Asymptotic method for investigation of multifrequency oscillations of quasilinear systems of second order

z is sm all positive parameter.f.. F is periodic* function of "k with period 277... It SubstiUitinp- the expressions ÍS... IKBAHHH KBA3II/mi-ini!HblX CMCTEM BTOPOrO llOPfl;j,KA B.

S C I E N T I F I C R E S E A R C H O F V I E T N A M

H A N O I 1987

PỈUVLEEDINGS OF ĩ ỉ ì l i NATIONAL CENTI

M e c h a n i c s

A s y m p t o t i c method f o r in v e s ti g a ti o n o f

m u i t i f r e q u e n c y oscillations o f q u a s i l i n e a r

systems of second o r d e r

N G U Y E N V A N D A O

I n s t i t u t e o f m e c h a n i c s

NGUYEN VAN DỈNII

H a n o i p o 1V! e f h n i c a i I n s t i t u u*

Si'MMAHY

Mosl o f m e c h a n i c a l p r o b l e m s a r e w r i t t e n in t h e f o r m o f d i f f e r e n t i a l

e q u a t i o n s o f s e c o n d o r d e r In t hi s a r t i c l e t h e a s y m p t o t i c m e t h o d f o r c o n s t r u c ­

t i o n o f 1 he s o l u t i o n o f t h e s e e q u a t i o n s in t h e g e n e r a l r e s o n a n t c a s e s is p r e s e n t e d

T h e 1’irst p a r a g r a p h is c o n c e r n e d w i l h t h e s y s t e m o f o n e d e g r e e o f f r e e d o m a n d

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

m e a n s o f tho n r o p o s e đ m e t h o d o n e c a n r i n d t h e a p p r o x i m a t e s o l u t i o n s OÍ c o n ­

s i d e r e d ( ( [ n a t i o n s w i l l i d e s i r a b l e ' e x a c t n e s s

Mosi ol i m ' c ha r.ica! probh'iYis ilft’ w r i t t e n in llic f o r m ol din C'lvniiai

liou.s s r c r i h ! o r d e r A s y m p t o t i c m e t h o d i'or m o n o f r e q u o n c y o s c i l l a t i o n s a n d SOI1U' s p i ‘(‘ial c a s r s oi' m u l t i f r o q u e n c y o s c i l l a t i o n s o f q u u s i i i n c a r s e c o n d o r d e r

s v s l o m s h a v e b e e n g i v e n i n • 1 2 Ti lV* m r l h c x i l o r SÍUCỈY O Í n u i l t i f r e q u e n e \

o s f i l l i i t i o n s or q u a s i ] i n c a r fi r st o r d e r s y s t e m s w a s p r e s e n t e d in — 5 i n ill!: p:i]K'i- [he a s y m p t o t i c m e t h o d f o r i n v e s l i g n l i o n o f m u l t i f r e q u r n c y o s c i l l a t i o n s o f

q u a s i l i n e a r S Y S Í C I U S o r s e c o n d o r d r in g e n e r a l r e s o n a n t c a s e s is d i s c u s s e d /(>/

i.vt I I S c o n s i d t T a (juasiiiiii a: v‘:j’lit 1 ion o f Si'comi o r d e r w i t h s l o w k

\ a r v i ii iị p a r a i i U ' k T s :

I l \ r n o n r c n o x

II SYS I KM w r m MNCiLK UKGHKK OK h'HKKDOM

-f f (-)

Ill

whore 'T = e\ is slo w ly v a ry in g time z is sm all positive parameter.

f.) = (0 1 0r) — = vk (t) F is periodic* function of (")k with period 277

(! i

! l i s s u p p o s e d i l i a ! 1ÌÌ ( t ) 0 ( t ) F s u f f i c i e n t n u m b , ' ! ' o f d e r i v a t i v e s

relative] V T 0 , \ — and that for a ll T in interval () < T L, the q u a n

-(it litics Hi ( T ) c ( t ) art1 positive, Th e re w ill be the fo llo w in g expansions :

F ( t 0 X e) - K, ( , 0 , X + « F , Ị T 0 X

h e r e i' j ( " X \ ) a ; v p o l y n o m i a l s o l X.

W e s h a l l c o n s i d e r t h e r e s o n a n t c as t' w h e n t h e r e e x i s t s t h e r e l a t i o n o f t y p e

p * co Í T ! + c * V ( T ) = II .

< : * = ( C l c r ) V < T ) = ( V , V , ) ( • ’>)

where* CO (T) = y 11 '~1 and i5* c * C Ĩ are integers, p* =ỹt= 0 If there

V m \ Z)

i s n o r o h i l i o n o f ÍYỊK* (.*>) \Y(k h a v e t h e n o n s e s o n a n t c a s e

In n o n r c s o n a n l case t h e s o l u t i o n o f e q u a t i o n (1) is f o u n d u n d e r t h e f o r m :

\ — a cos 4* + £• U1 a ’ lỉ'< ©) + u2 ,/T‘ a * ©' + ( I j

here u is neriodii* functions of 4", 0 k w ith period 2~ and q uantities a lF as

lunclioiv- or Ihne are (!e!ermin(‘(i from ecịnations :

-— £ -Vj ( T , c l ) ~ r £ “ A j ( t Li) +

(It

——— (-J ( : ) T £ il l ( ; , a ) T • • • • (5)

(It

S t i b s l i l u l i n i * ( ) ) i n t o ( 1 ) a n d c o m p a r i 11 ii l i i s f l y t h e c o e f f i c i e n t s o f e a n d

! h o 11 I Lie h a r m o n i c s w e o b t a i n :

A , ( t a ) = - - -— - — - I t s i l l ỉ (ỉ Ỉ ( i ( ) |

(i

f I* !■', c o s ' i Y

Bj ( t a) = - I * • 5 *; i r o s T i I M M B , ( ! ( - ) , , ( I ) )

m roa i 2 “ ) r + l

i(ị)‘j +1-H)

II, ( T a T (-)) = Y

Ự 2 ~ ) ' " ]m [c o — (pco + C'J

!>.(•

)_

f f I-, e - i ( p ' l ' K- 0 .

—

II II

Le i IIs c o n s i d e r n o w m o r e i n t e r e s t i n g re s o n a n t case, it is a s s u m e d t h a t

I h e r e is a r e s o n a n t r e l a t i o n :

Now (Ill* so lutio ji or equation (1) is presented as

\ — ; I c o s ( ' ■ *■ 'I ) + £ IIJ ( t cl c “Ị" , 0) ^ l u • * ■ ( ^ )

T h e a m p l i t u d e a a n d p h a s e '!' a r e d e l e n n i n e c l f r o m the- e q u a t i o n s

— — - = £ A; ( - SI in -4- e- A , ( - a T ) +

— - — = C.J (-) - Ll (z) + £ 14, a T ) - f £ - .

(It

SubstiUitinp- the expressions (ÍS) (9) into equation (1) and com paring tile

c o e f T i c i e n t s o f £ a n d t h o s e o r t h e h a r m o n i c s w e o h l : » i n :

■ J ill ( T ) Co ( t ) ( ! t i n ( t ) ( 2 - ) r ‘ 1 L -> (

ị LCO ( T )

-I — < 7 ~ p* ' - ' [ t o ( 7 ) — Ĩ 2 ( t ) I

') _

”* # "U *

I I-', e ? ơ| ’ 1 ' c o s (£ + '!') (I ( Ẹ r- 'Ỉ ) <iw, (K“)

II 0

2 CO ( T )

i Ị T )(T ' I )*■ [ (<)[ - ) - L2( T ) I

J • • J !■' (T O O X „ \ ) e i ơ | , * ! 1 s i n ( c + in (I ( Ẹ + M‘) c!0 (10r Ị

X

II 1}

E

! [ I ■(£+ ll ’H- (■<>; j» r - i [ J) (Ẹ 4- 4 ’) + C0]

I! II

ill))

I l l - S Y S T E M W I T H S E V E R A L D E G K K E S O F F R E E D O M

Co nsider the ]'ollo\vintỊ equations oi second o;-(k“r :

—“ Ị X I :liJ {':> cii Ị + X ! ìj,j lli = eí' íj (-)’ li‘ (,‘ e)‘

i = 1 i = 1

(j = 1 2 N)

i n ( r ( \ s o n a n l c a s e w h e n I h e r o e x i s t r e s o n a n l r e l a t i o n s

]]W Q ( T ) + c ! a ) M T ) ^ 0 ( 1 2 )

V = ( V , v r ) a = 1 1 ! 11 Ũ = 1 2 , £ > „ )

ilj arr f O j p = ( P , P N) c = (C c r).

i • . a re integers, ojj a r e n a tu ra l iro q u en c ie s the roots or e q u a tio n ><■*

a ’ *■ j ~ a i j t0* / / ~

■l-ỉn LÍ1 is C;:se t h e s o l u t i o n oi cijiUiiions (11) w i l l be r o u n d in I h e f o r m

X

( ‘ j = y z>[*> ÍI, c o s + ir s ) + e 1!<‘ > ( a Ị 4- T , 0 ) + e 2u<2> ( 1 3 )

a = ( n , a r r Í = a , ' X) r = ( li* ' I' x K Í-) = ( ( - ) , 0,.)

v r h o r C 9«* > a l l ' t h e n o r m a ! I ' i i n c t i o n s — t h e n o n i r i v i a ỉ s o l u t i o n o f iiu* C‘<] u u l i o n s

Y ' ( b i j — >‘ i j " ' ) - ? ’ ' ‘ = i J- ( ! • s - ■ • - X )

i = 1

u "") a r c i k e p e r i o d i c f u n c l i o n s o f ; + 4* 0 vviih p e r i o d 2 " a n d I ho aniT)i iitidc i

11 a n d o h u s o vr arc* d e t e r m i n e d fr o 111 e q u a t i o n s :

' = s A 1, ' ’ (T a T ) + E A (T 11 T ) + .

— —— = w , ( t ) — i.2s ( " ) + £ B , ( t .1 ;1 ) H ( 1 1 )

t i t

([ c

- - - i i sAJ'*' z J- v*-'*’ a - 0 (15;

<ii

By a n a l o p v j o (he rir sl p a r a g r a p h , f r o m e q u a t i o n s ( J i ) — (15) b v C

01Ì1DỈI-r i n ơ I IK' c o c i l i c i c i i i s o f £ a m i l l i o s e o ! Í Ì Ì O h i i n i i o n i v * ; , w o o b t a i n :

n

; V' Ơ' -' s ’ ••

•*-u" ■ ị )

V (toI — 1 2 ,) — — - ‘i a t o b 1/ ’ = — Ỷ

4 1) d ( + ) (Ifc) (

(1 (

i «o , — 1 2 |) a s - 4-12COsA.' =

r * 4 \

M

I V' Ơ'a ’■s ' T' • ’

Í

/V

X

II i =

i £ o ' ! ' 4r> j IL* ' “ 1

1 Q i e

-X s i l l • i; - f ' l ' j (I ( C | - f l l \ ) (1 ( - f 4 y ) ( ] © J ( ] 0 , .

>' * 4*

\ \

i [ ! ’ , Ũ , + r t )

EE?;'

I ).(• s

col — (Pcu + Cu)~] M s (27C)o , N + r

( )

i I P , ( Ị , + T ) cr 0 r]

(I ( S , + 0 , +

It i = I

\

52

■ u i = I

\ V s

M

'■ a i w v 11V ( CO f oj )

(I cp',v ( í I

1 ( I t

I V I

1 T I T s i n ( ; , + T v ).

E

l i m s I h ( ‘ ;I s \ í n [>Ỉo 1 ic* i i K‘th()(l p r e s e n t e d h e r e g i v e s t h e a p p r o x i m a t e s o l u ­

t i o n s OỈ t h e c o n s i d e r e d CM I n a t i o n s in c o m p l i c a t e d r r s o m m l c a s e s w i i Ỉ1 f h c* ( l o s i i v d (i (‘L j r c r o f (■ Xitel II t*ss.

1 ỉ l H B o r o n o f i o B , l O A ; \ \ i ) T p o n o / i b C K i i i i , A c H M r i T O T i m e c K H e MOTGUbi R T e o p i m

ìie iiìH e íiH b ix K o ieỐ a H H íí, V\0CK Ba, 1 9 6 3

2 i ( ) A U i i T p o n o i b C K n i i , 1 lp o ố ie M b i acH M H T O Tỉm ecK O M T e o p iỉM n e c T a u H O H a p H b ix

K o i e ố a H n i i , H a v k -a , 1 9 6 -4

?> Ỉ3.[] P yỐa HHI Í , Ko i e Õa HI IH KB33H.1HHeHHbIX CHCT6.M c 3aria3;ibIBaHHeM 113.1,

I l a v i c a Wo c Ki Ki , 1 9 6 9

1 A \ H K y a i y ì b , o IIOHTII IiepHOAHMeCKUX p e i I i e H l l à x KB33H.1HHÍÌHHbl.\ CHCT6M

Ii pi i M HOĩOKpaTHOM p e 3 0 H a n c e , 113B A H C C C P , O T H , M e x a H H K a H M a m i ĩ H o c -

r p ơ e n i i e N = I, 1960

5 M H K y m y i b , r l l A m i K e e B , ỉ l e C T a u HO H a pH b i e n p o u e c c h i n p n noMTH n e p n o - U P i e c k - n x K O i e ố a H H H x KBa3H auHeiiHbix CHCTPM K o i e ố a H H a H n p o H H O C T b n p H nepeMeHHbix Hanpíi>KeHHsrx, M3A Hayh-a, MocKBa, I960

(j Ngu ven V a n Đao, Nguven Van Đinh, A sv m p lo íic method fo r tho q u a s i-

lin c a r sy ste m s or sccond o rd er P ro ceed in g s of Hanoi P o lv lech n ical

I n s i i i u l c X 5 7 1975.

1 > H 3 1 0 M H

ACMWilTOTHqECKHH METO/J, 1 lCClEAOBAHMfl MH0r04ACT0Ti [bix

KO IKBAHHH KBA3II/mi-ini!HblX CMCTEM BTOPOrO llOPfl;j,KA

B u i m i o i i c r a r i o n p e v i a r a i i T C f l a c u M i i T O T H ' i e c K H i i MeTOA n c c a e ; i O B a m i a M H o r o -

l a c T O T H w c K o i e ố a i i i n i KBa3H i HHeí Ì Hbi x c u c T e M B T o p o r o n o p H A K a i! c T O K H b i x p e 3 0 H a H - : ' Hbi \ c.-iyii a f l x H e p B b i í í n a p a r p a c Ị ) I i o c n a m e n MCC.ieAOBaHHK) CHCTeMbi c O.HHOH C T e n e - Iibio C B o ố o t b i a B T o p o t t n a p a r p a ộ — CHCTCMbi c MHOTHMH CTeneHHMH CBOỐOAH r i p 0/1-

.'larao.Mwfi M e r o a n 0 3 B 0 i a e ' r ỈÍ3ÌỈTH rip H ố/íH > K ẽH H b ie p e u ie H H H p a c c M U T p H B a e M b ix y p a r t n c -

IIHỈI AO /KC i aeMOi i CTPITPim TO'IHOCTH.

Hcccivcd 7-.‘!-19S 1.