NONLINEAR RESPONSE OF A SPHERICAL BUBBLE TO A MULTI-FREQUENCY EXCITATION

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NONLINEAR RESPONSE OF A SPHERICAL BUBBLE TO A MULTI-FREQUENCY EXCITATION

A. Nayfeh, D. Mook

To cite this version:

A. Nayfeh, D. Mook. NONLINEAR RESPONSE OF A SPHERICAL BUBBLE TO A MULTI- FREQUENCY EXCITATION. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-310-C8-314.

�10.1051/jphyscol:1979855�. �jpa-00219561�

NONLINEAR RESPONSE OF A SPHERICAL BUBBLE TO A MULTI-FREQUENCY EXCITATION NAYFEH A.H. and MOOK D.T.

Engineering Science -and Mechanics- Department Virginia Polytechnic Institute and State University Blacksburg, Virginia - 24061 - U.S.A.

A b s t r a c t . - A n o n l i n e a r a n a l y s i s i s conducted f o r the r a d i a l o s c i l l a t i o n s o f a spherical gas bubble immersed i n a s l i g h t l y compressible f l u i d . The mass o f the gas bubble and the surface tension are assumed t o be n e g l i g i b l e . The r a d i a l v e l o c i t y o f the bubble i s assumed to be small compared w i t h the speed o f sound. The response o f the bubble to a plane wave t r a i n whose wavelength i s l a r g e compared w i t h the radius o f the bubble i s determined. The present analysis extends the e x i s t i n g s i n g l e - f r e - quency analyses t o the case o f pressure waves having m u l t i - f r e q u e n c i e s . The a n a l y s i s reveals the existence o f combination resonances f o r which a l a r g e bubble response e x i s t s . These resonances e x i s t i n a d d i t i o n to the primary and secondary resonances r e s u l t i n g from a s i n g l e - f r e a u e n c y e x c i t a t i o n . Mo- r e o v e r , the analysis reveals the existence o f simultaneous resonances which can considerably modify the response.

1 . I n t r o d u c t i o n . - We consider the n o n l i n e a r sym- m e t r i c response o f a spherical gas bubble immersed i n a s l i g h t l y compressible f l u i d t o an i n c i d e n t a c o u s t i c pressure wave. The pressure wave i s assu- med t o c o n s i s t o f m u l t i - f r e q u e n c i e s whose wave- lengths are l a r g e compared w i t h the radius o f the bubble.

The n o n l i n e a r symmetric response o f a s p h e r i c a l gas bubble to an i n c i d e n t a c o u s t i c pressure wave c o n s i s t i n g o f a s i n g l e frequency was s t u d i e d by E l l e r and Flynn / l / , Safer / 2 / , Lauterborn / 3 / and Nayfeh and Saric IM. E l l e r and Flynn / l / and Safar / 2 / analysed the generation o f subharmonic o s c i l l a t i o n s o f order o n e - h a l f by bubbles immersed i n an incompressible f l u i d . They determined a p e r i o d i c subharmonic s o l u t i o n and then i n v e s t i g a t e d i t s s t a - b i l i t y using a H i l l ' s equation t o determine the t h r e s h o l d o f t h e pressure wave f o r the onset o f the subharmonic o s c i l l a t i o n s . Using the method o f mul- t i p l e scales / 5 / , Nayfeh and Saric / 4 / analyzed primary resonances, subharmonic resonances o f order o n e - h a l f and superharmonic resonance o f order two.

I n t e g r a t i n g the equation governing the r a d i a l o s c i l - l a t i o n s o f a bubble, Lauterborn / 3 / found t h a t l a r g e

1 1 1 2 3 2 3 responses occur when fi/uo^r, - j , ^-, ? , 1, j , - p y .

I n t h i s paper, we extend these analyses t o the case o f a c o u s t i c waves c o n s i s t i n g o f m u l t i - f r e q u e n c i e s .

2 . Statement o f Problem. - The equation d e s c r i b i n g tne n o n l i n e a r r a d i a l o s c i l l a t i o n s o f a spherical gas bubble are / 6 /

where the dot denotes d i f f e r e n t i a t i o n w i t h respect t o t i m e , R* and R0* are the instantaneous and e q u i - l i b r i u m r a d i i o f the bubble, p0 and p0 are the undisturbed d e n s i t y and pressure o f the f l u i d , y i s the gas s p e c i f i c heat r a t i o , p* i s the f a r f i e l d pressure i n the f l u i d and y * i s the damping c o e f f i c i e n t due t o r a d i a t i o n and viscous d i s s i p a t i o n . Me introduce dimeasionless q u a n t i t i e s w i t h o u t a s t e r i s k s by using R* ,OJ0 = (SypVp'oRtf2) 2 a n d P * a s i"efe-

o rence quantities. Then, (1) becomes

JOURNAL DE PHYSIQUE Colloque C8, supplément au n°ll, tome 40, novembre 1979, page C8-310

Résumé. - On a effectué une analyse non linéaire des oscillations radiales d'une bulle de gaz sphé- rique dans un fluide peu compressible. La masse du gaz ainsi que la tension superficielle ont été négligées. La vitesse radiale de la bulle est supposée faible vis-âvis de la vitesse du son. On a dé- terminé la réponse de la bulle à un train d'ondes planes dont la longueur d'onde est grande devant le rayon de la bulle. Ce travail étend l'analyse existante correspondant à une fréquence unique, au cas d'ondes de pression multifréquences. L'analyse révèle l'existence de résonances combinées pour les- quelles la bulle répond intensément ; ces résonances s'ajoutent aux première et deuxième résonances résultant d'une excitation simple fréquence. De plus, l'analyse révèle l'existence de résonances si- multanées qui peuvent modifier considérablement la réponse.

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979855

JOURNAL DE PHYSIQUE

where y=fl/uo

.

L e t t i n g

A

F = € 2 f , .Fn = €fn, p = Ey (7)

R = 1

+

r,(t), pm = 1

+

3yE(t) (3)

S u b s t i t u t i n g ( 6 ) and (7) i n t o ( 4 ) and (5) and equa- and expanding t h e r e s u l t f o r small

n ,

we o b t a i n t i n g c o e f f i c i e n t s o f l i k e powers o f E

,

we have

+

r,, fmcos(wm~,

+ +

^f c o s w ~ ~

m=

1

⁽⁹⁾

The a c o u s t i c pressure wave i s assumed t o c o n s i s t o f mu1 m u l t i - f r e q u e n c i e s and we w r i t e i t as

where Do = a/aTO and Dl =

a / a

TI

.

The s o l u t i o n o f E ( t ) = F cos ut + M 1 Fmcos(wmt

+

-rm) ( 5 ) (8) can be expressed as

m= 1

M

TII = A(T1) exp (

ill

+,,,gl $,,exp(iwmTo) + cc (10) where the F ' s , w's and T ' S are constants. We assume

t h a t w l,corresponding t o a primary resonance/and a1 1 the wn are f a r away from 1.

where cc stands f o r t h e complex conjugate o f t h e I n t h i s paper, we use t h e method o f mu1 t i p l e preceding terms and

scales and o b t a i n a u n i f o r m asymptotic expansion o f the s o l u t i o n s o f equations ( 4 ) and (5) and we

1 ¹

compare some o f t h e r e s u l t s w i t h a numerical i n t e hm = -;ifm(u2m

-

1)- exp (,i rm) (11) g r a t i o n of equation ( 2 ) and (5).

3. An Asymptotic Expansion.

-

FolJowing t h e me- S u b s t i t u t i n g f o r n1 from (11) i n t o (9) g i v e s thod o f m u l t i p l e scales /5/, we seek a second

o r d e r s o l u t i o n f o r ( 4 ) and ( 5 ) i n t h e form 1

Dtne +712 =

-

2 i ( A 1

+

j ~ ) e x p ( i ~ , )

+

^{;?- f} e x p ( i T o )

1 M

+ Z f m e x ~ ( i r m ) ] exp[i (~+u,,,)TJ + A i 3&(y+l-~,,,) m= 1

where ^E i s a small dimentionless q u a n t i t y which i s

1 the order o f t h e amplitude o f o s c i l l a t i o n , T o = t

1 M + 7 f m e x ~ ( - i ~ m ) ] e x ~ [ i ( l - ~ ) T o ] + 7

c

A,,,

m,n=1 i s a f a s t s c a l e and

T I

= ~t i s a slow scale. More-

over, we o r d e r t h e damping and f o r c i n g [3(y+l+6+,wn)~

,,

f u n c t i o n s so t h a t t h e e f f e c t s o f t h e damping, non-

l i n e a r i t y and resonances appear i n t h e same per- 1

+ f n e x ~ ( i ~ n l ~ e x ~ [ i ( w , + ~ n ) T , ] + ~

c

A,,,

t u r b a t i o n equation. Thus, we l e t m,n=l

[3(y+l+wmun)

Tin

NAYFEH, and a l .

I n s p e c t i o n o f (12) shows t h a t , t o second order, a number o f resonances m i g h t e x i s t . These i n c l u d e ( i ) a primary resonance when w 31,

( i i ) a subharmonic resonance when w 2 ,

P 1

( i i i ) a superharmonic resonance when wkd

,

( 1 v) a summed combination resonance when wm+wm% 1, and

( v ) a difference combination resonance when

Wm

-

^Wn

=

1.

The f i r s t t h r e e o f these resonances are c h a r a c t e r i s - t i c s o f single-frequency e x c i t a t i o n s , w h i l e t h e l a s t two are c h a r a c t e r i s t i c s o f mu1 ti -frequency e x c i t a t i o n s . Another c h a r a c t e r i s t i c o f m u l t i - f r e - q,uency e x c i t a t i o n s i s the occurence o f simultaneous resonances ; t h a t i s , the frequency content of t h e e x c i t a t i o n i s such t h a t two o r more o f the above resonances might occur simultaneously. For example we l e t

( i ) a subharmonic resonance when up .* 3, ( i i ) a superharmonic resonance when wp* 1

,

2 3

( i i i ) u l trasubharmonic resonances when

uF3

^{o r 2,}

( i v ) a combination resonance wheni yn+wm?wk I, and ( v ) a sub-combination resonance when wn+wmS 2.

Thus, depend~ing on t h e c o n t e n t o f t h e e x c i t a t i o n , two o r more o f the second- and t h i r d - o r d e r rqsonan- ces m i g h t occur simultaneously. Next, we present some numerical examples i l l u s t r a t i n g t h e e f f e c t s o f mu1 t i - f r e q u e n c y e x c i t a t i o n s .

4. Results and Discussion.

-

Since s i n g l e - f r e - quency e x c i t a t i o n s were s t u d i e d e x t e n s i v e l y , we r e s t r i c t o u r discussion t o m u l t i - f r e q u e n c y e x c i t a - t i o n s . To a v o i d l e n g t h y algebra, we r e s t r i c t our discussion t o second o r d e r o n l y and s e t F = 0.

We s t a r t t h e discussion by t h e case o f combi- naison resonances o f t h e summed and d i f f e r e n c e t y - pe. Thus, we l e t

and assume t h a t Q, and 0 , a r e away from

1

and 2.

Then, e l i m i n a t i n g the secular terms from (12) y i e l d s If there are ^Other

(I4)

reduces

-

Hence, as Tl-r m, the s o l u t i o n o f (17) i s

-

A r , e x p ( i ~ ~ T ~ + i ~ ~ )

- r 3

,exp( io3Tl+i.r3+i.rb)

Where

Carrying o u t t h e expansion t o t h i r d o r d e r shows t h a t o t h e r resonances w i l l occur. These i n c l u d e

A

Thus, t h e s m a l l e r t h e values o f p and o 3 are, t h e l a r g e r t h e response i s , as shown i n F i g u r e 1. Conse- q u e n t l y , t h e n o n l i n e a r resonant response can consi- d e r a b l y exceed t h e l i n e a r response.

The a n a l y s i s f o r t h e case of a combination resonance o f t h e d i f f e r e n c e type can be obtained from from t h e above by simple changing t h e s i g n o f Q,

.

F i g . 2. shows t h a t t h e n o n l i n e a r response can consi- d e r a b l y exceed t h e l i n e a r response.

Next,we consider the case of a simultaneous resonance. We l e t

wp = 1.5, WI = 0.5 + s o l

JOURNAL CE PHYSIQUE

Fig. 3. shows t h a t t h e presence o f t h e simultaneous resonance i s t o increase t h e n o n l i n e a r response.

Fig. 1.

-

The e f f e c t o f a combination resonance o f t h e summed type on t h e response of the bubble f o r

u

⁼ 0.005, n 3 = 0.4, F1 = F2 = 0.02 ; ( a ) l i n e a r

,

^{( b )} Q,, = 0.6 and (c)Qb = 0.59.

Fig. 3.

-

The e f f e c t o f a simultaneous resonance on t h e bubble f o r - p - = 0.005, Ql = fl.5','Q2 =1.5, F1 = F2= 0.05 ; (a) l i n e a r , (b) F2 = 0 and ( c ) F2 = 0.02.

The magnitude o f t h e n o n l i n e a r p a r t increases w i t h

A

( ? a ) decreasing p and alas before.

As a l a s t example, we consider the case o f simultaneous sub- and super-harmonic resonances.

Thus (14) reduces t o

A

2 i ( ~ ~ + p A ) - 1 1 ~ ~ x ~ ( i a ~ ~ ~ + i ~ ~ ) - r ~ e x p ( 2 i a ~ ~ ~ + 2 i ~ 1 ) = 0 (21)

To s o l v e t h i s equation, we l e t

Fig. 2.

-

Effect of a combination resonance o f t h e 1 1

d i f f e r e n c e type on t h e response of the bubble f o r A = (Br + iBi) exp ( '2 ^{i02$ +}

7

iT2) (22) p= 0.005, Q 3 = 0.65, F1 = F2 = 0.02 ;

( a ) l i n e a r , (b) Q 4 = 1.65 and ( c )

n,

= 1.64.

where Br and Bi a r e r e a l i n (21), separate r e a l and imaginary p a r t s and o b t a i n

Then, (14) reduces t o

where

AS T!-+ W , t h e s o l u t i o n of (19) becomes

1 1

x

= (20,

- 2

an) T1 + 2 - r ~

- ²

^{T 2} (25)

1 ^A 1 ^*

-

¹

A= i r 1 ( u + 2 i a l ) - 1 e x p ( 2 i a l ~ ~ + 2 i n ) -

7

i r z ~ ( u - i ~ l ) x

The s o l u t i o n of t h e homogeneous problem i s of t h e

exp(-ialT1 + i ~

-

2 ^TI) (20) form

where

Hence , the particular solution will be the largest when

Thus, one can expect the superharmonic resonance to produce a large effect when

as shown in Fig.4.

F i g . 4. The e f f e c t OT simultaneous sub- and supers harmonic resonances f o r y = 0 . 0 0 5 , n j = 0 . 5 0 ,

fi2 = 2 . 0 , F2 = 0.05 ; (a) l i n e a r , (b) Fi = 0 . 0 2 , fi2 = 2.1 (c) Fj = 0.02, R2= 2.0 and (d)

F1 1 = 0, fi2= 2.0.

REFERENCES

HI E l l e r , A . I . , and F l y n n , H.G., J . Acoust. Soc, Am. 46 (1969) 722.

HI S a f a r , M.H., J . Phys. D. 3 (1970) 635.

17,1 Lauterbom, W., Acoustica 23, (1970) 73.

IM Nayfeh, A.H. and Saric, W.S., J. Sound Vib.

30 (1973) 445.

/5/ Nayfeh, A.H., Perturbation Methods (Wiley Interscience, New York, 1973).

/6/ Noltingk, B.E., and Neppiras, E.A. Proc.

Roy. Soc. Lond. B63 (1950) 674.