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VIBROACOUSTIC MODEL SCALING WITH DISSIPATION EFFECTS
L. Cheng, C. Lesueur
To cite this version:
L. Cheng, C. Lesueur. VIBROACOUSTIC MODEL SCALING WITH DISSIPATION EFFECTS.
Journal de Physique Colloques, 1990, 51 (C2), pp.C2-213-C2-216. �10.1051/jphyscol:1990251�. �jpa-
00230672�
ler Congres Frangals d'Acoustique 1990
VIBROACOUSTIC MODEL SCALING WITH DISSIPATION EFFECTS*l
>
L. CHENG*
2)and C. LESUEUR
Lafioratoire Vifirations-Acoustigue (L.V.R.). Bat. 303, INSR Lyon, 20 Avenue A. Einstein, F-69621 Villeurianne Cedex, Trance
Résumé - Cette étude examine le problème de similitude pour un système constitué d'une structure soumise à des excitations externes mécaniques et acoustiques, et couplée à une cavité.
Nous proposons une démarche générale pour tout type de structure et pour tenir compte des effets de dissipation (amortissements et absorption).
Deux exemples numériques sont proposés pour deux géométries particulières.
Abstract - This paper addresses the model scaling of vibroacoustic systems composed of vibrating structures excited by external mechanical and acous- tical excitations, and coupled to a cavity.
We propose a general approach for all type of structure and dissipative effect due to system damping and absorbent material are also taken into account. Two numerical examples are performed for two different geometry.
1- INTRODUCTION
The vibroacoustic similitude problem is also referred to as vibroacoustic model scaling. The aim is to develop scaling laws which enable us to predict vibroacoustic responses of a real structure by using the experimental results obtained on a small model. On the basis of the equations characterizing a structure coupled to a cavity, we give a systematic procedure for obtaining these laws and constraint conditions related to the fabrication of the model.
The dissipative effects, such as damping and absorption, lead to some harsh constaint conditions which can hardly be verified in practice. In this case, we furnish some approximate scaling laws give satisfactory results with allo- wable errors in engineering practice.
2 - BRIEF DESCRIPTION OF THE THEORETICAL MODEL
The model consists of a vibrating structure (AF)backed by a cavity (V) and excited by external loadings (sound pressure PE or mechanical point-force FE) . Some portions of the cavity (AA) are covered by layers of absorbent material and the rests (AR) are rigid. Internal sound pressure Pc and normal displace- ment of the structure w are governed by the equations established by DOWELL \l\
After taking into account the damping properties of each subsystem, these equa- tions can be summarized as follows :
oo
ZNA(<0).P- ( 0 , ) = ^ B1NMu,*qM(u,) (1) ,
00
W
W )* M
( W ) =, A
B2NM
?N(U) +V
W > (2) N=lwhere PN(ID) and qM(w)are respectively the generalized modal coordinates of cavity sound pressure and of structure_displacement. Z„A(oo) and ZMS(a)) contain dissipative factors of the system and QM(««) is excitation term. Pc and w can then be calculated by superposing these modal responses (N : cavity mode ; M : structural mode).
(1) Supported by Aerospatiale Company (Toulouse) Laboratoire d'Acoustique (2) Present address : G.A.U.S - Faculty of Applied Sciences - Universite
d e Sherbrooke - SHERBROOKE (Quebec) C A N A D A J 1 K 2 R 1 .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990251
COLLOQUE DE PHYSIQUE
3
-
GENERAL PROCEDUREI n what follows, s c a l e d model i s c h a r a c t e r i z e d by a s u p e r i o r index i n t h e equa- t i o n s . X i , d e f i n e d a s t h e r a t i o of a f u l l - s c a l e model c h a r a c t e r i s t i c s t o t h o s e of s c a i e d model, i s c a l l e d s c a l i n g r a t i o .
Three t y p e s o f c o n s t r a i n t c o n d i t i o n s a r e a s follows.
Type 1 : i f w N =g 1 ( h i ) , utN,- w M=g 2( A i). m ' M ( 3 )
then: gl ( h i ) =g2 (X i =g ( X i ) ( 4
Type TI: w=g3(Ai), W ' ( 5 )
with: g3(Xi) =g(hi) ( 6 )
I t can be seen t h a t t h e same frequency c o u p l i n g n a t u r e has been kept f o r b o t h systems.
Type 111:
- - -
By assuming p t N ( o ) = XPN(u); q B M ( w ) = yghM(o) and O n M ( u ) = ZfM(w) and by i n t r o - ducing t h e r e l a t i o n s h i p s among d i f f e r e n t parameters of two systems, (1) and
( 2 ) become : a
-
x i
1( A . 1 . Z ' ~ ~ ( W I ) = ~ f ( 1 . 1 L ~ ~ ~ ~ ~ ~ ' ~ q ~ ~ ( o ' )M = l ( 7 )
( 8 ) Then, t h e c o n s t r a i n t c o n d i t i o n s of t y p e s I11 a r e :
and t h e s o l u t i o n of X.Y.2 g i v e s t h e s c a l i n g laws.
4
-
SCALING RESULTS (Theory and numerical c a l c u l a t i o n s )a > Geometrical s c a l i n g :
Non d i s s i p a t i v e svstem : AG being s c a l i n g r a t i o w i t h r e s p e c t t o a l l geometri- c a l dimensions. I t can be shown t h a t t h e c o n d i t i o n s of type I and I1 r e q u i r e t h a t g(Xi)=AG and t h e t h i r d one i s v e r i f i e d a u t o m a t i c a l l y . So, we have :
w i t h : PE = P I E ( a c o u s t i c p r e s s u r e l o a d i n g ) FE = ( l / A G ) F2 ' ~(mechanical p o i n t - f o r c e )
One numerical r e s u l t r e l a t e d t o NR (Noise Reduction) i s given i n f i g u r e 1 concerning a f i n i t e s t i f f e n e d c y l i n d r i c a l s h e l l . The model i s two times s m a l ~ l e r than t h e r e a l one ()lG=1/2) and i s s t i f f e n e d by f o u r T shaped r i n g s t i f f e -
n e r s e q u a l l y spaced a l o n g l o n g i t u d i n a l a x i s . The a p p l i c a t i o n of t h e s c a l i n g laws g i v e s us t h e t h i r d curve c o i n c i d i n g t o t a l l y with t h e f i r s t one. From
(10) and (11)
,
it can be seen t h a t N R ( w ) = N R ' ( w ' ).
D i s s i p a t i v e svstem : With r e s p e c t t o forementioned r e s u l t s , two supplementary c o n d i t i o n s r e s u l t from damping and a b s o r p t i o n :
Where q ~ ( w ) i s c a v i t y modal l o s s f a c t o r and Z A ( w ) i s a c o u s t i c impedance of absorbeht m a t e r i a l . Usually, a b s o r p t i o n b r i n g s t h e e q u i v a l e n t damping e f f e c t s which exceed t h o s e of nN(w) (13) r e q u i r e s t h a t m a t e r i a l should be p r o p e r l y chosen, and t h e parameters may be o b t a i n e d by s o l v i n g t h e complex e q u a t i o n s
( 1 3 ) .
l y governed by a p a r a m e t e r 6 which i s d e f i n e d a s B = ~ ~ ( w , ' t , h ~ ) / Z ' ~ ( u ~ , ~ ' , h ' ~ ) ,
T b e i n g t h e f l o w - r e s i s t a n c e o f t h e m a t e r i a l a n d h t a t h e t h i c k n e s s . Based o n t h e f o r m u l a o f DELANY ( 2
1,
f i g u r e 2 shows t h e v a r i a t i o n o f 6 v e r s u s f r e q u e n - c i e s ( ~ = 1 0 ) . It i s known t h a t i t ' s t h e r e a l p a r t o f B whkch a f f e c t s t h e a c c u r a c y o f s c a l i n g r e s u l t s . The f a c t t h a t R e { B l i s d i f f e r e n t from 1 (1.2 ma- ximum) b r i n g s u s a n e r r o r o f a b o u t 2dB i n p,. F i g u r e 3 i l l u s t r a t e s a r e s u l t f o r a p a n e l - c a v i t y s y s t e m ( s e e 731) f o r which kG=1/2. ~ = 1 0 c . g . s . r a y l s / c m . The same c o n c l u s i o n c a n b e drawn.b) S c a l i n s w i t h d i s t o r s i o n ( q e o m e t r y a n d m a t e r i a l )
I f we want t o change t h e s t r u c t u r e t h i c k n e s s d i f f e r e n t l y from t h e o t h e r geo- m e t r i c a l p a r a m e t e r s a n d t o change t h e m a t e r i a l o f s t r u c t u r e i n t h e same t i m e , t h e e o n c r e t e s t r u c t u r e h a s t o be t a k e n i n t o c o n s i d e r a t i o n . T h i s l e a d s u s u a l l y t o t h e s p e c i f i c c h o i c e o f t h e m a t e r i a l .
Some d e t a i l i n f o r m a t i o n w i l l be g i v e n d u r i n g t h e t a l k . REFERENCES
11) E.H. DOWELL, G.F. GORMAN I11 a n d D.A. SMITH
-
1977, JSV 52, 519-542.121 M.E. DELANY a n d N. BAZLEY
-
1970, A p p l i e d A c o u s t i c s ( 3 1 , 105-116.131 L. CHENG a n d C. LESUEUR
-
1989, J o u r n a l d s A c o u s t i q u e n 0 2 ( 2 ) .0 200 400 600 800 1000 1200
FREQUENCY (Hz)
FIGURE 1 - T o t a l g e o m e t r i c a l s c a l i n g f o r a f i n i t e s t i f f e n e d c y l i n d r i c a l s h e l l . 1 : f u l l - s c a l e s h e l l .
2 : h a l f - s c a l e model.
3 : s c a l i n g r e s u l t .
COLLOQUE DE PHYSIQUE
1.5 I I I I I I I I I
1 2 3 4
0.5 -
I I I I
-0.5 I I I I I
0 200 400 600 800 1000
FREQUENCY (Hz)
F I G U R E 2
-
V a r i a t i o n o f B f o r d i f f e r e n t v a l u e s of- XG.1 : XG = 1/4 2 : XG = 1/3 3 : X G = 1/2 4 : X G = 2
0
0 400 800 1200 1600 2000
FREQUENCY (Hz)
F I G U R E 3
-
S c a l i n g f o r p a n e l - c a v i t y system with a Layer of a b s o r b e n t mate- r i a l .1 : f u l l - s c a l e model 2 : h a l f - s c a l e model 3 : s c a l i n g r e s u l t .
