Gear Rattling Noise Paper

8/8/2019 Gear Rattling Noise Paper

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368 K . K A R A G I A N N I S A N D F . P F E I F F E R( n ) . S uc h a de s c r i p t i on c a n be d e r i ve d a na l y t i c al l y r e s u l ti ng i n a s e t o f d i s c r e t e m a pp i ngs w h i c hg i ve a good ba si s f o r m o r e s t r uc t u r a l i nve s t iga t i ons o f r a t tl i ng [ 13] . M o r e ov e r , l ook i ng a t t her e s u l t s f o r r a t t l i ng p r oc e s s e s a nd t he i r r e s e m b l a nc e t o s t oc ha s t i c a ppe a r a nc e i t m a ke s s e ns e t oi n t r oduc e s om e m e a n va l ue s t o de s c r i be r a t t l i ng p r ope r t i e s .

T he r e e x i s t s s om e l i t e r a t u r e i n r e l a t e d f i e l d s , f r om w h i c h a s m a l l s e l e c t i on i s g i ve n .T he o r e t i c a l a s pe c t s a r e d i s c us s e d i n t he w e l l - know n books [ 1 , 7 ] , w he r e a s [ 8 ] g i ve s m a nyi n t e r es t i n g e x p e r i m e n t a l e x a m p l e s o f c h a o t i c v i b ra t io n s . T h e v e r y c o m p l e t e a n d f u n d a m e n t a l b o o kon dyna m i c s [ 9 ] i s one o f t he f i r s t one t o p r e s e n t i m pu l s i ve p r oc e s s e s m or e t ho r ough l y . G e ne r a lproblems of impac t sys tems a re d i scussed in [3 , 4 , 6 , 11 , 16-20] , whe re [6 , 19 , 20] i s somehowr e l a t e d t o t h e p r ob l e m s c on ne c t e d w i th r a t tl i ng . T he pa pe r [ 12 ] w a s t he s ta r t ing po i n t f o r t her a t t l i ng i nve s t i ga t i ons a s p r e s e n t e d he r e , i t e s t a b l i s he s a L a g r a ng i a n a pp r oa c h t o i m pu l s i vep r o c e s se s a n d a p p li e s it t o s o m e t o y w o o d p e c k e r d y n a m i c s , w h i c h r e s e m b l e s v e r y m u c h a r a t t li n gp r oc e s s . T he s e i de a s ha ve b e e n e x t e n de d i n t w o d i s s e r ta t i ons [ 2 , 15] , w h e r e t he f ir s t one [2 ]i nve s t i ga t es r a t tl i ng i n a ve r y f und a m e n t a l a nd c om pr e he ns i ve w a y a nd t he s e c o nd on e [ 15 ] g i vesn u m e r i c a l m e t h o d s t o d e t e r m i n e f ix p o i n t s o f n o n l i n e a r v i b ra t io n s a n d t o e v a l u a t e t h e i r p r o p e r ti e s .

2 . G e a r B o x M o d e l

2.1. Equations of MotionT h e p hys i c a l m o de l f o r t he i nve s t i ga t ions is d i sp l a ye d i n F i gu r e 1 . A g e a r m e c ha n i s m w i t h n s t a ge si s c o n s i d e r e d t a k i n g i n t o a c c o u n t o n l y r o t a t i o n a l m o t i o n s . T h e b o d i e s a r e e n u m e r a t e d i n s u c h a

~ s r i v i n. gh a f t r ~ .... D r i v e n S h a f t

I

P

@ @ @I

C o u n t e r s h a f t

F i g . 1 . M o d e l o f n s ta g e g e a r m e c h a n i s m .


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INVESTIGATIONS OF GEAR -RATTL ING 369w a y t h a t t h e c o u n t e r s h a f t c o r r e s p o n d s t o b o d y n u m b e r 1 a n d t h e l o o s e g e a r w h e e l s o n t h e o u t p u ts h a f t t o b o d i e s 2 . . . n .

I f t he re was no exc i t i ng f l uc tua t i on i n t he i npu t sha f t angu la r ve loc i t y the ge a r whee l s wo u ldr o t a t e w i t h t h e i r r e s p e c t i v e n o m i n a l c o n s t a n t s p e e d s . T h e r e f o r e i t is o b v i o u s l y u s e f u l to d e s c r i b et h e s y s t e m s t a t e u s i n g t h e d e v i a t i o n s f r o m t h e s e n o m i n a l v e l o c i t i e s . M a t h e m a t i c a l l y w e u s e t h ea n g u l a r c o o r d i n a t e s

, , = [ , . . . . .

a s g e n e r a l i z e d c o o r d i n a t e s . A c c o r d i n g t o th e a b o v e m e n t i o n e d s u b d i v is i o n o f t h e m o t i o n w ed i s t i n g u i s h i n t o f r e e f l i g h t w h e r e t h e w h e e l s d o n ' t h a v e a n y c o n t a c t a n d t h e i m p a c t p h a s e w h e no n e o r m o r e c o n s t r a i n t s a r e h i t. D u r i n g f r e e f li gh t o n l y t h e t o r q u e s t r a n s m i t t e d b y th e g e a rl u b r i c a n t o i l a p p e a r . T h e y a r e m o d e l l e d a s l i n e a r d a m p i n g a n d c o n s t a n t m o m e n t s . F r o m t h i s t h ee q u a t i o n s o f m o t i o n w r i t e

l~b + D~b + T, = 0 (1 )i f s j ~ ( - v j , 0 ) , j = 1 . . . . n , w i th

I = d i ag {I l , 12 . . . . I ,,} C R ..... t he ma t r i x o f t he m om en t s o f i ne r t i a ,D = d ia g {d ~ , d 2 , . . . , d ,,} e ~ " " t h e d a m p i n g m a t r ix ,T , . = { T r E , T ry _ . . . . T , ., ,} E ~ " t h e v e c t o r o f c o n s t a n t m o m e n t s ,v j t h e b a c k l a s h w i t h i n t h e j - t h g e a r m e s h ( g i v e n m a g n i t u d e ) ,s i t he re l a t i ve d i s t anc e i n t he j - t h ge a r mesh .T h e v e c t o r o f r e l a t iv e d i s t a n c e s i n t h e b a c k l a s h e s S = [ s~ , s 2 . . . . s , ] T E R *' i n t he fo l l owing

w i ll b e c a l l e d t h e v e c t o r o f p l a y c o o r d i n a t e s . T h e s e c o o r d i n a t e s w ill h a v e v a l u e s i n th e r a n g e s( - v , , 0 ) , j = 1 . . . . n . T h e y a r e r e l a t e d t o t h e a n g le c o o r d i n a t e s b y t h e r e g u l a r tr a n s f o r m a t i o n

S = Y~p + ve (2)w h e r e t h e r e g u l a r m a t r i x Y E [R ... a n d t h e v e c t o r v E R " d e p e n d o n t h e g e o m e t r i c p r o p e r t i e s o f th eg e a r b o x a n d e ( t ) = r e % ( t ) rep r ese n t s t he e xc i t a t i on o f t he i npu t gea r . r e i s t he b as i c rad iu s o f t hei n p u t g e a r w h e e l a n d % is t h e f l u c tu a t i o n o f t h e m o t i o n o f t h e i n p u t s ha f t. M e a s u r e m e n t s o f th ee x c i t a t i o n ~ , , s h o w e d t h a t i t m a y b e s u f f i c i e n t l y a p p r o x i m a t e d b y a s i n g l e h a r m o n i c f u n c t i o n% ( 0 = A s i n ( w t ) w i t h a m p l i t u d e A o f th e f l u c t u a ti o n s o f t h e a n g u l a r m o t i o n .

T h e s o l u t io n o f th e e q u a t i o n s o f m o t i o n ( 1 ) i s ( t ) : ( t o ) + B - ' [ E - e x p [ - B ( t - t o ) l ] ( ( ~ ( t o ) + B - ' c ) - B - ' e ( t - to) ( 3 )~ b (t ) = e x p [ - B ( t - t o ) ] ( ( t o ) + B - ~ c ) - B - L c ( 4 )

i f s j ~ ( v j , 0 ) , j = 1 . . . . , n wi th t he abb rev i a t i ons B = I -~ D and c = I -~ Tr . E i s t he i den t i t y ma t r i x ,E E ~ " . T h e c o n t a c t p h a s e is m o d e l l e d a s p a r t l y e la s t ic i m p a c t s i n c e t h e g e a r w h e e l s a r e m a d e o fh a r d e n e d s t e e l a n d t h e l o a d i s v e r y l o w . U s i n g i m p a c t t h e o r y o n e g e t s t r a n s i t i o n e q u a t i o n s f r o m


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370 K. KARAGIA NNIS AND F. PFEIFFERt he s y s t e m s t a t e be f o r e a n i m pa c t t o t he s t a t e a f t e r i t . F o r ba c k l a s h i t he t r a ns i t i on e qua t i ons a r e

t + = t -+

' - ' s pe c i f y i ng va l ue s j u s t be f o r e a nd " + ' s pe c i fy i ng t hos e j u s t a f t e r t he i m pa c t . U , E ~ ... . a nd~ i ~ ~ " a r e t he t r a ns i t i on m a t r i x a nd a t r ans i t i on ve c t o r , r e spe c t i ve l y , t ha t de p e nd on t he g e om e t r yo f t he ge a r box , t he c oe f f i c i e n t o f r e s t i t u t i on a nd t he i m pa c t i ng pa i r o f ge a r w he e l s . I n o r de r t oa c h i ev e a m o r e e v i d e n t m a t h e m a t i c a l f o r m t h e e q u a t i o n s o f m o t i o n a r e t r a n s f o r m e d i n t o t h e p l ayc oo r d i na t e s :

+ Y B y - I s = vY + Y B y - l v e - Y c (6 )i f s j E ( - v j , 0 ) , j = 1 . . . . n an d

S + = Y U / Y ~S + ( E - Y U i Y - t ) v e (7 )i f s i E { - v i , O } , s j E ( - v ~ , O ) , j = l , . . . , n , j # i .

T h e s y s te m o f t h e e q u a t i o n s ( 1 , 5 ) o r ( 6 , 7 ) m a y b e c a ll e d n o n - h o l o n o m i c , b e ca u s e t h e c o n s t r ai n tdu r i ng i m pa c t a ppe a r s i n t he f o r m o f a ve l oc i ty c ond i t i on ( e qua t i ons ( 5 ) o r ( 7 ) ) . T h e v i b r a t i onspos s e ss a s e l f - e xc it i ng c ha r a c t e r due t o s om e t yp i c al m e c ha n i s m s f o r t h is k i nd o f m o t i on . T heexc i ta t ion e ( t ) r e p r e s e n t s t h e e n e r g y s o u r c e , w h i c h t r a n s f e r s e n e r g y t o t h e d r i v e n w h e e l b y t h es w i t c h i ng be ha v i ou r o f t he i m pa c t i t s e l f . T he e ne r gy i s s t o r e d i n t he d r i ve n w he e l i n t he f o r m o fk i ne t i c e ne r gy a n d d i s s i pa t ed by o i l d r a g a nd b y i m pa c t l o s s es . T yp i c a l l y , a ba l a nc e o f t r a ns f e r r e de ne r gy a nd d i s s i pa ti on r e s u l t s i n a pe r i od i c a l s o l u t ion r e p r e s e n t e d b y a s t a b l e l im i t c yc l e , a t l e a s t int he pe r i od i c c a s e .2 . 2 . D i s c r e t e M o d e l

F r om e qua t i on s ( 6 ) a nd ( 7 ) i t i s obv i ous t ha t t he t i m e s hou l d be i n t r odu c e d i n to t he s y s t e m st a t es p a c e in o r d e r t o h a v e t h e d y n a m i c a l b e h a v i o u r o f th e s y s t e m c o m p l e t e l y d e s c r ib e d . N o wq = It , T ~ r ] T ~ [ 0, ~ ) ~ 2n s ta nds f o r t he s t a t e ve c t o r o f t he a ng l e c oo r d i na t e s w h e r e a sz = [t , S r , s T ] T @ [0 , ~ ) ~2n i s the s ta te vec tor in p lay coord ina te s . As the exc i ta t ion i s pe r iodict h e v e c t o r f i e l d o f t h e d y n a m i c a l s y s t e m h a s t h e s a m e p e r i o d a n d t h e r e f o r e t h e t i m e t m a y b eno r m e d w i t h i n t he i n t e r va l o f t he e xc i t a t i on pe r i od .

T he m o t i on i s e xa c t l y de f i ne d i f t he s e que nc e o f s y s t e m s t a t e s be f o r e o r a f t e r i m pa c t s i skno w n , I n a dd i t i on , t he t r a j e c t o r i e s be t w e e n i m pa c t s don ' t ha ve a ny i n f lue nc e on t he no i s ee m i s s i on t ha t i s c a us e d on l y by t he i m pa c t s t he m s e l ve s . W i t h r e ga r d t o t he s e c ons i de r a t i ons i t i sp r e f e r a b l e t o u s e a t i m e d i s c r e t e de s c r i p t i on i n s t e a d o f t he t i m e c on t i nuou s one . T h i s is ea s i ly doneby t he de f i n i t ion o f po i n t m a pp i ngs o n t o P o i nc a r ~ s ec t i ons (P . S . ). F o r t he s y s t e m s t a te s be f o r ei m pa c t s t he P .S . is ( v l og ic a l O R , A l og ic a l A N D ) :


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I N V ES TI G A TI O N S O F G EA R - R A TTLI N G 3 7 1

E : { q r s p . z ) / ( s t = - v ~ A ~ < 0 ) v ( S 1 : 0 A S 1 > 0 )v . . . v ( s , : - v /~ ,,O)}

w h e r e a s f o r t h e s t a t e s a f t e r i m p a c t s i t i s d e f i n e d a s+

= {q r s p . z ) / ( s I = - v , A ~ , - -> 0 ) V ( S , = 0 A ~ , - -< 0 )v ... v ( s , = - v , ,

T w o m a p p i n g s a r e d e f i n e d : t h e m a p p i n g d e s c r i b i n g t h e i m p a c t

F - 2 - + ~ ` E ~ 2"+1

i s d e f i n e d b y e q u a t i o n ( 5 ) a n d+

F , : Z - - , Z , : ' '+ 'r e p r e s e n t s t h e f r e e f l ig h t. T h e l a t te r o n e r e s u l t s f ro m t h e s o l u ti o n o f th e e q u a t i o n s o f m o t i o nd u r i n g f r e e f l ig h t a n d a n a d d i t i o n a l e q u a t i o n d e t e r m i n i n g t h e p o s i t io n a t t h e n e x t i m p a c t . N o w t h ew h o l e m o t i o n m a y b e w r i t t e n i n t h e f o r m

+ q~ ---~ q~ ~ q~+ l --~ qi+ ~ -~ q ,+~ T h e s t a b il it y o f t h e p r o c e s s m a y b e j u d g e d b y th e f u n c t i o n a l m a t r i c e s o f t h e m a p p i n g s . T h ef u n c t i o n a l m a t r i x o f t h e w h o l e m o t i o n is a c h i e v e d b y s e q u e n t i a l m u l t ip l i c a ti o n o f t h e m a t r i c e sc o r r e s p o n d i n g t o th e s e p a r a t e t r a n si t io n s . T h e f o r m o f t h e f u n c ti o n a l m a t r i c e s f o r t h e k - t h i m p a c ta n d t h e f o l l o w i n g f r e e f li g h t, r e s p e c t i v e l y , i s

O q ~ c ) q ; + ,M , , - 0 q 7 , M f ~ - 0 q~

I t i s i m p r a c t i b l e t o j u d g e o n t h e s t a b i l i t y o f p e r i o d i c s o l u t i o n s b y t h e i n v e s t i g a t i o n o f e i g e n v a l u e so f th e f u n c t i o n a l m a t r i x i n t h e n e a r o f fi x p o i n t s s i n c e t h e s e p o i n t s , i n g e n e r a l , c a n n o t b ed e t e r m i n e d a n a l y t i c a l ly . I n t h is c a s e t h e m o t i o n i s c a l c u l a t e d b y n u m e r i c a l e v a l u a t i o n o f t h em a p p i n g s s o th a t t h e s t a b il it y o f p e r i o d i c s o l u ti o n s b e c o m e s o b v i o u s . N e v e r t h e l e s s , w e d e t e r m i n et h e g r e a t e s t L j a p u n o v e x p o n e n t er a s a m e a s u r e f o r t h e r e g u l a ri t y o f t h e m o t i o n :

( 8 )w i t h

w,, = l~[ M I~M ., w , .k = I


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372 K . K A R A G I A N N I S A N D F . P F E IF F E R3 . Resul t s

3.1. Results for the One-Stage Mo delWith in sh i f t t r ansmiss ions the gea r whee l s no t unde r load show in many cases a one - s tage r a t t l ingb e h a v i o u r , b e c a u s e t h e mo t i o n i s a p p r o x i ma t e l y r e a c t i o n l e s s w i t h r e s p e c t t o t h e d r i v i n g wh e e l .M o r e o v e r , t h e o n e d e g r e e o f f r e e d o m s imp l i ci ty gi v es a g o o d i n s ig h t in t o t h e s t r u c t u r e o f t h ev ib ra t ions and the pa ramete r in f luences . F igu re 2 p resen t s such a s ing le s t age mode l .

Th e p a r a m e t e r s p a c e o f t h e o n e - s ta g e g e a r h a s s ma l l d i me n s i o n . Ev e n m o r e , u s i n g s imi l ar it yr u l e s it c an b e f u r t h e r r e d u c e d b y t h r e e . Th e e q u a t i o n s o f mo t i o n f o r t h e n o r m a l i z e d p l a yc o o r d i n a t e s a r e

g + / 3 s = 6 + 1 3 d + T if s ~ ( - 1 , O) l. +s - e s - i f s E { - 1 , 0 } ] (9 )

w i th e = c ~ s i n ( r ) w h e r e s=s/v , r=o~t , [3=d/ ( Io) ) , T=(Trr ) / ( Iw2v ) , a : ( r e A ) / v a re then o r ma l i z e d p l a y c o o r d i n a t e s t h e ms e l v e s , t h e t i me , d a mp i n g , c o n s t a n t mo me n t s a n d a mp l i t u d e s ,r e spec t ive ly . Der iva t ives in equa t ion (9 ) a re wi th r ega rd to the no rma l ized t ime r .

As m en t ion ed a bov e the vec to r z = ( r , s , s ) r E [0 , 27 r )x ~2 i s u sed wi th in the desc r ip t ion o ft h e d i s c re t e s y s te m, Th e ma p p i n g F~ o f t h e k - t h i mp a c t i s g i v e n b y

T k - - T k = 0+S k -- S k

. + . -

S k -- f f S k(10)

whereas fo r the k - th f r ee f l igh t phase the appropr ia te mapp ing Ffk i s

- Ttm~

m e s h p l a n e . /

e ( t) = - a s i n ( ~ t )

g+,F ig . 2 . M o d e l o f o n e - s t ag e r a t t l i n g .


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I N V E S T IG A T I O N S O F G E A R - R A T T L I N G 373_ 1--Sk+ t + Sk + c~(s in(r2+i ) - - s in O 2) ) + ~ (1 - exp ( - /30 -2+ l - ~+ ) ) )

( ~ ~ )' ~ "s ~ - o~ e o s O ' ~ ) - ~ ( ~ '~ . ~ - r ~ ) - - o~ c o s ( T ~ ; + , ) + S k - - c ~C O S O -~ .)-- " / e x p ( - 1 3 0 - 2 + , - ~ - 2 ) )

- s / , + l = 0+S [ + , - - S k E { - - 1 , 1 , 0 }

( l l )

T h e g r e a t e s t L j a p u n o v e x p o n e n t c a n b e c a l c u l a t e d b y e q u a t i o n ( 8 ) . A l t h o u g h t h e m a p p i n g sa l t o g e t h e r a r e e x t r e m e l y c o m p l e x t h e s u m o f t h e L j a p u n o v e x p o n e n t s m a y b e c a l c u l a t ed s i nc e t h ed e t e r m i n a n t o f t h e f u n c t i o n a l m a t r ic e s c a n b e g i v e n a n a l y ti c al ly . T h e d e t e r m i n a n t o f t h e m a p p i n gf r om be f o r e i m p a c t ( k ) t o t he s t a t e be f o r e i m p a c t ( k + 1 ) is

de t(Mt~M,~) : ._ ex p( - f i0 -~ + 1 - ~-~)) . . Sk+l

C o n s e q u e n t l y , t h e d e t e r m i n a n t o f t h e m a p p i n g f r o m b e f o r e t h e f i rs t t o b e f o r e t h e n - t h i m p a c tr e s u lt s f r o m t h e m u l t ip l ic a ti o n o f t h e d e t e r m i n a n t s o f e l e m e n t a r y m a p p i n g s.

d e t M r k M , . = d e t ( M r M ~ ) = e ' - " . ~ exp - /3 (~-~+1 r k) . (12)I k=l " " Sn+l k=l "

U s i n g t h e s e e q u a t i o n s t h e s u m o f t h e e x p o n e n t s c a n b e c a l c u l a t e d :" ) f~ ' + -~ + -3 = lim.-~ -nl In Ide t ( k[l~ ])~M~k= = 2 ln( e) - / 3 ( r / + 1 - ~-~ ) . (13 )

O ne o f t he e xp one n t s is ide n t i c a l ly z e r o a s i t c o r r e s pon ds t o t he de f i ne d m a pp i ng w i t h i n t he P .S .T he s um o f t he o t he r t w o i s ne ga t i ve w h i c h i s no t a s u r p r i s i ng r e s u l t s i nc e t he m a pp i ngs ha ve ad i s s ipa t ive c ha r a c t e r r e s u l ti ng i n a sh r i nk i ng pha s e s pa c e . N o t e , t ha t i n e qua t i o n ( 13 ) on l yd i s s ipa t ive t e r m s a ppe a r , i . e. t he c oe f f ic i e n t o f r e st i t u t ion s a nd t he no r m a l i z e d da m p i ngm ul t i p l i e d by t he m e a n f l i gh t t i m e w h i c h i s a m e a s u r e f o r t he m e a n s h r i nka ge du r i ng t he f l i gh tpha s e .

T h e n u m e r i c a l e v a l u a ti o n o f t h e m a p p i n g s a n d t h e c a l cu l a ti o n o f t h e g r e a t e s t e x p o n e n t h a sbe e n pe r f o r m e d on a d i g i t a l c om pu t e r t he r e s u l t s be i ng s how n be l ow . I n F i gu r e 3 ( a ) a b i f u r c a t i ond i a g r a m i s d i s p l a ye d w he r e t he pa r a m e t e r ~ ( a m p l i t ude o f e xc i t a t i on / ba c k l a s h ) is va r i e d . ( I n o r de rt o g e t a s t a t i ona r y s i t ua t i on t he m a pp i ng is pe r f o r m e d 5000 ti m e s o f w h i c h the l a s t 300 r e su l t s a r ed i s p l a ye d ) . F i gu r e ( 3a ) s how s t he d i a g r a m f o r t he r e l a t i ve ve l oc i t i e s w i t h i n t he ba c k l a s h , F i gu r e( 3b ) i s the r e s p e c t i ve d i a g r a m f o r t he t i m e p o i n t s r - i n the r a nge o f (0 , 27 r] a nd F i gu r e ( 3c ) s how st h e g r e a t e s t L j a p u n o v e x p o n e n t . T h e q u a li ta t iv e p r o p e r t i e s o f th e m o t i o n s t r o n g ly d e p e n d s o n t h epa r a m e t e r s o f t he s y s t e m . I n c e r t a i n pa r a m e t e r r e g i ons t he s o l u t i on i s pe r i od i c a nd t h r ough as e qu e nc e o f b i f u r c a ti ons be c om e s c ha o t ic . T h i s f e a t u r e r e pe a t s i ts e l f ove r a nd ove r . T h eb i f u r c a t i on i t s e l f i s t he de s t a b i l i z a t i on o f a pe r i od i c s o l u t i on r e s u l t i ng i n a no t he r s o l u t i on w i t ht w i c e t he pe r i od o r de r . T he d i s t a nc e be t w e e n t w o b i f u r c a t i ons de c r e a s e s r a p i d l y s o t ha t t heb i f u r c a t i on s e que nc e s oon r e s u l t s i n c ha o t i c be ha v i ou r .


8/21

374 K . K A R A G I A N N [ S A N D F . P F E I F F E R

1 .0

O.

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. . . . . . . . . . . .I { ' . . . . . . . . . . . . . . . . . . . . " . . . . . l l l l l i l ] l l ! l l : l i ~ ' : :

+ , , H i ! ~ I i , ! ; + I ! + : ; i ; l ' + " . . . . . . . i , l i : ~ i ~ : ! ! i : ! : ! ! . : '

. _ , : . ~ 0 ~: O . ' . . , . ; ] . 1 .5 + . , ; . . . . . ~ , . 2 .i i , i ~ : ! ! i . i ~ i i ' , ~ + ! i ~ ! + i ~ : i i ! ' : ~ i ! i i i i : i l ;' . . . . . ' : ' " " . . . . . . . . . . . . . . . . . . . . . . . . I ~ i ' i ! i t ' , ' : b : + i + ~ :i i i i i i i i i ! " . . . . ' ' @+ i i i ii ! t + " ' " ' + . . . . i ' . . . . . . . . . . . . . . . . . . . . . : . . . . . ' , l i ~ , i : i ; l ! + ! i . . . . .: . . . . . ! 1 , , I , ~ i ] l , , , t l -i I . ~ i : [ I i i ! ~ l : + . . .

6 . 0

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+ z " h : : : ' ; " ; ' " ' ' ; ' i ~ ~ .

. I 1 . , 7 ! " ' 1 . . . . ' , 1 J t , i ! ~ , h " : h l ~ , : , . , , i , , : I : : ! .

I ~ ' + ' , ; . . , . , , ' . ~ i : . ~ . : A . " ' + " -:. . . . J : l !l l l,

, ' ; i i i i l l i , 1 : i , I ; ~ : [ , ~ l l ] ,

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I I Q I j S D ~ 1 8 A ~ @ Q I O 6 0 Q I tO t O 0

. . . . - ~ i o o @

F ig . 3 . B i fu rca t io n o f o n e - s t ag e g ea r ; ( a ) b i fu rca t io n s o f r e l a t i v e v e lo c i ti e s w i th in b ack lash , (b ) b i fu rca t io n s o f n o rm a l i zedt im e , ( c ) g r ea t e s t L j ap u n o v ex p o n en t , t h e b i fu rca t io n p a ram ete r i s c~ an d th e o th e r p a ram ete r s a r e 13 = O . 1 , ~ = 0 . 1 ,

= 0 , 9 .

T h i s f e a t u r e i s s h o w n i n d e t a i l in F i g u r e 4 w h i c h m a g n i f i e s p a r t s o f F i g u r e 3 . F r o m t h isd i a g r a m t h e p a r a m e t e r s a j , a 2 , 4 3 a r e e x t r a c t e d w h i c h c o r r e s p o n d t o t h e f i rs t , t h e s e c o n d a n d t h et h i r d b i f u r c a t i o n , r es p e c t iv e l y . F r o m t h e s e a F e i g e n b a u m n u m b e r a s w e ll as t h e p a r a m e t e r , f o rw h i c h c h a o s a p p e a r s , c a n b e c a l c u l a t e d , T h e v a l u e s a r e

6 - a ~ - a t _ 5 (4 .6 69 )1

4 ~ = 6 , + I -/ - - - - - - ~ ( ~ - ~ ' ) = 1 . 5 5 6 ( 1 . 5 5 7 )


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I N V E S T I G A T IO N S O F G E A R - R A T T L I N G 375

1.0

-1.C

' . . . . . . . . . m l l, ~ . . . . . . . . . . . . . ; [ : : : ] : : : 1 : : . . . . . . . [ 1 1

i. ," - '

1 . 4 8 1 . 5 4 i i : i l } ! 1 . 60

............... . ......................... iii. . i . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ , ~ , I I l l i , i! 1 ! : i l i ~ ]

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1 . 5 4 . . ' " " "1 . 48 ' " : ' * ~ " . 1 . 6 0 a. . , * - , . ~ -

" ' , , . , * , . ,

F i g . 4. De t a i l s o f b i f u r c a t i o n d i a g r a m F i g u r e 3 : ( a ) De t a i l o f F i g . 3 a , ( b ) D e t a i l o f F i g . 3 c .

w i t h t he e xa c t t he o r e t i c a l re s u l ts g i ven f o r c om pa r i s on i n pa r e n t he s e s . T he d e ve l o pm e n t o f t hec o r r e s p ond i ng s t r a nge a t t r a c t o r i n t he P . S . i s g ive n i n F i gu r e 5 , w h i c h n i c e l y s how s t he l o ss o fpe r i od i c i t y a nd t he s p r e a d i ng o f t he a t t r a c t o r i n t he d i r e c t ion o f t he i n s t a b le m a n i f o l d . I n F i gu r e 6a f u ll y de v e l ope d s t r a nge a t t r a c t o r c a n be s e e n w i t h it s ge o m e t r i c s t r uc t u r e s be i ng d i s p l a ye d ve r ywell .3.2 . Resul ts for the Two-Stage Mod elT he m ode l l i ng ha s a l r e a dy be e n p r e s e n t e d a nd t he c om pl e t e de s c r i p t i on o f t he m o t i on i s pos s i b l eon l y i n ~ 5 . A s t he d i m e ns i on o f t he pa r a m e t e r s pa c e is ve r y h i gh i r re s pe c t i ve o f a poss i b ler e duc t i on by t h r e e by m e a ns o f s i m i l a r i t y r u l e s . T he r e f o r e i n t he f o l l ow i ng w e w i l l u s eu n n o r m a l i z e d p a r a m e t e r s .

Q ua l i t a t i ve l y t he r e s u l t s a r e t he s a m e a s in t he a bo ve c a s e o f the one - s t a ge g e a r , i , e . pe r i od i ca s w e l l a s c ha o t i c m o t i on w i t h a s e que nc e o f b i f u r c a t i ons i n be t w e e n . S i nc e t he pha s e s pa c e ha sm o r e d i m e n s i ons t he r e p r e s e n t a t i on o f r e s u l ts be c om e s m or e d i f fi c ul t. H e r e o n l y p r o j e c t i ons o fth i s space onto ce r ta in p lanes wi l l be shown.

F i gu r e 7 , f o r e xa m pl e , s how s d i f f e r e n t p r o j e c t i ons f o r a c ha o t i c m o t i on f o r a s pe ci fi cpa r a m e t e r c om bi na t i on . T he f ir st d i a g r a m ( a ) is t he p r o j e c t i on i n t o t he p l a ne o f ve l oc i ti e s , t hes e c ond ( b ) a p r o j e c t i on on t o no r m a l i z e d t i m e a nd t he ve l oc i t ie s i n ba c k l a s h 1 a f te r im pa c t s a nd t het h i r d ( c) a s im i l a r g r a ph f o r ba c k l a s h 2. T he i nne r s t r uc t u r e o f t he a t t r a c t o r a s c om pa r e d t o t hes i ng l e st a ge m od e l ha s va n i s he d a s a p r op e r t y o f t he p r o j e c t i on , a nd t h e i n t e r p r e t a t i on o f t her e s u l t s be c om e s m or e d i f f i c u l t . F i gu r e 8 s how s t he s a m e p r o j e c t i ons a s F i gu r e 7 f o r a pe r i od i cm o t i o n o f h i g h o r d e r .


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376 I


11/21

I N V E S T I G A T I O N S O F G E A R - R A T T L I N G 377

, +S 1

, . T + 7 7"

( s ~ + , - s k ) - - * - ( s a + , - s k ) ( 1 5 )e v e r y d i s t r i b u t i o n o f i m p a c t s w i l l b e p e r i o d i c w i t h p e r i o d ~-. O n t h e o t h e r h a n d , i f a m o t i o n i s n o ts y m m e t r i c w i t h i n t h e b a c k l a s h a t r a n s f o r m a t i o n o f th e m o t i o n b y e q u a t i o n ( 1 5 ) w i l l a l w a y s y i e ld ina c o r r e c t s o l u ti o n . T a k i n g t h e m e a n o f th e t r a n s f o r m e d a n d t h e u n - t r a n s f o r m e d m o t i o n r e g a i n s th em e n t i o n e d p e r i o d i c i t y o f th e m a r g i n a l d i s t ri b u t i o n . N o w l e t us a s s u m e t h e m a r g i n a l d i s t r i b u t i o nb e i n g k n o w n . E q u a t i o n s ( 1 4 ) a r e r e p l a c e d b y

(16)p ( r ) , r ~ [ 0 , 2 ~ ' )w h i c h i s a d i s c re t e s t o c h a s t i c e q u a t i o n . T i m e r is a s s u m e d a r a n d o m v a r i a b le w i t h s o m e p r o b a b i l i t yd e n s i t y p ( r ) c o r r e s p o n d i n g t o t h e d i s t r i b u t io n f o u n d a b o v e . F r o m e q u a t i o n ( 1 6 ) t h e m e a n v a l u e sc a n b e d e t e r m i n e d a n a l y t i c a ll y . C a l c u l a t i n g t h e e x p e c t e d v a l u e f o r e q u a t i o n 1 6) y i e l d s i n t h ef o l l o w i n g r e c u r s i v e f o r m u l a f o r t h e f i r s t o r d e r m o m e n t

f2 7 r

in k+ i = - e m l k + (1 + ) a J 1 w h e r e J i = p(r ) cos r d r and m i~ = E [~bk] .)


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380 K . K A R A G I A N N I S A N D F . P F E I F F E RH er e m~k i s t he exp ec t e d va lue o f t he ang u la r ve loc i t i e s a f t e r k repe t i t i ons . Th e t e rm J1 i n th i s ca sev a n i s h e s d u e t o t h e m e n t i o n e d p e r i o d i c i t y i n t h e m a r g i n a l d i s t r i b u t i o n p(r ) . T h e a b o v e i t e r a t i o nc o n v e r g e s t o z e r o i f E < 1:

m 1 = l i m = 0~ m l hT h e d e r i v a ti o n o f t h e s e c o n d m o m e n t is po s s ib l e b y q u a d r a t u r e o f e q u a t i o n ( 1 6 ) a nd d e t e r m i n a -t i o n o f t h e e x p e c t e d v a l u e s :

m z k + ~ = e 2 m e a " + ( 1 + e ) 2 a g = - 2 e ( 1 + e ) c ~ E [ 6 k c o s r ]f0 2~rw h e r e J 2 = p( r ) cos2r d r a n d m 2k = E [ 4 ~ I .

I n t h i s f o r m u l a m 2k s t a n d s f o r t h e e x p e c t e d v a l u e o f th e s q u a r e d a n g u l a r v e l o c i t ie s a f t e r ki t e r a t i o n s . O b v i o u s l y

E[G cos,]-~Oh o l d s s i n c e t h e v a l u e s ~ a r e s y m m e t r i c a n d t h e n o r m a l i z e d t i m e p o i n t s o f i m p a c t s a p p e a r i n t h ev i c in i ty o f ~ - / 2 a n d 3 7 r / 2. T h i s a p p r o x i m a t i o n r e s u l ts i n t h e f o l l o w i n g i t e r a t io n f o r m u l a f o r t h es e co n d m o m e n t

= 2 + ( 1 + e ) z a Q ./ f/2k + 1 E m 2k _.m2k can exp l i c i t l y be g iven a s a func t ion o f t he i n i t i a l cond i t i ons :

1 -- E 2k 2k- ( l + e ) a 2 J . + e m ~ . ( 1 7 )H e r e m 2, r e p r e s e n t s t h e i n f l u e n c e o f t h e i n it ia l c o n d i ti o n s . E q u a t i o n ( 1 7 ) c o n v e r g e s v e r y r a p id l y a s

< 1. I n p a r a l l e l, t h e i n f l u en c e o f t h e i n it ia l c o n d i t i o n s - o r o f t h e a s s u m e d d i s t r ib u t i o n o f t h ei n it ia l c o n d i t i o n s - d im i n is h e s. F o r k - - - ~ e q u a t i o n ( 1 7 ) c o n v e rg e s to t h e s e c o n d m o m e n t m 2 :

m , lim = J 2 I + E ,- a - ( 1 8 )T h e f a c t o r J 2 i n t h e a b o v e e q u a t i o n c o u l d b e d e t e r m i n e d b y n u m e r i c a l s i m u l a t i o n b u t i n s t e a d iso n l y e s t i m a t e d s i n c e a s a p r o p o r t i o n a l i t y f a c t o r i t s i n f l u e n c e i s s m a l l . L e t u s a s s u m e m a r g i n a ld i s t r ib u t i o n s o ( r ) , o n e f o r a p e r i o d i c a n d t h e o t h e r o n e f o r a c h a o ti c c a s e ( s e e F i g u r e 9 ) . F r o m

X

F i g . 9 . A s s u m e d m a r g i n a l d i s t r ib u t i o n p ( r ) .


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INVESTIGATIONS OF GEAR -RATTLIN G 381j / ' / ~ /

/ f / / / , /. / / / ~ / . / / /1 / . / / /

///

/ / I/ / / /

//. /

C t , :

d i s t r , a . . . . .d i s t r , b . . . .num. s im.

Fig. 10. S eco nd mo me nts from approxim ation (a and b) and sim ulation (c).

t hese d i s t r i bu t ions t he i n t eg ra l J~_ i s ca l cu l a t ed a nd in t rodu ced in to equa t ion (18 ) . F igu re 10 show st h e s e c o n d m o m e n t s r e s u l t in g f r o m t h is a p p r o x i m a t i o n f o r b o th o f t h e a s s u m e d m a r g in a ld i s t r i bu t ions o n the one ha nd and the num er i ca l s im u la t ion on the o the r . In sp i t e o f t he s im p l i c i tyo f t h e a p p l i e d m e t h o d t h e q u a l i t a ti v e a g r e e m e n t i s r e a s o n a b l e , R e s e t t i n g t h e n o r m a l i z e d p a r a m e -t e r s i n to u n n o r m a l i z e d f o r m s y i e l d s in t h e f o l lo w i n g re l a t io n b e t w e e n t h e s e c o n d m o m e n t a n d t h es y s t e m p a r a m e t e r s :

A-~w ~ (1 + e)E [ ~ 2 I = J ~ ( r / r ) 2 ( l - e ) ' ( 19 )

T h i s f o r m c l e a r l y s h o w s t h e i n f l u e n ce o f th e s i n gl e p a r a m e t e r s . T h e i n t e n si ty o f t h e a p p e a r i n gv e l o c i t i e s r i s e s q u a d r a t i c a l l y w i t h t h e e x c i t a t i o n f r e q u e n c y a n d a m p l i t u d e , d e c r e a s e s q u a d r a t i c a l l yw i t h t h e t r a n s m i s s i o n f a c t o r b u t f o r E < 0 . 8 s e e m s t o b e i n d e p e n d e n t o f th e n u m b e r o f r e s ti t u ti o n .

I t i s i n t e r e s t i n g t o c o m p a r e t h i s f o r m u l a e q u a t i o n ( 1 9 ) w i t h s o m e r e s u l t s o b t a i n e d f r o m t h ed i sc re t e m app ings a s de r ived fo r one -s t age ra t t l i ng and wh ich wr i t e [13 ]

A 2 ( .o 2N o i se - ~ b 2 - (1 + e ) ~ 1 ( r / r , ) Z ,

w h i c h r e s e m b l e s v e r y m u c h e q u a t i o n ( 1 9 ) w i t h s li g ht m o d i f i c at i o n s. T h e e q u a t i o n s a y s , t h a t n o i s eis p r o p o r t i o n a l t o t h e s q u a r e s o f t h e c o e f f i c ie n t o f r e s t it u t i o n e , o f t h e e x c i ta t io n f r e q u e n c y w a n dam pl i tud e A . I t is fu r the r i nve rse p ro por t i o na l t o t he squa re o f t he base c i rcl e rad ius r o f t hed r iven w hee l . I t i s p ro por t i ona l t o t he d r iven w hee l m om en t o f i ne r t ia 1 . Th i s fo rm ula i s o f ve ryl a rge p rac t i ca l s i gn i f i cance , because i t had and s t i l l has d i rec t connec t ion w i th t he des ign o f sh i f tt r a n s m i s s i o n s y s t e m s a n d , m o r e o v e r , o f c o m p l e t e d r i v e t r ai n s . T h e a m p l i t u d e A a t t h e g e a r b o xen t ra nce shou ld be sm a l l , wh ich m eans tha t w i th in t he op e ra t in g rang e o f a ca r i ts d r ive t ra inshou ld have no e igen fo rm s wi th a l a rge am pl i tude a t the gea r box en t ry . Th i s aga in g ives d i rec tr e c o m m e n d a t i o n s f o r d r i v e t r a i n d e s i g n . M o r e o v e r , a l a r g e w h e e l s h o u l d n o t d r i v e a s m a l l o n e d u eto t he t e rm (1/ r-~ ), wh ich m eans tha t t he upp er s t ages shou ld poss ib ly no t be sh i f t ed on the m a ins h a f t b u t o n t h e c o u n t e r s h a f t . M o s t m o d e r n g e a r b o x d e s i g n s ta k e t hi s f ac t in t o c o n s i d e r a t io n [ l 1 ,I 4 1 .


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382 K. KARA GIANNIS AND F. PFEIFFER4 .2 . P a r a m e t e r D e p e n d e n c e f o r t h e T w o - S t a ge M o d e lA l s o i n t h e t w o - s t a g e c a s e l e t u s i n v e s t i g a t e a s i m p l i f i e d m o d e l n e g l e c t i n g d a m p i n g a n d c o n s t a n tm o m e n t s . A s m e n t i o n e d i n S e c t i o n 3 . 2 w e w i l l u s e u n n o r m a l i z e d p a r a m e t e r s t r y i n g t o f i n da n a l y t ic a l r e l a t io n s b e t w e e n t h e e x p e c t e d m e a n v a l u e s a n d th e s y s t e m p a r a m e t e r s . A s i n t h eo n e - s t a g e c a s e d e p e n d i n g o n t h e p a r a m e t e r s t h e r e a p p e a r p e r i o d i c a s w e l l a s c h a o t i c s o l u t i o n s i nt h e v i c i n it y o f t h e f ix p o i n t s ( c o m p a r e F i g u r e s 7 a n d 8 ) . A g a i n t h e s y s t e m c a n b e d e s c r i b e dc o m p l e t e l y b y t h e i n v a r i a n t d i s t r i b u ti o n . H e r e w e c o n s i d e r o n l y t h e m a r g i n a l d i s t r i b u t i o n o f th ei m p a c t t i m e s w i th i n t h e e x c i t a ti o n p e r i o d . C o m p a r e d t o t h e o n e - s t a g e c a s e t h e s it u a t io n i s m o r ec o m p l i c a t e d a s i m p a c t s m a y a p p e a r i n e it h e r b a c k l a s h 1 o r 2. T h e P . S . a c h ie v e d f r o m t h en u m e r i c a l s i m u l a t i o n a c c o r d i n g t o S e c t i o n 3 . 2 s h o w s t h a t t h e i m p a c t s i n t h e t w o b a c k l a s h e s a r ec o r r e l a t e d . T h i s s u g g e s t s a s im p l e a p p r o x i m a t e r e l a t i o n b e t w e e n t h e d i s t r i b u t io n s o f t h e i m p a c t s inb ac k l a sh 1 an d 2 . I t is a s su m ed t h a t a f t e r l~ imp ac t s i n b ack l a sh 1 l 2 imp ac t s i n b ack l a sh 2 ap p ea r .T h e v a l u e o f t h e q u o t i e n t l l / L d e p e n d s o n t h e b a c k l a s h e s a n d m o m e n t s o f i n e r ti a a n d s h o u l d b ei d e n t i f ie d n u m e r i c a l ly . F o r t h e c a s e o f a p p r o x i m a t e l y e q u a l m o m e n t s o f i n e r ti a o f t h e l o o s e g e a rw h ee l s an d i f t h e q u o t i en t o f t h e b ack l a sh es i s i n th e ran g e o f 0 .5 -2 t h en l l ~ 12 ~- 1 . Th i s m ean st h a t a f t e r a n i m p a c t i n b a c k l a s h 1 a n i m p a c t i n b a c k l a s h 2 m a y b e e x p e c t e d . A s s u m i n g t h i s t o b eex ac t l y t h e ca se y i e ld s in an ap p ro x ima t io n o f th e rea l m o t io n wh ich s ti ll wil l su f f i c i en tl ya p p r o x i m a t e t h e a t t r a c t o r . L e t u s n o w a s s u m e t h e m a r g i n a l d i s tr i b u t io n o f i m p a c t t i m e s i nb a c k l a sh 1 w a s k n o w n a p r io r i . I n s t e a d o f t h e m a p p i n g s w e w i ll u s e t h e f o l l o w i n g d i s c r e tes t o c h a s t i c e q u a t i o n s

'k ~ + , = u l 4 ' ~ + ~"c o s ( t o t )t , ( t ) , t ~ [ 0 , - ~ )

( 2 0 )

~b2~ = (~ b, , ~b2)T ~ E 2 is t h e v e c t o r o f a n g u l a r v e l o c i ti e s o f t h e t w o g e a r w h e e l s a f t e r t h e k - t h i m p a c tin ba ck las h 2 an d ~blk+l = (~bl , ~b2)T E [~2 a re t h e an g u l a r v e lo c i t i e s a f t e r t h e (k + 1 ) - t h im p ac t i nb a c k l a s h 1 . T h e v e c t o r ~ = to ~l is i n t r o d u c e d a s a n a b b r e v i a t i o n . T h e t r a n s f e r m a t r i c e s U I , U 1 a n dt h e v e c t o r ~ h a v e a l r e a d y b e e n d e f i n e d i n S e c t io n 2 , s u b s c r ip t s 1 a n d 2 r e p r e s e n t t h e s t a g e i n w h i c ht h e i m p a c t o c c u r s . E q u a t i o n s ( 2 0 ) a s s u m e a p r o b a b i l i t y d e n s i t y f o r t h e i m p a c t s i n b a c k l a s h 1 .A f t e r e a c h o f t h e s e i m p a c t s o n e i n b a c k l a s h 2 f o l l o w s . U s i n g t h e s t o c h a s t i c m o d e l o f e q u a t i o n s( 2 0 ) o n e c a n c a l c u l a t e t h e e x p e c t e d v a l u e s o f t h e a p p e a r i n g v e l o c it i es . F o r t h i s p u r p o s e w ef o r m u l a t e t h e e q u a t i o n s i n t h e fo l l o w i n g w a y :

~ , ,+~ = U 2 U ,~ k 2 ,+ U 2 ~"c o s ( t o t )

p ( t ) , t ~ [ 0 , ~ )( 2 1 )

T h e f ir s t e q u a t i o n c o n t a i n s o n l y r e l a t io n s b e t w e e n s y s t e m s t a t e s in b a c k l a s h 2 a n d c a n e a s i l y b ep r o c e s s e d f u r t h e r . D e t e r m i n a t i o n o f th e e x p e c t e d v a l u e s y i el d s


17/21

= U ~ U t k t ~ + U ~ J 1J['~2~.. 2 - - -/ . t l~ , = U21t .%with J , , = (2~ ,~ p ( w t ) c o s " ( w t ) d r .d~ P

INVESTIGATIONS OF GEAR -RATTL ING 383

( 2 2 )

7#zk = E(~bt ' g~2) , is the exp ect ed valu e o f the v eloci t ies af te r the k - t h impac t i n back l a sh 2 and# 2 ~ , = E ( ~b ~, 2 ) T r e p r e s e n t s t h e e x p e c t e d v a l u e o f t h e v e l o c i ti e s a f t e r t h e ( k + 1 ) -t h i m p a c t i nb a c k l a s h 1. E q u a t i o n ( 2 2 ) c o n v e r g e s i f t h e a b s o l u t e e i g e n v a l u e s o f t h e m a t r i x U 2 U t a r e l e s s t h a no n e . F o r t h e i n t e r e s t in g p a r a m e t e r r e g i o n t h is h o l d s tr u e . T h e e x p e c t e d v a l u e s c a n e x p li c it ly b efou nd a s func t ions o f t he i n it ia l cond i t i ons and t he i r l imi ts can be g iven :

/ a~ = l i m b % = ( E - U , U I ) ' U ~ ' J 1( 2 3 )

E v a l u a t i n g t h is f o r m w i t h r e s p e c t to t h e p a r a m e t e r s o f t h e m o d e l y i e l dst~ , : i,t~_ : J ~ A w ( 1 / i , , 1 / ( i ~ i 2 ) ) r ( 2 4 )

r l l r 2wi th i I = ~ , i 2 =~ , , be ing t he t ransmiss ion fac to rs o f t he two s t ages . Th i s re su l t was t o b ee x p e c t e d a s t h e e x c i t a ti o n is a m p l if i e d b y t h e t r a n s m i s s io n f a c t o r s w e i g h t e d b y t h e v a l u e J l whichd e p e n d s o n t h e m a r g i n a l d i s t r i b u ti o n o f th e i m p a c t s A s i n c a l c u la t in g t h e f ir s t o r d e r m o m e n t s w ew i ll d is t in g u i s h b e t w e e n t h e t w o b a c k l a s h e s f o r t h e c a l c u la t i o n o f e x p e c t e d v a l u e s o f t h e s q u a r e s o f T ~ 2 , 2t he ve loc i t i e s . C2k = [ I g~2 ] ~ i s t he co r re l a t i on ma t r ix o f ve loc i t i e s a f t e r impac t s in back l a s h 2 . T ~ ]~ " "and k i t e ra t i ons o f t he p ro cess , and C~ ,~ , = [1~ o2] E - - i s t he co r re l a t i on ma t r ix o f t he ve loc i t i e sa f t e r impa c t s i n back l a sh 1 and k i t e ra t i ons o f t he p roce ss . Th e ev a lua t i on o f C~ i s don e bym u l t i p l y i n g t h e f ir s t o f th e e q u a t i o n s ( 2 1 ) w i t h it s t r a n s p o s e d a n d e x t r a c t i n g t h e e x p e c t e d v a l u e s .

C ~ = U 2 U I C 2 ( U 2 u I ) T + [ U 2 ~ t ~ ( U ~ U I ) T + U m U I ~ 2 ( U 2 ~ ) T ] j I + U 2 ~ ( U 2 ~ ) T j 2 ( 2 5 )T h i s e q u a t i o n i s a r e c u r s i v e m a t r i x e q u a t i o n w h i c h c o n v e r g e s u n d e r t h e c o n d i t i o n t h a t t h e a b s o l u t ee i g e n v a l u e s o f t h e m a t r i x U ~ U j a r e l e ss t h a n o n e . T h e l im i t is a s o l u t io n o f t h e a b o v e e q u a t i o n a n ds o a m a t r i x e q u a t i o n f o r i t s d e t e r m i n a t i o n a p p e a r s :

C~ = U = U , C2 (U ~ U , ) " + [ U ~ O ~ ( U y , ) " + U , U , t~ 2(U ~ ~ ) " ] J , + U ~ ' ( U ~ ' ) ' j 2wh ere C~ = l im C~ . (26 )

S i m i l a r ly . a c o r r e s p o n d i n g m a t r i x e q u a t i o n c a n h e d e r i v e d f o r t h e c o r r e l a ti o n m a t r i x o f t h eve loc i t i e s i n back l a sh l :

c , = U , U 2 C , ( U I U ~ ) r + [~ 'v h (U ~ U ~ ) rwh ere C 1 : l ir a C ~ .

~ - U 1 U 2 j t , L I j t~ l ~ T ] J 1 + ~TJ 2

The co r re l a t i on ma t r i ces can be de t e rmined ana ly t i ca l l y w i th t he re su l t s be ing a l i t t l e b i t l eng thy .L i k e i n t h e o n e - s t a g e c a s e t h e i n te g r a l J1 v a n i s h e s d u e t o t h e s y m m e t r y o f th e a t t r a c ti o n r e g i o n i fa ll poss ib l e so lu t i ons a re su m m ed u p . I f on ly a s i ng le a t t rac to r i s con s ide re d t he i n t eg ra l sh ou ld


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384 K . K A R A G I A N N I S A N D F . P F E I F F E Rst i l l have a smal l value. The elements of the correlat ion matr ix are

A2~o2 g j ( e , x ) A 2 0~ 2 g ~ ( e , x )C~'L- i~- l (1- e) g(e, x~ J~ C ~ -2 = 21i2~'(1- E) g (e, x~ J2

A2co2 g3(e, x)-- .9,~_ l]z 2 g( e, x) J~

andA2w 2 fl( e, x) A2w -' L(E, x)

C 2 " / 1 ( 1 - ~ ) f (E , x ) J " C ~ t~12- (1-E) f (E,x) J"A2 w ~- L (E , x) I21I

2 . . 2C 2 1 2 I i I 2 f (E , X) J " with x 12Th e funct ions f l , f2, f.~, f , gl , g2, g3, g are high ord er po lynom ials of e and x. Th e p ropert ies of thepropo sed formulas agree w el l wi th exper imenta l resu lt s. From the m ent ion ed expec ted va luesoth er characteristics can be der ived such as ene rgy dissipation by the impacts or the inten sity ofthe impacts . Again, these resul ts in terms of mean values are of great s ignif icance on theest imat ion of parameter inf luences with regard to gear box design, where detai led s t ructuralpropert ies l ike per iodici ty, chaos or bifurcat ion do not play an important role .

1 Dr iv e n wh e e l2 D r iv in g wh e e l3 E x c e n te r me c h a n i s m4 F ly wh e e l5 E d d y c u r r e n t b r a k e6 Spr ing7 Ad ju s tm e n t o f b a c k la sh8 ln v o lu tc to o th in g

1 8 7

6

9 I n d u c t iv e d i s p la c e me n t t r a n s d u c e r s1 0 F e e d e r1 1 Me a s u r e me n t d a ta p r o c e s s in g12 Signa l ana lyzer1 3 A n a l o g / D i a g i t a l c o n v e r te r1 4 C o m p u t e r

2

11

~ 4I 12/ /

A~D

131 4

Fig . 11 . Exper imenta l se t-up for one-s tage ra t t l ing [2] .


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INVESTIGATIONS OF GEA R-RAT TLING 3855 . E x p e r i m e n t a l R e s u l ts

T h e t h e o r e t i c a l m e t h o d s a s g i v e n a b o v e h a v e b e e n d e v e l o p e d p r i m a r i l y f o r p r a c t i c a l p u r p o s e s . T h ef u n d a m e n t a l i n si g ht s i n to p a r a m e t e r i n f lu e n c e s o n r a t tl in g a r e t h e b a s i s f o r i m p r o v e m e n t s o f g e a rb o x es an d o f d r i v e t ra i n sy s t em s a s was i n d i ca t ed i n Sec t i o n 4 .1 .

T o g e n e r a t e s o m e c o n f i d e n c e in t o t h e t h e o r e t i c a l m e t h o d s a n d t o b e s u r e o f a g o o dc o r r e s p o n d e n c e o f i d e a a n d r e a li ty e x p e r i m e n t a l i n v e s ti g a t io n s a n d c o m p a r i s o n s h a v e t o b ep e r f o r m e d . T h e r e f o r e , a t e s t s e t - u p f o r o n e - s t a g e r a t t l i n g h a s b e e n r e a l i z e d a c c o r d i n g t o F i g u r e 1 1 .A wh ee l wi th a si n g l e i n v o lu t e t o o th i s ex c i t ed b y v ib ra t i n g co n s t ra in t s . In ad d i t i o n , an e d d yc u r r e n t b r a k e s i m u l a t e s l i n e a r d a m p i n g f o r c e s w h e r e a s a w e e k s p r i n g a d d s a n a l m o s t c o n s t a n tm o m e n t u m . T h e v i b r a t i n g c o n s t r a in t s t h e m s e l v e s a r e d ri v e n vi a a n e x c e n t e r m e c h a n i s m b y ad i r e c t c u r r e n t m o t o r . A l l p a r a m e t e r s t h a t a r e i m p o r t a n t f o r t h e p r o b l e m c a n b e v a r i e d s u c h as t h ee x c i t a t i o n f r e q u e n c y a n d a m p l i t u d e , t h e b a c k l a s h , t h e d a m p i n g f a c t o r a n d t h e c o n s t a n t m o m e n t .

T h e f o l l o w i n g r a n g e s a r e p o s s i b l e :- e x c i t a t i o n a m p l i t u d e ( 0 . . . 9 ) m m- e x c it a ti o n f r e q u e n c y ( 0 . , . 5 0 ) H z- b a c k l as h ( 0 . . . 7 ) r a m- v is c ou s d a m p i ng ( 0 . . . 1 0 0 ) N m m s / r a d- c o n s t a n t d a m p i n g ( 0 . . . 5 0 0) N m m

M e a s u r e m e n t r e s u lt s a r e th e m o t i o n s o f t h e t w o w h e e l s , t h e t o o t h w h e e l a n d th e o n e w i t h t h ev ib ra t i n g co n s t ra in t s . Th e s i g n a l s a re t h en fu r t h e r p ro cessed o n a d ig i t a l co mp u te r .

F i g u r e 1 2 g i v e s a c o m p a r i s o n t o t h e o r y a n d m e a s u r e m e n t s f o r p e r i o d i c a n d c h a o t i c m o t i o n[ 2 ] . A s c a n b e s e e n , p e r i o d i c r a t t l i n g m o t i o n g e n e r a t e s a p e r i o d i c i m p a c t p a t t e r n w i t h i n t h eb a c k l a s h o f t h e g e a r m e s h e s , c h a o t i c m o t i o n g e n e r a t e s a c h a o t ic i m p a c t p a t t e r n . T h e b a c k l a s h e s i nth e two f i g u re s a re rep re sen t ed i n a p e r i o d i c fo rm in d i ca t i n g t h e ex t e rn a l p e r i o d i c ex c i t a t i o n .

6 . S u m m a r y

M a t h e m a t i c a l m o d e l s o f ra t tl in g a r e p r e s e n t e d o n t h e b a s is o f a p a t c h i n g p r o c e d u r e p u t t i n gt o g e t h e r f r e e f l ig h t p h a s e s a n d i m p a c t s e q u e n c e s a n d o n t h e b a s is o f d i s c r e t e m a p p i n g s d e s c r i b in gt h e r a t tl in g s t a t e b e f o r e s o m e i m p a c t i n d e p e n d e n c e o f th e s t a t e o f t h e p r e c e d i n g i m p a c t . D i s c r e t em a p p i n g s a r e w e l l s u i t e d f o r s t r u c tu r a l i n v e s ti g a ti o n s i n c lu d i n g s t a b il i ty , L j a p u n o v - e x p o n e n t s ,b i f u r c a t i o n s , F e i g e n b a u m n u m b e r s , a n d t h e y g i v e a s u i t a b l e b a s i s f o r e s t i m a t i n g p a r a m e t e ri n f l u en ces . Th e l a s t m en t io n ed a sp ec t i s p a r t i cu l a r l y imp o r t an t fo r p rac t ica l ap p l i ca t i o n s an d t h en o i s e r e d u c t i o n o f s h if t tr a n s m i s s i o n a n d d r i v e t ra i n s y s t e m o f ca r s . S i m i l ar r e s u l ts f o r p a r a m e t e ri n f lu e n c e s c a n b e o b t a i n e d b y c o n s i d e ri n g t h e m e a n v a l u e s o f t h e r a t tl in g p r o c e s s , a v e r y i m p o r t a n ts t e p t o a s t o c h a s ti c a p p r o a c h o f im p u l s i v e p r o c e s s e s .

T h e p r e s e n t e d m e t h o d s s h o u l d b e a b l e t o d e s c r i b e t h e r e a l w o r l d a n d t h u s r e a l g e a r b o x e s .T h e r e f o r e , a c o m p a r i s o n w i th m e a s u r e m e n t s h a d t o b e c a r r i e d t h r o u g h . A o n e - s t a g e ra t t li n g w a sr e a l i z e d , a n d t h e m e a s u r e d r e s u l t s c o m p a r e d w i t h t h e c a l c u l a t e d o n e s . G o o d a g r e e m e n t c o u l d b ea c h i e v e d , w h i c h g i v e s s o m e c o n f i d e n c e i n t h e t h e o r y d e v e l o p e d . A p p l i c a t i o n s o n s e v e r a l r e a l c a rc o n f i g u r a t i o n s h a v e b e e n p e r f o r m e d s u c c e s s f u l l y .


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386 K . K A R A G I A N N I S A N D F . P F E I F F E RM e a s u r e m e n t

I

' I V V V V V V V V V

a .@

T h e o r y

V V V V V V VV V V I

a) P e r i o d i c M o t i o n

M e a s u r e m e n t

'

T h e o r y

V l I V V V V VV I I

b ) C h a o t i c M o t i o nFig. 12. Comparison theory-measurements for one-s tage ra t t l ing [2] ,

Reference s

1 . Guckenhe imer J . and Holmes , P . , Nonlinear Oscillations, Dynamical Systems, a nd B ifurcations o f V ector Fields',Spr inge r Verlag , New York , Be r l in , He ide lbe rg , T okyo , 1983 .2. Ka ragiann is , K. , 'A naly se s tol3behafte ter Schwingungssysteme m it An wen dung auf Rasselschwingungen beim Zah -n r a d g e t r i e b e , Fortschritt Berichte VD1, Reih e l h Schwingungstechnik, N r. 12 5, VDI-Verlag Dfisse ldorf , 1989.3. K/icfikay, F. and Pfe iffer , F. , ' f ib er Rasselschwingungen in Kfz-S chal tgetr ieb en ' , lngenieur-Archiv 56, 1986, 25-37,4. Kfic i ikay, F. , Dynamik der Zahnradgetriebe, Spr inge r Ver lag , Be r l in , He id e lbe rg , New York , 1987 .5 . Kune r t , A . and Pfe i f fe r, F . , 'S tochas t i c mod e l fo r ra tt l ing in gea r -boxes ' , Proceed ings IUTAM Symposium onNonlinear Dynamics in Engineering System, Stut tgart 1989, Schiehlen, W. (ed. ) , Springer Verlag, New York, Berl in ,He ide lbe rg , T okyo , 1990 .6 . L i , G . X . , Rand , R . H . and Mo on , F . C . , 'B i fu rca tion and chaos in a fo rced zero -s ti f fness impac t os c i l l a to r ' ,International Journal Nonlinear Mechanics 25 (4), 1990, 417-432.


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I N V E S TI G A T IO N S O F G E A R - R A T T L I N G 3877 . L ic h te n b e r g , A . J o u r n a l a n d L ie b e r ma n , M. A . , Regular and Stochastic Motion, S p r in g e r Ve r la g , Ne w Yo r k ,

He id e lb e r g , Be r l in , 1 9 8 3 .8 . Mo o n , F . C . , Chaotic Vibrations, J o h n W i le y & S o n s , 1 9 8 7 .9 . Par s , L . A. , A Treatise on Analytical Dynamics, Ox Bow Press , 1979 .

1 (I . P f e i f fe r , F . a n d Kf ic i ik a y , F . . ' E in e e r w e i t e r t e me c h a n i s c h e S to Bth e o r i e u n d ih r e An w e n d u n g in d e rG e t r i e b e d y n a m i k ' , VDI-Z, 127 (9) , 1985 , 341-349 .

11 . Pfe if fe r , F . and P res t l , W. , 'Deco upl i ng measu res for ra t t l ing no ise in gearb oxes ' , Proe. 1 Mechanics E, C a m b i d g e ,1 9 9 ( / .1 2 . P f e i f f e r , F . , " Me c h a n i s c h e S y s te me mi t u n s te t ig e n Ub e r g / in g e n ' , lngenieur Archi~'54, 1 984, 232 240.1 3 . P f e i f f e r , F . , ' S e l t s a me At t r a k to r e n in Z a h n r a d g e t r i e b e n ' , h~genieur Archiv 58, 198 8, 113-125.14 . Pfe if fe r , F . , ~Dynamical sys tem s with t ime-vary ing or uns teady s t ruc tu re ' , ZAMM 7 1 , 1 9 9 1 , 4 , T 6 - T 2 2 .1 5. Re i th me ie r , E . , ' P e r io d i s c h e L 6 s u n g e n n ic h t l in e a re r S c h win g u n g s p r o b le m e - N u m e r i s c h e Be h a n d lu n g , S ta b il i t ii t ,

B i f u r k a t io n ' , D i s s e r t a t io n T U Mf in c h e n , L e h r s tu h l B f f ir Me c h a n ik , 1 9 89 .1 6. T o s h imi t s u S a k a i , Yu h j i Do i a n d Ke n ic h i Ya m a mo to , ' T h e o r e t i c a l a n d e x p e r ime n ta l a n a ly s i s o f ra t t l i n g n o i se o f

a u t o m a t i v e g e a r b o x ' , SA E Passenger Car Meeting, De a r b o r n , M ic h ig a n , J u n e 1 9 81 .17 . Sh aw, S . W. , "The dynamics of a harmonica l ly ex c i ted sys tem having r ig id amp li tude co ns tra in ts . P ar t 1: Sub harm onic

and loca l b ifurca t ions ' , Journal o f Applied Mechanics5 2 , J u n e 1 9 8 5 , 4 5 3 - 4 5 8 .18 . Shaw, S . W. , 'The dynamics of a harmonica l ly exc i ted sys tem having r ig id ampli tude cons tra in ts . Par t I f : Chaot ic

mo t io n a n d g lo b a l b i f u r c a t io n s ' , Journal of Applied Mechanics52, June 1985 , 459-464 .19 . Wood , L . A. and By rne . K. P ., 'An a lys is o f a r ando m re pea te d impac t p rocess ' , Journal (~[ Sound and Vibration78(3) , 1981 , 329-345 .2 0 . W o o d , L . A . a n d B y r n e , K . P . , ' E x p e r im e n ta l in v e s t ig at io n o f a ra n d o m r e p e a te d imp a c t p r o c e s s ' , Journal of Soundand Vibration 85 (1) , 1982 , 53-69 ,

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