Multifrequency excitation of parametric oscillations of dynamical systems

CÔNG HÒA XÃ HỘI CHỦ N G H ĨA VIỆT NAM

VIỆN KHOA H Ọ C VIỆT NAM

ACT A MATHEMATICA

VIETNAMICA

N " I

HÀ NỘI — 1 9 8 0

ACTA MATHEMÁTICA VIETNAMICA

T O M 5, N° 1 (1980)

M U L T I F R E Q U E N C Y E X C I T A T I O N O F P A R A M E T R I C O S C I L L A T I O N S

N G U Y Ễ N V À N Đ Ạ O

I n s t i t u t e of Mechanics, Hanoi

L e t US c o n s i d e r I h c p a r a m e t r i c o s c i l l a t i o n s o f n o n l i n e a r s y s t e m s d e s c r i b e d

b y d i i f c r e n l i a l e q u a t io n o f f o r m

X + w5 X + E 2 f k ( * ) sin0k + e Q (X, X) = 0 (0.1)

k 1

h e r e 0k = v k / + r k, k = 1, a n d v lt v r a r e p o s i t i v e i n c o m m e n s u r a b l e

n u m b e r s , p k ( X ) a r e p o l y n o m i a l s o f x t Q (£, x ) — n o n l i n e a r f u n c t i o n o f X f X

A lo t o f p h y s i c a l a n d t e c h n i c a l p r o b l e m s a r c l e d to i n v e s t i g a t i o n o f e q u a ­

t io n (0 1) F o r ' i n s t a n c e , o s c i l l a t i o n o f / a p e n d u l u m w h o s e s u s p e n s i o n p o in t a c c o m ­

p l i s h e s I h c c o m p l i c a t e d o s c i l l a t i o n s a n d t h e t r a n s v e r s a l o s c i l l a t i o n o f b e a m u n d e r

th e a c t io n o f c o m p l i c a t e d l o n g i t u d i n a l f o r c e a r e d e s c r i b e d b y e q u a t i o n (0 1)

I t i s s u p p o s e d t h a t t h e r e e x i s t s a r e s o n a n t r e l a t i o n o f t y p e

w h e r e Q > 0 , a n d c * , , c* a r c i n t e g e r s T h e n u m b e r n m is d i f f e r e n t f r o m z e ­

r o , b u t s o m e o f c* m a y b e d i s a p p e a r e d A r a t h e r s i m p l e p r o b l e m w h e n Q ( X , X) ES

33 0 p k (a :) = X h a s b e e n c o n s i d e r e d b y I s c h u c h v v a n d I a s c h u c h V T [ 1 ]

T h i s p a p e r c o n s is t s o f t h r e e s e c t io n s

I n s e c t i o n 1 a g e n e r a l t h e o r y o f c o n s t r u c t i o n o f a p p r o x i m a t e s o l u t i o n s o f

e q u a t io n (0.1) b y a s y m p t o t i c m e t h o d o f n o n l i n e a r o s c i l l c t i o n s is g i v e n

S e c t io n 2 is d e v o t e d to i n v e s t i g a t i o n o f m o n o f r e q u e n c y r e s o n a n t c a s e w h e n

th e r e l a t i o n

n* D + cm V = 0

s b

t a k e s p l a c e I t h a s b e e n f o u n d t h a t t h i s i s t h e m o s t i m p o r t a n t c a s e o f m u l t i f r e ­

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

t e r n a l e x c i t a t i o n w i t h r e s o n a n t f r e q u e n c y ( v s) h a s i n f l u e n c e o n t h e p a r a m e t r i c

o s c i l l a t i o n T h e i n f l u e n c e o f o t h e r n o n r e s o n a n t c o m p o n e n t s is f o u n d o n l y in the

s e c o n d a p p r o x i m a t i o n

In s e c t i o n 3 t h e m u l l i f r e q u e n c v r e s o n a n t c a s e w h e n

n* Q + c* Vj + + c* v r = 0

is t r u e w i l h m o r e t h a n o n e o f Cj, C2 c, is c o n s i d e r e d I n t h i s c a s e t h e i n f l u e n c e

o f e x l e r n a l e x c i t a l i o n o n p a r a m e t r i c , o s c i l l a t i o n is f o u n d o n l v in Ihc* s e c o n d a p ­

p r o x i m a t i o n

§ 1 C O N S T R U C T IO N O F A P P R O X I M A T E S O L U T IO N S

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

e q u a t i o n (0.1) is f o u n d u n d e r t h e a s y m p t o t i c e x p a n s i o n [ 2 ] :

X = Cl C O S ( ~ 4 * I f ) -4- U \ ( g z - f - v |\ 0 ) -Ị - E'U'J ( f l , c, Ỷ “t"

w h e r e

0 = (01 02 9***9 0r)«

a n d t h e a m p l i t u d e a a n d phase* a r c d e t e r m i n e d f r o m t h e e q u a t i o n s

d a

( 1.2)

rf/

d /

= eAi (a, ty) -f e2Am (fl, V) + e3 +

== to — Q -f- eJ5.(a 'lị') ~j- 6~z?2 ((li V) "4" "T •••

F u n c t i o n “ s a t i s f i e s t h e c o n d i t i o n

- ệ - = ĩì, n * i + c* e, + + c ; e, 3 0

al

( 1 3 )

(1.4)

S u b s t i t u t i n g Ihc e x p r e s s i o n (1.1) i n t o (0.1) a n d c o m p a r i n g t h e c o e f f i c i e n l s

o f E F.5 w c o b t a i n t h e f o l l o w i n g e q u a t i o n s

QJ- - « 2 i o v k - 4 - / V i V ^

(h) + Q ) fl -p 2 u ).-li] s i n ( I + yf) — (u) — Q ) -— - — 2 CO aB

r

C = 1

Ĩ

2u) v k — “ — — + 2 2 VjVk —— ^ - h ID u 2 =

^ r i O0ka ( | + T|)) k4 ^ j O0j 9 6u

I flS] n oB] , n , n , ii-Bj •

c o s (g + \t0 —

(1.5)

A, Ỉ à l + B ,

òa

ồ Cl

Ồ A ] flip

ồ\p

a B \ - 2u, aB2 + (u) - Q) iA2

s i n ( I + r|>) —

01]>

ờtị>

c o s ( I + 1|1) +

(1.6)

w h e r e

R°2 = U )

-t-ỔX.

2 A, 2 V

k-1

i4|C0s ( r 4 - T|v) — nB]s in ( ; 4- \ị>) 4- to

a 2ỉij

Ỡ -Iil , ễ) t

k -h 2u ) / li

— - - + 2 v k

« (E + V ) k- 1 Ồ0k j ồ2ỈIl

òF\

àX'

+ (d) — Q ) ò B i ỜU1

~p í(J) — Q)

ô ( ; + i p ) ô a Í T Ỉ Ô M ( | + * | > ) ô d - H 1)' 8.4, ÍU , li F 0 = _ V p k ( x 0) S in 0 k - Q ( x 0 i 0)

(1.7)

ô\ị) ô (s+ T ị> )

x0 = a c o s ( ; + \|ĩ) x0 = — U) a s i n ( I + yp).

L e t u s e x p a n d p k ( £ 0), 0(.To, x 0) in t h e F o u r i e r s c r i e s W e f i n d

P k ( x 0) = P k [ a c o s ( ỉ + V ) l = 2 ,j0km ( a ) c o s m ( Ê + V ) ’ Q ( x0 * o ) =

m ^ o

= Q [acos(; - f v>) — uiasin(; + v ) ] = y \ [ f m( a ) c o s m ( l + y ) + g m( a ) s i n m { l + V ) ) , (l.S)

here,

Dk

27Ĩ

0

P k(acosQ>)dO,

2 7 C

o

27T

27T

f m(O-) — Q(acosO, — U)asin0)cos77i0do

o

20Ĩ

g m( f l) = - ỉ — f Q ( a c o s O , — a )a s in O )s in /7 7 0 d < I>

* J •

o

W e shall find the function U\ in ex p an d ed fo rm with u n de fin ed coefficien ts

“ 1 = 2 1 u l n c i c r s i n [ n ( s + + Cl01 + - + C r 0 r l +

T o d c le r m in e the unknow n cocl'ficicnls Hlnci t , ulnci -c and functions A'l, B\ WE

substitute exp ression s (1.8) (1.10) into (1.5) and we com pare the co e ffic ie n ts of

sin [ n( l + lị1) + Cj6i + + c,0r], COS [n(; + Ip) + C)01 + C ọ 0 2 -f + c,.0r] By su bs­ tituting (1.8) a n d '(1.10) into (1.5) we got

z fa)2 - ( * « * > + 2 ci v ')2] Í u lnc, cr sin [*»(£ + V) + - + cr «r] + t>lnc,

n * C l C r i = l

Cr

c o s [ n ( | + T|0 - f Cifl] + + cr0r] I = ( ill — Q ) a — — 2 u ) A j

Ỗ1|5

s i n ( i + ty)

(,„ _ Q ) _ 2m flB COS (; + 1|’) — 2 [/m (a )eosn i<;+ijO + <7„,(fl)sinm(| + i|>)J—

m^>0 1

2 2 A m (ư) j sin Ị m ( | + t ị ' ) -}-0,] — s in [m ( ; + t|') - 0,] I (1.11)

i = l m i o

§ 2 M O N O F R E Q U E N C Y R E S O N A N T C A S E

In t h i s s e c t i o n w e c o n s i d e r t h e r e s o n a n t e a s e w h e n t h e r e e x i s t s a r e s o n a n t

r e l a t i o n

w h e r e /?*, c* , a r c i n t e g e r s , 1 < s < r, n * c* < 0 C o m p a r i n g w i t h (0 2) w e h a v e

T h e la s t t e r m s in ( 1 1 1 ) w i l l c o n t a i n s in ( I 4- 1|') , c o s ( | + \Ị)) i f

( m + 1) ( I + \|’) 4- 8i = X.

O n t h e o t h e r h a n d f r o m (1 3) w e h a v e

n*(l + V) + c* 0S = n*vp.

Ti n- c o m p a r i s o n o f f o r m u l a e (2 2) a n d (2 3) g i v e s

( 2 2 )

(2.3)

o r

m = 4- 1 -4- a n * i = 5, 4- 1 = Ơ c , X = ơ n * \ị'

s

(2 4)

S in c c D sm = 0 i f m <c 0 a n d n * c * < 0, D n* = 0 , t h e n w e h a v e t h e

s Sj ■— ■ — ]

f o l l o w i n g t e r m s c o n t a i n i n g s i n ( I + rị>), c o s ( I + \p) in (1 11) :

> D («u - Q ) a + 2 wAi

ÔTị)

sin(Ê + Tị)) — (u ) _ Q ) - Ì í L l _ 2 t aaB1

a Ip

= /’i ( a ) c o s ( i + r j j ) + 3i ( a ) s i n ( ! + 'l’) + _ L Z) n

2 D>, n + 1 s i n i ( | + r | ) ) - — TịỊ Í c* J

2 s ' - T - + 1

+ 0 D s _ J i

si n

c o s ( | + ự) =

( I - H 0+ V

c s -J

/ V \ n *

(* + v>) + , V

F r o m h e r e w e o b t a in

2 (I)A] + (to — Q ) 0 -

*-ờ\ị)

= 0l(« o + “ T / ỡ s " • + / ) n* - 0 n * \ c o s - ^ - T | \

■ c V ' 1 ** ' c * “ 1 /

2to flB j — ((.) n ) — —1 «=

ft\p

= /*l ( f l ) + - - / D n * -f- Z) n * + D n * \ s in — — Ap (2 5 )

I f (1) — Q = 0 ( e ) t h e n w e h a v e

(2.G)

uv

U)

B v c o m p a r i s o n o f Iho h i e h c r h a r m o n i c s in ( I n ) w e g e t

» M = 0 Cr = « - - 5 - ( 0 V " + 1

-c, = « = - ' " <0) " + '

J Z L A n ( f l ) ( i + 5)

- ( J 1 U ) + v , ) - ] u l n c i = 0 Ci_ , == 0 Ci = l C I + I = 0 Cr - 0 2

= — Din (a) ■(* s)

— (n u) — V i)-] u ln c i _ 0 Ci_) = 0, Ci = - 1 , c - r 0, Cr — 2

= - — Ds „( a)f n =/= Ị -4— ĩ “ )

Ị u > ” — H u ) - f v s ) " l H l n C l = 0 , C r -1= 0 C s = 1 Cli+l — 0 C r — 2 \

Ịu)'J — (n il)— Vs) ] u ln C ) = 0 CS-1 = 0 C i = —l.c * + l = 0 , C r - 0 2 \ c t >

(2 7 )

A n d t h e r e f o r e w c h a v e

J o / V I s - ' i -Ĩ o n ( a ) s i n n ( Ỉ + ^ ) + fn(Cf) COS n ( ; + ^ ) ]

1 " ^ £ > in ( g ) s in [ n ( | + ^ ) + Bi I +

+ T ^ ' (nu> + Vi)2 — u>:

i = 1 n> 0

i=£=s

J ^ D\n(o) s i n [/?(! -4- ty) — 6i]

2 i = l n > 0 0) - ( n c o — Vi)

i=/=s

, J y - ' D s n ( f l ) s in [ n ( | + \p) + 9s]

n=7=l H—

c

n > 0 s

+

+ - - Z - - _ -ỳ - i í 1 -— (2.8)

n «= — IL _ q i 1

c

T o d e t e r m i n e t h e f u n c t i o n s ^ 2 H> \vt* c o m p a r e t lie c o e f f i c i e n t s o f s in ( ; + \|>)

c o s ( l + ty) in ( 1 6 ) F i r s t w e w r i t e /?•> in th e e x p a n d e d f o r m

•K = 2 f ^ ° i n c i c c o s Í n ( * + 'íO + C1 0 1 - f ••• “i~ c r0, ] +

n c i c r

^ ° ° n c i (*, s *n f n ^ + V) + c,e, + + Crflr] ị (2.9)

w h e r e

23T

fi°io o = " 2 jT I* J R " d ( ỉd + 'I’)

o 27T

0

Zi9i n r = ~ f t COS [ n ( | + ẹ ) + c , e , + d ( | + \|5) de, der

r>0

22nC]«**Cr

0

27T

= J R° s i n [ n ( t + If) + Cj6 + c , 0] d ( l + T|))de, d0r

0

t V io I p r m c P f i n t Ct ir» i n Ơ ( * _i_ I h \ c i n ( b -L n n i T P c n

I n (2 9) t h e t e r m s c o n t a i n i n g COS ( ; 4- \|>), s i n ( I + lị)) c o r r e s p o n d to t h a t n

a n d Ci, cr l o r w h i c h

n (l + ẹ ) + c,0,+ + cr0r = ± ( i + T|0-+ a

O n t h e o t h e r h a n d h a v e

n* (l+'p ) - 4 - cmes = ns • \p.

F r o m h e r e it f o l l o w s t h a t

/I + 1 = n /1#, Cj = c2 = = Cs- 1 = Cs+1 = = Cr = 0 ,

ịi = , a = |i n* lị).

c #s

T h u s w e c a n w r i t e R ° (2 9) i n t h e f o r m

u°2 = c (a, \|>) c o s (t + \|>) + s (a, <p) s in (|+ t |> ) + (2.10)

w h e r e d o t s d e n o t e t h e t e r m s w h i c h d o n o t c o n t a i n s i n (i + 'lO, COS ( t + v ) a n d

c (a, rp) = 2 Í ( fi02M !+ + R ° ỉr ] cos » n# V + (R0? 2 + + >sin I*

S i a i p ) = 2 [ a < ° r - H ° £ + ) s i n ^ V + i R ' E * - K ° £ - ) c o s i | > / ! > ] ;

R O Ị i : _ _ w o

21 2 1 t l + P-n , C| = 0 C5_ 1 = o, Cs = H r * , C»41 = 0 c r = 0 , RO

21 21 , - 1 + JJ n * , Cl = 0 c.s— l = 0, Cs = ụ ĩ C&+1 = 0 , Cr = o .

= R °

22 2 2 , 1 - f f-ifl* , Cl = Ò Cs- 1 = 0, Cs = Cs-fl = r f r — o

/^O ỊX

2 2 2 2 , — 1 + JA 7Ì , C ] = 0 c S_ 1 = 0 , C s = r ? f s - H = 0 , , C y = o . (2.1 J)

S u b s t i t u t i n g ( 2 1 0 ) in t o ( 1 6 ) a n d c o m p a r i n g c o e f f i c e n t s o f C O S ( I + ty)

s in w e o b t a i n

2 (i) A 2 + (£2 — U)) a

2a <t> iJ2 + (£2—“>) ■

ò tị'

I f (0 — 0 = 0 ( e ) t h e n w c h a v e

• > 1 D I A a ^ i u ồ

= 2 * 1 ,5 ] + o A i — 1— h Q

a,4 j Ồ Aj

- = 4- i j

-ồ 4- S ( a , \ | ) )

B, _ a n ? - C(a,x10

a 2 = Ị

-2ci) 2A\B] -f- CL'l] dBj

0 \|5

4- a B j -B fi,

a\(j

= 1

2 ail) [a , 8 A lA a + B ,

Ổ.4.1 Btp

a B \

T h u s i n t h e f i r s t a p p r o x i m a t i o n \ vc h a v e

/ c s

X = a COS ( — — es + V)

n*

(2.13)

h e r e a a n d Tị! a r c d e t e r m i n e d b y e q u a t i o n s :

' + D _

— = — 0] ( a ) +

c o s —— \|) ,

c s dtị)

dl

, - — — 1 s, — — — + 1 I

/2

sill —— rj- c*

(2.14)

T h e r e f i n e m e n t o f t h e f i r s t a p p r o x i m a t i o n is

X a c o - ( — p J - t t W e l V f f n ( « ) s in n ( E + ^ ) - f / ~ n ( a ) c o n / i ( E + ^ )

I _Ị_ D m ( f l ) s i n [ n d - M l O + f l j ] J ]_ Z)jn( Q ) s i n [ n ( | 4 - ty ) — flj]

1 D sn( f l ) s i n [ / i ( | + I ]_ 7) ,s n ( f l) s iD [ / ? ( |4 - t |0 — fls] t

( n it ) + \ \ s)2— Cf)2 2 / ) = - ! +

/1 ^ 0

y A „ ( n ) s i n [ /1(E + Tị>)— B,] } ( 2 1 5 )

* u>2 - (n u )— v s) s

« = f = - - V ± i

C5

n ^ 0

in w h i c h a a n d \p s a t i s f y t h e e q u a t i o n s (2 14).

I n t h e s e c o n d a p p r o x i m a t i o n f u n c t i o n X is o f f o r m (2 15) b u t a a n d \|) a r c

d e t e r m i n e d b y t h e f o l l o w i n g e q u a t i o i i S

= - i - g , ( a ) + (Z), J l l _|_ J — —— 1 - / J , , _ + , ) c o s rp

e

2(0"

2Ai Bi + a A j + a B

d\p

dt

Cl)- Q + _ £ _ / - l ( a ) + _ I _ ( D „

c:

+ 0

+ s (a , yp)

, + D

+

2 a U)

ồAị

ỒQ + B , ± iL _ air _ c (a, v>)

V n*

)sin-— 115-f

c ?

(2.16)

A s p e c i a l c a s e

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

t h e p a r a m e t r i c o s c i l l a t i o n o f n o n l i n e a r s y s t e m o f t y p e

w h e r e Xi a r e c o n s t a n t s

u s e 01 t n e g e n e r a l i n e o r y d e v e l o p e d in t ni s

^ i l l a t i o n o f n o n l i n e a r s y s t e m o f t y p e

r

' Ì + + E I 2 Xi s i n 0j + e Q (X, x ) = 0

1 - 1

;tants.

(2.17)

L e t u s c o n s i d e r t h e s i l b h a r m o n i c r e s o n a n t c a s e , f o r w h i c h

2Q — v s = 0, u>* = Q2 - f e A

h e r e A — d e t u n i n g o f f r e q u e n c i e s

Bv s u b s t i t u t i n g n* = 2, c* = — 1, P s( x ) = XRx,' P s ( a c o s O ) = X5 a c o s O i n (1.8)

s

w e g e t

D,,1 = x 5a „ D 6,0 = 1K, U = 0, n > 2,

a n d t h e r e f o r e t h e f o r m u l a e (2.6) b e c o m e

:U>

B] = — Ỉ— f i ( a ) - — x ss i n 2\|).

F o l l o w i n g f o r m u l a e (2.13), in t h e f i r s t a p p r o x i m a t i o n w e h a v e

X = acos — ( v st + r s) + T|)

h e r e a a n d \p a r e d e t e r m i n e d f r o m e q u a t i o n s ( 2 1 4 ) :

2 7T

= — -— I s i n (l) Ọ(acos<1\ — li iasinO) d o -t— — Xsa c o s 2 r | \

u

12 7T

— UJ - — -f - -— f cưs(1> Ợ(acos4>, — m a s i n O ) d<|) - — Xs

(2.18)

(2.19)

T h e r e f i n e m e n t o f t h e f i r s t a p p r o x i m a t i o n is oi' l o r m (2.15) w i l n I = 0„

X = a c o s 0S -f H’j - f fc I — / o ( a ) y , / ' n C o n n ( ; + \p) -Ị- g n s i n n ( I - f v>)

n > 2 «*<n«-l> “

r

i = l

s i n 0, + 0, + V'j

V i ( v s + V J

r Xi s i n ị - ị - 0S — ©i - h

i = l

i H = s

V i'( V K — V i)

(2.20)

in w h i c h a andiị ; s a t i s f y e q u a t i o n s (1.5).

§ 3 M ULTI F RE QUE NCY R E S O N A N T CASE

I n t h i s s e c t io n i l is s u p p o s e d t h a t m o r e t h a n o n e o f c * c* a r e d i f f e r e n t

F r o m z e r o , SC) t h a t w e h a v e a r e s o n a n t r e l a t i o n

I n t h is c a s e it is e a s y to s h o w t h a t th e l a s l s u m s in ( 1 1 1 ) d o n o t c o n t a i n s i n (£ + t|>),

c o s ( ; + 1Ị)) I n f’a c l, t h e s e t e r m s c o r r e s p o n d to t h e n u m b e r m s a t i s f y i n g r e l a t i o n

+ ty) d r 0i = + ( I 4 “ ty) + ^ ( 3 2 )

B u i o n t h e o t h e r h a n d f o l l o w i n g (1 4 ) w e h a v e

n*(X "t" V ) H“ C ] fli + ••• c r =