NONLINEAR PROPAGATION OF WAVES INDUCED BY AN INFINITE VIBRATING CYLINDER

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NONLINEAR PROPAGATION OF WAVES INDUCED BY AN INFINITE VIBRATING CYLINDER

Ali Nayfeh, Samuel G. Kelly

To cite this version:

Ali Nayfeh, Samuel G. Kelly. NONLINEAR PROPAGATION OF WAVES INDUCED BY AN IN- FINITE VIBRATING CYLINDER. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-8-C8-13.

�10.1051/jphyscol:1979802�. �jpa-00219507�

JOURNAL DE PHYSIQUE CoZZoque C8, suppzdment au n o 11, tome 40, novembre 1979, page c8- 8

NONLINEAR PROPAGATION OF WAVES INDUCED BY AN ZNFINITE VIBRATING CYLINDER A l i H. Nayfeh and Samuel G . K e l l y

,?ngineering Science and Mechanics Department, Virginia PoZytechnic I n s t i t u t e and State University BZacksburg, Virginia 24061 U . S. A.

Resume.- On u t i l i s e l a methode de r e n o r m a l i s a t i o n pour determiner un developpement uniforme au premier o r d r e pour l a propagation non l i n e a i r e d'ondes i n d u i t e s p a r l a v i b r a t i o n d ' u n c y l i n d r e de s e c t i o n c i r c u - l a i r e e t de longueur i n f i n i e . Les v i b r a t i o n s t r a n s v e r s a l e s du c y l i n d r e o n t une expression gen6rale com- p o r t a n t des modes e t des frgquences quelconques. On presente des exemples montrant l ' e f f e t des d i f f e r e n t s paramet-es adimensionnels s u r l a d i s t o r s i o n de l a pression e t de l a v i t e s s e . On odonne en o u t r e des exemples i 1 lu s t r a n t 1 ' i n t e r a c t i o n d'ondes de frequences d i f f e r e n t e s .

Abstract.- The method o f r e n o r m a l i z a t i o n i s used t o determine a f i r s t - o r d e r u n i f o r m expansion f o r the n o n l i n e a r propagation of waves induced by t h e v i b r a t i o n o f an i n f i n i t e c i r c u l a r c y l i n d e r . The transverse v i b r a t i o n s o f t h e c y l i n d e r a r e general and c o n s i s t of multimodes and m u l t i - f r e q u e n c i e s . Examples a r e presented showing t h e e f f e c t o f t h e d i f f e r e n t dimensionless parameters on t h e d i s t o r s i o n o f t h e pressure and v e l o c i t y components. Mereover, examples a r e presented showing t h e i n t e r a c t i o n o f waves w i t h d i f f e r e n t frequencies.

1. I n t r o d u c t i o n .

-

The problem considered i s t h a t o f n o n l i n e a r waves produced i n a f l u i d by t h e p u l - s a t i o n s o f an i n f i n i t e c i r c u l a r cylinder.The f l o w f i e l d i s assumed t o be i n v i s c i d so t h a t i t can be described by a p o t e n t i a l function. The c y l i n d e r i s assumed t o execute g e n e r ~ l motions c o n s i s t i n g o f mu1 timodes and mu1 t i - f r e q u e n c i e s . The method o f r e n o r m a l i z a t i o n C1, Sec. 3.41 i s used t o o b t a i n a u n i f o r m expansion by n o r m a l i z i n g t h e pressure and v e l o c i t y components.

Using t h e method of m u l t i p l e scales [ I , chap. 6

1,

Nayfeh and Kluwick 121 showed t h a t a u n i f o r m expansion f o r a one-dimensional non-disper- s i v e wave can be o t a i n e d o n l y i f t h e s e c u l a r terms a r e e l i m i n a t e d from t h e p r i m i t i v e v a r i a b l e s (i.e., pressure, v e l o c i t y components, s t r e s s , s t r a i n ) . Ginsberg [3] used a dual t r a n s f o r m a t i o n t o o b t a i n an expansion f o r t h e n o n l i n e a r i n t e r a c t i o n o f an a c o u s t i c f i e l d w i t h a f l a t p l a t e under a harmonic e x c i t a t i o n . He used t h e method o f r e n o r m a l i z a t i o n t o remove t h e s e c u l a r terms from t h e p o t e n t i a l f u n c t i o n . C a l c u l a t i n g t h e pressure and v e l o c i t y from t h i s p o t e n t i a l , he found t h a t they a r e n o t

The purpose o f t h e p r e s e n t paper i s t o extend t h e a n a l y s i s [4] t o t h e case o f a c y l i n d e r execu- i n g general motions c o n s i s t i n g of multimodes and mu1 t i - f r e q u e n c i e s . Ginsberg

[ 6 ]

s t u d i e d t h e case o f a c y l i n d e r executing a harmonic motion.

2. Problem f o r m u l a t i o n . - The problem i s formulated by using a p o l a r coordinate system (F,B) such t h a t r i s t h e d i s t a n c e as measured from t h e c e n t e r o f t h e c y l i n d e r and 8 i s t h e p o l a r angle. The f o l l o - v i n g dimensionless q u a n t i t i e s a r e i n t r o d u c e d :

-+ -+

r = r / R , t = c t , p = p/p,, v = v/c

where R i s t h e undisturbed r a d i u s o f t h e c y l i n d e r , p, i s t h e pressure i n t h e undisturbed s t a t e , $ i s t h e f l u i d v e l o c i t y and c i s t h e speed o f sound i n t h e undisturbed f l o w .

The f l u i d motion i n assumed t o be i n v i s c i d and i r r o t a t i o n a l so t h a t i t can be described by a dimensionless p o t e n t i a l f u n c t i o n 4 (r,e,t) such t h a t

$ r

V@. The equations d e s c r i b i n g t h e f l u i d motion i n dimensionless q u a n t i t e s a r e 1 3 , 4 ) . u n i f o r m and had t o i n t r o d u c e a new t r a n s f o r m a t i o n 2 2

t o remove t h e s e c u l a r terms from t h e pressure and

v

@-

@,,-

^(Y-1)

^@tv

⁴

^-

^2V@

^-

^mpt

.

^,

v e l o c i t y components. Nayfeh and K e l l y [4] obtained ⁺ = O a uniform expansion f o r t h i s problem by n o r m a l i z i n g Y-1

d i r e c t l y t h e pressure and v e l o c i t y components. Using ⁺ P)

7

'

-

( y - l ) [@t+

v@

v41 ( 3 ) a numerical example, Ginsberg [5]pointed o u t t h a t

the dual t r a n s f o r m a t i o n i s i n a c c u r a t e near t h e The kinematic c o n d i t i o n t h a t any p a r t i c l e on shock f r o n t s . t h e c y l i n d e r remains on t h e c y l i n d e r ( i . e . t h e

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:1979802

JOURNAL DE PHYSIQUE

component of t h e r e l a t i v e v e l o c i t y o f t h e f l u i d w i t h r e s p e c t t o t h a t o f t h e c y l i n d e r i s tangent t o t h e c y l i n d e r ) gives

where r = 1

+

w(B,t) i s t h e instantaneous r a d i u s o f t h e c y l i n d e r . The v e l o c i t y p o t e n t i a l i s a l s o chosen t o s a t i s f y t h e r a d i a t i o n c o n d i t i o n ; th a t i s , t h e motion of t h e f l u i d must be an outgoing wave o f s u i t a b l y decaying amplitude f o r l a r g e distances from t h e c y l i n d e r . The dimensionless motion o f t h e c y l i n d e r i s taken t o be

w = E C Amncos(Qmnt

+

v )cos (no

+

+,) (5) mn

where ^E i s a small dimensionless q u a n t i t y Amn, vmn and T~ a r e constants.

3. S t r a i g h f o r w a r d Expansions.- Expansions f o r t h e p o t e n t i a l f u n c t i o n and pressure a r e sought i n t h e form.

S u b s t i t u t i n g ( 5 )

-

^{( 7 )} i n t o (2) and (4) and equating c o e f f i c i e n t s o f l i k e powers o f E, we o b t a i n

Order E 2

V

$ 1

-

^{@ltt =}

⁰

2

Order E

The s o l u t i o n o f (8) and (9) t h a t s a t i s f i e s t h e r a d i a t i o n c o n d i t i o n i s

where

S u b s t i t u t i n g f o r i n t o (10) and (11) we o b t a i n an inhomogeneous problem f o r

( P ~ .

TO determine a f i r s t - o r d e r u n i f o r m expansion, we do n o t need t o s o l v e e x p l i c i t y f o r $ 2 . I t i s s u f f i c i e n t t o e x h i b i t i t s behavior f o r l a r g e r where t h e nonuni- f o r m i t y e x i s t s . To t h i s end, we f i n d t h a t

cos (ne+rn)

- - ^I

BmnBkai$,,,

IT m,n,k,R cos (n8+rn)

1

cos(R0+rR) (.Qmn+QkR)- CO~(R,,,~+Q~,) ( t - r )

4. Renormalized Expansion.- We note t h a t poten- t i a l f u n c t i o n c o n t a i n s s e c u l a r terms. However, as i n t h e one-dimensional case 127 one does n o t know w i t h o u t c a r r y i n g o u t a s o l u t i o n buy u s i n g t h e method o f m u l t i p l e scales which p a r t o f t h e s e c u l a r terms belong t o t h e f i r s t - o r d e r expan- s i o n and hence does n o t produce a n o n u n i f o r m i t y and which p a r t belongs t o t h e second-order expansion and hence produces a n o n u n i f o m i t y . Thus, t o determine a u n i f o r m expansion, one cannot use t h e method r e n o r m a l i z a t i o n t o norma- l i z e a p o t e n t i a l f u n c t i o n unless ke knows t h e p a r t o f t h e s e c u l a r terms t h a t t o a n o n u n i f o n n i t y Nayfeh and Kluwick C2

1

showed t h a t a u n i f o r m expansion can be obtained i f t h e method o f

C8- 10 A l i H. Nayfeh and Samuel G . K e l l y

remormalization i s used t o normalize p r i m i t i v e v a r i a b l e s such as t h e pressure and v e l o c i t y . Thus, t o determine a u n i f o r m expansion, we normalize the r a d i a l v e l o c i t y which we c a l c u l a t e next. I t f o l l o w s from (14) t h a t

3/2 l/2

C O S ( ~ ~ + T ~ )

-

^{E 2} (Y + I )

I

R

BmnBkenmn ke

TI m,n,k,ll cos (n8+rn) x

c o s ( e e + ~ ~ ) s i n [Rmn(t-r)-timn +L(n +L)TI] x

2 2

3 / 2

cos

[S~~~(~-~)-~~~+L(II+L)T]+

O ( E ~ Y E r )

2 2

a s r - + m (15

We n o t e t h a t v norluniform f o r l a r g e r. To determine a Uniform expansion, we use t h e method of renormal i z a t i o n and l e t

S u b s t i t u t i n g f o r r from (16) i n t o (15), expanding f o r small ^E w i t h q k e p t f i x e d and choosing

r

t o e l i - minate t h e terms t h a t produce t h e nonuniform?ty i n v, we o b t a i n

l/2

r = - ( y + l ) E "%n cos

[am,,

( t-q)+amn m,n Bmn (+

+

L

( n + L ) r l cos (no + T ~ ) +f(n,e,t)

2 2

(17

where f

5

0 (1) as q -+ m. Then (15) and (16) become

l / 2

v =

-

^{C Bmn(-) %n cos p.,,},,(t-n)+brnn m,n TI^

Hence,

Comparing (18) and (19), we note t h a t (18) can be r e w r i t t e n as

2

r = n+ ( y + l ) q v + ~ f ( q , 8 , t ) + O ( ~ ) ₍₂₁ where f ( 0 ( 1 ) as n ^-+ m . We choose f so t h a t r =n when r = 1. Hence, (21) becomes

Therefore, t o t h e f i r s t approximation, v i s g i v e n by (20) where

n

i s d e f i n e d by (22). Moreover, t h e pressure and c i r c u m f e r e n t i a l v e l o c i t y component become

5. Examples.- For g i v e n values o f 8 and t, t h e pressure and v e l o c i t y components may be c a l c u l a t e d by s e l e c t i n g a value o f q and c a l c u l a t i n g v, u and p from (20), (24) and (23). Then, (22) i s used t o s o l v e d f o r t h e corresponding value o f r. T h i s pro- cedure may be continued u s i n g small increments of q t o y e i l d t h e v a r i a t i o n o f v, u and p w i t h r a d i a l d i s t a n c e along l i n e s o f constant 8 a t any i n s t a n t t.

The e f f e c t of t h e n o n l i n e a r i t y i s t o steepe- n t h e wavefronts f o r i n c r e a s i n g r a d i a l d i s t a n - ces from t h e c y l i n d e r . T h i s steepening e v e n t u a l l y r e s u l t s i n t h e f o r m a t i o n o f a shock wave a t t h e f i r s t l o c a t i o n where t h e wave p r o f i l e s a t t a i n a v e r t i c a l tangent. T h i s i s t h e f i r s t l o c a t i o n

JOURNAL DE PHYSIQUE

P

where av/ar = m. However, (20) and (22) seem t o be t o o complicated t o a r r i v e a t an a n a l y t i c a l expression f o r t h e shock l o c a t i o n f o r m u t i - frequencies o r multimodes. Hence, t h e l o c a t i o n o f t h e shock wave i s obtained numerically.

Figures 1-8 present waveforms f o r v and p f o r d i f f e r e n t o s c i l l a t i o n s o f t h e c y l i n d e r . A l l

l r

A

p l o t s presented a r e along €I= 22.5O f o r t = 0,

E = .07 and y = 1.4. Figures 1 and 2 show t h e comparison o f v and p f o r two d i f f e r e n t s i n g l e frequency o s c i l l a t i o n s . These f i g u r e s show t h a t the shock forms a t an e a r l i e r l o c a t i o n f o r t h e

h i g h e r frequency o s c i 1 la t i o n s as expected. ^r

Fig. 2 : Pressure p r o f i l e s ; a) w =

-

Ecos t c0s28 and b ) w =

-

~ c o s 2 t cos2e

r

I

Fig. 1 : Radial v e l o c i t y p r o f i l e s ;

a) w =

-

^{E cos} t cos 28, b ) w = -E cos 2 t cos20

F i g . 3 : Radial v e l o c i t y p r o f i l e s ; a) w =

-

³⁶

cos2t cos20,b) w ⁼ - € ( c o s t

+

^.3cos2t)

Figures 3 and 4 show t h e e f f e c t o f adding a cos28 subharmonic frequency t o a s i n g l e frequency o s c i l -

l a t i o n o f t h e c y l i n d e r . The shock forms a t a

Figures 7 and 8 compare a two-frequency s l i g h t l y e a r l i e r l o c a t i o n i n t h e case o f t h e two-

o s c i l l a t i o n w i t h three-frequency o s c i l l a t i o n s . frequency o s c i l l a t i o n .

These f i g u r e s show t h a t . t h e steepening increases Figures 5 and 6 compare a single-frequency as t h e number o f frequencies i n t h e o s c i l l a t i o n o s c i 1 l a t i on w i t h two frequency o s c i l l a t i o n s . These of t h e c y l i n d e r increases. Moroever, t h e higher f i g u r e s show t h a t t h e shock f o r m a t i o n d i s t a n c e t h e frequency o f an added o s c i l l a t i o n i s t h e decreases w i t h i n c r e a s i n g t h e added o s c i l l a t i o n 1 a r g e r t h e n o n l i n e a r steepening i s and hence t h e

frequency. s m a l l e r t h e shock formation d i s t a n c e i s .

A.H. Nayfeh and S.G. Kelly

C8- 12

Fig. 4

:

Pressure profiles

;

a) w

=

- ^.3~cos2t

~ 0 ~ 2 8 , b) w

=

-~(cost+.3cos2t)cos28

v

Fig. 5

:

Radial velocity profiles

;

a) w

=

&cost

~0.~28, b) w

=

-~(cost+cos 5 ^t)cos28

c) w

=

-~(cost+cos 7t)cos2i

B

Fig. 6

:

Pressure profiles

;

a) w

=

-&cost ~ 0 ~ 2 8 ,

b )

w

=

-&(cost +5 ^t)cos28,

^C)

w

=

-

^E

4

(cos t+coslt) cos

28

v

I 4

Fig. 7

:

Pressure profiles

;

a) w

=

-~(cost+cos~t)cos2e,

Fig. 8 : Pressure p r o f i l e s ;

a) w

,

, ~ ~ c o s t + c o s ~ t ) c o s 2 8 ,

References

/1/ Nayfeh, A.H., P e r t u r b a t i o n Methods, Wiley I n t e r s c i e n c e , 11Y/3).

/2/ Nayfeh, A.H. and Kluwick, A., J. Sound Vib.

48, (1976), 293.

-

/3/ Ginsberg, J.H., J. Sound Vib.

-

40 (1975), 351.

/4/ Nayfeh, A.H. and K e l l y , S.G., J. Sound

Vib.,

( i n press).

/5/ Ginsberg, J.H.,

J.

Sound Vib., ( i n press).

/6/

Ginsberg, J.H., J. Acoust.Soc. Am. ( t o be pub1 ished)

.