Non-linear oscillations of third order systems

b If the characteristic equation has a negative root and a pair of imaginary roots f]sO then we have the critical case.In addition,if Vr pQ/q here p,q are integers,then we have the criti

Proceedings of the

V I I I u

International

Conference on

Nonlinear

Oscillations

PR A G U E 1978

l a t l i t u t * o f T h c r m o m c c h a o i c *

C z e c h o s l o v a k A c f td c my of S c i c a c t *

N O N -LIN E A R O S C IL L A T IO N S

OF T H IR D ORDER SYSTEMS

Ng uyen Van Dao

N ational Ce nt er o f S c i e n t i f i c Re se a r c h

of V i e t n a m , Vien k hoa hoc V i e t n a m - Hanoi

f r e e o s c i l l a t i o n , s e l f - e x c i t e d o s c i l l a t i o n , f o r c e d o s c i l l a t i o n an d p a r a ­

m e t r i c o n e o f t h e d y n a m i c a l s y s t e m d e s c r i b e d by t h i r d o r d e r d i f f e r e n t i a l equation (1) in the critical case are examined systematically by asymp­

Introduction

A l o t o f m e c h a n i c a l and p h y s i c a l p r o b le m s l e a d t o s t u d y o r o s c i l ­

l a t i o n i n t h e 3 y ste m g o v e r n e d by t h e e q u a t i o n o f t h i r d o r d e r

The following definition is accepted

* 2 = - rị + i Q t ^2 = -rj-in ,then we have the non-crỉtical case

b) If the characteristic equation has a negative root and a pair of

imaginary roots (f]sO) then we have the critical case.In addition,if Vr pQ/q here p,q are integers,then we have the critical resonant case

The non-critical case has been investigated in many publications

(see,for example [1-6] ) but the critical case has not,to the author#s knowledge,been examined hitherto.In this work we atudy the non-linear oscillations of the third order system (l) in the critical case.We shall find the family of two parameters particular solution of equation (1) which has strong stability property.To assess the validity of the theory

a series of experiments on the analog computer have been made.The theore­ tical results show a good agreement with the analog computer ones

§1 Non-linear Oscillations in third order autonomous Systems

I»et us consider the following equation

We shall find a partial 'two parameters solution of this equation in form

517

<p « n t *4/ Here a find ^ r r c f u r r t i o ’ip 3 f i t I s f y i r p , tbc f o l l o v i n r , e q u a t i o n s

£Y~ = £ A ^ ( a ) 4 € 2 A ^ ( » ) + C ^ i n ) + e ? B ^ ( a ) f ( 1 3 )

S u b s t i t u t i n g ( 1 > ) , ( ! 3 ) i n t o Í 1 1 ) Zi\r\ c o m p ^ r i n c t h e t e r m s c o r -

tal.nỉn.-r £ rrit1 r ircp , co~vp ’.ve jrt

c a ' ( 4 2+ Q2)' ’ “ 1 ? n , - ’ ( ỉ ? + a 2 )

' - _ ^ 1 Tj 1 D — -‘ 1 - — J I

■|= " n ( | a+n*) ’ ? a i , ( ^ +

A ( 1 - n - M l + m n ) n.2 ( 1 - T , ? ) ( | ? H tn V )

w h e r e r , h , u „ _ , V , r r t h e r o f f f i c i m t r o f ĩ > i » r i ế > r CVỈV n r : ‘ o r : - :

R ( a c o s c f , - i l a s i n < f , - i l ' « ' C C i < p ) s > H h P i n n <f ) , ( Ì ^ )

u I » — rr ( U- oosrrcP H V : : 1 n n < P ) in ^ * n

F o r t h e P u f f i n ; c.-ire R a ~ f i y'^ ,ve r p v e

dt = 2 f ’ fli ? a p 1 ’ p = — -ji— .

r 4 ( 1 + n 2 )

I f fi > 0 t h e 'imp] i t uric o f OGCÍ J.lrt.1 on i n c r e - i f £ ir t l x n e T f 6 < n th(r

" j p p l j t u f ’ e J e c ^ e r s e a

F o r t h e Van d e r To l c.-ise ii = ( 1 - x ^ ) A t h e e q u a t i o n s o f t h e f i r s t approximations are

X * a c o s <p -*■ - 3 “ - R—( 3 0 0 3 ? ^ - i s i n ? ( p ) , ( 1 * 7 )

The stationory amplitude is p- s 2.The oncillation d1f.0 rnms on ftiialoi; - computer are presented on fi£ ! for ( a A s 1 , 6 s ' 1

$2 N o n - l i n e a r O s c i l l a t i o n s i n t ' l i r d o r d e r n o n-cT utononous S y s t e m s

Now v:e s t u d y t h e o s c i 11/ i t i o n s o f t h e fjyiitem g o v e r n e d by e q u a t i o n

X -»• fc * + n ? x + Ị a ? X s t \ ( x f 5 : ,x ) + i P c i m f t , ( 2 1 )

n = s f €6~

The p a r t i ' l l p e r i o d i c s o l u t i o n w i t h p e r j o d 27C / S o f e q u a t i o n (2.1)

i s f o u n d i n t h e form

X « r c o s ( * t + tị») + g u - t i * , + , i t ) + e2u ^ ( a , tị/ , S t) , ( 2 2 )

v/he>*e u ( a , 4 » t i t ) i r e p e r i o d i c f u n c t i o n s w i t h p e r i o d 21T r e l a t i v e l y

end 'St r n d n , Ỷ fire d e t e r m i n e d from t h e e q u a t i o n s

~ * e A1 ( , 4/ ) + £ 2 A2 ( p , 4>) + , e 3 , ( p , 4» ) + £ 2 fl2 ( a , <|> ( 2 3 J Ô.V substituting (2.2), (2 3) into (2.1) and comparing the coeffici-

o n t r o f £ '.rid f l i n cp , COS cp we o b t a i n a f t e r s i m p l e c a ] c u l G t i 0 n 3 :

r ^ s i n f - i c o s ỷ ) - » r n d L r 21 t B H i a * <0 0« l l z i r , 1± í ĩ 21 J ( 2 4 )

u - _ l l l m -'n<rI Hi 2 >> " o 0 2m v 1m- mifro o 1m 1m — o 4 * r2m 9 9 o o •

h e r e r i1f r0 , u tV- r e i - o u r i e r c o e f f i c i e n t s :

n u , 8 - x - f r x = y ( r 1 n c o s n < p + r ? n s i n n < f ) , ( 2 5

n s c

I ^ s 2 _ ^ ( u 1;noos>n<? + v ^ n i n n u p )

-'he s t n t t o n r w y s o l u t i o n o f e q u a t i o n s (2.3) i n t h e f i r s t a p p r o x i m a ­

t i o n Ì 3

7 * r i i + r h - - •

519

-+ i r 2 1 ) ] > , > 0 (2.6)

F o r t h e D u f f i n g c a s e R z - p x ^ v/e h a v e t h e f o l ] o v ; i n £ e q u a t i o n o f

r e s o n a n c e c u r v e

The stability condition of the stationary solution ia of form

V = 1 + ± £ a 2 ± %/ i - jL a 4 , P r — * p = -0 ~r~ ' ( 2 - 7 ;

a 2 r V * n 2 a ( fc2+ a ) r M i W )

and t h e n e c c e s c r y c o n d i t i o n f o r t h e s t a b i l i t y o f s o l u t i o n i s 0

F i g u r e 2 i s p r e s e n t e d f o r f r 1 U , f l * 1 , p * x - 1 0 ^ a nd ( c u r v e 1) , p*= 1 , 2 5 1 0“ k ( c u r v e 2 ) The f a t l i n e s c o r r e s p o n d t o t h e s t a b l e s t a t e o f

o s c i l l a t i o n

For the Van der Pol case Rs s(1-x^)x the equation of resonance curve is

V = 1 + -- J -

A*1 , v 2= 1) » Ja4/81 (curve 1), J a 4 / 2 7 (curve 2 ) , J s 8 / 2 7 (curve 3 ) The unstable branches are presented by hatching

ự ị - u - 1 ) 2

Js

c )

( | 2+ a2)p2

4 a 2 I 2

( 2 . 8 )

520

-The p r o c e s s o f f r e q u e n c y e n t r a i n m e n t h a s b e e n i n v e s t i g a t e d and

p r e s e n t e d g r a p h i c a l l y i n f i g 4 T h i s f i g u r e show s t h e i n t e r a c t i o n b e t ­

w e e n t h e f o r c e d and s e l f - e x c i t e d o s c i l l a t i o n s i n t h e n e a r r e s o n a n c e

z o n e F o r e x a m p l e , o b s e r v i n g th e o s c i l l a t i o n diagram v/hen i n c r e a s i n g th e

e x c i t d r v f r e q u e n c y Í we s e e f i r s t t h e b e a t I n t h e i n t e r v a l

t h e brat d io a o p e ^ r s :\nd t h e r e i s o n l y t h e h a rm o n ic o s c i l l a t i o n v / i t h

e ' v c i t i n ~ f r e q u e n c y I n c r e a s i n g m o re y we s e e t h a t t h e b e a t a p p e a r s

r.~a i n A s i t i s Sfcf.n i n t h e f i g u r e 4 t h e f r e a u e n c y e n t r a i n m e n t z o n e ỈS

n r r r o p r t h e r t h r t i n t h e s e c o n d o r d e r s y s t e m Ị l ừ Ị T h i s phen o m en o n

war: o b s e r v e d on n n a l o ' ' - c o m p u t e r t o o

fig 4 r

3 Non-autonomous Systems (continued)

Parametric Ocnillotion of third order non-linear System

? h e f o l l o w i n g e q u a t i o n

i X - Q2 X + l a ' x + e [ k x 3+ h7?+ R ( x , s , x ) - C X C 0 3 t f t ] s O , ( 3 1 )

£A * n 2 (1- f|2 ) , r ỵ s t / ĩ a hns b een investigated.The partial two parameters

s o l u t i o n o f ( 3 1 ) i s f o u n d ỉ n t h e s e r i e s

X = -COof | t + i | o + e u , { n 9 | t ) + «2u2(n, 4 > , | t ) + ( 3 2)

521

ẫ t :7 ^ [8(k- ỉ - o c o ^ - §f l : , i n?+H n 3 , ( %3)

& ■ Ĩ T ĩ r ^ - [ í ( í 2 ^ 2jA“ 4 f , « l : + Q - V ) r ^ r , i r 2* - g | c o c ? Ỷ +R , ] ,

H1 » <K0C0!5cf> + | l <E0::in4>> , r?2 » ự < v * 0 = * > - < V s i ” *F> •

HQ s H(acor «f , - «^o.Rin<p ^ ^C03<f ).

Tr r i - 5 t h e a'Tip'J' t i l de c u r v e '»£: p i " tic.*? f o r t h e n , * i u 2 0 , i s i l « 1 j

c « O C5 1 k s -0 1 iinri h s C ( c u r v e 1) , ‘'l * • r> (c irv G 2) /’.rid h * c 1

( c u r v e 3) 1 h e r e k s e ’ / n , h s e h / a ; , c * C c / s i .■

v/h*re a, Ỷ satisfy

T h e i n f l u e n c ei ỉ i ; i J J .'.uv:.V /V * */ -J u L u ' u oulo^b 'Tri.ct.ionI J J v> 1 « u 11 u 1 1 r.i;~nx on O'-«’%rv“i?t?,i c

I t ’t i c n was s t u d i e d , i n t h i c c.-tco t h e r e s o n a n c e c u r v e h i ư c 1 r r e rorr»

S e e r : l £ j b f o r t * f l s 1 , h s 0 l 5 , k s - 0 , 1 , c = 0 ' o .'VV? ! j * a 2 , 5

( c u r v e l ) , h * s 5.10*; ( c u r v e 2 ) t y * s Z \ / t ! •

a

0.3

ft 2

Ỡ./

f i g 6

5 22