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Nonlinear Dynamics 5: 3-23, 1994.@ 1994 KluwerAc ademic Publishers. Printed in the Netherlands.

Nonl inear F lexura l -F lexura l -Tors iona l Interac t ions inB e a m s I n c l u d i n g t h e Ef fe c t o f To r s io n a l D y n a m i c s.I: P r i m a r y R e s o n a n c eM . R . M . C R E S P O D A S IL V A an d C . L . Z A R E T Z KYDepartm ent o f Mech anical Engineering Aeronautical Engineering and M echanics, Rensselaer PolytechnicInstitute, Troy, N Y 1 2180-3590, U.S.A.(Received: 9 March 1992; accepted: 22 June 1992)

Abstract. Nonlinear coupling between torsional and both in-plane and out-of-plane flexural motion is examinedfor inextensional beams (or beam-like structures) whose torsional and flexural eigenfrequencles are of the sameorder. The analysis presented here is based on a consistent set of nonlinear differential equations which containboth curvature and inertia nonlinearities, and account for torsional dynamics. Response characteristics, includingstability, are determined for canti lever beams subjected to a lateral periodic excitation. The beam's response in thepresence of a one-to-one internal resonance involving a torsional frequency and an in-plane bending frequency isinvestigated in detail.Key words : Beams, nonlinear oscillations, flexural-torsional dynamics, torsional response, nonlinear resonance.

1 . I n t r o d u c t i o nT o d a t e , t h e an a l y se s o f n o n l i n ea r d y n am i cs o f b eam s t h a t h av e b een p re sen t ed i n t h e l it e r a tu reh a v e d e a l t w i t h t h e c a s e w h e r e t h e t o r s io n a l f re q u e n c i e s o f th e b e a m a r e m u c h h i g h e r t h a ni t s b en d i n g f r eq u en c i e s . F o r su ch a c a se , t h e t o r s i o n a l i n e r t i a h a s n o s i g n i f i c an t e f f ec t o nt h e m o t i o n o f t h e b eam an d , th u s , t h e to r s i o n a l d e fo rm a t i o n is d e t e rm i n ed d i r ec t ly b y t h en o n l i n e a r c o u p l i n g b e t w e e n i n - p la n e a n d o u t - o f - p la n e b e n d i n g . M a n y a p p l ic a t io n s i n v o l v i n gt h e d y n am i cs o f s t ru c t u ra l e l em en t s f a ll i n to t h i s i m p o r t an t c l a ss o f p ro b l em s . A n a l y t i c a l an dex p e r i m en t a l i n v es t i g a t i o n s fo r su ch ca se s a r e ex em p l i f i ed b y t h e w o rk s r ep o r t ed i n [1 -8 ] .

F o r b ea m s h av i n g a c ro s s s ec t i o n w i t h h i g h a sp ec t r a ti o , fo r ex am p l e , t h e fi rs t t o r s i o n a ln a t u ra l f r eq u en cy i s o f t h e o rd e r o f a l o w er b en d i n g n a t u ra l f r eq u en cy . T h i s m ay a l so b eo b s e r v e d i n l o n g s tr u c tu r a l m e m b e r s th a t m a y b e m o d e l e d a s a b e a m , s u c h a s c o m p o n e n t so f sp ace s t ru c t u re s o r s a t e ll it e s . I n t h is c a se , t h e n o n l i n ea r co u p l i n g b e t w ee n t o r s i o n a l an db e n d i n g m o t i o n s m a y c a u s e a n e x c h a n g e o f e n e r g y b e t w e e n s u c h m o t io n s .T h e r e a r e tw o m y t h s w h i c h a r e c o m m o n a m o n g e n g i n e e r s w h o s e t ra i n in g h a s n o t i n c lu d e dn o n l i n ea r an a l y s i s t e ch n i q u es ; (1 ) " k eep t h e am p l i t u d es o f ex c i t a t i o n sm a l l , an d a l i n ea ran a l y s i s sh o u l d b e su f f i c i en t " ; (2 ) " am p l i t u d es o f n o n l i n ea r m o t i o n s a r e so l a rg e i n b eams t ru c t u re s th a t t h e s t ru c t u re w o u l d b e d e s t ro y e d l o n g b e fo re its am p l i t u d e g ro w s t h a t l a rg e .T h u s , a s t u d y o f n o n l i n e a r p h e n o m e n a in s u c h s tr u c tu r e s is u n n e e d e d " . T h e r e s ul ts p r e s e n t e di n P a r ts I an d I I o f th i s w o rk , an d i n s ev e ra l o t h e r p ap e r s t h a t h av e b ee n p u b l i sh ed i n t h et ech n i ca l l i te r a t u re , c l e a r l y s e rv e t o d i sp e l su ch m y t h s .

In t h i s p ap e r , n o n l i n ea r co u p l i n g b e t w een t o r s i o n a l an d f l ex u ra l m o t i o n s i n in ex t en s i o n a lb e a m s i s e x a m i n e d b y t a k in g i n t o a c c o u n t th e t o r s io n a l d y n a m i c s o f t h e b e a m . H e r e , t o r si o n a lan d f l ex u ra l n a t u ra l f r eq u en c i e s a r e o f t h e s am e o rd e r , t h u s a l l o w i n g fo r t h e o ccu r ren ce o fa n u m b e r o f n o n l i n e a r p h e n o m e n a th a t i n v o l v e s su c h m o t io n s . T h e d i ff e re n t ia l e q u a t io n sfo rm u l a t ed i n [2 , 3 ] an d i n [4 ] a r e u sed fo r su ch i n v es t i g a t io n . T h o se eq u a t i o n s co n t a i n a l lt h e g eo m e t r i c n o n l i n ea r i t i e s t h a t a r i s e w h en an i n ex t en s i o n a l b eam i s d e fo rm i n g i n t h ree -

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4 M .R .M . Crespo da S i l va and C . L . Zare t zkyd i m e n s i o n a l s p a c e . T h e b e a m i s s u b j e c t e d t o a la t er a l p e r i o d i c e x c i ta t io n w i t h f r e q u e n c y n e a ra b e n d i n g n a t u r a l f r e q u e n c y w h i c h , i n tu r n , i s n e a r t h e f ir s t t o r s i o n a l n a t u r a l f r e q u e n c y o f th eb e a m . I n P a r t I I o f th i s w o r k , a c o m b i n a t i o n r e s o n a n c e w h e r e a t o r si o n a l n a tu r a l f re q u e n c y i sn e a r t h e s u m o f t w o b e n d i n g n a t u r a l f r e q u e n c i e s is a d d r e s se d .2 . Equa t io ns o f M o t io nT h e s y s t e m a n a l y z e d h e r e i s s h o w n i n F i g u r e 1 . I t c o n s i s t s o f a c a n t il e v e r o f l e n g t h L a n ds p e c i f i c ( i. e ., p e r u n i t l e n g t h ) m a s s r e ( s ) , s u b j e c t e d t o a d i s t r i b u t e d p e r i o d i c f o r c e Q v ( s , t ) =qv(s) c o s ( f ~ t ) , o r t o a b a s e d i s p l a c e m e n t q v c o s ( f ~ t ) , a p p l i e d i n t h e i n e r t ia l y - d i r e c t i o n . T h ei n d e p e n d e n t v a r ia b l e s i s t h e a rc l e n g t h m e a s u r e d a l o n g t h e b e a m , a n d t d e n o t e s ti m e . T h e s m a l le f f e c t s d u e t o e x t e n s i o n a l i t y , w h i c h a r e ta k e n i n t o a c c o u n t in [ 4 , 9 , 1 0 ] , a n d s h e a r d e f o r m a t i o na re n e g l e c t e d . T h e d i f f e r e n ti a l e q u a t i o n s , a n d t h e b o u n d a r y c o n d i t i o n s , t h a t g o v e r n t h e f l ex u r a l-f l e x u r a l - t o r s i o n a l m o t i o n s f o r i n e x t e n s i o n a l b e a m s w e r e f i rs t f o r m u l a t e d i n [ 2] . F u r t h e r d e t a i lso f th e f o r m u l a t i o n a r e a l s o g i v e n i n [ 4]. T h e e x p a n d e d v e r s i o n o f t h o s e e q u a t i o n s , w h i c h a r ea m e n a b l e t o a p e r tu r b a t i o n a n a l y s i s o f t h e m o t i o n , a r e g i v e n b e l o w f o r t h e c a s e w h e n t h eb e a m i s s u b j e c t e d t o t h e d i s tr i b u t e d fo r c e i n d i c a t e d a b o v e . T h e f ir st t w o e q u a t i o n s b e l o w a r eE q u a t i o n s ( 4 6 b ) a n d ( 4 7 b ) i n [ 4 ], w h i l e t h e t h i rd e q u a t i o n i s E q u a t i o n ( 1 l d ) i n [2 ].

m 4) + c ~ ,i ~+ ( D ( v" )" = { -D ~ w " ( O " + v " w ' ) + w ' Q o ~ - v ' [ D ( v ' v " + w ' w " ) ] '+ ( D n - D ) w ' v ' t w " + [ ( D , - D ) ( O ~ w " - 0 2 v " ) ] '+ 3~w ' (o~ + ~0',~ ') + 3 U ( , /~ ' + w '~ ' )- [ 0 ~ - ~ , ) ( o S - e > ' + w ' , / w ' ) - ~ , ~ ' ]

v l S m S ~ 1f f(~,2 wt2) . .d8 d8 -}- qv (s)c O s(~ t) . ( l a )2 J, / v ,L 0m g , + c ~ ( v + ( D , T w " ) " = { D ~v "(O " + v " w ' ) - w ' [ D ,T v'v "+ D n w 'w " ] '

+ [ ( D , 7 - V ) ( O . v " + O ~ w " ) ] ' - 3 ~ i /( 0~ + i / w ' )+ w ' ( 3 J , z ' + a ~ ' ~ ' ) - [ ( 3 ~ - 3 ~ ) ( o ~ ' + o ~ ,z ') - 3 J ]

S 8 I f/ ( "v 2 J r - w " 2 ) " d 8 d s .2 L 0 ( l b )- ( j ~ - j ~ ) [ ( + ' 2 - ~ ' ~ ) o ~ - ~ ' ~ ' ] Qo~. (lc)

I n th e a b o v e e q u a t i o n s , ( ) ' a n d ( ) d e n o t e , re s p e c t i v e l y , p a r t ia l d i f fe r e n t i a t i o n w i t h r e s p e c tt o s a n d t . T h e q u a n t i t i e s D ~ ( s ) a n d D ( ( s ) a r e t h e p r i n c i p a l f l e x u r a l s ti f f n e s s e s o f th e b e a m ,

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T o r s i o n a l- F l e x u r a l D y n a m i c s : P r i m a r y R e s o n a n c e 5^Y

I /. . . . . . . . . . . . . . _ 1 , ' w ( s , t )s + u ( s , t )

F i g . 1 , Coordinate systems and elastic deflections for the beam.

w h i l e D ~ ( s ) i s t h e t o r s i o n a l s t i f f n e s s ; 3 o ( s ) , 3 ~ ( s ) a n d j { ( s ) = 3 o ( s ) + 3 g ( s ) a re th e d i s tr i b u t e dm a s s m o m e n t s o f i ne r ti a . T h e p r i n c i p a l a x e s o f t h e b e a m ' s c r o s s s e c t io n a t lo c a t i o n s , % ~ a n d ~ ,a r e s h o w n i n F i g u r e 1 ( w h e r e c a r e t s ( ^) a r e u s e d t o i n d i c a t e u n i t v e c t o r s ). T h e q u a n t i t i e s v ( s , t )a n d w ( s , t ) a r e t h e b e a m ' s d e f l e c t i o n s a l o n g t w o i n e r t ia l d i r e c t i o n s , a s s h o w n i n F i g u r e 1 , w h i l e0 = ( s , t ) i s a n o r ie n t a t i o n a n g l e t h a t w a s i n t r o d u c e d t o d e s c r i b e t h e o r i e n t a t io n o f t h e c r o s ss e c t io n a t l o c a t io n s . T h e e x p a n d e d f o r m o f th e b e a m ' s t o r s i o n is e q u a l t o "7' = 0 " + v " w ' ,w h e r e 7 ( s ) i s t h e a n g l e o f t o r s io n o f t h e b e a m . T h e q u a n t i t ie s c ~ a n d c w i n E q u a t i o n s ( l a , b ) a rev i s c o u s d a m p i n g c o e f f ic i e n t s th a t a re i n t r o d u c e d t o m o d e l f l e x u ra l d a m p i n g i n th e s t ru c t u re .T h e b o u n d a r y c o n d i t io n s f o r th e a b o v e e q u a t i o n s a re

v ( o , 0 = = = w ' ( O , t ) = 3 ( o , t ) = o ( 2 a )v " ( L , t ) = v ' " ( L , t ) = w " ( L , t ) = w ' " ( L , t ) = 7 ' ( L , t ) = O . ( 2 b )

E q u a t i o n s (1 a - c ) a re n o n l i n e a r p a r t ia l i n t e g r o - d i ff e r e n ti a l e q u a t i o n s d e s c r i b in g t h e c o u p l e df l e x u ra l - fl e x u r a l- t o rs i o n a l r e s p o n s e o f a n i n it ia l ly s t ra i g h t i n e x t e n s i o n a l b e a m p o s s e s s i n g b o t hv a r y i n g c r o s s s e c t i o n a l d i m e n s i o n s a n d n o n u n i f o r m m a t e r i a l p r o p e r ti e s . T h i s i n c l u d e s t a p e r e da n d s t e p p e d b e a m s a n d l o n g s l e n d e r s tr u c t u re s th a t m a y b e a p p r o x i m a t e d a s a b e a m . B ya c c o u n t i n g f o r t h e to r s i o n a l d i s tr i b u t e d m a s s m o m e n t o f i n e rt ia i n th e s e e q u a t i o n s , r e s p o n s e si n v o l v i n g d y n a m i c c o u p l i n g b e t w e e n t o rs io n a l a n d f l ex u ra l m o t i o n s m a y b e a d d r e s s e d .

E q u a t i o n s ( l a -- c ) w i l l b e s p e c i a l i z e d to a u n i f o r m h o m o g e n e o u s c a n t il e v e r b e a m f r o m t h i sp o i n t f o r w a r d . F o r c o n v e n i e n c e , E q u a t i o n s (1 a - c ) a re w r i t te n i n t e r m s o f n o n d i m e n s i o n a l q u a n -t i t i e s d e f i n e d a s s * = s / L , v* = v / L , w * = w / L , t* : t v / D , / ( m L 4 ) , ~ * = F ~ L 2 v / m / D , ,/3 y = D g / D ~ , /3~ = D e / D ~ , a ~ = 3 o / ( m L 2 ) , 3"~ = 3 / ( m L 2 ) , c*~ = c v L a / v / - - m D ~a n d c ~ = c w L 2 / x / - - m D ~ . B y m a k i n g u s e o f t h e a n g le o f to r s i o n f o r t h e b e a m , w h i c hi s d e t e r m i n e d a s [ 3 , 4 ] ~ ,( s, t ) = 0 x ( s , t ) + f o v " w ' d s = 0,~(s, t ) + v ' w ' - f ~ v ' w " d s ,a n d b y i n t r o d u c i n g a sm a l l d a m p i n g t e rm i n E q u a t i o n ( l c ) , w h i c h i s a p p r o x i m a t e d a s

- c 0 x 0 x a = - 3 ~c -yOx ~ - 3 {cTVy , t h e n o r m a l iz e d f o r m o f E q u a t i o n s ( l a - c ) m a y b ew r i t te n a s g i v e n b e l o w . T h e s u p e r s c r ip t s a re d r o p p e d f o r c o n v e n i e n c e in n o t a t i o n

o_ +

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6 M . R . M . C r e s p o d a S i l v a a n d C . L . Z a r e t z k y+ V ! [ - - ~ y V t V t tt - - W I W tt! . . ]/ /+ 3 ~ ) lv + 3 O i b tw - ~ (v 12 + w l 2 ) " d s d s

1 0( i )3 ~ ~ / + ( v ' w " ) ' d s - v ' ( v ' ( v ~0[ ( )- [ - (3~? - - 3C) i ) t 7 2 - - ( V '7 - - ( V ' v ' w " d s + 3 ~ i /

0+ v ' (3~5 2 + 3~@ 2' - 3 ~ c T w ' ~ / } ' + q v ( S ) C O S ( f ~ t ) . ( 3 a )

@0

- - W I ( V I t2 - Jv W I t 2 )

[ ' J ]W t - - f l y V t V t tt - - W t W m + 3 ~ ? J tv t + 3 T l ~ )t W t - - - ~ ( V 2 + w t 2 ) " d 8 d s1 0

( J )3 ~ ~ + ( v ' w " ) d s - v ' ( v ' ~ '0- - [ ( 3 0 - - 3 ( ) ( ( v t " Y 2 - t- i ~ t'~ - i / i v t t w t d 8 ) - 3 ~ ? ( vt] "+ W t ( f ~ i J t2 + 3 , q / b t 2 ) } t . ( 3 b )

J ~ + 3 ~ - 9 ~ 7 " = - ( 1 - m y ) [ ~ ( v " 2 _ ~ , , 2 ) _ , ~ , , ]I ; ]3 ~ ( v ' ~ " ) d ~ - ~ ' ~ ' + ( 3 ~ - 3 ~ ) [ ( ~ ' ~ - ~ ' 2 ) ~ - ~ ' w ' ] 1 .0

(3c)

I n t h e n e x t s e c t i o n t h e e i g e n f u n c t i o n s a s s o c i a t e d w i t h t h e l i n e a r i z e d c o u n t e r p a r t o f t h ea b o v e e q u a t i o n s a r e d e t e r m i n e d . E q u a t i o n s ( 3 a -- c) a re t h e n t r a n s f o r m e d i n to a s e t o f o r d i n a r yd i f f e r e n t i a l e q u a t i o n s w h i c h a r e u s e d t o a n a l y z e t h e c o u p l e d m o t i o n s w i t h a p e r t u r b a t i o nt e c h n i q u e .

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T o r s i o n a l- F l e xu r a l D y n a m i c s : P r i m a r y R e s o n a n c e 73. Eigenfun ct ions Assoc iated wi th the Linearized Eq uat ionsT h e s o l u t i o n t o t h e u n d a m p e d l i n e a r iz e d d i ff e r e n ti a l e q u a t i o n s o f m o t i o n w i ll b e u s e d a s t h es t a r t i n g p o i n t f o r a p e r t u r b a t i o n a n a l y s i s b a s e d o n E q u a t i o n s ( 3 a - c ) . T h o s e s o l u t i o n s a r e o ft h e f o r m

v = F ~ ( s ) v t ( t ) ( 4 a )w = F w ( s ) w t ( t ) ( 4 b )7 = F ~ ( s ) ' y t ( t ) ( 4 c )

w i t h v t ( t ) = c o s ( t o r t + B y ) , w t ( t ) = c o s ( w w t + B w ) a n d 7 t t = c os (w.y t + B.~). In ge n e ra l ,t h e e i g e n f u n c t i o n s F ~ ( s ) , F w ( s ) a n d F.~(s) a r e d e t e r m i n e d b y s o l v i n g t h e l i n e a r i z e d c o u n -t e rp a r t t o E q u a t i o n s ( 3 a - c ) n u m e r i c a ll y . F o r a b e a m w i t h c o n s t a n t d i s tr i b u t e d p r o p e r ti e s , th ee i g e n f u n c t i o n s F v ( s ) a n d F .~ ( s ) s a t is f y t h e d i f f e r e n t i a l e q u a t i o n s

ptttt 2 2r. ttv * v - c G F ~ + 3 ( c G r ~ = 0 ( 5 a )9~F4, 2+ 3~co.yF. = 0 (Sb)

w i t h t h e b o u n d a r y c o n d i t io n F v ( 0 ) = F ' ( 0 ) = F .y ( 0 ) = 0 a n d F ~ ' ( 1 ) = F ~ ( 1 ) = 0 . T h ee i g e n f u n c t i o n a s s o c i a t e d w i t h t h e l i n e a r i z e d p a r t o f w ( s , t ), F w ( s ) , i s o b t a i n e d b y s e t t i n gf ly = 1 , ~o~ = ~w a nd 3 = 3,7 i n t he e xpr e s s i o n ob t a i ne d fo r F~( s ) .

T h e s o l u t io n s t o E q u a t i o n s ( 5 a , b ) a r e o b t a i n e d a s

c o s , 2 , ( 6 a )1 2 ] n = 1 , 2 , . . . (6 b)

w h e r e1 % =

a n d

r ~ c o s h r l + r ~ c o s r 2r 2 s i n h r l + f i r 2 s in r2 ( 6 c )

( 6 d )

T 2 = ( 6 e )

~ = ( 2 ~ - 1 ) 7 ~ . (60

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8 M . R . M . C r e s po d a S i lv a a n d C . L . Z a r e t zk y

T A B L E IN a t u r a l f r e q u e n c i e s ~ w v e r s u s 3n

3 ~ w ~ ( f ir s t m o d e ) w ~ ( s e c o n d m o d e ) w ~ ( t h ir d m o d e )0 1.8752 ~ 3.51 6 4.6952 ~ 22.0 43 7.8552 ~ 61.70 . 0 0 2 5 3 . 5 1 9 8 2 1 . 6 7 6 5 8 . 4 3 20 .005 3 .5236 21 .331 55 .633

T h e q u a n t i t i e s r l a n d 9" s a t i s f y t h e c h a r a c t e ri s t ic e q u a t i o n g i v e n b e l o w , w h i c h i s o b t a i n e d b yi m p o s i n g t h e c o n d i t i o n F ~ " ( 1 ) = 0 o n t h e f u n c t i o n F ~ .

r~ + r 4 + 2 r ~ r ~ ( c o s h r l ) c o s r 2 + r l r z ( r ~ - r 2 ) ( s i n h r l ) s i n r 2 = 0 . (7)T h e c o n s t a n t s C ~ , C w a n d C - t h a t a p p e a r in t h e e x p r e s s i o n s f o r Fv , Fw and F .~ , r e spe c t ive ly ,

a r e a r b it r ar y . F o r l a t e r c o n v e n i e n c e , t h e s e c o n s t a n t s a r e c h o s e n s o t h a t1

0

1 1

0 0T h e e f f e c t o f t h e d i s t r i b u t e d m a s s m o m e n t s o f in e r t ia o n t h e n a t u r a l f r e q u e n c i e s ~ow

( o r ~ v / x / ~ u ) i s il lu s t ra t e d i n T a b l e I f o r s e v e ra l v a l u e s o f 3~ (o r 3 0 . F o r a h o m o g e n e o u sb e a m w i t h a s q u a r e c r o s s s e c t i o n , f o r e x a m p l e , o n e h a s 3 ~ = ( b / L ) 2 / 1 2 , w h e r e b i s a c r o s ss e c t i o n d i m e n s i o n . E v e n t h o u g h t h e e f f e c t o f t h e d i s tr i b u te d m a s s m o m e n t s o f i n e rt ia o n t h ef r e q u e n c i e s ~ , a n d ~ w i s m o r e p r o n o u n c e d f o r t h e h ig h e r m o d e s , s u c h e f f e c t i s n e g l i g ib l ef o r t y p i c a l v a l u e s o f 3,7 a n d 3 ~. F o r a h o m o g e n e o u s r e c t a n g u l a r c r o s s s e c t i o n b e a m o f l e n g t hL a n d c r o s s s e c t i o n d i m e n s i o n s a a n d b f o r e x a m p l e , o n e h a s 3~ = ( b / L ) 2 / 1 2 . T h i s g i v e s3,7 ~ 0.0 01 if L i b = 9 . Va lues o f 3,1 and 3~ sm al l e r t han 0 .001 a r e t yp ica l i n p r ac t i ce .

T h e d i s t r i b u t e d m a s s m o m e n t s o f i n e rt i a a l s o c o n t r i b u t e to a n u m b e r o f n o n l i n e a r t e r m si n E q u a t i o n s ( 3 a - c ) . T o e s t im a t e t h e i r c o n t ri b u t i o n s , a n d c o m p a r e t h e m t o t h o s e a r i s i n g fr o mt h e s t i f fn e s s t e r m s , G a l e r k i n ' s m e t h o d i s a p p l i e d t o E q u a t i o n s ( 3a - -c ) w i t h v ( s , t ) , w ( s , t ) a n d" y (s , t ) a p p r o x i m a t e d a s a t = F ~ ( s ) a t ( t ) w i t h a t ( t ) = c o s [ w ~ t + B ~ ] ( a = v , w , 7 ) . T h et e r m s a s s o c i a t e d w i t h t h e d i s t r i b u t e d m a s s m o m e n t s o f i n e r ti a a n d th e s t if f n e s s t e r m s p r o d u c es i m i l a r G a l e r k i n c o e f f i c i e n ts . A s a n e x a m p l e , t h e t e r m s /3 u v ~2 v m a n d 3~v~2~)~p r o d u c e a s i m i l a rv t t e r m i n th e r e d u c e d e q u a t i o n s . T h e G a l e r k i n c o e f f i c i e n t s f o r s u c h t e r m s w e r e e v a l u a t e du s i n g f i r s t b e n d i n g a n d t o r s i o n m o d e s , a s w e l l a s t h i r d b e n d i n g a n d f i r s t t o r s i o n m o d e s . T h en u m e r i c a l v a l u e s o f t h e c o e f f i c i e n t s a s s o c i a t e d w i t h t h e d is t r i b u t e d m a s s m o m e n t o f i n e rt i aw e r e f o u n d t o b e n e g l i g i b l e w h e n c o m p a r e d t o t h e v a l u e s f o r t h e c o e f f i c ie n t s o f t h e s i m i l a rt e r m s a s s o c i a t e d w i t h t h e d i s t r ib u t e d s t i ff n e s s e s . F o r th e s e r e a s o n s , t h e 3n, 3~ and 34 -----3n + 3~t e rms in Equa t ions (3a , b ) and the 3 ,~ and 3 t e rms in Equa t ion (3c ) wi l l be neg lec t ed , t huss i m p l i f y in g t h e a n a ly s i s. W h e n t h e n a tu r a l f re q u e n c y ~ 7 i s c o m m e n s u r a b l e w i t h th e b e n d i n gn a t u r a l f r e q u e n c i e s , t h e t e r m 3~;~ i n E q u a t i o n ( 3 c ) is a s s o c i a t e d w i t h t h e l i n e a ri z e d c o u n t e r p a r t

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T o r s i o n a l- F l e x u r a l D y n a m i c s : P r i m a r y R e s o n a n c e 9o f th a t e q u a t i o n a n d , t h u s c a n n o t b e n e g l e c te d . I n s u c h a c a s e , e n e r g y m a y b e e x c h a n g e db e t w e e n t o r s i o n a l a n d b e n d i n g m o t i o n s . S u c h m o t i o n s a r e i n v e s t i g a t e d i n t h is p a p e r.4 . P e r t u r b a t i o n A n a l y s i s o f t h e M o t i o nA p p r o x i m a t e s o l u ti o n s f o r th e m o t i o n m a y b e o b t a i n e d b y u s in g G a l e r k i n ' s m e t h o d t o r e d u c et h e e x p a n d e d d i f f e r e n t i a l e q u a t i o n s o f m o t i o n , Eq u a t i o n s ( 3 a - c ) , t o a s e t o f o r d i n a r y d i f f e r e n t i a le q u a t i o n s . H e r e , a o n e m o d e a p p r o x i m a t i o n i s u s e d , a n d n o n l i n e a r i n t e r a c ti o n s i n t h e c o u p l e dv ( s , t ) ~ F ~ ( s ) v t ( t ) , w ( s , t ) ~ F ~ ( s ) w t ( t ) a n d 7 ( s , t ) ~ F T ( s ) " / t ( t ) m o t i o n s a r e t h e ni n v e s t i g a t e d . A o n e m o d e a p p r o x i m a t i o n i s j u s t i f i e d w h e n t h e f r e q u e n c i e s a s s o c i a t e d w i t ho t h e r m o d e s t h a t a r e n o t n e a r t h e e x c i t a t io n f r e q u e n c y a r e n o t c o m m e n s u r a b l e . I n t h is c a s et h o s e m o d e s " d i e o u t " d u e t o d a m p i n g i n t h e s y s t e m [1 1]. W h e n t h e d i s tr i b ut e d m a s s m o m e n t so f i n e r t ia t e r m s a r e n e g l e c t e d , a s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n , t h e fo l l o w i n g o r d i n a r yd i f f e r e n t i a l e q u a t i o n s o f m o t i o n a r e o b t a i n e d f o r t h e q u a n t i t ie s v t ( t ) , w t (t ) a n d 7 t (t ) :

+ + : + + +

- ~ - f [ ~ V 5 ( ' 0 2 ) " V t~ - O { v 6 ( W 2 ) " 'O t " ~- I v C O S ( f i t ) (8a)~ iJ t + C w @ t q -C 0 2 W t = Olw1 TtVt q-O~ w2W t"/2 q- O gw3Wt 2 -1-Ctw4w 3

q - O~ w s (W 2 )" W t q - O g w 6 (~ U 2 t" W t (8b)5 t q- CT ;yt q- O 32,yt = O@ ~/t?) -.}- O@2~/tw2 -t- Ct73~tll)t -~- [O@4(~3tWt)" q- Ct75Vt~3t]" . (8C )

I n t h e a b o v e e q u a t i o n s , f v A f l F v ( s ) q v ( t ) d s , a n d t h e c o e f f i c i e n t s a v ~ , a w ~ an d a-r~ (i = 1, 2, . . ) a r e l i s te d i n A p p e n d i x A .

T o a n a l y z e t h e m o t i o n g o v e r n e d b y t h e t h r e e c o u p l e d n o n l i n e a r o r d in a r y d i f fe r e n t ia l E q u a -t i o n s ( S a - c ) , t h e m e t h o d o f m u l t i p le ti m e s c a l e s [ 1 1] w i l l b e u s e d . Th r e e t im e s c a l e s t o = t ,t l = e t and t2 = 62t a re in t rod uced , and v t , w t a n d 7 t a r e e x p a n d e d i n t e r m s o f e a s

a t ( t o , t l , t 2 ; e ) = e a t ~ (t o , t l , t 2 ) + e 2 a t 2 ( t o , t l , t 2 ) + e 3 a t3 ( to , t l , t 2 ) a = v , w , ' y . (9 )T h e c a s e w h e n t h e e x c i t a ti o n f r e q u e n c y , f~ , i s n e a r o n e o f t h e e i g e n f r e q u e n c i e s o f t h e u n d a m p e dm o t i o n , W v, a n d w h e n w v i s n e a r w 7 i s a d d r e s s e d . I n t h is c a s e t h e b e a m e x h i b i ts a n o n l i n e a rp r i m a r y r e s o n a n c e , a n d t h e r e is e n e r g y e x c h a n g e b e t w e e n t h e v t and "Yt mo t ions cau sed byn e a r - i n t e g e r c o m m e n s u r a b i l i t y b e t w e e n t h e n a t u ra l f r e q u e n c i e s o f t h e s y st e m B y t r a n s fe r r in gt h e d a m p i n g a n d t h e e x c i t a ti o n t e r m s o u t o f t h e O ( e ) a p p r o x i m a t i o n a s

c~ = e2 Cv2 , cw = e2 cw2 , c-r ~ - - f 2 C 7 2 , f v = e3 v , ( 1 0 a - d )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 s a r e o b t a i n e d f o r e a c h o r d e r o f a p p r o x i m a t i o n w h e n t h ee x p a n s i o n s d e f i n e d b y Eq u a t i o n ( 9 ) a r e s u b s t i t u t e d i n t o Eq u a t i o n s ( 8a -- c) :

d g v t , + W 2 v Vtl = O . ' .v t l = A ~ ( t l , t2) co s [ W v to + B v ( t l , t2)] & A v co s qSv (1 l a )d 2 w t l + COwWt2 = O .'.w t, = A w ( t l , t 2 ) c o s [W w tO + B , ~ ( t l , t 2 ) ] =A A w c o s q 3 w ( l l b )

2d 2 7 t , + co.yTt = 0. ' .% = A - y ( t l , t2) co s [co-~to + B . y ( t l , t2) ] =~ A ~ co s qS.~ (1 lc )

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1 0 M . R . M . C r e s p o d a S i l v a a n d C . L . Z a r e t z k y2 )

d 2v t2 + c o2 vt2 = - 2 d o d , v t , + a . ," / t, w t , = 2 w ~ [ ( d , A v ) s i n ~ + A ~ ( d , B v ) c O s . ]+ a v ~ A v o A 7 [ c o s ( 7 + ~ ) + c o s ( 7 - ~ , ) ] / 2 ( 1 2 a )

d 2 w ta + c o 2w tz = - 2 d o d l w t ~ + a w ," /t lV t , = 2 W w [ ( d l A w ) s i n e w + A w ( d , B w ) c o s e w ]+ a w l A v A 7 [ c o s ( 7 + C v ) + c o s ( 7 - C v ) ] / 2 ( 1 2 5 )

d2"yt2

o ( , 3 )

+ c o 2 % 2 = - 2 d o d l T t , + a 7 3 v t , w t , + a 7 4 d ~ ( v t , w t , ) + o [ ~ / sd o (V t ld O ~ t l )= 2 w . [ ( d l A . r ) s i n ' r + A ~ ( d l B T ) c o s ' r ]

2 2+ [ a .~ 3 - ( w ~ + z ~ ) a 7 4 - w ~ a T , ] [ c o s ( ~ + ~ ) + c o s ( v - ,1 ,)] A . A ~ / 2+ w . c o w ( 2 a 7 4 + a .r , ) [ c o s ( v - e w ) - c o s ( . + e w ) ] A r A b ~ 2 ( 1 2 c )

d2vt3 + co2vt3 - 2 d o d l v t 2 - ( d 2 + 2 d o d 2 ) v t l + a v ~ ( T t 2 w t , + w t z 'Y t ,)- ~ O L V 2 t t ' f f t l - ~ - O L V 3 t l q J ) $ 1 ~ O g v 4 V t l "3 V- cv2dovt~ + fv3 c o s ( Q t 0 ) ( 1 3 a )

d g W t 3 " Jl-w g w t 3 : - - C w 2 d o ~ l J t 1 - - 2 d o d l w t 2 - - ( d 2 -~ - 2 d o d 2 ) w t , + a ~ , ( " Y t 2 V t l - }- V t 2 " ~ t l )2 3q- aw 2 Wtl ~t21 -q- aw 3 Vtl Wtl -k- aw4 Wtl

-~ - O l w 5 W t l d 2 ( W 2 1 ) -~ - O L w 6 W t l d 2 0 (V~I ) ( 1 3 b )

d2o,t3 q- w 2+ a . r3 ( v t lw t 2 + w q v t 2 ) + a - r , ( 2 d o d l ( v q w t t ) + d 2 [v t2 w tl + w t z v t l ] )+ o e - r s { d l (v t t d o w t l ) + d o [ v t 2 d o w t ~ + v t ~ ( d lw t ~ + d o w t 2 ) ] } ( 1 3 c )

I n th e a b o v e e q u a t i o n s , t h e n o t a t io n d ~ ( ) = O ( ) / O t n ( n = 0 . l , 2 ) i s u s e d . T h e s o l u t i o n st o t h e O ( e ) d i f fe r e n t i a l e q u a t i o n s , w h i c h a r e a l s o g i v e n i n E q u a t i o n s ( l l a - c ) , w e r e u s e d inE q u a t i o n s (1 2 a - c ) .

T h e O ( e 2 ) d i f f e r e n t i a l e q u a t i o n s e x h i b i t t h e c o m b i n a t i o n r e s o n a n c e s c o~ ~-. l eo 7 w ,, .] .w~, ,-~ ]w 7 q - w , ] a n d w 7 ~ t w ~ : k w , I. H e r e , t h e c a s e w h e r e t h e v a l u e s o f t h e n a t u r a l f r e q u e n c i e s

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T o r s io n a l - F le x u r a l D y n a m i c s : P r i m a r y R e s o n a n c e 1 1a re " a w a y " f ro m th e s e r e s o n a n c e s i s co n s id e r e d . In s t e a d , t h e m o t io n w h e n Wv ~ co7, w h ic h i sa r e s o n a n c e c o n d i t i o n e x h ib i t e d a t t h e O (e 3 ) l e v e l, i s i n v e s tig a t e d . T h e c a s e w h e n ~ 7 i s n e a rWv + ww i s a n a ly z e d in [1 2 ]. F o r t h e c a s e c o n s id e r e d in t h is p a p e r , th e fo l lo w in g c o n d i t i o n sfo r e l im in a t io n o f s e c u l a r t e rm s a t t h e O (e 2 ) l e v e l, a n d s o lu t io n s t o E q u a t io n s (1 2 a -c ) , a r eo b ta in e d :

d l A ~ = 0 ; d l B ~ = 0 ( a = v , w , @ (14a, b)c % A , r A w [ co s( 7 + Cw) cos(qS-r - Cw) ]

v t 2 - 2 [ c o l : ( -7 ~ , ~ -7 7 ) 2 + c o ~ - - - ~ - 7 ~ - ~ ) 2 J (15a)

W t 2 - - c t w l A T A v [ C 0 S ( 7 + C v ) c o s ( ~ - ~ )C O 2 - - ( C O , ) , - - C d v ) 2 (15b)

7 t 2 = A ~ A ~ [ k v ~ cos(~ + era) + k-~2cos(v - Cm)] (15c)w h e r e

kVl = O Z 3 , - - O @ 4 ( C O v q - C O w ) 2 - - O Z , , / SCO w(CO w + C O y )2 [ w 2 - ( C O v + W m ) 2] (16a)

(16b)

N o te t h a t t h e s e c o n d o rd e r t o r s io n a l c o m p o n e n t o f t h e b e a m r e s p o n s e i s o n ly d e p e n d e n t o nth e f i r st o rd e r i n -p l a n e a n d o u t -o f -p la n e b e n d in g m o t io n s.W ith the O (e) and O(e 2 ) so lutions g iven by Equations (11a-c ) and (15a-c) , the 0 @ 3)d i f f e r e n t ia l e q u a t io n s , E q u a t io n s (1 3 a -c ) , t a k e t h e fo l lo w in g fo rm s :

dgvt3 + a)2vt3 = fv3 c o s (~ to ) - b w v ( Z d z A v + c v 2 A v ) s i n C v~ 3 2 k 6 A v A 2 ] ~2 cos(v - 27 ) + [ 2 w v A , d 2 B~ + k 2 A v + k 3 A v A 7 + co sk l A v A 7

+ A v A 2 a ~ 1 72 + ~ c~ 3 - w ~ a , 6+ h ig h f r e q u e n c y t e rm s (1 7 a )

g~ w ~ 3 2 7 - - w~o 2d2Am + cmzAm) sin OmJ r- C d w W t 3. 2 2+ A m [ 2c v w dz B m + k 7 A 2 + k s A ~ + k 9 A v ] cos w

+ AmAav ~ml k~2 + ~w ~ -~vam~

+ ~ m 2 ( ~ m - ~ v - ~ ) ( < ~ + ~ v - ~ )+ h ig h f r e q u e n c y t e rm s (1 7 b )

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12 M. R . M. Crespo da S i lva and C. L . Zare tzky

d2 ~ t3 +

+ a -7 2 -5g( 71o-/-b-7)

02 2 coT(2d2A, + c72A7) sin 95"r~ t 3 =2k 4 A T A 2 cos(29 5~ - 95.y) + [2coTA.rd2B7 - k s A T A 2 - k m A , r A ~ ] c o s 7

@ 0:72}x A T A ~ c o s( 2 ~ o - 7 ) + h i g h f r e q u e n c y t e r m s . ( 1 7 c )

Eq u a t i o n s ( 1 7 a - c ) e x h i b i t t h e i n t e r n a l r e s o n a n t c o n d i t i o n w 7 ~ Wv. Th e cou p led mo t ionsi n t h e p r e s e n c e o f t h i s r e s o n a n c e c o n d i t i o n , a n d w i t h f~ ~ a ~v , a r e n o w i n v e s t i g a t e d w i t h t h eo b j e c t i v e o f d e t e r m i n i n g h o w t h e f l e x u r a l a n d t o r s i o n a l m o t i o n s i n t e r a c t . To w a r d t hi s e n d , t w od e t u n i n g p a r a m e t e r s e 2a 2 a n d e 2 A 2 a r e i n t r o d u c e d a s

f t = wv(1 + ~20"2) ( l g a )Wv = co7(1 + e 2 A 2 ) . (18b)

By d e f i n i n g th e f o l l o w i n g q u a n t i t ie sA# = 2 A z w u t 2 + 2 ( B~ - B .~ ) ~ 2 A 2 w ~ t 2 + 2 ( B~ - BT) ( 1 9 a )

C f ~ COv~r2t2 - B v (19b)an d by n ot ic in g tha t ~2t0 = 95~ + 95I, 295v - q57 + # an d 2957 - 95~ = 95~ - #, th e fo ll ow in gc o n d i t i o n s f o r e l i m i n a t i o n o f s e c u l a r t e r m s a t t h e O (e3 ) l e v e l ar e o b t a i n e d f r o m E q u a t i o n s ( 1 7 a -c):

22w~d2A~ + Wvcv2Av + k l A ~A 7 sin # - f~3 sin 95y = 0 (2 0a )2 3 2 22 w v A v d2 B v + k l A v A 7 c o s # + k 2 A v + k 3 A v A 7 + k 6 A v A w + f v 3 C O S 9 5 f = 0 (20b)

2d2Aw + cw2Aw = 0 ( 2 0 c )Aw [2ww d2B w + k7 A 2 + k8A2v + k9 A2] = 0 (20d)2wTdzA7 + CTzWTA~ + k4ATA2~ s in # = 0 (20e )Av[2 wvd 2B 7 - k4 A2 c o s I x - k5A2v - kloA2w] = 0. (20 f)

Th e e x p r e s s i o n s f o r t h e c o e f f i c i e n t s k l t h r o u g h k s a r e l is t e d be l o w , w h i l e t h o s e f o r k 6 t h r o u g hk l 0 a r e l i st e d i n A p p e n d i x B .1 [ o w l ] l

3 2k2 = -~O~v4-- WvOivs

(21 a)

( 2 1 b )

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O / ' v 2 O g v lO g w l [ - '- ~ w13= -~-- - -- -- -- --~- +To r s i on a l -F l exu r a l Dyn a m i cs : Pr i ma r y Res o n a n ce 1 3

'1o 2 - 4 c o 2 ( 2 1 c )k4 = ~w'(cov2~74 - oL%) oe71 (2 1 d )4co2 4k 5 = C t W l [ W 2 ( 2 O z 7 5 + O L .y 4 ) - - O g ,- ),3 ] - } - O g W l O L T 4 (.a .2 - - O L ,T 3 O L - y , (21e)

4o :2 - 4co2 4o:2 2E q u a t i o n s ( 2 0 a - f ) a r e c o u p l e d n o n a u t o n o m o u s o r d i n a ry d i f f er e n ti a l e q u a t io n s in t e r m s

o f t h e v a r i a b l e s A~, A~, , A-r , Bv , Bw a n d B .r . S ince d2B~ = cov0-2 - d2 / and d 2 B .r =(o-2 + A2 )cov - d 2 I - d 2 # / 2 , t h o s e e q u a t i o n s a r e a u to n o m o u s i n t e r m s o f t h e v a r i a b le sA v , A ~ o , A . r , I , # a n d B ,o . Th e a u to n o m o u s e q u a t i o n s a d m i t th e e q u i l i b r iu m s o lu t io n A ~ =co nst an t = A~ , (c~ = v , w, % wi th Aw~ = 0 and f l = cons tan t = f ie ( /3 = Cy, p) .

A s d i s c lo s e d b y Eq u a t i o n ( 2 0 c ) , A ~ - 4 0 a s t - -+ o c. I t s h o u ld b e n o t e d t h a t t h e c o n c lu s io nA w = 0 i n d i c a t e s t h a t t h e o u t - o f - p l a n e b e n d in g c o m p o n e n t o f t h e r e s p o n s e i s o f h ig h e r or d e r .T h e s a m e c o n c l u s i o n m a y a l s o b e o b t a in e d f r o m t h e e x p r e s s i o n fo r t h e ~ c o m p o n e n t o f t h eb e a m ' s c u r v a tu r e v e c to r w h ic h i s g iv e n a s [2 , 4 ]

p v = T v " - w ' 1 + + 7 2 w ' + O ( e 4 ). (22 )A s t h i s c o m p o n e n t o f th e c u r v a tu r e t e n d s to z e r o , w h ic h im p l i e s t h a t t h e b e n d in g s t i f f n e s s D,~i s e f f e c t i v e ly i n f in i t e, o n e t h e n o b t a in s w " = v" 'y , w h ic h a l s o d i s c lo s e s t h a t w = O ( e 2 ) . Th es a m e c o n c lu s io n m a y a l s o b e o b t a in e d w h e n Eq u a t i o n ( 3 b ) i s w r i t t e n in t h e f o l l o w in g f o r m int e r m s o f d i m e n s i o n a l t im e :

m L 4 ./ O - - + + w mlD , c w w v - D ~ -

= 7% " + 1 - ~ S T v W 0 @ 3 ) . (23 )Th e a b o v e d i f f e r e n t i a l e q u a t i o n a l s o d i s c lo s e s t h a t w " - 4 ~ / '7 a s D , 7 - 4 o o . Th u s , f o r t h ereso nan t mot io n co r r esp ond ing to coo ~ w. , and f~ ~ coy, t h e c o n c lu s io n t h a t th e w - m o t io n i so f h ig h e r - o r d e r g o e s "h a n d - in - h a n d " w i th t h e c o n c lu s io n t h a t t h e 7) c o m p o n e n t o f t h e c u r v a tu r eis small .

Ex p l i c i t e x p r e s s io n s f o r t h e e q u i l i b r i u m v a lu e s o f t h e i n - p l a n e b e n d in g a n d t o r s i o n a la m p l i t u d e s a s f u n c t i o n s o f t h e d e t u n i n g p a r a m e t e r s 6 2 0 - 2 a n d 6 2 Z ~ 2 , d a m p i n g p a r a m e t e r se2cv2 a n d e 2 c 72 a n d t h e e x c i t a t i o n s t r e n g th e3 fv3 c a n b e o b t a i n e d f r o m E q u a t i o n s ( 2 0 a - f ) .Th o s e e q u a t i o n s a d m i t tw o e q u i l i b r i u m s o lu t io n s . O n e e q u i l i b r i u m s o lu t io n , r e f e r r e d t o a s E l ,c o r r e s p o n d s t o eA.r, = 0 , wh i le fo r the secon d equ i l ib r ium so lu t ion , re fe r red to a s E2 , cA-r ,i s n o n z e r o . Th e a m p l i t u d e - f r e q u e n c y r e s p o n s e c h a r a c t e r i st i c f o r E l i s o b t a in e d a s

- k 2 ( e A v ~ )3 5:: V/(e3 fv , ) 2 - -co2(e2cv2)2(eAv~)220"2 = (24 )2 w 2 e A ~Th i s i s t h e s a m e a s t h e a m p l i t u d e - f r e q u e n c y r e s p o n s e c h a r a c t e r i s t i c o f a c l a s s ic a l D u f f i n gosci l la tor .

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14 M . R . M . C r e s p o d a S i l v a a n d C . L . Z a r e t z k yFo r equi l ibr ium E2 the s ta te var iab les #e an d CA can be e l imina ted f rom Equ at ions (2 0a-f ) to obta in the fo l lowing two express ions which impl ic i t ly re la te the s teady s ta te in-p lane

bend ing an d t o r s iona l amp l i t udes , e A ~ , and e A T s , t o t he de tun ing and dam ping pa rame te r s ,and to the exc i ta t ion s t rength :[ 2 w v w , ~ ( o 2 + A 2) - k s A 2 ] 2 + [c-r2wT] - [ k 4 A 2 1 2 = 0 (25a)

- k s A 2 , } 2 w Z a 2 A v , 3 2 ] 2k 4 A v ~ + + k 2 A v ~ + k 3 A v e A v e

k lC72WTA27~ ]2+ c ,~ 2 w v A ~ , ~ ] - f ~ 2 3 = 0 " (25b)

Equ i l ibr ium E2 exis t s on ly w hen rea l so lu t ions for both A.~ and A.y~ are obta ined f romEqua t ions (25a , b).Equat ion (25a) d isc loses tha t for E2, the in-p lane bending ampl i tude , Ave , i s dependentonly o n the tors iona l dam ping , e2c72, and on the f re quen cy de tunin g parameters e2(r2 andeZA 2 . That i s , to f ir st o rder , the in-p lane m ot ion i s indepe nden t of both the exc i ta t ion s t rength ,~ 3 f v 3 and o f t he f l exu ra l damping coe f fi c ien t , J C v 2 . Since the exc i ta t ion s t rength has n o e f fec ton t he am p l i tude o f t he d i r ec tl y exc i ted i n -p lane bend ing com ponen t o f t he beam ' s r e sponse ,th is cor resp onds to a sa tura t ion phen om enon , as descr ibed in [11, 13] .

Th e s tab i l ity of the s teady s ta te mo t ion for equi l ibr ium E2 ma y be ascer ta ined by per turb ingthe eq ui lib riu m state as z_(t) = z__ + Z__s(t , w h e r e z__ = [ A v , A T , # , f ] T . By l i nea r i z ingEq uation s (20a, b, e , f ) in the perturb at ion state z__ , the stabi l i ty of the pe rturbe d m otion canbe de te rm ined by a pply ing the R outh-H urwi tz s tab i li ty c r ite r ion to the d if fe ren t ia l equa t ion s?c = e2d2z__s = Az__ . The e l emen t s o f t he 4 x 4 ma t r i x a r e r ead i ly ob t a ined f rom Equa t ions (20a ,b , e , f) . In the process of l inear iz ing Equa t ion (20f) about the equi l ibr ium E2, the fo l low ingequat ion for p , i s a lso obta ined . This provides a condi t ion for the ex is tence of E2 w hen thecon straint [ cos #e [ -< 1 is im pose d.

2WVWT(Cr2 + A 2) - k5A2vcos #e = k 4 A 2 ~ (26)

The s tead y s ta te respon se is de te rm ined by equi l ibr ium E1 outs ide the range whe re E2 exis ts .

5 . R e s u l t s a n d D i s c u s s i o nThe r e su lt s ob t a ined i n t he p r ev ious s ec ti on a r e now p re sen t ed i n the fo rm o f s eve ra l amp l i tude -f r equency r e sponse p lo ts . F o r t h i s, a hom ogene ous can t i l eve r w i th a r ec t angu la r c ro s s s ec t i oni s u sed a s an exam ple i n o rde r t o choose app rop r ia t e va lue s fo r t he nond imen s iona l pa r ame te r s/3v ,/37 an d 3~. Fo r rec tan gular c ross sec t ion beam s the non dim ens ion a l to rs iona l s t i ffness /37ma y be expresse d as [14] :

A G K 3 G 2

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Torsional-Flexural Dyn am ics: Prim ary Reson ance 15T A B L E I I

Va lu es o f k l th ro ugh ks fo r co~, ~ co,b b ~ 16- - 4 8

k l -2 0 1 . 3 2 -2 0 3 . 0 8k2 739 .83 6658 .9k3 -401 .41 -394 .95k 4 1 9 6 8 0 0 2 0 3 8 7 0k s 3 9 2 4 4 0 3 9 7 0 4 0

w h e r e h /b = v / ~ u i s th e r a t io o f t h e c r o s s - s e c t i o n a l h e i g h t to t h e w i d t h o f t h e c r o s s s e c t i o n ,a n d G / E i s th e r a t i o o f t h e s h e a r i n g m o d u l u s t o t h e e x t e n s i o n a l m o d u l u s o f t h e b e a m m a t e r ia l T h e n o r m a l i z e d m a s s m o m e n t o f i n e r ti a 3{ i s d e t e r m i n e d as :

3 e = ] ~ 1 + (28)T o e x e m p l i f y t h e r e s o n a n t m o t i o n f o r b e a m s p o s s e s s in g a o n e - t o - o n e in t e rn a l r e s o n a n c e ,

a b e a m f o r w h i c h t h e t h i r d i n - p la n e f l e x u r a l e i g e n f r e q u e n c y is n e a r th e f u n d a m e n t a l t o r s io n -a l f r e q u e n c y is "d e s i g n e d " a s i n d i c a t e d b e l o w . By e q u a t i n g t h e e x p r e s s i o n s f o r t h e s e tw of r e q u e n c i e s , a s o b t a i n e d f r o m E q u a t i o n s ( 6 d , f) a n d ( 7 ) a s f o ll o w s :

r r ~ hco, = -~ ~ ~Ov = (7 .8 54 8) 2 (29)a n d b y c h o o s i n g a v a l u e f o r G / E a n d f o r h /b = v /-~u , the co r resp ond ing va lues fo r L/b, 3.~a n d 3~ a r e t h e n d e t e r m i n e d b y m a k i n g u s e o f t h e a b o v e e q u a t i o n s . Th e v a l u e G / E = 0 .4 wi l lb e u s e d . Re s u l t s w i l l b e s h o w n f o r b /h = 48 , and fo r b /h = 16. Fo r b /h = 4 8 o n e o b t a i n sL /b ~ 9 an d oar ~ co. ~ 1 .285, w hile for b /h = 16, L/b .~ 9 .16 and cov ~ co. ..~ 3. 85 7 (f orL /b = 9 one ob ta ins coy ~ 3 .857 and co. ~ 3 .787) . The num er ic a l va lues ob ta ined fo r thecons tan ts k l th rough ks a re l i s ted in Tab le I I .

F i g u r e s 2 t h r o u g h 4 d i s p l a y t h e e f f e c t o f v a r y i n g t h e e x c i ta t i o n s t r e n g t h o n t h e f r e q u e n c yr e s p o n s e f o r t h e b e a m w i t h b /h = 4 8 , a n d f o r w h i c h f ~ i s n e a r th e t h ir d i n - p l a n e b e n d i n ge i g e n f r e q u e n c y , coy3, with co~3 n e a r t h e f i rs t f u n d a m e n t a l t o r s i o n f r e q u e n c y . In t h e s e f i g u r e s ,b o t h t h e p l a n a r r e s p o n s e ( e q u i l i b r i u m E l ) a n d t h e c o u p l e d r e s p o n s e ( e q u i l i b r i u m E2 ) a r ep l o t t e d f o r 6 _ 2 c v 2 ~ - - 0 . 0 0 0 8 , ~ 2 c7 2 = 0 .0 1 8 a n d e 2 A 2 = 0 . I t c a n b e o b s e r v e d f r o m F i g u r e 2 ,f o r a n e x c i t a t i o n s t r e n g t h o f e3fv3 = 2 x 1 0 - 5 , t h a t b e l o w a c r i t ic a l v a l u e o f d e t u n i n g o fa p p r o x i m a t e l y 0 .01 o n l y e q u i l i b r i u m E l e x i s ts . A s t h e d r i v i n g f r e q u e n c y is i n c r e a s e d f r o mt h e r e g i o n s l i g h t l y b e l o w t h i s c r i ti c a l v a l u e o f d e t u n i n g , t h e a m p l i t u d e o f t h e b e n d i n g m o t i o nf o l l o w s t h e p l a n a r r e s p o n s e c u r v e u n t il p o i n t A w h e r e e q u i l ib r i u m E1 b e c o m e s u n s ta b l e . A tt h is p o i n t t h e m o t i o n i s g i v e n b y e q u i li b r iu m E 2 , w i t h t h e b e n d i n g a m p l i tu d e j u m p i n g d o w n t op o i n t B a n d t h e t o r si o n a l a m p l i t u d e j u m p i n g f r o m z e r o to p o i n t C . A s t h e d r iv i n g f r e q u e n c y i si n c r e a s e d fu r t h e r , t h e b e n d i n g a m p l i t u d e f o l l o w s t h e s ta b l e b r a n c h B- D t o p o i n t D w h i l e t h ea m p l i t u d e o f t h e t o r si o n a l m o t i o n p r o c e e d s f r o m p o i n t C t o p o i n t E . A s t h e d ri v i n g f r e q u e n c yi s i n c r e a s e d w i t h i n t h e r a n g e w h e r e e q u i l i b r i u m E2 e x i s t s , b o t h t h e b e n d i n g a n d t o r s i o n a la m p l i t u d e s i n c r e a s e e v e n t h o u g h t h e d e t u n i n g i s i n c re a s i n g . A s t h e d e t u n i n g i n c r e a s e s b e y o n dp o i n t D o n t h e b e n d i n g r e s p o n s e c u r v e a n d p o i n t E o n t h e t o r si o n a l r e s p o n s e c u r v e , e q u i l i b r i u m

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Fig. 2 . Am pl i tude- frequency respo nse fo r a bea m wi th b /h = 4 8 and for which ~ : c , 2 = 0 . 0 0 0 8 , ~ 2 c7 2 = 0 . 0 1 8 ,e 2 A 2 = 0 a n d 3fv 3 : 2 x 1 0 - 5 .

0 0 2 5 - - s t a b l e

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F ig . 3. Am pl i tude- frequency respo nse fo r a beam wi th b / h = 4 8 and for which E2cv2 = 0 . 0 0 0 8 , e 2 c7 2 = 0 . 0 1 8 ,c2A 2 = 0 a n d e 3 f v 3 = 1 . 4 5 x 1 0 - 5 .

E 2 c e a s e s t o e x i s t a n d t h e b e n d i n g a m p l i tu d e j u m p s t o p o i n t F w h i l e t h e t or s io n a l a m p l it u d ej u m p s t o z e r o a s t h e d r i v i n g f r e q u e n c y i s in c r e a s e d f u r th e r.

F i g u r e 3 s h o w s t h e e f f e c t o f lo w e r i n g t h e e x c i t a t i o n s t re n g t h to 1 . 4 5 1 0 - 5 . I n t h i s c a s e ,t h e r e g i o n w h e r e e q u i l ib r i u m E 2 e x i st s i s n o w c o n s id e r a b l y r e d u c e d . T h e j u m p f r o m p o i n tA t o p o i n t B s h o w n i n F i g u r e 2 d i s a p p e a r s i n t h i s c a s e . I n s t e a d , t h e b e n d i n g r e s p o n s e c u r v e sf o r t h e e q u i l ib r i u m s o l u t i o n s E 1 a n d E 2 a re e s se n t i a l l y t a n g e n t t o o n e a n o t h e r a t p o i n t A a n dt h e a m p l i t u d e - f r e q u e n c y r e s p o n s e c u r v e f o r t h e t o r s io n a l m o t i o n ( i . e ., t h e c u r v e m a r k e d A 7i n F i g u r e 3 ) i s a c l o s e d l o o p . D e p e n d i n g o n t h e i n it ia l c o n d i t i o n s o f t h e m o t i o n , t h e i n - p l a n eb e n d i n g m o t i o n f o l l o w s e i t h e r b r a n ch A - C o f t h e o p e n l i n e fo r e q u i li b r iu m E 2 o r b r a n ch A - Eo f e q u i l ib r i u m E 1 a s t h e d r i v i n g f r e q u e n c y f ~ i s i n c r e a s e d b e y o n d t h e v a l u e c o r r e s p o n d i n g t op o i n t A .T h e e f f e c t o f a f u r t h e r d e c r e a s e i n t h e v a l u e o f 6 3 f v 3 i s s h o w n i n F i g u r e 4 . T h e b e n d i n gr e s p o n s e c o r r e s p o n d i n g t o e q u il ib r i u m E 2 b e c o m e s d e t a c h e d f r o m t h e r e sp o n s e c u r v e r e p r e -

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Fig. 4 . A m p l i t u d e - fr e q u e n c y r e s p o n se f or a b e a m w i t h b / h = 4 8 a n d f o r w h i c h e 2 c v 2 = 0 . 0 0 0 8 , E 2 c7 2 = 0 . 0 1 8 ,~2A2= 0 a n d e 3 f v 3 = 1 . 2 x 1 0 - 5 .

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Fig. 5. Amplitude-frequencyresponse for a beam w ith b/h = 16 and for wh ich e2cv 2 = 0 .0 02 , c~ -c72 = 0 .0 1 ,e3f~,~ = 1 . 7 5 x 1 0 - 4 a n d E 2 A 2 = 0 .

s e n t i n g t h e b e n d i n g m o t i o n f o r e q u i li b r i u m E l . A f ur th e r d e c r e as e i n th e e x c i ta t i o n s t r e n g thl e a d s t o t h e e v e n t u a l d i s a p p e a r a n c e o f e q u i l ib r i u m E 2 .

T he sens i t iv i t y o f t he f requency re spo nse t o t he in t erna l de t uning e2 A2 i s i l lus t ra t ed inF i g u r e s 5 t h r o u g h 7 f o r a b e a m w i t h b / h = 1 6 . Fo r t hese f i g ures , e 2 c v 2 - 0 . 0 0 2 , e 2 c 7 2 - 0 . 0 1a n d E a f v 3 - - 1 .7 5 1 0 - 4 . F o r J A 2 - - 0 , t h e f r e q u e n c y r e s p o n s e s h o w n i n F i g u re 5 r e se m b l e st ha t s h o w n i n F i g u r e 2 . T h e j u m p c h a r a c t e ri s ti c s a r e id e n t i c a l to t h o s e i n F i g u r e 2 a n d o n l y t h es h a p e o f t h e r e s p o n s e c u r v e s f o r e q u i l i b r i u m E 2 i s s e e n t o b e a f f e c t e d b y t h e d i ff e r en t s h a p ep a r a m e t e r s f o r t h i s b e a m . A s t h e v a l u e o f e 2 A 2 i s i n c r e a s e d t o 0 . 0 2 , h o w e v e r , t h e n a t u r e o ft h e c o u p l e d r e s p o n s e ( e q u i l i b r i u m E 2 ) , s h o w n i n F i g u r e 6 , c h a n g e s . T h e c u r v e s r e p r e s e n t i n gt h e b e n d i n g c o m p o n e n t f o r e q u i li b r i u m E 2 n o w i n te r s e ct th e E 1 p l a n a r r e s p o n s e c u r v e a t fo u rp o i n ts . A j u m p p h e n o m e n o n , w h i c h i s n o w f r o m E l t o E l , c a n b e o b s e r v e d at p o i n t A a s t h eb e n d i n g a m p l i t u d e j u m p s f r o m p o i n t A t o p o i n t B w h e n t h e d e t u n i n g i s in c r e a s e d p a s t t h ev a l u e c o r r e s p o n d i n g t o p o i n t A . J u m p s fr o m E 1 tO E l , a n d f r om E 2 t o E 2 , a r e n o w p r e d i c te d .

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18 M. R. M. Crespo da Silva and C. L Zaretzky0 . 1 2 5 0 . 0 0 8

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F i g . 6 . A m p l i t u d e - f r e q u e n c y r e s p o n s e f o r a b e a m w i th b / h = 1 6 a n d f o r w h i c h e 2 c v2 = 0 . 0 0 2 , e 2 c . y ~ = 0 . 0 1 ,c 3 f~ 3 = 1 . 7 5 x 1 0 - 4 a n d e 2 A 2 = 0 . 0 2 .

0 . 1 2 5 0 . 0 0 8. , l - - s t a b l e. . . . . . . . . . . . . unstableo . 1 ; . , , 0 . 0 0 60 0 7 5 " : ~ : ' "

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0 0 2 5 ~ ~ . ~ 0 . 0 0 20 i f i 0- 0 . 0 2 0 0 1 0 2 0 .1 0 4 0 . 0 6 0 . 0 8 0 . 1 0

E 0 2Fig. 7 . A m p l i t u d e - f r e q u e n c y r e s p o n s e f o r a b e a m w i t h b /h = 1 6 a n d f o r w h i c h e 2 c .2 = 0 . 0 0 2 , e 2 c. r2 = 0 . 0 1 ,e 3 f . 3 = 1 . 7 5 x 1 0 - 4 a n d e 2 A 2 = - 0 . 0 2 5 .

F i g u r e 7 s h o w s t h e e f fe c t o f re d u c i n g 2A 2 to 2A 2 = - 0 . 0 2 5 . T h e a m p l i t u d e - f r e q u e n c yr e s p o n s e c u r v e s n o w s h o w i s o la t e d " is l a n d s ", w h i c h e v e n t u a l ly d i s a p p e a r as 62A2 i s d e c r e a s e dfur ther .T h e e f f ec t o f v a ry i n g t h e o n e - t o -o n e in t e rn a l r e so n an ce d e t u n i n g p a ram e t e r e2 /k 2 o n t h ef r eq u en cy r e sp o n se d i sp l ay e d i n F i g u re s 5 t h ro u g h 7 can b e s een t o b e s i m i l a r t o t h e e f f ec t o fv a ry i n g t h e ex c i t a ti o n s t r en g t h i n F i g u re s 2 t h ro u g h 4 . F o r n eg a t i v e v a l u e s o f t h i s d e t u n i n gp a ram e t e r t h e f r eq u en cy r e sp o n se i s s i m i l a r t o t h a t f o r sm a l l ex c i t a ti o n s t r en g t h s i n t h a t th er e s p o n s e c u r v e s f o r t h e c o u p l e d m o t i o n a r e i s o la t e d a n d s o m e t i m e s f ra g m e n t e d . F o r p o s i t i v ev a l u e s o f E2A2 t h e r e sp o n se i s s i m i l a r to t h a t f o r h i g h e r ex c i t a ti o n s t r en g t h s i n t h a t th e co u p l edr e s p o n s e c u r v e s i n te r s e c t t h e p l a n a r b e n d i n g r e s p o n s e c u r v e w h i c h r e s u l ts i n m u l t i p le ju m p sb e t w e e n t h e c o u p l e d a n d u n c o u p l e d r e s p o n se s . F o r c e r ta i n v a l u es o f e 2 A 2 ( a s i n F i g u re 6 )t h e " c l a s s i ca l " j u m p p o i n t w i t h i n t h e u n co u p l ed i n -p lan e r e sp o n se m ay b e a l t e red so t h a t th e

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T o r s io n a l -F l e x u r a l D y n a m i c s : P r i m a r y R e s o n a n c e 1 90 0 5 . . . . . . 0 0 0 4

- - ~lllble _ ," -. . . . . . . . . . . . . u n s l a b l e . . - - " "

( . . . . . . . . . . . . . . . . . . - -

0 o .o 6o 14 o od02a 0 .0d042 00d0 56 0 .0007 3 f v : l

0 0 4

0 0 3

0 0 2

0 0 1

o .

0.003

0 . 0 0 2

0 . 0 0 1

eA,,e

Fig. 8 . V a r i a t i o n o f t h e r e s p o n s e a m p l i t u d e s w i t h 3fv 3 f o r a b e a m w i t h b / h = 1 6 a n d f o r w h i c h E 2 C v 2 ~ - 0 . 0 0 2 , 2c ,y 2 = 0 . 0 1 , 2 A 2 = 0 . 0 2 a n d 2 c r 2 = 0 . 0 0 5 .

0 . 0 4 0 . 0 0 6

- - s l a b l e - '"

. . . . . . . .. . . . . . . .. . . ; i i i i i i i . . . . . - " f ? . . . .- . . . . . .. . . . . 0 . 0 0 4cA - r e 0 . 0 2 ", A' , : : > e A o ~

o . o l ~ ; ~ ............A A j "" . . . . . . . . . . . . . . . . .

o f " : , ~ , , o0 0 . 0 0 0 1 5 0 . 0 0 0 3 0 0 0 0 4 5 0 . 0 0 0 6

63 v3Fig. 9. Variation of the res ponse amplitudeswith e3f.3 for a beam with b/h = 16 and for wh ich e2c.2 = 0.002. 2C y2 = 0 . 0 1 , 2 A 2 = 0 . 0 2 and 2 0 - 2 = - - 0 . 0 0 5 .

i n - pl a n e b e n d i n g a m p l i t u d e w i l l e x p e r i e n c e a m o r e p r o n o u n c e d j u m p t h a n t h a t p re d i c t e d f ro ma n a n a l y s i s t h a t n e g l e c t s t h e p o s s i b i l it y o f b e n d i n g - t o r s i o n a l c o u p l i n g .

Th e c i r c l e s t h a t a p p e a r i n F i g u r e 5 r e p r e s e n t th e r e s u l ts o f t h e n u m e r i c a l i n t e g r a ti o n o fEq u a t i o n s ( 8 a - c ) w i t h t h e s a m e p a r a m e t e r v a l u e s u s e d t o g e n e r a t e t h e p e r t u r b a t i o n r e s u l t s .Th i s f i g u r e w a s a r b i t r a r i l y c h o s e n t o c o m p a r e t h e r e s u l t s o f t h e n u m e r i c a l i n t e g r a t i o n w i t ht h o s e o b t a i n e d f r o m t h e p e r tu r b a t i o n a n a ly s i s . S i n c e A . = 0 is a n e q u i l i b r i u m s o l u t i o n toEq u a t i o n s ( 2 0 a - f ) , n o n z e r o i n i t i a l c o n d i t i o n s f o r t h e t o r s i o n a l m o t i o n h a d t o b e c h o s e n f o rt h e n u m e r i c a l i n t e g r a t i o n f o r t h e b e a m t o e x h i b i t t h e c o u p l e d r e s p o n s e g i v e n b y e q u i l i b r i u mE2 . O n l y t h e r e s p o n s e c o r r e s p o n d i n g to e q u i l i b r iu m E l i s e x h i b i t e d i f t h e t o rs i o n a l m o t i o n i si d e n t i c a l l y z er o . Th u s , u n l e s s t h e r e i s a n i n it i al c o n d i t i o n i n v o l v i n g t o r s i o n , t h e b e a m ' s s t e a d ys t a te r e s p o n s e w i l l c o n s i s t o n l y o f t h e d i r e c t l y e x c i t e d i n - p l a n e b e n d i n g m o t i o n .

F i g u r e s 8 a n d 9 s h o w t h e e f f e c t o f v a r y i n g t h e e x c i ta t i o n s t r e n g t h , at a c o n s t a n t d r i v i n gf r e q u e n c y , o n t h e b e n d i n g a n d t o r si o n al r e s p o n s e o f a b e a m f o r w h i c h b / h = 16 . Bo th f igu res

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2 0 M . R. M. Crespo da Silva and C. L. Zaretzkysh o w t h i s e f f ec t f o r eaCv2 = 0 . 0 0 2 , e2c72 = 0 .01 and e Z A 2 = 0 . 0 2 . F i g u re 8 , w h i ch w asg en e ra t ed fo r e2 cr2 = 0 . 0 0 5 , d i s c l o se s t h a t a s t h e ex c i t a t i o n s t r en g t h is i n c rea sed f ro m ze ro ,t h e b e n d i n g am p l i t u d e i n c rea se s s t e ad i ly u n t il i t r e ach es a v a l u e co r r e sp o n d i n g to p o i n t A .T h i s p o i n t r ep re sen t s a t r an s i t i o n b e t w een s t ab i l i t y an d i n s t ab i l i t y fo r eq u i l i b r i u m E l . A st h e ex c i t a t i o n s t r en g t h i s i n c rea sed fu r t h e r , t h e b en d i n g am p l i t u d e r em a i n s a t t h e co n s t an tv a l u e c o r r e s p o n d i n g t o p o i n t A w h i l e t h e t o rs i o n a l a m p l i tu d e j u m p s f r o m z e r o t o p o i n t Can d i n c rea se s t o t h e v a l u e co r r e sp o n d i n g to p o i n t E . T h i s i s c le a r l y an ex am p l e o f s a t u ra t i o ns i n ce a t t h e ex c i t a t io n s t r en g t h co r r e sp o n d i n g to p o i n t A , t h e b en d i n g am p l i t u d e " sa t u ra te s "a n d a n y f u r t h e r e n e r g y p u m p e d i n to t h e s y s t e m is t r a n sf e r re d t o t h e t o r si o n a l c o m p o n e n t v iat h e i n t e rn a l o n e - t o -o n e r e so n an ce . A t t h e ex c i t a ti o n s t r en g t h co r r e sp o n d i n g to p o i n t s B an dE o f t h e b e n d i n g a n d t o r s i o n a l c o m p o n e n t s o f e q u i li b r iu m E 2 , r e s p e c ti v e ly , e q u i l ib r i u m E lb eco m es s t ab l e ag a i n w h i l e eq u i l i b r i u m E2 b eco m es u n s t ab l e . T h u s , t h e re i s a j u m p i n t h eb e n d i n g a m p l i tu d e f r o m p o i n t B t o p o i n t D w h i l e th e t o r s io n a l a m p l i tu d e j u m p s d o w n f r o mp o i n t E t o z e ro . F o r h i g h e r v a l u e s o f t h e ex c i t a ti o n s t ren g t h , o n l y eq u i l ib r i u m E l is s t abl e . I t isi n t e r e s t in g t o n o t e t h a t a p o r t i o n o f t h e r e sp o n se cu rv e sh o w n i n F i g u re 8 t h a t co r r e sp o n d s t osa t u ra t io n i s u n s t ab l e fo r t h e v a l u e s o f t h e p a ram e t e r s i n d i ca t ed in t h a t f i g ure . O n l y eq u i l i b r iu mE1 i s s t ab l e i n t ha t reg ion .A n o t h e r j u m p p h e n o m e n o n m a y b e o b s e r v e d in F i g u re 8 . T h i s j u m p , w h i c h o c c u rs w h e nt h e ex c i t a t i o n s t r en g t h is l o w ered b e l o w t h e l ev e l a t w h i ch eq u i l i b r i u m E 2 ex i s t s , co n s i s t s o fa j u m p d o w n o f th e b e n d i n g a m p l i tu d e f r o m p o i n t G t o p o i n t H w h i le t h e t o rs i o n a l a m p l i tu d ej u m p s d o w n f r o m p o i n t F t o a v a l u e o f z er o .

F i g u re 9 sh o w s t h a t fo r a v a l u e o f d e t u n i n g o f 6 2 0 2 ~ - - -0 . 0 0 5 eq u i l ib r i u m E 2 i s u n s t ab l ew i t h in t h e r a n g e o f e x c it a ti o n s t re n g t h s s h o w n . T h e b r a n c h A - C o f t h e b e n d i n g r e s p o n s eco r r e sp o n d i n g to eq u i l i b r i u m E 1 i s a lso u n s t ab l e w h i ch r e su lt s in a j u m p o f t h e b en d i n gam p l i t u d e f ro m p o i n t A t o p o i n t B a s t h e ex c i ta t i o n s t r en g t h is i n c rea sed f ro m ze ro . I f o n l yi n -p l an e b en d i n g h ad b een co n s i d e red i n t h e r e sp o n se an a l y s i s , i . e . , i f t h e p o s s i b i l i t y o fb en d i n g - t o r s i o n co u p l i n g h ad b een n eg l ec t ed a p r io r i , t h e r e sp o n se cu rv e o b t a i n ed fo r th ep l a n a r b e n d i n g a m p l i tu d e w o u l d r e s e m b l e t h a t o f t h e b e n d i n g m o t i o n g i v en b y e q u i l ib r i u mE 1 w i t h t h e ex cep t i o n t h a t s ec ti o n A -C o f t h e r e sp o n se cu rv e w o u l d b e s tab l e. I n t h a t c a se ,t h e j u m p u p o f t h e b e n d i n g a m p l i tu d e w o u l d o c c u r a t p o i n t A ', r a t h e r th a n a t p o i n t A , a st h e ex c i t a t io n s t r en g t h w as i n c rea sed . T h u s , w h i l e eq u i l i b r i u m E z i s u n s t ab l e t h ro u g h o u t t h ereg i o n o f ex c i t a t io n s t r en g t h s sh o w n i n F i g u re 9 , t h e e f f ec t o f t h e n o n l i n ea r b en d i n g - t o r s i o nco u p l i n g s t il l r e f le c t s i ts e l f o n t h e j u m p p h en o m en o n fo r t h e i n -p l an e r e sp o n se .

6. SummaryT h e f l ex u ra l- f l ex u ra l- t o r s io n a l r e sp o n se o f b eam s , i n c l u d i n g t h e e f f ec t o f t o r s i o n a l d y n am i cs ,w as i n v es t i g a t ed b y t ak i n g i n t o acco u n t a ll t h e g eo m e t r i c n o n l in ea r i t ie s in t h e d i f f e r en t i a le q u a t i o n s o f m o t io n . T h e b e a m w a s s u b j e c te d t o a n i n - p l a n e re s o n a n t b e n d i n g e x c it a ti o n i nt h e p r e s e n c e o f a o n e - t o - o n e in t e rn a l r e s o n a n c e b e t w e e n a n i n -p l a n e b e n d i n g f r e q u e n c y a n da t o r s i o n a l fr eq u en cy . T h e r e sp o n se i n v es t i g a t ed h e re co n s i s t s o f s in g l e m o d e ap p ro x i m a t i o nf o r b e n d i n g a n d f o r t o r s io n , w h i c h i s v a li d w h e n t h e re i s n o c o m m e n s u r a b i l it y b e t w e e n t h ef r eq u en c i e s o f o t h e r m o d es . A n a l y t i c a l ex p re s s i o n s fo r th e s t e ad y s t a t e r e sp o n se ch a rac t e r i s ti c sw e re o b t a i n ed b y a p e r t u rb a t i o n an a l y s i s . N u m e r i ca l i n t eg ra t io n o f th e g o v e rn i n g d i f f e r en t ia leq u a t i o n s w as a l so p e r fo rm ed i n o rd e r t o v e r i fy t h e accu racy o f t h e p e r t u rb a t i o n an a ly s i s .

T h e s en s i t iv i t y o f th e f r eq u en cy r e s p o n se t o b o t h t h e ex c i t a t i o n s t ren g t h , an d t o t h e i n t e rn a lan d ex t e rn a l r e so n an c e d e t u n i n g s , h a s b een an a l y zed i n d e ta i l. T h e r e su l t s o b t a i n ed h e re sh o w

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T o r s i o n a l - F l e x u r a l D y n a m i cs ." P r i m a r y R e s o n a n c e 2 1t h a t th e r e sp o n se o f t h e b ea m ex h i b i ts v a r i o u s fo rm s d u e t o t h e d i f f e r en t n o n l i n ea r i t i e s in t h ee q u a t i o n s , e a c h h a v i n g a s s o c i a te d w i th i t a u n iq u e s e t o f j u m p p h e n o m e n a b e tw e e n t h e c o u p l e dan d u n co u p l ed s t e ad y s t a te m o t i o n s . A l so , w i t h i n ce r t a i n r eg i o n s o f th e ex c i t a t i o n s t r en g t h ,t h e i n - p l an e b e n d i n g c o m p o n e n t o f th e c o u p l e d r e s p o n s e s a tu r a te s s o t h a t a n y f u r t h e r e n e r g y" p u m p e d " i n t o t h e sy s t em i s t r an s fe r r ed t o t h e to r s i o n a l m o t i o n " v i a " t h e i n t e rn a l r e so n an ce .W h e n d e s i g n i n g s t r u ct u ra l s y s t e m s , t h e e n g i n e e r s h o u l d b e a w a r e o f s u c h p h e n o m e n a , w h i c ha re cau sed b y t h e n o n l i n ea r i t i e s t h a t a r e a l w ay s p re s en t i n th e d i f f e r en t ia l eq u a t i o n s o f m o t i o nfo r t h e s t ru c tu re .AcknowledgementT h e a u t h o r s a r e i n d e b t e d t o t h e r e v i e w e r s f o r th e i r c o n s t ru c t i v e c o m m e n t s , w h i c h h e l p e di m p ro v e t h e q u a l i t y o f t h e p re sen t a t i o n o f th i s w o rk .Appendix AT h e G a l e rk i n co e f f i c i en ts fi rs t ap p ea r i n g i n E q u a t i o n s (8 a -c ) a r e d e f i n ed a s fo l lo w s :

1

c t~ l = J [ -/ 3 7 F ~ ( / r ;F '+ : ) ' + ( l - J v ) F ~ ( F , ~ F T ) " ] a s0

(A 1 )

1

~ v ~ - ( 1 / 3 . ) i F ~ ( F ; ' 2 , ,- F 4 ) a .0

( A 2 )

/ { [ / 1= F ~ , F ; d sv 3 ( 1 - 9 y ) L F ; ~ ' ' "0 0l I t 2 l l l I l l l- / 3 y F ~ ( F D F ; ; ) - F , , ( F D F ; F ; ) } d s

1i # # l / I fC~v4 = - J y F ~ [ F D ( F v F ) ] d s0

Olv 5 1 [ F v t t2= - - F ~ d s d sJ20 1 0

d s

( i i )i~v6 = - 7 F ~ F ~ F , ~ d s d s0 1 0

d s

1

= i [ 9 . . . ( . ; / ' / ' + ( , - . .0

(a3)

(A 4 )

(A5)

( A 6 )

(A 7 )

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22 M . R. M. Crespo da Silva and C. L. Zaretzky1/ ~ (p t t I~2]t t

0( A 8 )

0 : w 3 1 { , i s ] ,= - o f 1 - f y ) F ~ ,t f p , p ,t- ~o- v ds0t l / 2 /+ F w ( F ; F ; ) + f vF w ( F ,'F ; F ~ v" ) } d s ( A g)

1

= - f [ F~ ,( F;F ~ ) ': w 4 F w i ' " ' d s0 ( A 1 0 )

' /w , : - ~ F w F ~ d s d s0 1 0 d s ( A l l )0 : ~ 6

1 /= - ~ F ,~ F "2 d , a s0 1 0

d s ( A 12)

O: 71 - -

1

( 1 - f y ) f 2 "2~ F v d s3~ 0( A 13)

0 : . 2 - -

1

3 j w0

( A 14)

0 5 . 3 - -

1(1 - ~ y) f ~ ~,,~, ,3~ J * . * w * , o d s0 ( A 15)1 $

0 0( A 16)

1

0 :., = f F .F ' F " d s0

( A I 7 )

Appendix BThe fo l lowing are so me of the k i coef f ic ient s f ir st appear ing in Equat ions (17a-c) . Th e o thersare def ined in the t ex t. No te tha t, to the o rder of the equat ions in w hich these coef f ic ient s

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Tors iona l -F lexu r a I D ynam ics: P r im ary R esonan ce 2 3appear, wv = w-r in the equa t ions below .

k6 = 4COw(COw + 2COy)- 4 - O g v l [ 0 5 T 5 G U W ( G U W - 0 . 3 /) ) " t - O ~ , ),4 (C O - - C U r ) 2 - - 0 9 7 3 A t - O { V (B1)

4ww(ww 2 w v ) 23 2k 7 = ~O~w4 -- CtwSa)w (B2)

O g W , [ O L " /, W t O ( O - J W " { - ~ t )V ) + O g ' , /4 C O Z y - ~ - C O y ) 2 - - O g 7 34 w ~ ( w ~ + 2 w ~ )

+ - O v ) + - _ + 0 3 3 )4 c o w ( c o w - 2 C O v ) 2k 9 - ~ w 2 ~ vlO ~ wl ( B 4 )2 2 ( w 2 - 4 w 2 )

2 0~.,/3 ) 2 Ct23+ ( B 5 )4 C O w (C O w - 2 co ~ ) 4cow(cow + 2 c o y ) 2R e f e r e n c e s

1. Haight, E. C. and King, W. W., 'Stabil i ty of nonlinear oscillations of an elastic ro d' , Journal o f the AcousticalSociety of Am erica 52, 1971, 899-911.2. Cresp o da Si lva , M. R. M. and Glynn , C. C. , 'Nonl inear flexura l- f lexural - torsional dynam ics of inex tension albeams. I : Equa t ions o f mo t ion ' , Journal o f Structural Mechanics 6, 1978, 437--448.3. Cre spo da Silva, M R. M. and Gly nn, C. C., 'Non linear flexural-flexural-torsional dyn am ics o f inextensional

beams. I I : Fo rced mot ions ' , Journal o f Structural Mechanics 6, 1978, 449-461.4. Cres po da Silva, M. R. M. , 'Equat ions for nonl inear analysis o f 3D mot ions of beam s' , Applied MechanicsReviews 44, 1991, 51-59.5. Cresp o da Silva, M . R. M. and Zaretzky, C. L . , 'No nl inear modal coupl ing in planar and non-plan ar responsesof inextensional bea ms ' , International Journal of Non-L inear Mecha nics 25, 1990, 227-239.6. Nay feh, A. H. , M ook, D. T ., and Sr idhar , S. 'Nonl inear analysis of the forced response o f s t ruc tura l e leme nts ' ,Journal of the Acoustical Society of America 55, 1974, 281-291.7. Na yfeh, A. H. , M ook, D. T ., and Lobi tz , D. W., 'Num erica l per turbation method for the nonl inear analysisof structural vib ration s' , AIAA Journal 12, 1974, 1222-1228.8. Haddow , A. G., Barr, A. D. S., and Moo k, D. T. 'Theoretica l and experimental study of mo dal interaction ina two-degree -o f - f r eedom s t ruc tu re ', Journal of Sound and Vibration 97, 1984, 451-473.9. Cre spo da Silva, M. R. M., 'No nline ar flexural-flexural-torsional-extensional dy nam ics of beams. I. Fo rm u-la t ion ' , International Journa l o f Solids and Structures 24, 1988, 1225-1234.10. Cr esp od a Silva, M. R. M. , 'No nl inear flexura l- f lexural -torsional -extensional dyna mics of beams. I I : Resp onseana lys i s ' , International Journa l o f Solids and Structures 24, 1988, 1235-1242.11. Nay feh, A. H. and M ook, D. T . , Nonlinear Oscillations, Wiley-Interscience, Ne w York, NY, 1979.12. Zare tzky, C. L. and Cresp o da Silva, M. R. M., 'N onlin ear flexural-flexural-torsional interactions in beam sinclu ding the effects of torsional dynamics. 11: Com bination reso nan ce' , Nonlinear Dynamics 5, 1994 (forth-coming) .13. Nay feh, A. H. , M ook, D. T ., and Marshal l, L . R., 'Non l inear coupl ing of pi tch and rol l modes in ship mot io ns ' ,Journal of Hydronautics 7, 1973, 145-152.14. Ro ark, R. J. and Young, W. C., Formulas fo r Stress and Strain, McG raw-Hi l l, New York, NY, 1975 .