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Sddhand, Vol. 9, Part 4, December 1986, pp. 255-269. L~) Printed in India.

N o i s e r e d u c t i o n w i t h p e r f o r a t e d t h r e e - d u c t m u f f l e r c o m p o n e n t s

K NARAYANA RAO* and M L MUNJALDepartment of Mechanical Engineering, Indian Institute of Science,Bangalore 560 012, India::'Present address: Mechanical Engineering Department, UniversityCollege of Engineering, Osmania University, Hydcrabad 500 0~7 India: Present address: Department of Mechanical Engineering, University ofCalgary, 2 5{)0 University Drive NW, Calgary, Alberta,Canada T2N IN4MS received 21 March 1986; revised 28 May 1986Abstract. This paper describes the authors' dzstributed parameterapproach for derivation of closed-form expressions for the four-poleparameters of the perforated three-duct muffler components. In thismethod, three simultaneous second-order partial differential equationsare first reduced to a set 9f six first-order ordinary differentialequations. These equations are then uncoupled by means of a modalmatrix. The resulting 6 6 matrix is reduced to the 2 x 2 transfer matrixusing the relevant boundary conditions. This is combined with transfermatrices of other elements (upstream and downstream of this perfo-rated element) to predict muffler performance like noise reduction,which is also measured. The correlation between experimental andtheoretical values of noise reduction is shown to be satisfactory.Keywords. Mufflers; noise control; transfer matrix method.

I. IntroductionThe majority of the commercial mufflers used on internal combustion enginescontain one or more perforated tubes. These may interact with the surroundingannular cavity as in a Helmholtz resonator, or may conduct the gases as in plugmufflers and three-duct cross-flow or reverse-flow mufflers. Unfortunately,however, analysis of such perforated-element mufflers was not known until a fewyears ago, with the result that the design of commercial mufflers was based on trialand error.Recently, the authors developed and presented a generalized decoupling methodfor the analysis of perforated-element mufflers with and without mean flow,wherein the mean flow Mach numbers are allowed to have their actual (unequal)values (Rao & Munjal 1984). But the method was demonstrated for two-ductelements only. This method makes use of the distributed parameter approach in


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256 K Narayana Rao and M L Munjalcon t r a s t t o Su l l ivan ' s s egm e n t a t i on app r o ach ( Su l li van 1979a, b ) , w he r e ape r f o r a t ed c l em en t i s no t i ona l l y d i v i ded i n t o a num ber o f s egm en t s .

T h e p r e s e n t w o r k e x te n d s t h e d i s tr i b u te d p a r a m e t e r a p p r o a c h t o t h r e e - d u c tm u f f l e r e l em en t con f i gu r a t i ons by r em ov i ng t he am bi gu i t ie s in eva l ua t i on o f t hee i genva l ues o f t he cha r ac t e r i s t i c m a t r i x . The t h r ee coup l ed pa r t i a l d i f f e r en t i a lequa t i ons f o r w ave p r opaga t i on have been so l ved us i ng t he au t ho r s ' gene r a l i z edd e c o u p l i n g a p p r o a c h w i t h a n d w i th o u t m e a n f l o w i n c o rp o r a t in g t h e r e l e v a n t e x a c tbounda r y cond i t i ons . Exp l i c i t exp r e s s i ons have been de r i ved f o r f ou r - po l epa r am e t e r s o f t he c r os s - f l ow expans i on cham ber a s w e l l a s t he r eve r se - f l owexpans i on ch am ber , show n i n fi gu r e 1 . The t r ans f e r m a t r i c e s so de r i ved have b eenc h e c k e d e x p e r i m e n t a l ly b y c o m p a r i n g t h e p r e d i c t e d v a l u e s o f n o i se r e d u c t i o n w i t ht h o s e o b s e r v e d e x p e r i m e n t a l l y .

2. The coupled governing equationsI m p l i c i t i n t he f o l l ow i ng equa t i ons , and t he so l u t i on t he r eo f , a r e ce r t a i na s sum pt i ons t ha t a r e t yp i ca l o f au t om ot i ve m uf f l e r ana l ysi s , v iz ,( i) w or k i ng in t he f r equ enc y dom a i n , a ll a cous t i c va r i ab le s a r e m ade t o haveh a r m o n i c t i m e - d e p e n d e n c e e v e r y w h e r e ;( ii) a t the f r eque nc i e s o f i n te r e s t , on l y p l ane w aves w ou l d p r opaga t e ; h i gh e r o r de rm o d e s c o u l d b e i g n o r e d ;( i i i ) mean f low ve loc i t i es a re smal l enough; the f low i s incompress ib le ;( iv) hea t t ransfe r and v i scos ity e f fec t s a re n egle c ted ; a ll processes a re i sen t ropic ;

mic rophoneM! dl~ !2 icrophone! ' I --.--.~M5

I ,a i2 [ - i : ' ~ h['~- Ii t , )m i c r o p h o n e

M I d 1. [m i c r o . p h o n e I- q 0 - - - -M 3 d 3 [

t 3

Figure I . T hree -duc t muf f l e r conf igura ti ons :b . r eve r se - f l ow expans ion cham ber .

a . c ross -f l ow expans ion cham ber ,


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N o i s e r e d u c ti o n w i th p e r f o r a te d m u f f l e r c o m p o n e n t s 2 5 7( v ) m e a n f l o w g ra d i e n t s a l o n g a n d a c r o s s t h e d u c t a r e n e g l e c t e d . T h e s a m e h o l d sf o r m e a n p r e s s u r e a n d m e a n d e n s i t y ;( v i) v e l o c i t y o f c r o s s f l o w is a s s u m e d t o b e u n i f o r m a ll o v e r t h e p e r f o r a t e . T h i sa l l o w s u s t o a s s u m e a u n i f o r m a c o u s t i c i m p e d a n c e f o r th e p e r f o r a t e .F o r t h e th r e e - d u c t m o d e l a s s h o w n in f i g u re 2 , t h e m a s s c o n t i n u i t y e q u a t i o n s m a yb e w r i t te n a s ( S u l li v a n & C r o c k e r 1 9 78 )

C ] W I C ~ D l 4 O p jP(1-- + W o , + - - p o U l , 2 - - ( 1 )~ z c ) z d ~ c )tf o r d u c t 1 o f d i a m e t e r d ~,

0 1 4'2 ,9 p 2 4 d l 4 d 3P o - - + W o , p o U l , 2 +az Oz (d_~ - d~ - d~ ) (d~ - d~ - d~)apeO t

f o r d u c t 2 o f d i a m e t e r d 2 , a n dOw 3 01)3 4 Op3

P(1 -- + W , , , - - Po u2.3 -O z O z d 3 3 t

p o u 2 , 3

( 2 )

( 3 )f o r d u c t 3 o f d i a m e t e r d 3. T h e n o t a t i o n i s g i v e n in a p p e n d i x D .

T h e c o rr e sp o n d i n g m o m e n t u m e q u a t io n s a re( c ) % ) - O p j j = 1 ' '~ 3 "o w , + w o , - z a zPo O---t " - " ( 4 ) , ( 5 ) , ( 6 )

T h e r a d ia l m o m e n t u m a t i n t e r f a c e s o f d u c t 1 a n d d u c t 3 r e s u lt i n t h e e q u a t i o n su l , z ( z , t ) = [ p ~ ( z , t ) - p 2 ( z , t )] /( p o co ~ 1 ), ( 7 a )

a n du 2 . 3 ( z , t ) = [ p 2 ( z , t ) - p 3 ( z , t )] / (p , , c o ~ z ) , ( 7 b )

r e s p e c t i v e l y .A s s u m i n g t h a t t h e f lu i d is a n i d e a l ga s a n d t h e p r o c e s s i n v o l v in g w a v ep r o p a g a t i o n i s i s e n t r o p i c ,

p j = p j c ~ , j = 1 , 2 , 3 . (8 )F o r h a r m o n i c t i m e d e p e n d e n c e ,

p j ( z , t ) = p i ( z ) e i ' t , j = 1 , 2 , 3 ,a n d

( 9 a )

e , j = l , 2 , k = 2 , 3 . ( 9b )i , k ( z , t ) = U j . k ( Z ) i,o ,

S u b s t i t u t i n g ( 7 a ) , ( 7 b ) , ( 8 ) , ( 9 a ) a n d ( 9 b ) i n t o ( 1 ) t o ( 6 ) a n d e l i m i n a t i n g P l , P 2 ,f 13 , / /1 , 2 , / ,/ 2 ,3 , W l , w 2 a n d w 3 , y i e l d s t h e f o l l o w i n g t h r e e c o u p l e d d i f f e r e n t i a le q u a t i o n s f o r d u c t s 1 , 2 a n d 3 , r e s p e c t i v e l y ( M u n j a l 1 9 8 6 ),


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2 5 8 K N a r a ya n a R a o a n d M L M u n j a l

w h e r e

a n d

[ d 2 i M , ( k ] + k 2 ) d k 2 ]+ - - p ~ ( z )d z 2 1 - M 21 k d z 1 - M 2

" 1 - . . . .I -M d z a - M

I iM 2 k ~ - k 2 d k 2 - - k 2 I p l ( z )1 - M--~ J

+ 2 2 2d iM 2 ( k ~ + k 2 ) d k b + k c - k ]+ p 2 ( z )d z 2 1 - M 2 k d z ' l ~ M 2 ~ J[ iM2 ( k c 2 . - k 2 ) d k 2 c - k 2 1

+ t 1 - M 2 \ k d z ] -- ~ 2 22 j p 3 ( z ) = 0 ,

iM____2_ k 2 - dl _ M 2 ( /~ k 2 ) d zk 2 - k 2 ]

l _ M 2 p 2 ( z )

+ d z 2 1 - M 2 k dzz + - I - M 2 p 3 ( z ) = O ,

O)k -C o

k 2 = k 2 _ _

k 2 = k 2 -

k ~ = k 2 - - -

Wo, Wo- - - , M 1 - , M 2 - : , M 3 - ,CO CO Co

i 4 k i 4 k d l, k 2 = k 2 -d , s , ( d 2 - d 2 - d2 ) S rl 'i 4 k d 3

( d 2 - d ~ 2 '- - d 3 ) ~ " 2i 4 k

d3~"

( 1 0 )

( 1 1 )

( 1 2 )

( 1 3 )

3. Decoupling analysisT h e a n a l y s i s r u n s e x a c tl y a s f o r t h e t w o - d u c t p e r f o r a t e d e l e m e n t m u f f l e rs ( R a o &M u n j a l 1 9 8 4) . T h e t h r e e s e c o n d - o r d e r d i f f e r en t i a l e q u a t i o n s ( 1 0 ), (1 1 ) a n d ( 12 ) a ref ir s t r e d u c e d t o s ix f i r s t -o r d e r l i n e a r e q u a t io l a s . T r e a t i n g t h e d i f f e r e n t i a l o p e r a t o ra s a n a l g e b r a ic v a r ia b l e , t h e s e s ix e q u a t i o n s a r e d e c o u p l e d t h r o u g h a n e i g e n v a l u ea n a l y s i s a n d t h e u s e o f p r i n c i p a l c o o r d i n a t e s . F i n a l l y a c o u s t i c s ta t e v a r i a b l e s a tz = 0 a r e r e l a t e d t o t h o s e a t z -- l t h r o u g h a m a t r i x , w h i c h is t h e r e q u i r e d t r a n s f e rm a t r i x .


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N o i s e r e d u c ti o n w i t h p e r f o r a t e d m u f f l e r c o m p o n e n t s 2 5 9

D e n o t i n g d / d z b y p r i m e ( ' ) , a n d d e f i n in gP l = Y | , P 2 = Y e , P 3 = Y 3 , P l = Y 4 ,P 2 = Y 5 an d P 3 = Y 6 ,( 1 0 ) , . ( 1 1 ) a n d ( 1 2 ) m a y b e w r i t t e n i n t h e f o r m

- - 1 0 0 D 0 00 - 1 0 0 D 00 0 - 1 0 0 DD 0 0 0 : l D + a 2 0:3 D + 0:.) 00 D 0 a s D + 0 : 6 0 :7 D + 0: ~ a g D + a l ( i0 0 D 0 0 :11D +0: )2 0 :13D+0:14

Y lY 2Y 3Y 4Y5I,. Y%

( 1 4 )0 ]00

1( 1 5 a )

00

o r

w h e r e[ ~ l { y } = { 0 } , ( 1 5 b)

1: 1 - - ~ , 0 : - , - - , 0 : 3 - - - - - ,1 - M i k - 1 - M ~ 1 - M ~ kk ] - k 2 iM 9 [ k ~ - - k 2 ] ( k ~ , - k 2)

" 0 : 4 - - .~ , 0 : 5 - - - k 7 - ) ' 0 : 6 - -I - M i I - M ~ I - M ~

0:l"7- 1 - M?~ " 0:~ - ~ ' 0:~ -1 - M ~

a n d

- (k c - k 2) iM 3 [ k~ - k2 ~ k ,e t - k 2- - - - " ~ k ) , 0 : 1 2 - ,0:to 1 - M ~ , 0 : 1 1 1 - M ~ k f - M 2

k 2- i M 3 k 2 + k 2 ~ , a ,4 - , ( 1 6 )a 1 3 - l _ M 2 k ] 1 - M ~

dD - - - . ( 1 7 )d zM a k i n g u s e o f p r in c i pa l c o o r d i n a t e s , t h e s e e q u a t i o n s m a y b e d e c o u p l e d .

T h e n e w v a r i a b l e s , i . e . , t h e p r i n c i p a l c o - o r d i n a t e s F ~ , F e , [ `3 , [` 4, r '5 a n d I"6 a r er e l a t e d t o v a r i a b l e s y j , Y 2 , Y 3, Y 4 , Y 5 a n d y , , t h r o u g h t h e m o d a l m a t r i x [ ~ ] a s


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2 6 0 K N a r a y a n a R a o a n d M L M u n j a lw h e r e

a n d

~ l , j = 1 .0 ( s a y ) ,' t ' 2 , j = - ( ~ + 0 /l & + 0 / 2 ) / ( 0 / 3 & + 0 /4 ) ,

( /3 ~ + 0 / , / 3 j + 0 / ~ ) ( t ~ + , ~ t 3 j + , ~ s ) - ( 0 / . , t ~ j + 0 / ~ ) ( 0 / 5 / 3 j + 0 / ~ )' q - t3 , =~ 4 , j = 1 " 0 / / 3 i ,~II ,j = ~It2.j/ [~j ,

( 0 / 3 ~ j + 0 / 4 ) ( 0 / 9 ~ j + O~10)

q r6 4 = ~ 3 , j / [ 3 j , ( 1 9 )a n d s u b s c r i p t j t a k e s t h e v a l u e s 1 , 2 . . . . 6 . /3 's a r e r o o t s o f t h e s i x th d e g r e ep o l y n o m i a l

] a l = 0 , ( 2 0 )t o b e f o u n d n u m e r i c a ll y o n a c o m p u t e r b y m e a n s o f o n e o f th e s t a n d a r ds u b r o u t i n e s .

T h e g e n e r a l s o l u t i o n s t o f ir s t o r d e r e q u a t i o n s ( 2 0 ) c a n b e o b t a i n e d a sr j ( z ) = C j e ~ 'z , ( 2 1 )

w h e r e s u b s c r i p t j t a k e s v a l u e s 1 , 2 . . . . 6 .N e x t , ( 4 ) , ( 5 ) a n d ( 6 ) m a y b e u s e d t o o b t a i n e x p r e s s i o n s f o r w l ( z ) , w 2 ( z ) a n d

w 3 ( z ) . F i n a l l y , w e m a y w r i t ep 1 ( z )p 2 ( z )p 3 ( z )p o c o w l ( z )poCoW2(Z)poCow3(z )

= D ]C 2C~

GC 6

( 2 2 )

v e l o c i t y a t z = 1 ; t h a t i s ,

w h e r e t h e

p , ( 0 ) p , q )p2( O) p2 ( l )p3(0) -- [TI p~(t)m eo w , ( 0 ) p , : o w l ( 1)poCo w2 (0) poco w2 (1)poCoW3(O) poCoW3(l)t r a n s f e r m a t r i x i s g i v e n b y

[ T ] = [ .4 ( 0 )1 [ A f t ) ] - ' .

( 2 3 )

( 2 4 )

w h e r e e l e m e n t s A i , j i n v o l v in g o n l y M , k a n d l a r e g i v e n b y e x p r e s s i o n s n o t e d i na p p e n d i x A .N o w t h e p r e s s u r e a n d v e l o c i t y a t z = 0 c a n b e r e l a t e d t o t h e p r e s s u r e a n d


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N o i s e r e d u c t i o n w i t h p e r f o r a t e d m u f f l e r c o m p o n e n t s 2 6 l4 . T r a ns f e r m a t r i x f o r c r o s s - f l o w e x pa ns i o n c ha m be rTo e l imina t e fou r o f t he s i x s t a t e va r i ab l e s i n t he fo r ego ing gene ra l so lu t i on , ( 23 ) ,w e m a k e u s e o f f o u r b o u n d a r y c o n d i t i o n s . I n t h e c a s e o f c r o s s - f l o w e x p a n s i o nc h a m b e r w i t h r i g i d e n d t e r m i n a t i o n , t h e b o u n d a r y c o n d i t i o n s a r e ( T h a w a n i &Jaya raman 1983)

z = 0 : p o c o w 2 ( O ) = - i t an (k l , ) p2 (0 )z = 0 " p o c o w 3 ( O ) = - i t a n i k 1 , , ) p . ~ ( O )z = 1 : p o c o w l ( 1 ) = - i t a n ( k l l , ) p L ( l )z = l : p o c o w 2 ( l ) = - i t a n ( k l b ) p 2 ( l ) . (25)

Sk ipp ing t he e l imina t i on de t a i l s , t he f i na l exp re s s ions fo r t he ups t r eam s t a t eva r i ab l e s and t he downs t r eam va r i ab l e s c an be wr i t t en a s (Rap 1984)

poCoWl ( 0 ) T~. T , I po co w 3(l)The exp re s s ions fo r T a r e g iven i n append ix B .

5 . T r a ns f e r m a t r i x f o r r e v e r s e f l o w e x pa ns i o n c ha m be r

The fo l l owing changes have t o be no t ed i n t h i s e l emen t :( a ) mean f l ow in duc t t h r ee i s nega t i ve w i th r e spec t t o t he r e f e r ence d i r ec t i on ;(b ) t he bounda ry cond i t i ons a r e d i f f e r en t ; t he se a r e

z = 0 : p o c o w 2 ( O ) = - i t an (k t . ) p2 (O)z = l : p o c ~ j w l ( l ) = - i t a n ( k l t , ) p l ( l )z = I : p o c o w 2 ( 1 ) = - i t an(k lh) p2( l )

= l : p o c o w 3 ( I ) = - i t a n ( k 1 1 , ) p 3 ( l ) . (27)Sk ipp ing t he a lgeb ra i c de t a i l s o f t he e l imina t i on p roces s , t he t r ans f e r m a t r ix

r e l a t i on fo r t h i s e l emen t may be wr i t t en a s (Rap 1984)L , T t ,[ ' 3 ' ' Ioocow3(O)~c~w~ 0 )

and t he r e su l t i ng exp re s s ions fo r T , , , T b , T c and T,t a re g iven in appen dix C (R ap1984).

6 . Exper imenta l ev idenceIn o rde r t o t e s t t he va l id i t y o f t he m a them a t i ca l m ode l a t va r ious mean f l owc o n d i t i o n s , n o i s e r e d u c t i o n m e a s u r e m e n t s w e r e m a d e o n t y p i c a l t h r e e - d u c tp e r f o r a t e d e l e m e n t m u f f l e r c o n fi g u r a ti o n s o f f ig u r e 1.


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262 K N a r a y a n a R a o a n d M L M u n ja lNoi s e r educ t i on , NR, is t he d i f f e r en ce i n s ound p r e s s u r e l eve l s a t two a r b i t r a ri l y

s e l ec t ed po i n t s i n t he exhaus t p i pe and t he t a i l p i pe , l oca t ed ups t r eam andd o w n s t r e a m o f t h e m u f f l e r p r o p e r , r e s p ec t iv e l y . R e f e r r i n g t o f i g u r e 1 ,NR = S e L l - S P L 3 = 20 logm I P ~ / P 3 [ , (29)w h e r e p d e n o t e s t h e r o o t m e a n s q u a r e v a l u e o f t h e a c o u s t ic p r e ss u r e p e r t u r b a t i o n .Fo r t h e pu r pos e o f c a l cu la t i on , s ince p~ is no t k now n d i r ec t l y , one c an w r i teP , /P 3 = ( P l /P o ) " P o /P 3 ) , (30)

w h e r e P 0 is t h e a c o u s t ic p re s s u r e a t t h e r a d i a t io n e n d . A p p l y i n g t h e w a v er e l a t i ons h i ps f o r t he p i pe s ec t i on o f l eng t h le , one ob t a i n s ( Mun j a l 1986)

(P3 /Po) = cos k o l e + ( i g 3 / z o ) sin kole , (31)w h e r e Z o i s t h e r a d i a t i o n i m p e d a n c e , po /v o . (32)T h e o t h e r p r e s s u re r a t io P~/Po m a y b e o b t a i n e d f r o m t h e t r a n s f e r m a t ri x r e la t io n

[ P l l = [l l v l T:I T 1 2 ]P I T z 2o ' (33)whe r e v deno t e s acous t i c m as s ve l oc i t y , p o A u , an d Ti i ( i , j = 1, 2) ar e th e fou r-po l e p a r am e t e r s o f t he ov e r a l l t r ans f e r m a t r i x r e l a t i ng the s t a t e va r i ab l e s a t po i n t s1 and 0 . Th i s m a t r i x i s a p r oduc t o f t he t r ans f e r m a t r i c e s o f t he ups t r eam p i pe ,p e r f o r a t e d e l e m e n t , a n d d o w n s t r e a m p i p e .

F r om ( 32 ) and ( 33 ) we ob t a i n( P l / P o ) = Tll @ T 1 2 1 Z o (34)

Equ a t i ons ( 29 ), ( 30 ) , (31 ) and ( 34) m ay be com bi ne d to ob t a i nr . + r ,2 / Z o [NR = 20 log10 . (35)Ios kole+ i Y3 /Zo ' s in k o le

7 . R e s u l t s a n d d i s c u s s i o nG e n e r a l F o r t r a n p r o g r a m s h a v e b e e n d e v e l o p e d fo r c o m p u t a t i o n ( o n DEC 10c o m p u t e r ) o f n o i s e r e d u c t i o n f o r t h r e e - d u c t e l e m e n t c o n f i g u r a t i o n s s h o w n i nf ig u r e 1 o v e r t h e f r e q u e n c y r a n g e c o v e r e d b y p u r e p l a n e w a v e p r o p a g a t i o n , in s t e p so f 2 0 H z . T h e f o ll o w in g p e r f o r a t e d i m p e d a n c e s w e r e i n c o r p o r a t e d i n t h e n o i s ereduct ion predic t ion (Sul l ivan 1979) ,

~ = ( 6 1 0 - 3 + i 4 .8 1 0 - 5 f ) / o -, f or M = 0 ,~r = (0-5 14 d M / l o - + i 4 . 8 10-5 f ) /~ r , for M >I 0 .05 .

F i gu r e 2 s hows t he t yp i ca l cu r ves o f no i s e r educ t i on ve r s us f r eq uen cy o f t h r ee - du c tc r os s -f lo w m u f f l e r r e p r e s e n t e d i n f i g u re l a f o r a s t a t i o n a r y m e d i u m ( M = 0c o n d i t io n ) . N e g l e c t i n g t h e t e m p e r a t u r e g r a d i e n t a n d t h e t u b e w a l l t h i c k n e ss , t h e r eis g o o d a g r e e m e n t b e t w e e n t h e p r e d i c ti o n s o f t h e p r e s e n t i n v e s t ig a t io n a n d t h o s e o fexpe r i m en t a l r e su l t s. I n f i gu r es 3 an d 4 a s i m i la r exe r c is e is r epe a t ed f o r i n l e t fl ow,M a c h n u m b e r M = 0 .1 0 a n d M = 0 .2 . T h o u g h t h e r e a r e s m a ll d i s c re p a n c i e s, y e tt h e a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t i s g e n e r a l l y g o o d .


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Noise reduction w ith perforated muffler com ponents 263d B12 0

1 0 0

8 0

6 0 A4 02 0

I 1 I I ! m t o s u r c m e N t ~ s

p l " e d l c t t o n s

- 2 0 m i0 0. 5 1 . 0 I I I1.5 2 .0 2 .5 3 .0 k H z

f r e q u e n c yN o i s e r e d u c t i o n o f t h r e e - d u c t c r o s s - f l o w e x p a n s i o n c h a m b e r o f f i g u r e l a f o r

d B120 j , i ,

/ m e o s u r e m e n t $1 0 0 L predict ions

t-O

moe-.

F ig u r e 3 .M1 = 0-1.

80

60

4 0

2 0

F i g u r e 2 .M~ = 0

-2C I I 1 I t0 0 . 5 l O 1 -5 2 . 0 2 . 5 3 . 0 kH zf r e q u e n c y

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


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264 K Narayana Rao and M L Munjald 81 2 0

I

I O O

8 o

- 2 0 i0-5

6C _

~ q

" 20

I I I I

m e a s u r e f n e N t E_ _ predictions

i I I I1.0 1.5 2-0 2.5f r e q u e n c y 3 . 0 k H z

Figure 4. Noise reduction of three-duct cross-flow expansion chamber of figure ia forM1 = 0-2.

N o i s e r e d u c t io n a t z e r o m e a n f l o w o f th e r e v e r s e - f l o w e x p a n s i o n c h a m b e r o ff ig u r e l b is s h o w n i n f ig u r e 5 . T h e a g r e e m e n t b e t w e e n t h e o r e t i c a l p r e d i c ti o n s a n de x p e r i m e n t a l o b s e r v a t i o n s i s g o o d i n t e rm s o f p e a k l o c a t io n s a s w e l l a s m a g n i t u d e s .F i g u r e s 6 a n d 7 p r e s e n t c o m p u t e d a n d e x p e r i m e n t a l n o i s e re d u c t i o n c u r v e s f o r t h esame muf f l e r w i th M = 0 .1 and M = 0 -2 r e spec t i ve ly . He re t oo , t he t heo re t i c a lr e su l t s ma t ch t he expe r imen ta l r e su l t s we l l .

I n g e n e r a l, e x p e r i m e n t a l v a l u e s o f n o is e r e d u c t i o n a r e i n g o o d a g r e e m e n t w i t hthe p r ed i c t ed r e su l t s , t he r eby subs t an t i a t i ng t he v a l i d it y o f the t r ans f e r m a t r i c e sde r ived and u sed i n t he p r ed i c t i ons .

T h e s e t r a n s f e r m a t r i c e s m a y b e u s e d a l o n g w i t h t h o s e f o r o t h e r n o n - p e r f o r a t e de l e m e n t s l i k e a t u b e , s u d d e n c o n t r a c t i o n , s u d d e n e x p a n s i o n , e x t e n d e d i n l e t ,e x t e n d e d o u t l e t , f l o w - r e v e r sa l c o n t r a c t io n a n d f l o w - r e v e r s a l e x p a n s i o n ( M u n j a i1975 ; Pan i ck e r & M un ja l 1981 ), to p r ed i c t t he ove ra l l pe r fo rm anc e ( ir t t e rm s o fi n se r t i on l o s s , t r ansmi s s ion l o s s , l eve l d i f f e r ence o r no i s e r educ t i on ) o f a g ivenm u f f l e r. T h e s e p e r f o r m a n c e c u r v e s f o r t y pi c al d i m e n s i o n s m a y b e u s e d f o rt ab l e -de s ign o f an e f f i c i en t muf f l e r , keep ing i n mind o the r non -acous t i c a lcons ide r a t i ons l ike ove ra l l s i z e, we igh t , co s t o f f ab r i ca t i on , back p r e s su re on t heeng ine , du rab i l i t y e t c . (Mun ja l 1986 ) .

T h e f ir st a u t h o r w o u l d l ik e to t h a n k t h e a u t h o r i ti e s o f t h e O s m a n i a U n i v e r s i ty f o rt h e s t u d y l ea v e g r a n te d . T h e s e c o n d a u t h o r a c k n o w l e d g e s w i t h t h a n k s t h e f i n a n ci a ls u p p o r t r e c e i v e d f r o m t h e S t i f t u n g V o l k s w a g e n w e r k , H a n n o v e r , G e r m a n y , f o r ap r o j e c t i n c o l l a b o r a t i o n w i th P r o f e s s o r M H e c k l a n d P r o f e s s o r M H u b e r t o f t h eTechn i sche Un ive r s i t i i t Be r l i n .


11/15

Noise reduction w ith perforated m uffler comp onents 265d B1213

10C

80

~o 6O

~- ~03C 20

--20

I I I I I

r n e o s u r e m e n [ sp redic aons

1 t o oL I0.5 1.0 L I I1.5 2.0 2.5 3-0 kH zf r e q u e n c y

Figure 5. Noise reduction of three-duct reverse-flow expansion chamber of figure lb forM] = 0.

o

oc

d 81"20

100

80

I l [ I I m e o s u r e m e n ~ s- - p r e d l c Q o n s

-2 0

F i g u r e 6 .Mi = 0.1.

6O4 0

2 0

0

I i J I L0.5 I -0 1-5 2.0 2.5 3.0 k H zf r e q u e n c y

Noise reduction of three-duct reverse-flow expansion chamber of figure lb for


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266 K N a r a y a na R a o a n d M L M u n j a l

F i g u r e 7 .

t-Ot)

t noc

d B12 0

100

80

60

/.0

20

l I I l m e o s u r @ r n c ~ n t s- - p r e d i c U 0n s

-2 0 I I 1 I t0 0 .5 1 .0 1 .5 2 .0 2 .5 3 . 0k H zf requency

N o i s e r e d u c t i o n o f t h r e e - d u c t r e v e r s e - fl o w e x p a n s i o n c h a m b e r o f f i g u r e l b f o r M 1 = 0 - 2.

Appendix AT h e f o ll o w i ng r e la t io n s h o l d f o r b o t h t h e t h r e e - d u c t c r o ss -f lo w e x p a n s i o n c h a m b e ra n d t h e t h r e e - d u c t r e v e r s e f l o w e x p a n s i o n c h a m b e r .

A 1 j = ~ b n,j e~ jz ,A2, i = ~5, je t3 j~ ,A3 , j = dd6,/eOj ,Z 4 , j = - e ~ j Z / ( i k + M l / 3 j) ,A s j = - ~ 2 d e O j Z / ( i k + M 2 ~ y ) ,A 6 , j = -~ 3 , j e t~ jz ( i k + M 3f l j ) .

Appendix BT h e f o l l o w i n g r e l a t i o n s h o l d f o r t h r e e - d u c t p e r f o r a t e d e l e m e n t c r o s s - f l o we x p a n s i o n c h a m b e r :

T , = T T I , 2 + A 3 C 3 ; Tb = T T I , 4 + B 3 C 3 ;7",. = T T 3 , 2 + A 3 D 3 ; T d = T T 3 , 4 + B 3 D 3 ;A 3 = ( T T 2 , 2 X 2 - T T 4 , 2 ) / ~ ; B 3 = ( T T 2 . 4 X 2 - T T4 ,4 )/F 2;


13/15

N o i s e r e d u c t i o n w i t h p e r f o r a t e d m u f f l e r c o m p o n e n t sC3 = T T I , I + X I T T I , 3 ; D3 = T T 3 , 1 + X 1 T T 3 , 3 ;F 2 = T T 4 . I "I - X 1 T T 4 . 3 - - X 2 ( T T 2 . 1 + X I T T 2 .3 ).

[ T T ] is a n i n t e r m e d i a t e 4 x 4 m a t r ix d e f i n e d a s

w i t h

a n d

m

p 2 ( O ) P 3 ( / )poCowl (0) = [ T T]4 x4 poCow2(lpocow2 (0 ) O~cow2(l) _

T T l . 1 = A 1 A 2 + T I , 2 ;T T 1 , 3 = C 1 A 2 + T 1 ,5 ;T T e . 1 = A 1 B e + i re ,2T T 2 . 3 = C 1 B 2 + T2 .5 ;T T 3 . 1 = A ~ C 2 + T 4 ",;T T 3 . 3 = C 1 C 2 -t- T4.5;T T 4 . 1 = A I D 2 + T s.2 ;T T4.3 = C1D2 + T5.5 ;A 1C1A 2C2F1x1

T T 1 , 2 = B I A 2 + T~.3;T T I . 4 = D I A 2 + Tl,6 ;

; TT2 .2 = B 1 B 2 + T2,3;T T 2 , 4 = D 1 B 2 + T2,6;TT3 .2 = B1C2 + T4.3;T T 3 , 4 = D 1 C 2 + T4.6;T T 4 . 2 = B I D 2 + T5.3;T T 4 . 4 = D 1 D 2 + T 5 .6 ;

= ( T 3 , 2 X 2 - T 6 , 2 ) / F 1 ; B 1 = ( T 3 , 3 X 2 - T 6 ,3 )/ F 1;= ( T 3 . 5 X 2 - T 6 , 5 ) / F I ; D 1 = ( T 3 . 6 X 2 - T 6 . 6 ) / F I ;= T I . I + X I T 1 . 4 ; B 2 = T 2 . 1 + X I T 2 .4 ;= T 4 . 1 + X I T 4 . 4 ; D 2 = T s . l + X 1 T 5 . 4 ;= T 6,1 + X I T 6 , 4 - X 2 ( T 3 , 1 + X 1 T 3 , 4) ;= i ta n ( k l b ) ;

X 2 = - i t a n ( k l a ) .

2 6 7

A p p e n d i x CT h e f o ll o w i n g r e l a ti o n s h o l d o n l y f o r t h r e e - d u c t p e r f o r a t e d e l e m e n t r e v e r s e - fl o we x p a n s i o n c h a m b e r

7 ",, = B l . I D l . l + B I . 2 D 2 . 1 + B I . 3 D 3 . 1 ,T t, = B ~ , I D I . 2 + B ~ . 2 D 2 , 2 + B I , 3 D 3 . 2 ,7",. = B4 ,1D t,I + B4,2D2,1 + Ba.3D3 .1,Td = B4 ,1D I .2 + B4 ,2D2,~_+ B4,3D 3.2 ,


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2 6 8 K Narayana Rao and M L M un j a l

A p p e n d i x

B i l , i 2 = T i l ,i 2 + X 1 T / 1 , ( i 2 + 3 ) ,i 1 = 1 , 2 . . . . . 6 ,i2 = 1 , 2 , 3 ,) (1 = i t a n (k /b ) .Dl,1 = C l . l D 2 . 1 + C l . 3 D 3 . 1 ,D I,z = C1,1 D2,2 + C1,2 D3,2,1) 2 , 1 = C 3 , 2 /F4; D 2 , 2 = - C 2 , 2 / F 4 ,D 3 ,1 = _ C 3 , 1 / F 4 ; D 3 , 2 = C 2 , 1 / F 4 ,F 4 = C 2 , 1 C 3 , 2 - C 2 , 2 C 3 , 1 ,C I, 1 = ( B 5 , 2 - X 2 B 2 , i ) / F 3 ; C 1 , 2 = ( B s , 3 - X 2 B 2 , 3 ) / F 3 ,C2,1 = B3 , z+C I , I B3 , 1 ; C2 , 2 = B3 , 3+Cl , 2B3 , ~ ,C 3,1 -~ B 6 , 2 + C l , l B 6 , 1 ; C 3 , 2 = B 6 , 3 + C I , z B 6 , 1 ,F 3 = B 2 , 1 X 2 - B 5 , 1 ,X 2 = - i tan (k la ).D . N o t a t i o n

A 1 , A oC odfik/MM igoPoPtt e m pU I , 2 , U 2 , 3

WowYzpoOo )~ 1 , ~ 2

i n t e r n a l a r e a s o f i n l e t a n d e x i t p i p e s , r e s p e c t i v e l y ,v e l o c i t y o f w a v e p r o p a g a t i o ni n t e r n a l d i a m e t e r o f p i p ef r e q u e n c y ,i o t a , X / ~ -1 ,w a v e n u m b e r , (tO~Co),l e n g t h o f p i p e ,M a c h n u m b e r , (Wo/co) ,i n l e t M a c h n u m b e r ,e x i t M a c h n u m b e r ,p r e s s u r e o f t h e u n d i s t u r b e d f l u i d ,f l u c t u a t i n g p r e s s u r e ,t i m e c o - o r d i n a t e ,t e m p e r a t u r e ,r a d i a l f l u c t u a t i n g v e l o c i t i e s a t 1 , 2 a n d 2 , 3 i n t e r f a c e s o f t h e c o n -t r o l v o l u m e , r e s p e c t i v e l y ,v e l o c i t y o f t h e u n d i s t u r b e d f l u i d ,f l u c t u a t i n g v e l o c i t y ,c h a r a c t e r i s t i c i m p e d a n c e , co~A,a x i a l c o - o r d i n a t e ,d e n s i t y o f u n d i s t u r b e d f l u i d ,f l u c t u a t i o n i n d e n s i t y ,c i r c u l a r f r e q u e n c y ,a c o u s t i c a l i m p e d a n c e s a t 1 , 2 a n d 2 , 3 i n t e r f a c e s , r e s p e c t i v e l y .


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Noise reduction w ith perforated mu ff ler comp onents 269References

Munjal M L 1975 J . S o , n d V i b . 39: 1115-I 19Munjal M L 1986 A c o u s t t c s o f d u ct ,s a t t d m u f f l e r s (New York: John Wiley) (in print)Panickcr V B, Munjal M L 1981a J . I n d i a , h t ~ ' t . S o L A63:1-19Panicker V B, Munjal M L 1981b J . h u l i a n h z s t . S c i . A63:21-38Rao K N 1984 P r e d i c t t o n a n d v e rl if ic a ti o n ~ f t h e a e r o - a c o u st i c p e r f o r m a n c e o f p e r f i )r a t e d e l e m e n t

;m(l~flers . Ph.D. thesis, Indian Institute of Science, BangaloreRao K N, M unjal M L 19 84 A gencralized decoupling method for analysing perforated eleme nt

mufflers, Proceedings o f the Ne lson Acoustics Conference, Mad ison, USASullivan J W 1979a J . A c o u s t . S o c . A m . 66:772-778Sullivan J W 1979b J . A ~ o u s t . S o c . A m . 66:779-788Sullivan J W, Crocker M J 1978 J . A c o u s t , S o c . A m . 64:2(17-215Thawani P T, Jayaraman K 1983 J . A c o u s 7 . S o c A m . 73:1387-1389

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