequency ; t o c o n t r o l f r e q u e n c y w e ( n o n r o t a t i n g ) $ 1 0 of 1 . 3 1 and a chordwise mass c e n t e rl o c a t i o no f4 2 . 5 %o fc h o r da f t o ft h el e a d i n ge d g e ,g i v i n g a valueof 0 . 1 3 9 f o r e q u i v a l e n t mass c e n t e rl o c a t i o na sd e f i n e di nR e f e r e n c e 1 0 . T h i sc a s e was chosen b e c a u s et h ed a t a as p l o t t e d i n Figure 8e ofReference 1 0 i n d i c a t e d t h e r es h o u l db e a r e l a t i v e l yl a r g ec h a n g e i n s t a b i l i t y w i t h ad- v a n c er a t i o .N a t u r a lf r e q u e n c i e s and mode s h a p e sf o rt h ef i r s t three coupied modes of t h e b l a d e were c a l c u l a t e d f o r a v a l u eo f t h er a t i o w / C 2 = 4 . 3 7 ,a n dF o u r i e rc o e f f i c i e n t so f t h e c o e f f i - 0 0 c i e n t s o f t h e e q u a t i o n s o f m o t i o n were c a l c u l a t e d f o r v a l u e s o f a d v a n c er a t i o p of 0.0, 0.09,. 0 . 1 7 5 , 0 . 2 4 and 0.30. The F o u r i e r c o e f f i c i e n t sf o re a c hv a l u eo f p were s u p p l i e dt ot h em a i n com- puterprogramandthesystemcharacteristicvaluescomputed.

S u b s e q u e n t t o t h e s e c a l c u l a t i o n s , it was determinedthrough communicationswithoneoftheauthorsofReference 1 0 t h a t t h e d a t ao fF i g u r e - 8 eo ft h a tr e f e r e n c e were m i s l a b e l l e d . The symbols f o r v a l u e s of w of 1.31 and 0.63 were r e v e r s e d . The e x p e r i - $ P 0 0 m e n t a l l y d e t e r m i n e d f l u t t e r b o u n d a r y a c t u a l l y c o g r e s p o n d i n g t o t h e c a l c u l a t i o n sp e r f o r m e d ,c o n s i s t i n go f a p l o to f w / a v e r s u s p , 8 0 takenfromFigure8eofReference 1 0 , i s r e p r o d u c e di nF i g u r e 6 .

The p o i n t s a t which c h a r a c t e r i s t i c v a l u e s were c a l c u l a t e d are i n d i c a t e d by a s t e r i s k s on t h e l i n e drawn a t w /O = 4.37.

8 0 As can be seen from F i g u r e 6 , t h e mislabelling of the data has led t o a rather unsatisfactory comparison of theory w i t h experiment. T h e experimental data s h o w s very l i t t l e change w i t h p.

Hence, f o r a comparison w i t h t h i s , c a s e , t h e characteristic values should have been determined a t several d i f f e r e n t values of w / Q , 8 0 f o r f i x e d p , rather t h a n a t f i x e d w / Q f o r various values of p.

8 0 Unfortunately, t i m e l i m i t a t i o n s prevented carrying o u t more ex- t e n s i v e c a l c u l a t i o n s . However, some information can still be derived from the c a l c u l a t i o n s which were performed.

T h e characteristic values o b t a i n e d f o r each advance r a t i o are as follows: p = 0.0 p = 0 . 1 7 5

I p = 0 . 0 9 1

. ..

x R x R x R ~~ A I _ ~ . ~"

- 0 . 0 6 2 7 3

- 0 . 0 6 2 7 1 0 . 0 2 0 2 8

0 . 0 2 8 0 0 - 0 . 0 6 2 6 6

0 . 0 2 8 7 1

- 0 . 0 6 2 7 3 - 0 . 0 2 0 2 8

- 0 . 0 6 2 7 1

- 0 . 0 2 8 0 0

- 0 . 0 6 2 6 6

- 0 . 0 2 8 7 1

- 0 . 3 3 0 5 2

- 0 . 3 3 0 5 5 0 . 2 0 1 6 3

0 . 2 0 1 8 5 - 0 . 3 3 0 6 8

0 . 2 0 2 6 7

- 0 . 3 3 0 5 2

- 0 . 2 0 1 6 3

- 0 . 3 3 0 5 5 - 0 . 2 0 1 8 5

- 0 . 3 3 0 6 8

- 0 . 2 0 2 6 7

- 0 . 4 7 0 1 5

- 0 . 4 7 0 1 6 0 . 0 8 8 7 4

0 . 0 8 9 5 4

- 0 . 4 7 0 1 2

0 . 0 9 1 5 2

- 0 . 4 7 0 1 5

- 0 . 0 8 8 7 4

- 0 . 4 7 0 1 6

- 0 . 0 8 9 5 4

- 0 . 4 7 0 1 2

- 0 . 0 9 1 5 2

..".

A I x R

I x R

0 . 0 3 0 6 0

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

- 0 . 0 6 2 6 8

- 0 . 0 3 0 6 0

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

0 . 2 0 4 8 9

- 0 . 3 3 2 2 7

- 0 . 3 3 1 2 0 0 . 2 0 3 6 6

- 0 . 3 3 2 2 7 - 0 . 2 0 4 8 9

- 0 . 3 3 1 2 0 - 0 . 2 0 3 6 6

- 0 . 4 7 0 1 3 0 . 0 9 5 1 0

- 0 . 4 7 0 1 0 0 . 0 9 3 4 2

- 0 . 0 9 5 1 0

- 0 . 4 7 0 1 3

- 0.47010 - 0 . 0 9 3 4 2

6.0

5.0

4.0

3 . 0 t

2.0

I .o

0 0.1 0.2 0.3 0.4

u Figure 6 . E x p e r i m e n t a lf l u t t e rb o u n d a r yf o r a model r o t o r blade. R a t i oo fn o n r o t a t i n gp i t c h n a t u r a l f r e q u e n c y w t o r o t a t i o n a l s p e e d 0 0

-

v e r s u sa d v a n c er a t i o u , for ;91/we = 1 . 3 1

and x /C = 13 9 (fromReference 1 0 ) . P o i n t s

e a t w h i c h c h a r a c t e r i s t i c v a l u e s were c a l c u l a t e d a r e i n d i c a t e d by a s t e r i s k s .

The r e s u l t s o f t h e c a l c u l a t i o n s i n d i c a t e , f i r s t , t h a t t h e s t a b i l i t y of t h e r o t o r f o r t h e c o n t r o l s t i f f n e s s s e l e c t e d i s e s s e n t i a l l y i n - d e p e n d e n to fa d v a n c er a t i oo v e rt h er a n g eo f LI considered. The r e l a t i v e i n s e n s i t i v i t yt oa d v a n c er a t i oc h a n g e s is c e r t a i n l y i n a g r e e m e n tw i t ht h ep l o to fF i g u r e 6 . Secondly, it shouldbenoted t h a t t h e r o t o r i s p r e d i c t e dt ob ev e r yn e a r l yu n s t a b l e . The v a l u e f o r X R of - 0 . 0 6 3 i n d i c a t e s t h a t a b o u t t e n r o t o r r e v o l u t i o n s are r e q u i r e dt o damp themotionby a f a c t o r e-’. The l i m i t e dc a l c u l a - were p e r f o r m e d ,t h e n ,a r e a t l e a s t i n q u a l i t a t i v e t i o n s which a g r e e m e n tw i t ht h ee x p e r i m e n t a lr e s u l t s . A d e f i n i t i v eq u a n t i t a t i v e comparisonmustawait more e x t e n s i v e c a l c u l a t i o n s .

CONCLUDING REMARKS A m e t h o dh a sb e e nd e v e l o p e df o ra n a l y z i n gt h ea e r o e l a s t i c s t a b i l i t yo fh e l i c o p t e rr o t o r s i n f o r w a r df l i g h t . The method e m p l o y sf o r m u l a t i o n sf o rc a l c u l a t i n gt h ec h a r a c t e r i s t i cv a l u e s o ft h ep e r t u r b a t i o ne q u a t i o n so fm o t i o n ,t h e l a t t e r being a coupled set o f s e c o n d - o r d e r , l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h p e r i o d i cc o e f f i c i e n t s . The c h a r a c t e r i s t i cv a l u e s , which a r et h e zerosof an i n f i n i t ed e t e r m i n a n t ,a r ec a l c u l a t e d fromanequiva- l e n t a n a l y t i c form f o r t h e i n f i n i t e d e t e r m i n a n t c o n s i s t i n g of a f i n i t e sum of t r i g o n o m e t r i c f u n c t i o n s .

A d i g i t a l computerprogram w a s preparedwhichimplementsthe method f o rt h r e ed e g r e e s of freedom.Calculationsofcharac- t e r i s t i c v a l u e sc a r r i e do u tf o r a t e s t c a s ed e m o n s t r a t e dt h e p r a c t i c a l i t y o f t h e m e t h o d , w i t h a n t i c i p a t e d r e s u l t s g e n e r a l l y o b t a i n e d t o f o u r s i g n i f i c a n t f i g u r e s .

C a l c u l a t i o n s were c a r r i e d o u t f o r c o m p a r i s o n w i t h r e s u l t s o f a d i r e c t t i m e i n t e g r a t i o n on a d i g i t a l computer of t h ee q u a t i o n s f o r a r i g i dr o t o rb l a d ew i t hf l a p p i n g andlead-laghinges. The r e s u l t s were i n q u a l i t a t i v ea g r e e m e n t .Q u a n t i t a t i v ed i f f e r e n c e s were a t t r i b u t a b l e t o t h e n o n l i n e a r e f f e c t s which were i n c l u d e d i n t h e d i r e c t t i m e i n t e g r a t i o n .

L a s t l y ,l i m i t e dc a l c u l a t i o n s were performedforcomparison w i t he x p e r i m e n t a l l yd e r i v e df l u t t e rb o u n d a r i e sf o r a model r o t o r b l a d ew i t ht h r e e degrees of freedom. The c a l c u l a t i o n si n d i c a t e t h a t t h e r o t o r i s o n l y m a r g i n a l l y s t a b l e a t t h e c o n t r o l s t i f f n e s s s e l e c t e d and t h a t t h e s t a b i l i t y i s r e l a t i v e l y i n s e n s i t i v e t o ad- v a n c er a t i o ,i na g r e e m e n tw i t ht h ee x p e r i m e n t a l results.

4 2 " APPENDIX A R e l a t i o n s h i p Among t h e S o l u t i o n s of O r i g i n a l a n d D i f f e r e n t i a t e d S y s t e m s L e t X 0 be a s o l u t i o n of d3xO dX 0

+ c - + D X 0 = 0

dz d Z a s w e l l as of d4X0 dX 0 + E - + F X O = O .

dz dz From Eq. ( A - 2 1 ,

( A 2 ) - (" + B - A 2 ] g - dX0

dz dz dz

But Eq. ( A - 1 ) , d i f f e r e n t i a t e do n c e ,g i v e s

If (A-4) i s s u b s t i t u t e d i n (A-31, a number o f terms canceland a common f a c t o r c a n b e e x t r a c t e d , w i t h t h e r e s u l t t h a t d2Xo

- E - A g

- + A - dX 0

dA + B

+ = 0

dz dz

-

Thus, X0 m u s t be a s o l u t i o no ft h eo r i g i n a le q u a t i o n , Eq. ( 1 0 ) .

It i s , of c o u r s e ,a l s ot r u et h a te v e r ys o l u t i o n of Eq. (10) i s a

s o l u t i o no f Eq. (A-1) and of Eq. ( A - 2 ) . Thus, X0 is a s o l u t i o n o f Eq. ( 1 0 ) i f andonly i f it i s a s o l u t i o no fb o t h Eq. (A-1) and Eq. (A-2) .

A P P E N D I X B

Listing of Basic Computer Program - d u d

The program was coded i n FORTRAN I V . A CDC 6 4 0 0 d i g i t a l computer was employed f o r a l l c a l c u l a t i o n s .

C C I: C C C c GET THE 6 T H O R D E R 1 0 3 T S C C c c C c 5 1 I I D O 50 K 1 r N F I J K 3 ( I * L f - L J l * J ) * L K + K A R ( I J K ) 0

A I ( 1 J K ) = 0

R R ( I J K ) = 0 R I ( I J K ) = 0 5 0 C O N T I N U E

N I N T = 20

A N I N T = N I N T

.G = (.97-.2)/ANINT

H r ( l , - r Z ) / A N I N T P I = 3,1415926536 P I 2 = PI/2, NF2 = N F * 2 A N F a N F R N F = l./ANF P H I = ( P I * A N U ) / A N F M U I = 1, + MU P 1180 = P I / l B U , N D E G = 3601/13tG N I N T = N I N T + l D O 1 N P H I = 1 , N D E G A N P H I = N P H I - 1 P H E Y E ; A N P H I + D E I ; PHI = A N P H I + P I 1 8 0 * U S G C O S P H I = C O S ( P H 1 ) S I N P H I = S I N ( P H 1 ) M U S I P H = M U s S I N P H I

MUCOPH = M U + C O S P H I

N I N T l = N I N T + l " CALL Q S F ( H ~ Y Y , Y Y I N I Y T ) CALL Q S F ( H I Y I Y I N I Y T ) CALL Q S F ( G r W W , H W , N I N T ) CALL Q S F ( G I Z Z I Z Z I N I V T ) CALL Q S F ( G I W I W ~ N I N T ) CALL Q S F ( H t Z 1 r Z l r Y I V T ) A P H I l ( N P H I 1 = Y Y ( N I V T ) = M u C O P ~ * Y ( Y I Y T ) R P H I l ( N P H 1 )

= 0 . 4 2 8 6 8 7 4 * ( N W ( V I N T ) 4 Z Z ( N I N T ) )

A P H I 2 ( N P H I ) H P H I 2 ( N P H I ) = 0 . 4 2 8 6 8 7 4 + ( W ( V I N T ) + Z ~ ( N I N T ) ) +MUCOPH 1 C O N T I N U E D O 15 K = l r N F K l l ( L I - L J l + l ) * L K * K K 2 2 = ( L I * Z - L J 1 * 2 ) + L < + f i A K 1 K - 1

P I A K I = P I * A K 1

V A L U E l = 0 V A L U E 2 = 0 V A L U k 3 = 0 V A L U E 3 = 0 V A L U E 4 = 0 V A L U k 5 = 0 V A L U E 6 = 0 V A L U E 7 = 0 V A L U k 8 = 0

1 1 0 1 0 L = l r N F 2

A N U * L - 1 P I K l N U = ( P I A K I * A N U ) / A N F C O S P I = C O S ( P I K l N U ) S I N P I = S I N ( P I K 1 N U l V A L U E l = V A L U E 1 + A P 4 1 1 ( L ) + C O S P I = VALUE3 + B P d I l ( L ) COSPI VALUE3 = VALUE4 * B P i I l ( L ) * S I V P I VALUE4 s V A L U E 5 + A P H 1 2 ( L ) * C O S 3 1 VALUE5

= V A L U E 6 + A P H 1 2 ( L ) * S I N ~ I

VALUE6 = VALUE7 +BP;.112(L) * C O S P I VALUE7

= VALUE8 + B a H I 2 ( L ) * S I N P I

VALUE8 1 0 CONTINUE = VALUEl*RN;*GAYMAP A R ( K 1 1 ) = VALUE5'RNF * G A M M A P A R ( K 2 2 ) = -VALUE2*RVF *GAYM4P A I ( K I 1 ) = -VALUE6*RVF*GAMYAP A I ( K 2 2 ) = 2 . + V A L U E 3 * R N F *GAYMAP H R ( K 1 I ) = 2 . * V A L U E 7 * H V F *[;AMYAP H R ( K 2 2 ) = - 2 * * V A L U E 4 * 4 N F *GAMMAP H I ( K 1 1 ) = - 2 * * V A L U E 9 * d N F * G A M M A P R I ( K 2 2 ) 15 C O N T I N U E K 2 2 1 = ( ~ * L I - L J ~ + ~ ) * L K + ~ R R ( K 2 2 1 ) = H R ( K 2 2 1 ) * 4 , ~111 = ( L I - L J ~ + ~ ) * L K * ~ R R ( K 1 1 1 ) = B R ( K I . 1 1 ) + ,3157896 = ( 3 + L I - L J 1 + 3 ) * L < + l K 1 3 3 = ( 3 * L I - L J 1 + 3 ) * L < + 2 K 2 3 3 A R ( K 1 3 3 ) = % * B R ( K 1 3 3 ) = 2 , A R ( K 2 3 3 ) = 1.0 R E T U H N E N D

I

QSF 0 4 7 QSF QSF QSF 0 5 2 Q S F 0 5 3 Q S F 0 5 4 QSF 0 5 5 QSF (3SF 0 5 7 O S F 0 5 a QSF 0 5 9 QSF 060 Q S F 0 6 1 Q S F 0 6 2 OSF E63 Q S F 0 6 4 Q S F 0 6 5 Q S F 3 6 6 Q S F O h 7 QSF 0 6 8 Q S F 0 6 9 Q S F 0 7 0 0 5 F 0 7 3 - Q S F 0 7 2 Q S F 9 7 3 O S F 0 7 4 O S F 0 7 5 Q S F 076 Q S F 0 7 7 Q S F 0 7 8 Q S F 0 7 9 QSF 0 8 0 QSF Z ( N D I H ) = A U X 2 Q S F oe.

R E T U R N Q S F 08.

6 Z ( N D I M - l ) = S U M 2 QSF 0 8 .

Z ( N D I M ) = A U X l Q S F OS: R E T U R N Q S P 0 8 C E N D OF I N I E G R A T I O N LOOP QSF 0 8 C Q S F 0 8 Q S F C QSF 0 9 C Q S F 0 9 8 Q S F 0 9 .

Q S P 0 9 QSF 0 9 QSF 0 9 Q S F 0 9 Q S F 0 9 Q S F 0 9 QSF 0 9 QSF 1 0 9 QSF 1 0 Q S F 1 0 Q S F 1 0 1 0 QSF I O Q S F 1 0 Q S F 1 0 C OSF io N D I M IS E Q U A L T O 3 t : Q S F 1 0 11 SUMl'HT+(lt25+Y(l)+Y(2)+Y(2).*25*Y(3)) Q S F 1 0 S U Y 2 = Y ( 2 ) + Y ( 2 ) QSF 1 1 S U M 2 = S U M 2 + S U M 2 QSF Z ( 3 ) = H T + ( Y ( l ) + S U M 2 + Y ( 3 ) ) QSF 1 1 Z ( l ) = O * Q S F Z ( 2 ) = S U M l Q S F 11 12 R E T U R N Q S f END Q S F 1 1 c - C C C c: C C r: c C C c C C C C 'C C I: C e e .

C

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

c - C E Q U A T I O N S FOR C O M P J T A P J O N S e L J = L J + l = L J - 1 L J l LK=NF N 2 F 1 = 2 * NF -1 N F I 0 NF + 1 KL = 2 * N F - 1 C D O 5 I = l r L I C D O 5 J = l r L J 1

c

C N O T E ,.* THE L A S T SUM O N N IS O M I T T E D FOR K = N F

C O N S T 2 =

CONST5 = 1 7 , 17, 1 6 1 F t N F - K ) 16 N F K a N F - K c D O 20 N = l r NFK I L N l = I L * N + 1 L J N K ” J L + N + K I L N K = IL + N + K L J N l = JL + N +1 A R I L N l = A R ( I L N l ) R R L J N K = B A H t L J N K 1 A I I L N ~ = A I ( I L N ~ ) B I L J N K = B A I ( L J N K ) A R I L N K = A H ( I L N K ) H R L J N l = B A R ( L J N l ) AIILNK=AI(ILNK) R I L J N l = B A I ( L J N l ) APPI.E=ARILN~*BRLJYK PEAR = A I I L N l * B I L J N # , P E A C H = A R I L N K * B R L J V l PLUM = A I I L N K I B I L J N l CONST2 = C O N S T 2 + A P P L E + P E A 2 + P E A C H + P L U M e 1 B I L J N l C C O N S T 2 = C O N S T 2 + A H I L N l * ~ R ~ ~ N K + A I I L N l * ~ l L J N K + A R ~ L ~ K * B R L J N l + A I l L ~ ~ c CONST2 = C O N S T 2 + A q ( I L V 1 I ) B A R ( L J N K 1 + A I ( I L N 1 ) * B A I ( L J N t 0 + C 1 A R ( I L N l 0 t B A R ( L J V 1 ) + A I ( I L N K 1 B A I ( L J N 1 ) A P P L E = A R I L N l * E I I L J N K = A I I L N l * B H L J V K PEAR

= A R I L N K * B I L J V l

PEACH 1 7 C O N S T R = C O N S T H + C O V S T l + C O V S T 2 C O N S T I = C O N S T I + C 3 N S T 4 + C 3 N S T 5 7 C O N 1 I N U E C R ( 1 J K L ) s C R ( 1 J K L ) - C O N S T R C I ( I J K L ) = C I ( I J K L ) - C O N S T I 55 C O N T I N U E C C C D O 9 K = N F l , N 2 F 1 C O N S T R = O 1 O C O N S T I = 0.0 I J K L = I K L + K e U O 8 L = l r 3 I L = ( L I + I - L J + L ) * L K JL = ( L I * L - L J + J ) * L K C O N S T 3 = O * O C O N S T 6 = U t 0

K 1 N = K + 1 - NF

C 2 2 D O 3 0 N = K 1 N ,NF

I L N = I L + N

L J K l N :: JL + K + 1 - N A R I L N = A R ( I L N ) B R L J K l = B A H ( L J K I N ) A I I L N = A I ( I L N ) R I L J K l = B A I ( L J K l N ) C O N S 1 3 = C O N S T ~ + A R I L N * B ~ L J K ~ - A ~ I L N * ~ ~ L J K ~ C O N S T 6 = C O N S T 6 + A R I L N * B I L J K l * A I I L N * 9 R L J K l C C O N S 1 3 = C O N S 1 3 + A q ( I L U ) * 3 A R f L J K l N ) - A I t I L N ) B A I ( L J K 1 Y ) C C O N S T 6 = C O N S T 6 + A q ( 1 I . V ) I J A I ( L J K 1 N ) + A I ( I L N ) B A R t L J K l V ) 30 C O N T I N U E C 21 C O N S T R = CONSTR + C O N S T 3 8 C O N S T I = C O N S T I + C O N S T 6 C T ( I J K L ) = - C O N S T I

C R ( I J K L ) = - CONSTR

9 C O N T 1 NUE C 5 C O N T I N U E C L J = L J - l RETURN E N D SUQRGUTINE P R O O ( X C ~ F I M , ~ O O T ~ ~ R O O T I , C O F , N U M I I E R ) D I M E N S I O N ~ C O F ~ 1 ~ ~ C 3 F ~ l ~ r ~ ~ 0 T R ~ ~ ~ ~ R O O T I ~ l ~ DOUBLE P R E C I S I O N X O ~ X ~ , Y O , Y O , X I Y , X P R , Y P R I U X I U Y , V I Y T , X T . U OOUULE P R E C I S I O N x ~ ~ , Y T Z , S U M S Q , D X , D Y , T E M P , A ~ P H A C COMPUTES THE HEAL AND C 0 r l p ~ E X e O O T S OF A P O L Y N O M I A L C U S I N G T H E NkWTON R A P H S O V I T E R A T I O N T E C H N I Q U E C PARAMETERS I: XCOF VECTOR OF M + l C 3 E F F I C I E N T S OF T H E P O L Y N O M I A L O H U E S E D F R O M S M A L L E S T T O L A R G E S T P O W E R C C COF A W O R K I N G V k C T 3 H O F S I Z E M + l c M THE O R D E R 0.’ THE P O L Y N O M I A L C ROOT I H E S U L T A N T V E C T 3 R 3 F L E N G T H M OF I M A G I N A R Y P A R T S R O O T I ( 1 ) 1s THE I Y I T I A L VALUE OF THE Y GUESS I M A G I N A R Y PART .

c

c ROOTR R E S U L T A N T J E S T O R SF LENGTH M OF R k A L R O O T S

C H O O f R ( 1 ) Is THE I Y I T I A L VALUE OF THE Y G U E S S R E A L P A R T

C I E H E d K O H C O D E c I E H = 0 N 3 E R R 3 R e S U R R O U T I N t P O L R T ( X C ~ F I C ~ F , M , ? O O T R , R O O T I , I E R ) c I E R = 1 Y IS L E S S T H A N 1 c

I E R = 2 Y IS GREATER THAN 3 6

e I E R = 3 J N ~ B L E T O DETERMIYE ROOTS I N 5 0 I T E R A T I ~ N S

c I E H = 4 c(I3H O q D E R C O E F F I C I E N T IS ZERO C PHOGHAMMED 3~ r( G H L E M E L C S , A * S p A * I N C .

ROCdESTER M . Y . 7 1 6 I C

c R O C H E S T E 3 \1, Y. 716 271 3450

C S U H R O U T I N t P O L R T ( X C ~ F , C 3 F , Y , ~ O O T R , R O ~ T I , I E R ) I F 1 1 0 V = M I E R = O I F ( X C O F ( N + I ) ) l O v 2 5 , 1 0 1 0 IF(N)15,15032 1 5 I E R = l 2 0 RETURN 2 5 I E R = 4 G O T O 20 30 I E R = 2 Q O T O 2 0 32 IF(N-36) 3 5 8 3 5 r 3 0 3 5 N X s N NXX N + l N 2 z 1 K J 1 = N + l D O 4 0 L = l r K J l

M T = K J l - L + l

4 0 C O F ( M T ) = X C O F ( L ) 4 5 X0 = , 0 0 5 0 0 1 0 1 Y O s 0 , 0 1 0 0 0 1 0 1 I N = 0 50 x = x 0

x0 = - l O , * Y O

Y O = - l O , + X x = x o Y = Y O I N = I N *1 G O T O 5 9 55 I F I T =1 X P R = X Y P R = Y 59. I C T = 0 6 0 UX = 0 9 0 UY = 0 , o v 2 0 . 0 Y T = 0 . 0 X f = 1.0 U = L ' O F ( N + l ) I F ( U ) 6 5 r 1 3 0 r 6 f j

6 5 D O 70 I = 1 , N

L = N - I + l T E M P = C O F ( L ) X T 2 = X + X T - Y + Y T

Y T 2 = X t Y r + Y * X T

V = V + T E M P * Y T 2 u = U + T t M P * X T 2 F I = I U X = U X + F I * X T * T E M P U Y = U Y - F I * Y T * T E M P X T = X T 2 7 0 Y T = Y T 2 S U M S O = U X * U X + U Y + U Y IF(SUMSQ1 75,110, 7 5 7 5 D X = ( V * U Y = U + U X ) / S U H S 3 X o X + D X

D Y 3 - ( U * U Y + V * U X ) / S U Y S Q

Y = Y + D Y 78 I F ( D A B S ( D Y ) ~ D A H S ( D X ) ~ l , ~ - l 2 ) 1 0 0 , 8 0 , 8 0 80 I C 1 3 I C T *1 IFCICT-500) 60,85,8f 8 5 I F ( 1 F I T ) 1 0 0 , 9 0 ~ 1 0 0 9 0 l F ( I N - 5 ) 5 0 r 9 5 ~ 9 5 9 5 I E R = 3 R E T U R N 1 0 0 D O 105 L = l r N X X M T 0 K J l -1*1

TEMP = X C O F ( M T )

X C O F ( M T 1 = C O F ( L ) 1 0 5 COF(i.1 = TEMP ! T E M P = N N t N X

N X = I T E M P

I F ( I F I 1 ) 120 D 55,120 116 I F ( I G I T ) 1 1 5 ~ 5 0 , 1 1 5 115 X I X P R

Y = Y P R

120 I F I T = 0 I F ( X ) 122,125,122 I F ( O A B S ( Y / X ) - l , U - l O ) 1358125,125 1 2 2 1 2 5 ALPHA = X + X S U M S 0 J X + X + Y + Y hl t N - 2 G O T O 1 4 0 1 3 n x t 0.0 NX = NX -1 N X X s NXX -1

135 Y = 0 . 0

S U M S D = 0 . 0 ALPHA = X N Iri-I.

341) C O F ( 2 ) = C O F ( 2 ) + A L P H A + C O F ( 1 ) K = L 1 4 5 D O 150 L = 2 , N K = K 150 C O F ( L + l ) = C O F ( L + l ) + A L P ~ A * C O F ( L ) - S U M S Q * C O F ( L - l ) 1 5 5 R O O T I ( N 2 ) = Y H O O T H ( N 2 ) = X N 2 t N 2 + 1 I F ( S U M S Q ) 1 6 0 r l 6 5 r 1 6 0 160 Y = - Y S U M S Q = O , G O T O 155 265 I F ( N ) 2 0 r 2 O r 4 5 END I L N l = I L + N + l L J K N = J L + K + N CONSTI=CONSTI+CR(ILYl)*AR(LJ<N) * C f ( I L N l ) + A I ( L J K N )

C O N S T 7 = C H ( I L N 1 ) 4 I ( L J K N ) - C I ( I L N I ) + A R ( L J K N ) * CONST7

15 C O N T I N U E 3 CONST2=OeO CONSTBzO N F ' l r N F - 1 D O 2 0 N= l # N F 1 ILKNCIL+K+N L J N l = J L + N + l C O N S T Z = C R ( I L K N ) * A ? ( L J V l ) + C I ( I L K N ) * ' A I ( L J N 1 ) * CONS12 C O N S T B = C O N S T B - C q ( l L A N ) + A I ( L J N 1 ) + C I ( I L K N ) + A R ( L J N l ) 2 0 C O N T I N U E C O N S T 3 = 0 CONST9 = O D O 2 5 N = l r K I L K l N = I L + K + l - N LJN= JL+N

CONST3 = CONST3 * C 3 ( I L I ( l N ) * A R ( L J N 1 - C I ( ( L K 1 N ) * A I ( L J N )

CONST9- CONSTO + C R ( I L K 1 N ) A I ( L J N ) + C I ( I L K 1 N ) + A R ( L J N ) 2 5 C O N T I N U E CONSTR=CONSTR+ C O Y S r l + C 3 N S T 2 + C O N S T 3 C O N S T I = C O N S T 7 + CONSTB + CONST9 + C O N S T I 55 C O N T I N U E E R ( I J K ) = E R ( I J K ) - Z O V S T R E I ( I J K ) = l i I ( I J K ) - C 3 N S T I 4 5 C O N T I N U E e N F l = N F + l N F 2 1 = 2 + N F - l D O 6 0 K = N F l r N F 2 1 I J K z I J + K I J K 2 = I J 2 + t ( A K S K + K - 2

E I ( I J K ) = A K * A C R ( I J K 2 ) + O I ( I J K 2 )

E R ( I J K ) = - A K * A C I ( I J K 2 ) + D R ( I J K 2 )

e! f I ( I J K ) = A K * A C R ( I J 4 ) + O I I I J K I

C E R ( 1 J K ) = - A K * A C I ( I J K ) * D R ( I J # )

C O N S T R = O C O N S T I t O D O 6 5 L = l r 3 NN:2*NF-l-K

I L = ( L I + I - L J * L ) * L K 2 N F

JL D ( L I + L - L J + J ) * L K Y F I L = ( L I + I - L J + L ) * L K C C J L 7 ( L I + L - L J + J ) + L K C O N S T 4 ~ 0 . 0 C O N S T l O = O . O ! F ( N N ) 6 6 , 6 6 r 6 7 6 7 D O 70 N 5 I r N N I L K N = I L + K + N L J N l = J L + N + l C O N S T l O = - C R ( I L K N ) * A I ( L J N 1 ) * C I f I L K N ) * A R ( L J N 1 ) * C O N S T 1 0 C O N S T 4 = C R ( I L K N ) A R ( L J Y 1 ) + CI(1LKFI) A I ( L J N 1 ) C O N S S O C O N T I NUE 7 0 6 6 C O N S T 5 = O e O C O N S T l l = 0 C C C O N S T R = C O N S T R + C O N S T ~ + C O V S T ~ C O N S T I = C O N S T I + C O N S T ~ O + S O V S T 1 1 6 5 C O N T I N U E

E R ( I J K ) = ER('IJK) - c O N S T R

E I ( I J K ) = f I ( I J K ) 0 C O Y S T 1 60 C O N T I N U E

c

N F 2 = 2 * N F N F 3 2 = 3 * N F - 2 D O 1 0 0 K s N F 2 r N F 3 2 I J K = I J + K I J K 2 = IJ2+K CONSTR = 0 C O N S T I = O C D O 1 0 5 L=1,3 C l L = ( I * L I - L J + L ) * L K I L t ( I * L I - L J + L ) * L K 2 N F C JL P ( L I * L - L J * J ) * L K

JL = ( L I * L - L J * J ) * L K V P

COhlST6z 0 C O N S T 1 2 = 0

NN = K - 2 * N F 2

I F ( N F - N N ) 1 0 1 , 102, 1 0 2 C

1 0 2 D O 1 1 0 N = NN , N F

I L K l N : I L + K + l - N L J N = S L + N

CONST6 = C R ( I L K 1 N ) A R ( L J N ) - c I ( I L K ~ N ) A I ( L J N )

C O N S 7 1 2 = C R ( I L K l N 1 A I ( L J N ) 4 C I ( I L K 1 N ) A R ( L J N )

1 1 0 C O N T I N U E c 1 0 1 GONSTR= CONSTR .c C O Y S T 6 .

C O N S T I = C O N S T I + C 3 N S T l 2 1 0 5 C O N T I N U E

E R ( I J K ) = - CONSTR

E I t I J K ) - C O N S T I

1 0 0 C O N T I N U E C 5 C O N T I N U E C L J = L J - 1 R E T U R N END *END I I R N N = H ( N N ) + X X I X I R = B ( N N ) - X X I A A = A ( N N ) XAA=XXR+AA E X A A = E X P ( X A A ) E M X A A = E X P ( - X A A ) I C O S H Y = ( E X A A + E M X A A ) * . 5 D O 2 0 N = L n N K s 2 A A = A ( N ) XXAA=XXH+AA R N = B ( I\ 1 X I B N = X X I * H N B N X I = t l N - X x I F X X A A = E X P ( X X A A ) E M X X A A = f X P ( - X X A A ) S I N W Y = ( E X X A A - t M X X A A ) * . 5 C O S H Y = ( E X X A A + E M X X A A ) + . 5 7 0 2 2 F O R M A I ( + ivOP = 1 5 ) D O 2 1 N = N l P N O X X C N = X X R + C ( N ) E X C N = E X P ( X X C N ) E M X C N = E X P ( - X X C N ) S I N H Y = ( E X ~ M - E I ~ X C N ) * . ~ C O S H Y = ( E X ~ N + E M X C N ) + , S U ( N ) = s I N H Y / ( C O S H Y - C O S ( X X I ) ) 2 1 C O N T I N U E D O 39 N = I r N O M N = I M M - l ) * N O * N 30 C C ( M N ) = U ( N ) 2 5 C O N T I N U E D O 125 M M = 2 s N K , 2 X X R r - P I * Y H ( M M ) X X I = * P I * Y I ( M H ) D O 13U N = l r N K 1 , 2 X X H A = X X H + A ( N + l ) E X X R A = E X P ( X X R A ) E M X X R A = f X P ( - X X R A ) S I N H Y = ( f X ~ R A l t M X X R A ) 4 . 5 C O S H Y = ( E X X R A * t M X X R A ) * . 5 B N X I = E I ( N + l ) - X X I C O S H Y = ( E X R A N + E M X R A N ) + . 5 1 6 0 C C ( M N ) ~ S I N H Y / ( C O S H Y - C O S ( B ( N ) ) ) D O 165 N'NlrNO MNsM+N X R C N s ( X X R + C ( N ) ) * @ S E X R N e f X P ( X R C N ) E M X R N = E X P ( - X R C N ) S I N H Y = E X R N - E M X R N C O S H Y = E X R N + E M X R N C C ( M N ) = C O S H Y / S I N H Y C O N T I N U E 1 6 5 C O N T I N U f D O 40 N l , N K 1 , 2 M N = N O + ( N O - I ) + N

M N 1 = M N * I

C C ( M N 1 ) = 1.

C C ( M N ) = (J.0 C O N T 1 NUE 4 9 D O 7 0 N=NK sND M N t N O * N O l + N C C ( M N ) = . l e U 7 0 C O N T 1 NUE D O 50 M = l s N Y l , 2 B I Q A ( M ) = U Y R ( M ) - I , R I G A ( M * l ) = - U Y l ( M * l ) 5 0 D O 60 M = N l , N O l 60 H I G A ( M ) = D Y R ( M ) - l * R J G A ( N O ) = U R E T U R N END * E N D D O 5 MR 3 MS,ND R = M R

NDMk = ND-MR

N D Y R 3 = NDMR*NDHR+NDMS MRND+ MR*ND MRND3 = MRND+HRND+H?ND M R M S l = M R - M S + l D O 5 M E l r L M

I = MRND3+M

C I = 3 * ( M R + N D ) + M

c THE I I N D E X R E Q U I R E S ONLY MI? AND M

c rluDIcEs A R E I N T H R E E DIMEYSIONS

C N I R M F 3 * ( - M R + N D ) + M

N I R M = N U M H 3 + M

B A z B E T ( N I H M 1 F R N E G I J = G A M M ( N I R M ) F I N E G I J = D E L T ( N I R M ) N I R M l = ( N I H M - l ) + L J l C I R M r 3 * ( M R + N D ) + M F F R I = F F R ( I )

F F I I = F F I ( 1 )

A B t A L P H ( 1 )

I R M = I

I R M l = ( I R M - l ) * L J l D O 5 N = l r L N M N = ( L I * M * L J * N ) * L K C

c M N R S l = M N + M R - M S * i

M N + M R M S l MNRSlz N J S N t MSND3+N I: N J S N = 3 * ( - M S + N D ) + N c N E G I J = ( N I H M - l ) + L J l + N J S \ I N E G I J = N I H M l + N J S ; J JSN G N D M S 3 + N C J S N = 3 * ( M S + N D ) + N c 8 8 8 7 9 9 9 9 C C K K o ( K - I ) * M + K

GRKK = G R ( K K )

G I K K = G I ( K K ) P I R t G R K K * P I R N - G I K Y * P I I A '

P I 1 = P I I N C G R K K + G I K K + ? I S N

P I R N s P I R 103 P I I N = P I I C L J = L J - 1 RETURN C U N L I K E L Y E V E N T A ( Y M ) * * 2 7 1 NA=M-KM c D O 7 3 LL=l,NA L U V M = ( L l . - l ) * M + J I F ( G H ( L U V M ) - D E L T A X ) 9 1 r 9 1 , 9 2 9 1 I F ( G I ( L U V M ) - D E L T A X ) 7 3 3 , 7 3 3 , 9 2 7 3 C O N T I N U E C 7 3 3 W R I T E ( N P , 1 0 0 2 ) D E L T A X , D I Y 2 J 1 0 0 2 F O R M A T ( 5 4 H * * * * U N A B L E T 3 F I N O AN ALPHA O R BETA LARGER THAN DELTA / 1 2 1 H I N S U 8 H O U T I N E D E L Y / 2 / O H D E L T A X = E Z O e 8 , 1 7 H O L D V A L U E U S E D = E 2 0 t 8 n l O H COLLlYbl 1 1 5 ) I F ( D ) 7 2 r 9 9 9 r 7 2 9 2 D O 7 7 J = l r N A N A J = ( N A - l ) * M + J # J = ( L L - l ) * M + J G R ( N A J ) = G R ( N A J ) + G R ( t ( J ) G I ( N A J ) = G I t N A J ) + GI(I<J) NANA = ( N A - l ) * M + N 4 D = G R ( N B N A ) c G H ( N A N 4 ) + G I ( ~ A N A ) ~ ~ I ~ ~ A N A ) 7 7 C O N T I N U E e 4 0 0 7 F O R M A T ( 2 1 H A D J U S T H E Y T O N COLLJMN 1 5 ) W R I T E ( N P n 4 O O 7 ) M G O T O 7 2 9 9 9 W R I T E ( N P I ~ O O ~ ) 1 0 0 9 F O R M A T ( l H 0 5 G H N E C E S S 4 R Y T O A B 3 R l DUE TO S I N G U L A R I T Y S T O P I I E L T A X = 0 , O L J Z L J - 1 4 W R I T E ( N P , 1 0 0 5 ) RETURN END SUBROUTINE U P ( P C n N D r N S T A R T 0 N P ) D I M E N S I O N P C ( l 8 )

c S U B R O U T I N E T O E S T A e L I S H C S I T E q I O N FOR CONVERGENCE

C THE L A S T T H R E E D E T E R H I N A N T S M U S T BE MONOTONIC, CONVERGINGc

EPSIL = , 0 7 5

E P 2 G 2 . + E P S I L 5 1 N D I t N D - 1 o N D - 2 N D 2 a P C ( N D l I - P C ( N D 2 ) PA

PD = P C ( N D ) - P C ( N D l )

P A 6 5 z P A P A 7 6 z P D P A 1 = A B S ( P A ) P D l = A B S ( P D ) I F ( P D 1 - P A i l B r 5 5 0 5 5 8 K = K I F ( P A ) 1,202 1 P A ; - I .

G O T O 3 2 P A = 1 e 3 I f ( P D ) 40505 4 P D = * l , G O T O 6 5 P D = l , 6 I F ( P A + P D ) 7 , 5 5 # 7 7 P D = P D l / P C ( N U l ) P A = P A l / P C ( N D 2 ) P A = A B S ( P A ) P D = A B S ( P D ) I F ( P A - E P S I L ) 1 7 r 1 7 r 5 5 1 7 I F ( P D - E P S I L ) 18,19,55 1 8 K t K R A T I O = P A 7 6 / P A 6 5 IF(RATI0)55,55,19 1 9 I F ( R A T I O - . 8 ) 5 4 , 5 5 , 5 3 5 4 C A L L P R E ( N D , P C , N P ) N S T A H T = 0 R E T U R N 5 5 C O N T I N U E N S T A R T = 1 R E T U R N END S U B R O U T I N E P R E ( N D , P C . , q P ) D I H E N S I O N P C ( 1 8 ) C P R E D I C T THE C O N V E H G E E D V A L U E S B A S E D ON THE L A S T THREE C l 3 E T E H M I N A N T S REAL K 2 K l r K Z K 3 r M U 1 K l r ~ 2 r K 3 D E L T A = . 0 0 0 0 1 C 1 = P C ( N D - 2 )

C2 = P C ( N D - 1 )

C3 F P C ( N D ) K l m N l I - 2 K2 5 N D - 1 K 3 G ND K2K3= K2/K3 K 2 K l = K 2 / K l C W R I T T E N F O R THE I N F I N I T E D E T E R Y I N A N T S U B R O U T I N E W I T H A S Y M P T O T I C c L I M I T S ,

P C ( 2 ) = 0

MU = ( C l - C 2 ) / ( C 2 - C 3 ) P = , O O O O 1 F P = M U * ( l , ~ K 2 K 3 * * P ) ~ Y Z ~ l * * P ~ l A A B S ( F P ) / F P C T H I S G E T S N E G A T I V E O R P O S I T I V E ONE

A M = 1

AP = 1 M2001 M 1 0 0 = 1 0 0

M1 = 1

1 3 K = K D O 1 M M 2 0 , M 1 0 0 , M l AP = K 2 K l * A P A M t K 2 K J * A M FP = MU-MU*AM - A P + I

El = A B S ( F P ) / F P

1 F ( A + B ) 1 , 1 0 1 1 C Z E R O IS C H A N G f OF S 1 3 N 1 C O N T I N U E P C ( 2 ) = 3 F?ETUWN 11 K = K 1 0 M 5 a = M + 5 0 1 R PN=M 19 FA = M U * ( ~ ~ - K ~ K ~ * * P V ) - K ~ K ~ * * ’ I ~ ~ ~ D O 1 5 L = l r 5 0 P N K 2 K 3 = K 2 K 3 * * P N P N K Z K l = K 2 K l * * P M S U B R Q U T I N E S S A ( O U T , 4 K r I ~ r K , M d R D E R , F ~ , P A , P C I I E R R ) I N T E G E R H21 I N T E G E R R 1 I N T E G E R R r R 2 r R 2 2 D I M E N S I O N ~ I G A ~ 1 3 ~ 1 3 ~ ~ A ~ l 3 ~ 1 3 ~ ~ B ~ l 3 r 1 3 ~ ~ C ~ l 3 ~ 1 D I M E N S I O N B I G A P ( 1 3 r 1 3 r 6 ) r B P ( l 3 , ~ 3 , 6 ) r C P ( l 3 r l ~ r 6 ) r A P ( l 3 , 1 ~ , 6 ) D I M E N S I O N F ( l J ) , A C ( 1 3 ) , A K ( 1 3 ) I I I M E N S I O N A R ( l J ) , A S ( l 3 ~ ~ P ( 1 3 ) ~ Q ( ¶ 3 ~ , P C ( ~ 3 ) , P A ( l 3 ~ , ~ ~ ~ 1 3 ~ C C C T H E P O L Y N O M I A L C O E F F I Z I ' E N T S OF THE H I G H E R O R D E R SYSTEMS.

C THE 9 THE AND 1 2 T H O S D E R S Y S T E M S H A V E A S S O C I A T E D c W I T H THEM C H A R A C T E R I S T I C ? O L Y N O M I A L S OF DEGREE 9 AND 1 2 C R E S P E C T I V E L Y , T H E C ~ E F F I C I E N T S OF T H E S E P O L Y N O M I A L S A R E N E E O E Q T O c D E F I N E THE S H A R A C T E ~ I S T I C E Q J A T I O N S O F THE SYSTEHS8 AND C ARE COMPUIEU AS FOLLOIJS,,.

C T H E C O E F F I C I E N T S D E S I 3 E D 4 R E K l , , , , K 9 C N P = 6 D O 4 5 M = l , K M 2 = 2 * M M 2 1 z M 2 - 1 PBM2 = P B ( M 2 ) C O S B 2 M = C O S ( P B M 2 ) P A M 2 = P A ( M 2 ) E A 2 M = E X P ( - P A M 2 )

F2M = F ( M 2 )

F 2 A 2 M = E X P ( - 2 , * P A M 2 ) F 2 M l = F ( M 2 1 ) A R ( M ) z 2 . * E A 2 M + ( F 2 q l * S I N ( P B q 2 ) + F 2 M * C O S B 2 M ) A S ( M ) = - 2 v * F 2 M * E 2 A 2 M

P ( M ) = - 2 , * E A 2 M * C O S 3 2 9

Q ( M ) = E 2 A 2 M W R I T E ( N P , 2 0 0 0 ) P ( M ) r 3 ( ~ ) , A ~ ( M ) r A S ( M ) 4 5 C O N T I N U E K l = K * 1 D O 1 0 M=Kl,MORDER M2 = 2+M

M21 = M2-1

PCM2 = - P C ( M 2 )

EC2M = E X P ( P C M 2 ) P C M 2 1 = - P C ( H 2 1 )

E C 2 M 1 = E X P ( P C M 2 1 )

F2M = F ( M 2 ) I11111111111 I I I I I I I I I I 11111111 II i I II I 9 1 C C 1 0 0 0 C l O O O C c C 1 4 4 6 2 7 C C C 26 1 0 1

c

6 9 7 0 Clr301, 1 0 0 1

A12 = K 7 * H l l + K8 + K5/MU1 - K6*MUHU - KHAT8

L 5 = Hi1

L 3 = KHAT3 - K3 - l ( l * H 1 0 + H l l * ( K l * K I - K 2 )

A 1 3 = K 7 * ( H I O - K l * Y 1 1 ) + K B * H l l + K 9 -+ K 6 / M U 1 - KHAT9

A 2 1 = K5 + K 2 / M J 1 - l(3*MUMU +K4*MUUU/MU1 -KHAT5

A22 = K 5 * H l l + K6 + K 3 / M U 1 - K4*MUMU - KHAT6

A23 = K 5 * ( H 1 0 - K 1 * H l 1 ) + K S * H l l + ~ ~ ~ K 4 / M U l - K ~ A ~ 7

A 3 1 = K3 + 1 / M 3 1 - Y l * M U M U + ( K 2 * M U U U ) / M U l - KHAT3

A32 = K3*H11 + K 4 t K I / M U P - K2eMUMU- KHAT4

A33 = K 3 * ( H l O - K 1 * 4 1 1 ) + K 4 * H l l + K5 + K 2 / M U 1 - KHAT5

H 1 = K H A T 1 0 - Y 7 * L J K8 L 4 - K 9 * L 5

02 = K ~ A T ~ - K B - K ~ * L ~ - K ~ * ~ ~ - < Y * L ~

H3 = KHAT6 - K6 K 3 * L 3 - < 4 * L 4 - K 5 * L 5

c FROM CHAMkRS R U L t . , , C D l = D E L T A l / D , D2 = DELTA2/D, 0 3 = DELTAJ/D

A 2 2 5 3 = A 2 2 * A 3 3 - A32 * A23

A 1 2 3 3 = A 1 2 * A 3 J - A 3 2 * A 1 3

A 1 2 2 3 = A 1 2 * A 2 3 - A22 * A 1 3

n 5 A I I ~ A Z ~ ~ ~ - ~ 2 1 + ~ 1 2 3 3 + ~ 3 1 + ~ 1 2 2 3

I F ( D 1 1 5 r b r 1 5 1 0 0 2 F O R M A T ( 1 H l r 4 6 H * * * D E V O ' 4 I ~ A T O R U F O R C R A Y E R S RULE IS Z E R O E O J 5 W R I T E ( N P , 1 0 0 2 1 S T O P 15 C O N T I N U E

D E L T A 1 = B I * A 2 2 3 3 - 8 2 + 4 1 2 3 3 + 8 3 * A 1 2 2 3

A 2 1 3 5 = A 2 1 * A 3 3 - A 3 1 + A 2 3

= A l l * A 3 3 - A 3 1 + A 1 3

A 1 1 3 3

A 1 1 2 3 = A l l * A 2 3 - A 2 1 * A 1 3

DELTA2 = - 8 1 * A 2 1 3 3 + 82 A 1 1 3 3 - 83 * A 1 1 2 3

A 2 1 3 2 = A 2 1 * A 3 2 - A 3 1 e A 2 2

A 1 1 3 2 = A l l * A32 - A 3 1 * A 1 2

4 1 1 2 2 = A l l * A 2 2 - A 2 1 * A 1 2

DELTA3 = I 3 1 * A 2 1 3 2 - 8 2 Q A 1 1 3 2 + 83 * A 1 1 2 2

D l = D E L T A l / D

D2 = D E L T A 2 / D

03 = DELTA3/D C THE C O E F F I C I E N T S T H E N A2E G I V E N B Y . e .

SS(l)=Kl-D3 SS(2) = K Z 1 D 2 - D 3 * S S ( l ) S S ( 3 ) r: K J o D 1 - D 2 * S S ( 1 ) - 0 3 * S S ( 2 )

S S ( 4 ) = K 4 - U l * S S ( 1 ) - D2 * S S ( 2 ) - 0 3 * S S ( 3 )

SS(5) = K5 - 01*SS(2) - 02*SS(3)-03 * S S ( 4 )

S S ( 6 ) = K6 - D l * S S ( 3 ) = 0 2 * S S ( 4 ) - D 3 * S S ( ~ ) RETURN E N D A P P E N D I X C Method f o r E x t r a c t i o n o f t h e Common PolynomialFactor From t h e Two Higher-Degree- Polynomials Given t h e t w o polynomials where it i s known a p r i o r i t h a t N and T b o t hc o n t a i n a s i x t h - d e g r e e f a c t o r S , it i s r e q u i r e dt od e t e r m i n e 0 1 , 0 2 , .., 0 6 i n terms o ft h ec o e f f i - c i e n t s kn and kn.

N o t e , f i r s t , t h a t t h e r a t i o o f T t o N mustreduce t o t h e r a t i o of a s i x t h - d e g r e ep o l y n o m i a lt o a c u b i c , t h e common s e x t i c f a c t o r c a n c e l l i n g o f f :

T ( z ) Z 6 + X1z5 + X2z4 + ... + X 5 Z + X L

( C - 1 ) . z ) - =

2 3 + u 1 ~ 2 + v 2 ~ + u 3

canbeformed,the Now, a set o fn i n el i n e a ra l g e b r a i ce q u a t i o n s s o l u t i o n o f w h i c h g i v e s t h e v a l u e s o f t h e X's and P I S . T h i s set

i s o b t a i n e db ym u l t i p l y i n g Eq. (C-1) through byN(z) [ z 3 + p y z 2

+ u 2 z + p 3 1 a n dg r o u p i n gt h ec o e f f i c i e n t s of l i k e powersof z.

There is considerableredundancy,with a t o t a lo ff o u r t e e ne q u a - t i o n si nt h en i n e unknowns. A convenient set o fn i n e are ( w i t h t h e a s s o c i a t e d power o f z f r o m whicheach w a s o b t a i n e d i n d i c a t e d on t h e l e f t ) A 2 0 : k 1 2 p 3 = k9X6 I f t h e f i r s t , t h r e e and l a s t t h r e e of t h e s ee q u a t i o n sa r ep r o p e r l y combined and s u . b s t i t u t e d i n t h e f o u r t h , f i f t h and s i x t h e q u a t i o n s , t h e n a set of t h r e ee q u a t i o n s for t h e c o e f f i c i e n t s u 1 , 1.12 and u g i s o b t a i n e d : i = 1 , 2 , 3 : ( C - 2 ) where while and E q s . (c-2) are readily scllved f o r P I , ~2 and ~ 3 , using C r a r n e r ' s ruie. The coefficients of the sextic now follow immediately, since from which REFERENCES ZluWaldt, F., Gates, C., and P i z i a l i , R., " I n v e s t i g a t i o n of 1.

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NASA-Langley, ?967 - 32 CR-1332