Bài tập cơ học giải tích

NONLINEAR OSCILLATIONS OF THE THIRD ORDER SYSTEMS PART m... Nguyen Van Dao4.

Journal o f Technical Physics, J Tech P h y s 21, 2, 253 -2 6 5 , 1980.

Polish Academy o f Sciences, Institute o f Fundamental Technological Research, W arszawa.

NONLINEAR OSCILLATIONS OF THE THIRD ORDER SYSTEMS

PART m PARAMETRIC OSCILLATION

N G U Y E N V A N D A O ( H A N O I )

In tro d u c tio n

"he t h e o r y o f 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 t h e s e c o n d - o r d e r s y s t e m h a s b e e n i n -

v e s t g a t e d in a l o t o f p u b l i c a t i o n s F o r a l o n g t i m e i t h a s p l a y e d a n i m p o r t a n t r o l e i n t h e

th eory o f n o n l i n e a r o s c i l l a t i o n s R e c e n t l y , i n s o m e p r o b l e m s o f t h e d y n a m i c s o n e c a n

m ee 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 t h e t h i r d - o r d e r s y s t e m [7], t o t h e s t u d y o f w h i c h t h i s

charter is devoted (cf [11, 12]).

n t h e fir s t S e c t i o n t h e a p p r o x i m a t e s o l u t i o n o f t h e m o t i o n e q u a t i o n is c o n s t r u c t e d

T h e s t a t i o n a r y s o l u t i o n s a r e s t u d i e d

The s e c o n d S e c t i o n i s c o n c e r n e d w i t h 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 o s c i l l a -

t i o i T h e R o u t h - H u r w i t z c r it e r i a a r e t a k e n o u t

The i n f l u e n c e o f t h e C o u l o m b f r i c t i o n 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 is c o n s i d e r e d i n

t h e S e c t 3 I n t h i s c a s e t h e r e s o n a n c e c u r v e h a s t h e d o s e d f o r m

n S e c t 4 t h e i n f l u e n c e o f t h e t u r b u l e n t f r i c t i o n 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 i s s t u d i e d

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

c h a i g e i n t h e r i g i d i t y o f t h e s y s t e m i n v e s t i g a t e d

U n d e r t h e i n f l u e n c e o f t h e c o m b i n a t i o n f r i c t i o n ( S e c t 5 ), t h e r e s o n a n c e c u r v e is

s i m l a r i n q u a l i t y t o t h a t i n t h e c a s e o f t h e C o u l o m b f r i c t i o n I t is o f c l o s e d f o r m a s w e l l

1 C o n s tru c tio n o f A p p ro x im a te S o lu tio n

L e t u s c o n s i d e r 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 t h e s y s t e m d e s c r i b e d b y t h e t h i r d o r d e r

d i f e r e n t i a l e q u a t i o n o f t h e f o r m :

( 1 ) x ' + Ệ x + Q 2x + Ệ Q 2x + e [ k x 3 + h x 3 + R ( x , x , x ) - c x c o s y t ] = 0 ,

w b r e Ệ , Q , k , h , c , y a r e c o n s t a n t s a n d R ( x , x , x ) is t h e f u n c t i o n c h a r a c t e r i z i n g t h e

n c i l i n e a r f r i c t i o n

W e a s s u m e t h a t t h e r e is a r e s o n a n c e r e l a t i o n

eA = Q 2(l -Tj2), ri = ^ Q

-T h e n , E q ( 1 1 ) c a n b e w r i t t e n a s :

Ỉ54 Nguyen Van Dao

^ h e n

1 3 ) f ( x , X , x ) = A X + Ệ Ạ x + k x 3 + h x 3 + R ( x , X , x )

A p a r t i a l t w o - p a r a m e t e r s s o l u t i o n o f ( 1 2 ) i s f o u n d i n t h e s e r i e s :

1.4) X = ỠCOS + + EU1 ị a , ip, - y / j + e2u 2 ị a , ip, - y f j +

n w i i c h us (a, y , 0) a r e p e r i o d i c f u n c t i o n s o f 6 a n d w i t h - t h e p e r i o d 271, a n d a, ip a r e

u n c t o n s o f t im e d e t e r m i n e d f r o m t h e s e t o f e q u a t i o n s :

; i 5 )

^ = e B L( a , ỳ ) + e 2B 2( a , y>) + .

1 ) d e t e r m i n e t h e f u n c t i o n s US, A S, B S, f i r s t w c c a l c u l a t e :

~dt = 2

d 2x

a s m ( p + s \ A X c o s ( p —a B 1 s m ( p + + £ • • 9

— a c o s c p + —y A! s i n(p — y a B 1 COSẹ ? + - Ỵ T I + 8

1.6)

- ^ - 3- = a s m 9? + £ I — ~ ^ - y M 1 cosẹ> + fl-Sj SÌ1199 + —^ 3 I + £ • • • ’

<p = y f + y

S i b s t i t u t i n g E q s ( 1 4 ) , ( 1 6 ) i n t o ( 1 2 ) 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 t s o f e w i t h e q

ỉ e g r t e s , w e o b t a i n :

1 7 )

+ sin ọ ? = - / o + ữ c o s ^ c c o s

/ o = / ị a c o s ọ ) , — - y ứ s i n ọ ? , — - ~ a c o s ẹ > Ị

N ) W , w e e x p a n d t h e f u n c t i o n / o i n t h e F o u r i e r s e r i e s :

00

; i 8 ) / o = J j ? [qm( a ) c o s m ( p + p m(a )s i nm (p ] ,

m = 0

Nonlinear oscillations o f the third order systems Part III 255

l e r e

2.71

q 0 = — Ị f i a c o s q ) , — - y ứ s i n ọ ? , — ^ ị - ứ c o s ọ ? ! d(p,

271 I

.9 ) qm = — J" / Ị a c o s ẹ ? , — ~ - a s i n ẹ > , — ^ - a c o s ọ ? ! cosmcpdcp,

o '

271

p m = — Ị / Ị a c o s ạ ) , — - y ứ s i n ọ ? , — —- a c o s ^ Ị smmĩpdcp.

T h e f u n c t i o n « ! s a t i s f y i n g E q ( 1 7 ) w i l l b e f o u n d in t h e f o r m

1 1 0 ) ^ [G„(a, yj)cosn<p + B n( a , y )s in m < p ]

/ ith t h e a d d i t i o n a l c o n d i t i o n t h a t i t c o n t a i n s n o r e s o n a n c e t e r m s I t w i l l b e s e e n l a t e r

hat this c o n d itio n is eq u iv a len t t o the fo llo w in g : t h e function Wi does n o t co n ta in COS99,

in 99.

B y s u b s t i t u t i n g ( 1 8 ) , ( 1 1 0 ) i n t o ( 1 7 ) , w e h a v e 1.10 ) i n t o ( 1 7 ) , w e h a v e :

| y y G n- ! / / „ j s i n A ! 9 5 - ỈƠ„Ị cosn ọ?

-Ị - ^ - v 4 1 + y £ a 5 i -Ị c o s 9 9 + -Ị^ 2 ~ ứjBi - y M i j s i n g j = c a c o s ỹ c o s y í

00

- ) (ạmc o s r n ( p + p msinm<j

- y , (< 7 m C o sw ẹ> + /> m s i n w ẹ > )

m = 0

B y c o m p a r i n g t h e h a r m o n i c s sinọ?, COS op, o n e o b t a i n s :

( 1 12 )

y Ệ A i - ^ - a B i = - - ^ - s i n 2y + P i

B y c o m p a r i n g t h e o t h e r h a r m o n i c s , w e g e t :

n g u y en van ư ao

On s o l v i n g E q s ( 1 1 3 ) , w e h a v e :

(1 — 2, , - - ị - a c c o s 2 v > j ổ 3,

c " =

- y + £/>„ - | - j - r t ứ c c o s 2 y > + - ^ - a c s i n 2 y I ỗ 3n

=

r

(„*_!) Ị^+rl^Ị

F r o m ( 1 1 2 ) , w e h a v e

£ < / 0 sin 9 9 > + í 2 < / 0 c o s ẹ > > — - Ị - a c COS 2 ^ - * Ạ-OCỆs i n 2 y

(1.1:)

£ < / o c os<p> — í 2 < / 0 sinẹ>> + - ^ - a c s i n 2 y > — ^ - f l c c o s 2 y >

w h e i e ( F ) i s t h e o p e r a t o r o f t h e a v e r a g i n g f u n c t i o n F o n t i m e B y p u t t i n g i n E q ( 1

fo f n m E q s ( 1 3 ) a n d ( 1 7 ) a n d c a l c u l a t i n g , w e h a v e t h e f o l l o w i n g e q u a t i o n s o f t h e Í

a p p r o x i m a t i o n :

( 1.10

d t = Ệ 2 + Q 2 [ 8

dt = a ( i 2 + Q * )

ĩ-^ -(fc — £ Q 2h ) a 3 — -^ -ứ cc o s2y — -^ -^ s in 2 ^ y + i? ! j ,

Ị^— ( Ệ 2 + & 2) A a + - ^ - ( Ệ k + Q * h ) a 3 + ^ - s i n 2

xp-a c .

■ ~ — Ệ Q o s 2 t p + R

2 Y

w h e c

2Ệ

R i = < JR0 COSỌỊ>4- — < /? 0 sino9>,

y

2Ẽ

' l n R 2 = — ( R 0 c o s ( p ) - ( R 0 sinq)'),

y

T t 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 w e h a v e a p a r t i a l s o l u t i o n o f E q ( 1 1 ) i n t h e f o r

L 2

Nonlinear oscillations o f the third order systems Part ỈII 257

iere a and Iffare t h e s o l u t i o n o f E q s ( 1 1 6 ) T h e r e f i n e m e n t o f t h e first a p p r o x i m a t i o n i s :

9) .Y = a c O S

X C O S /7ọ? +

Qnq„ + Ệp„ — ị ^ - n a c cos2ĩf + y - ứ c s i n 2 ^ Ị < 5 3n

th a a n d y> b e i n g t h e s o l u t i o n o f E q ( 1 1 6 )

T h e s t a t i o n a r y s o l u t i o n o f t h e s e t ( 1 1 6 ) is 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 :

.2 0)

^ - a 0 s i n 2 ĩ p + ^ - c o s l i f = - — { k - Ệ Q 2h ) a ị + R y,

B y e l i m i n a t i n g t h e p h a s e y), w e o b t a i n t h e e q u a t i o n f o r t h e a m p l i t u d e a0 :

h ere

2 2 ) W ( a 0 , y) :

4

R e l a t i o n ( 1 2 1 ) is p l o t t e d in F i g 1 f o r t h e c a s e R = 0, -Ệ = ũ = Ỉ, c* = 0 0 5 , kị, —0.1 and /7* = 0 (curve 1), /7* = 0.05 (curve 2) and / 7* = 0.1 (curve 3) F r o m this

cure, it is s e e n that with increacing h, the m a x im u m o f the am plitudes d ecreases and

l e nonlinear s y s t e m b e c o m e s h a r d e r I n F i g 2 t h e r e s o n a n c e c u r v e s a r e p r e s e n t e d f o r

i e case R = 0 , Ệ = o = \, c* = 0 0 5 , /?* = 0 1 a n d k * = 0 ( c u r v e 1), k* = - 0 0 5

:urve 2), /c* = - 0 1 (curve 3) W ith decreasing k , the m a x im u m o f the a m p litu d e

e c r e a s e s a n d t h e n o n l i n e a r s y s t e m b e c o m e s s o f t e r Ịả'* = - Q ĩ k , / 7 * — c * — p T c

j-F i g 1

Nguyen Van Dao

B \ contrast w ith th e param etric oscillation in the well k n o w n seco n d -o rd er system ,

ie riiidity o f the n o n lin e a r system and the m a x im u m o f the am plitudes o f o scilla tio n

ĩr e t e p e n d o n t h e c o m b i n a t i o n o f t h e p a r a m e t e r s h a n d k. T h e s y s t e m c o n s i d e r e d is

h a rt s y s t e m i f T = £ k + Q Ah > 0 a n d a s o f t o n e i f T < 0 I f Q = ỆQ2h — k is p o s i t i v e ,

l e n tie m a x i m u m o f a m p l i t u d e s d e c r e a s e s w i t h i n c r e a s i n g Q.

2 S t a b ilit y o f S ta tio n a ry O s c illa t io n

F i s t w e shall c o n s i d e r t h e s t a b i l i t y o f t h e s t a t i o n a r y s o l u t i o n a0 Ỷ 0 o f E q s ( 1 1 6 )

ubsttuting in them

'iíh í0 , xp0 being th e s o lu tio n o f Eqs (1 20), we have the fo llo w in g variation al eq u a tio n s:

ỊỊ^ -(& -£ í2 2/7)ứẳ + aoỊ— Ị j < 5 c - (Ậ2 + Q 2) A a 0

2 y

cỉôyj

- 2 ỗ y > Ị ,

I Ả ( Ệ k + Q V i ) a 0 + Ị * ì Ị ] ô a + ị j { k Ệ Q 2h ) a l + ~ R ^ ỏ v }

Tie characteristic eq u a tio n o f this system is:

2.2)

yheri

2.3)

z =

5 =

ễ 2 + Q 2 r ~ ( i ' - ỉ ữ 2/ i ) « ỗ + — (0 0* 1) ' ]

4 Q 2(Ệ2 + Q 2) X y

+

2Q 2

+4 í ă í ă i

+

Ệ 2 + w

+

a 0 \ a 0

Nonlinear oscillations o f the third order systems, Pari III 2 5 !

(2.‘ )

r h e e x p r e s s i o n z c a n b e a l s o w r i t t e n in t h e f o r m :

d W

e 2a 0

4 Q 2 ( Ệ 2 + Q 2) d a 0

h e r w is o f t h e f o r m ( 1 2 2 ) C o n s e q u e n t l y , t h e s t a b i l i t y c o n d i t i o n o f s t a t i o n a r y s o l u

t i o i is:

(2.3 3 ( k - Ệ Q 2h ) a l + 2 ( a 0R i y < 0,

Ô ÌV

(2.0

ô a 0 > 0 .

M o w , le t u s c o n s i d e r a s p e c i a l c a s e o f t h e s t a b i l i t y o f e q u i l i b r i u m a = 0, w h e n th

s y s e m ( 1 1 6 ) h a s t h e f o r m :

(2 ')

i/ r £ 2 + £ 2 8

<7

dtp

h i

— fls in 2 v > — - y - f c o s 2 ^

(2.r

o r

In t h i s c a s e w e p u t a = ỗ ữ , y = ^ o + ổ y a n d t h e v a r i a t i o n a l e q u a t i o n s a r e :

d ò a

dt

—— = -1 ■ ■_ sin (2 u ’o 4- v ) 00,

2y ( Ệ 2 + Q 2) - (Ệ 2 + Q 2) A - \/ ệ 2 + Q 2 c o s ( 2 tPo + O) ỗ a ,

0 = a r e t e V

2Ệ •

T h e s e c o n d E q ( 2 9 ) y i e l d s :

c o s ( 2 tpo + 0) — —- A y i 2 + Q 1

s in ( 2 Y’o + ớ) = ± ~ \ / c 2 — 4 ( Ệ 2 + £ } 2) A 2

a m t h e r e f o r e t h e first E q ( 2 9 ) is o f t h e f o r m 1:

d ò a

~ d t~ ^ ỹ | = f » V - 4 T F ’ + i F ) 2 * í «

H e n c e , t h e r e f o l l o w s t h e s t a b i l i t y c o n d i t i o n o f e q u i l i b r i u m a = 0

c

M l >

2] / ệ 2 + Q 2

8*

Nguỵetì Van Dao

2.10) rị1 < 1 - , ĩ]2 > 1 H - — = = = = , ĩ] = y/2í2.

2í2 2 | / | 2 + £ 2 2& 2> / f2 + í2 2

In t h e f i g u r e s p r e s e n t e d t h e s t a b i l i t y c o n d i t i o n s a r e s a t i s f i e d o n t h e l i n e s in b o l d fa c e

3 T h e In flu e n c e o f C o u lo m b F r ic tio n

L e t u s c o n s i d e r t h e c a s e

3 1 ) R ( x , X , x) = ÌĨQs i g n * ,

v h ere h0 is a p o s i t i v e c o n s t a n t ,

• + 1 i f À ' > 0 ,

0 if X = 0.

In t h i s c a s e i t is e a s y t o v e r i f y t h a t

3 3 )

71 h 0 i f Ũ ^ 0 ,

<i?0sinọ9) =

<i?0 c o sọ )) = 0 fo r all a

N o w , E q s ( 1 1 6 ) , ( 1 1 7 ) , ( 1 2 2 ) a r e o f t h e f o r m : f o r a 0 :

~dt = Ệ2 + Q

3 4 )

Ợc — ỆQ2h ) a 3 ^ -ứ cos2y — ^£sin2y> — p ~ ^ o | »

-3 5 )

c a f- ", 2

~ f c o s 2v +

n I I

R-> = —- h o

71

3 6)

T h e e q u a t i o n w — 0 y i e l d s :

;3.7) r 1 + ~4 Ỵ ị2 + -Q2) (Ệk * + Q *h *) a 2 + n t f 2 + Q'2) a 1

* £ 2 + £ 2

1 -X / 1 2 + Q 2

-0 ( f / i J|(i2 2- A - lls) f l 2 + - ^ - / i S ,

7 1 ( 2

( 3 8 ) yt = ố‘/c /; = e] \ /;* = E^ °

£C

2Q ■

Nonlinear oscillations o f the third order systems Part III 261

In F i e 3 t h e d e p e n d e n c e o f a0 o n i f is p r e s e n t e d f o r t h e c a s e Ệ = Q = 1, /z* = 0 0 5 ,

¥ — — 0 1 , c* = 0 0 5 a n d h0 = 2 5 - 1 0 - 3 ( c u r v e I ) , / ? 0 = 5 • 1 0 - 3 ( c u r v e 2). H e r e t h e : s o n a n c e c u r v e h a s a c l o s e d f o r m H o w e v e r , o n l y t h e u p p e r b r a n c h l i m i t e d b y t h e

ỉ r t i c a l t a n e e n c i e s c o r r e s p o n d s t o t h e s t a b i l i t y o f t h e s t a t i o n a r y c o n d i t i o n ( 2 6 ) T h e

l e r e a s e in h 0 l e a d s t o t h e n a r r o w i n g o f t h e r e s o n a n c e c u r v e W i t h s u f f i c i e n t l y h i g h v a l u e s

f h 0 , t h e r e is n o s t a t i o n a r y o s c i l l a t i o n

T o f i n d t h e e x p r e s s i o n s ( 1 1 4 ) fir s t w e e x p a n d :

00

4 V I 1 sign sin 09 = — > ——-—— sin(2;?7+ 1) 09.

7r 2 m + 1

m — 0

N o w t h e f o r m u l a e ( 1 9 ) a r e o f t h e f o r m :

Po = P i = 4s = 0 , i # l , 3 ,

q 1 = ỆA a + ~ k a z,

Pl = - A Q a - Ặ h Q 3a 3,

4

P in , - 0 , p 2m+1 - ^ i + T r ’ m ^

T h e r e f o r e , t h e e x p r e s s i o n s ( 1 1 4 ) a r e :

H o = G0 = 0 ,

" 3 = W t f ‘ + 9 Q 2 ) I - ^ - 0 + ■ § - (3fr + n 3 - 3 £ c a c o s 2 y - ĩ c a s i n 2 y I ,

2m+1 n Q m ( m + l ) [ f 2 + ( 2 m + l ) 2i 3 2] ’ ^ 2m ’ ^

2m+1 7 T Í32w ( w + l ) (2m + 1 ) [ £ 2 + (2m + 1)2Í22] ’ 2m

Nguyen Van Dao

4 T h e In flu en ce o f T u rb u le n t F ric tio n on P a ra m e tric O s c illa tio n

N o w , w e tu rn to th e s t u d y o n t h e c a s e o f t h e t u r b u l e n t f r i c t i o n , w h e n R ( x , X , x) h a s

2 f o r m :

le r e h2 is a p o s i t i v e c o n s t a n t It is e a s y t o s e e t h a t :

( / ỉ o S Ì n ọ ? ) =

< R 0 c os c p) = 0 ,

d th e r e f o r e E q s ( 1 1 6 ) t a k e t h e f o r m :

fo r a =£ 0 :

I ? _ 2

h 2 y a ,

da

dt

dtp

Ệ2 + Q 2 (Ệ Q2h —k ) a 3 + ~ a c o s 2 t p + - ^ - £ s i n 2 y > + ^ - h 2 Q a 2 ,

dt a( Ệ2 + Q 2) 2 Q^ 7 ( f24 - £ ?2) z l a 4 - (£Ả ' + í2 4/0 ữ3 + - ^ - s i n 2y>

Ệ c o s 2 w + - ^ - A 2 i 3 2a 2| 4Í2

B y c o m p a r i n g w i t h E q ( 1 1 6 ) , w e h a v e :

* ,

-.4 )

R 2 = -Z — h 2 Q za 2.

07Z

C o n s e q u e n t l y , th e e x p r e s s i o n ( 1 2 2 ) i s :

\ 2

U Ỉ

F ig 4.