The nonlinear system under consideration in this paper has a specification r’hich can be stated as an interaction between the first order of smallness nonresonance larametric excitation
Trang 1V i e t n a m J o u r n a l o f M e c h a n i c s , N C N S T o f V i e t n a m T X X , 1998, No 2 (11 - 17)
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
A N D P A R A M E T R I C E X C I T A T I O N S W I T H
D I F F E R E N T D E G R E E S O F S M A L L N E S S
Ng u y e n V a n D a o
V i e t n a m N a t i o n a l Uni v e r s i t y , H a n o i
A B S T R A C T The nonlinear system under consideration in this paper has a specification r’hich can be stated as an interaction between the first order of smallness nonresonance larametric excitation and the second order of smallness resonance forced excitation In
he first approximation these excitations have no effect However, they do interact one /ith another in the second approximation
The equations for the amplitude and phase of oscillation are found by means of the symptotic method The stationary oscillations and their stability are of special interest
h e e q u a t i o n o f m o t i o n a n d a s y m p t o t i c s o l u t i o n s
et us c o n sid e r à n o n lin e a r syste m governed by the d iffe re n tia l e q u a tio n
X -f IJ 2 X — e p x c o s u t + e 2 [ A x — 2 h x — / ? £ 3 -f r cos(cj£ — 77) ], (1.1)
s m a ll d im e n sio n le ss p a ra m e te r, 1 is n a t u ra l fre q u en cy, A is d e tu n in g p a
-er, p , h , /3 , r, ĩ ], U)are co n sta n ts and overdots denote d iffe re n tia tio n w it h
t to tim e t.
/e lo o k for the so lu tio n of the eq u atio n (1.1) in the form :
X ~ a cos 0 -Ị- eu1 ( a , r p , 0) H- e 2u 2 (a, xị), 0) -f , ( 1 3 )
0 = U)t + rị), xit( a , x j j , 0 ) are p e rio d ic fu n c tio n s w ith p e rio d 2 n w ith re sp ect
h a n g u la r v a ria b le s 0 and Ớ, and a and tỊ) are fu n c tio n s of tim e w h ic h w ill ,e rrn in e d fro m the eq u a tio n s:
~ = e A ị ( a , x p ) + e 2 A 2 [a,xl>) \
~ = E l 3 ị { a , x p ) -I- E 2 B 2 ( a , i p ) +
Trang 2, h e s e e q u a t i o n s A i ( a , r ị ')), a r c p e r i o d i c f u n c t i o n s o f t h e a n g u l a r v a r i a b l e / i t h p e r i o d 2 n
S u b s t i t u t i n g t h e e x p r e s s i o n s ( 1 3 ) a n d ( 1 4 ) i n t o t h e e q u a t i o n ( 1 1 ) a n d c o m
i n g t i l e c o e f f i c i e n t o f £ l w e o b t a i n
— 2 i ư A i s i n ớ — 2 o ơ a B \ COS 6 - f cư2 ^ ' Q Q2 + u 1^ — a P c o s ( 0 — 0 ) COS 0. ( 1 - 5 )
t n p a r i n g t h e h a r m o n i c s in ( 1 5 ) g i v e s :
A i = B x = 0,
p a
« 1
2 w
1
c o s rị) — - COS( 2 0 — i/>)
3
(1.0)
C o m p a r i n g th e coefficients of e2 in (1.1) we have
— 2 u M 2 s in 0 — 2 u a B i COS 0 + CƯ2 ^ + u 2 'j = p u 1COS U)t + A d COS 6
- f 2 h u > a s i i i 6 — / ? a 3 COS3 Ớ - f r c o s Ị ớ — ( 0 + ^ ) Ị ( 1 7 )
u a t i n g t h e coefFicients of the first har monics s i n ớ and COS 6 in (1.7) w e o b t a i n
I
_ 2 _
y l 2 ( a , t/>) = - h a - s in 2 ip — - ~ s i n ( 0 + r7),
ỉ h ( a ^ )
- f ~ a — c o s 2 t p - — c o s ( tp + v ) ,
(1.8)
! 1.
T h u s , in t h e s e c o n d a p p ro xi m a t io n on e has
X = a cos 0 -f I L
2w2 cos xp — - c o s(20 — t/>)
3
l e r e a a n d tỊ) sa t i sf y th e following difTerential e q u at io n s
d a
d t
dxị)
d t
2u>
2tƯ
2
2/iwa H - sill 2 0 -f- r sin(V> -f 7/) ,
4
A + - — a + — cos 2xị) - COS( 0 + r/)
( 1 9 )
( 1 1 0 )
/ 0
Trang 3t a t i o n a r y s o l u t i o n s
)enoti Iig
c = A + — - - P a l , D = — , H = 2 h w , ( 2 1 )
ve the following e qu at io ns for stationary values do, 00 satisfying th e relations:
d
/ = H a 0 + D a 0 s in 2xpo + r s in (0 o + Í?),
0 = C a 0 + D a 0COS 2 0 0 + r co s(0 o + *l) ■
Ve t r a n s f o r m e q u a tio n s (2 2 ) in to two eq u iv ale n t ones:
/ cos Ipo - g sin -00 = { D - C ) a 0 sin Ipo + H a 0 COS ipo + r s in 77 = 0,
/ s in 100 + <7 cos V'o = sin 0 0 + [ D - f c) a0 COS rpo + r COS r; = 0
o n d it io n for r e a lit y of s in i/>0 an d COS 00 is [2, 3 |:
sin v>0 =
cos xpo =
[D2 — (H2+ C 2 ) ] a o
r [ i f s i n TỊ — (D — c) COS 77]
[.D 2 - ( i 7 2 + C 2 ) ] a 0 ” ■
(2.3)
(2.4)
«2 [(£> - Ơ ) 2 + / / 2] > r2 s i n 2 n, (2.5)
a.Q [ ( D + c ) 2 + H 2] > r 2 COS2r]. (2.6)
) S u p p o s i n g that
ve f r o m e qu a ti o n s (2.4):
r [H cos 77 - ( D + C ) sin rj]
(2.8)
Climirnating 00 we obtain:
: a 2 0 [D2- (//2 + c 2)] 2 - r2 [//2 f D2+ c 2 - 2D C COS 2r? - sin 277] (2.10)
Trang 4b) if
if t h e r e s o n a n c e curve takes the form
c = ± V D 2 - IP ,
1 b y (2.8 ) one sh o u ld have
TVi = I I cos t] — (D -f c ) s in TJ = 0, 7V2 = H s in TỊ — ( D — c ) COS r/ = 0,
•quivalent-ly,
7VX cos T] -f 7V2 s in 7/ = 0, TVj sin T7 — yv2 COS 77 = 0
ĩse rela tions give:
/ / = D s i n 2 r/, c = D c o s 2 r7
js t it u t in g these v alu e s in to (2.5 ) an d (2.6) we o b ta in the fo llo w in g re s tric tio n the a m p lit u d e at)'
- Ỉ > J 5 5 • p 1 3 )
it e A s it w ill be seen la te r, the cu rv e ( 2 1 1 ) serves as the b o u n d a ry of the
b ilit,y zone
S y s t e m w i t h o u t f r i c t i o n
N o w , let us c o n sid e r a sp e c ia l case w hen /1 = 0 an d the e q u a tio n s (2.4) h ave
5 fo rm :
{ D - C ) a 0 s i n i p 0 = - r s i n r ; ,
( D -f c)a0 cos tpo — - r COS r/
a) I f D - c Ỷ 0 a n d D + C / 0 , then the reson ance cu rve c 1 is d e te rm in e d the e q u a tio n of typ e (2.10) w ith H — 0:
lere
W { u 2 , a 20 ) = a 20 { D 2 - c 2)2 - r 2 (L>2 -f c 2 - 2 D C c o s 2 r / ) (3 3 )
Trang 5[n a p a r t i c u l a r c a s e , w h e n T] = 0 , 7T t h e r e s o n a n c e c u r v e c 1 d e g e n e r a t e s i n t o
1) T h e c u rv e c Ị : D — c (d oub le)
2) T h e c u rv e Cj* : á ị [ D + C ) 2 — r 2 = 0
n i | = ' | — the resonance cu rv e C \ degenerates into
3) T h e c u rv e C j : D — —C (double)
4) T h e curve C j : a ị ( D — c ) 2 — r2 = 0.
b) If D — c = 0 (the resonance c u rv e c 2), then fro m ( 3 1 ) we h ave
0.a0 s in tpo = —r s in TÌ => s in 77 = 0 =$■ TỊ = 0,7T,
2 / } a n cos 00 = —r cos TỊ = ± r =>■ xf>0 = a rcco s^ ± ~ ^ => a fj > r 2
c) If D + c = 0 (the resonance cu rve c3), then fro m ( 3 1 ) we h ave
7T 37T
0 a 0 c o s 0 0 = — r c o s 77 => COS t] — 0 = > T] = — , — ,
2 D a 0 s i n t />0 = — r s i n rj = ± r =>• tpo — a r c s i n ^ ± — — J => a,Q > •
S t a b i l i t y o f s t a t i o n a r y o s c i l l a t i o n s
W it h th e n o ta tio n ( 2 1 ) the e q u a tio n s ( 1 1 0 ) can be w ritte n in the fo rm :
ị S i — ị j Ị a 2 ) a s i n 2xịj -f- r s i n ( v > 4 - 77 ) I ,
a — = - C a 4- D a COS 2xị) + r COSI0 4- 77 )
ỉt u d y th e s t a b ilit y of s t a tio n a r y o s c illa tio n s w ith a m p litu d e a 0 a n d p h a se xpo
:rm in e d fro m e q u a tio n s (2.2 ) or (2.4) we in tro d u ce the v a r ia tio n s :
a = a - a 0 , ip = Ip — xjjo.
s t it u t in g these v a lu e s in to (4.1) we o b ta in
= - — j ( / / -f D s \ n 2t/)0)rt + [‘2 D a 0 COS 2t/)0 -|- r cos(V>0 + v ) } (4-2)
= - — j ( ơ + c ' a 0 + D c o s 2 r p o ) a - [ 2 /) a 0 sin 2t/j0 -I' r s ill (-00 + f?)]*/'j>
Trang 6e ơ = - ^ « 0.
a o A 2 + — H * A — — - S = 0,
2 U) 4 i o l
e A is c h a r a c te r is t ic n u m b e rs,
T h e c h a r a c t e r i s t i c e q u a t i o n for last two e q u a t io n s is
(4.3)
i*7+ = a 0 [ ỉ ỉ — D sin 2 0 0 - r s i n ( 0 o + r?)] = 4/iwao > 0, (4.4)
s = ( H -f D sin 2 0 0 ) [2D a 0 sin 20 0 + r sin(V>0 + r/)] (4-5)
-f [ c + c ' a0 -f D c o s2 0o ) [ ^ ^ a o c o s2t/'o + rc o s (t/>0 + 77) ]
T h e e x p re s s io n for s ca n be w ritte n as
s = a 0 { D 2 - H 2 - c 2 - a ơc c " ) + a 20 C ' D COS 2xp0 (4.6)
stitutiing here the e xp res si o ns cost/>o and sill 00 from (2.8) we obta in
L>a0 cos 20 0 = - C a 0 - TTjr— r 7^ r ( c - D c o s 2 i ] )
d o \ U L — l i L — (JL)
s, we h ave
2(jD»2 - H 2 - C 2 ) 5 = 2 d o ( D 2 - H 2 - c 2 ) 2 - 4 a 2 0 C C ' { D 2 - 7/ 2 - c 2 )
ớ i y
- 2 r 2 C C ' + 2 r 2 D Ơ COS 2 r / = ,
<7 d o
T h u s , t h e s t a b i l i t y c o n d i t i o n o f t h e s t a t i o n a r y s o l u t i o n s d o a n d xpo t a k e s t h e
n
d W
ơ ã Q
3 resotnance c u rv e ( w = 0) d iv id e s the p la n e (d o ,cư) into re g io n s, ill each of ich t h e e x p re s sio n w hag a d e fin ite sig n ( + or —) If m o v in g up a lo n g the
Trang 7it lin e p a r a lle l to tlie a x is d o , we pass from a region w < 0 to a reg io n w > 0,
t the p o in t of in te rse ctio n betw een the s tra ig h t lin e and the reson ance c u rv e
r iv a t iv e d w / d a o is p o sitiv e So, th is p o in t co rre sp o n d s to a sta b le sta te lla t io n if M > 0 an d to an u n stab le one if M < 0 O n the c o n tra ry , if
is fro m a regio n w > 0 to a region w < 0, then the p o in t of in te rse ctio n ponding to a sta b le sta te of o s c illa tio n if M < 0 and to an u n sta b le one if
h is w o rk w as fin a n c ia lly su p p o rte d by the C o u n c il for N a t u r a l S cie n ce s of
i m
R e f e r e n c e s
lit r o p o ls k i Y u A , N g u y e n V a n D a o A p p lie d a s y m p to tic m e th o d s in n o n - fiear o s c illa t io n s K lu w e r A c a d e m ic P u b lis h e rs , 19 9 7
guyen V a n D a o In te ra c tio n of the elem ents c h a ra c te riz in g the q u a d ra tic
o n lin e a r it y an d forced e x c ita tio n w ith the o th er e x c ita tio n s J of M e c h a n ic s
o 4, 19 9 7
g u ye n V a n D a o In te ra c tio n betw een the elem ents c h a ra c te riz in g the forced
nd p a r a m e t r ic e x c ita tio n s V ie t n a m J o u rn a l of M e c h a n ic s , N o 1 , 19 9 8
R e c e i v e d N o v e m b e r 1 5 , 1 9 9 7
T Ư Ơ N G T Á C G IỮ A C Á C K ÍC H Đ Ộ N G T H Ô N G s ố V À C Ư Ỡ N G B Ứ C
CÓ B Ậ C B É K H Á C NHAU
ự tư ơ n g tác g iữ a k íc h động thô ng số kh ô n g cộng hư& ng có độ bé bậc m ột
ch d ộ n g cư ỡ n g bứ c cộng h ư ờ n g có độ bé bậc h a i đ ã đư ợc k h ả o sá t ơ x ấ p
r n h ấ t c á c k íc h động n à y k h ô n g g â y ra h iệu quả Song c h ú n g tư ơ n g tá c lẫ n tro n g x ấ p x ỉ thú' h a i C á c dao động dừ n g v à sự ổn đ ịn h c ủ a c h ú n g đ ã được iệt q u a n tâ m n g h iê n cứ u