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Free vibrations of a corrugated membrane with a fixed, inner circular boundary
Journal of Sound and Vibration (1988) 120(3), 622-625
FREE VIBRATIONS OF A CORRUGATED MEMBRANE WITH A FIXED, INNER CIRCULAR BOUNDARY 1. I NTRODUCTION
The present study deals with the analysis o f transverse vibrations o f t h e doubly connected membrane with fixed boundaries shown in Figure 1. The fundamental frequency coefficient o f such a system is determined by two independent methodologies: an approximate analytical approach based on the approximate conformal mapping transformation of the given domain on to a circular annulus in such a manner that the boundary conditions can be satisfied identically [1]; a finite element approach in which the shape functions contain a k optimization exponential parameter which allows for minimization of the desired eigenvalue [2, 3]. Since the analytical methodology is based upon the construction of co-ordinate functions in the transformed (~:) plane and they do contain a y exponential parameter which also allows for minimization o f the frequency coefficient it is concluded that both approaches, the analytical and the computational method, are based on Rayleigh's minimization criterion [4].
~-plone
Figure 1. Doubly connected annular membrane o f outer corrugated boundary subjected to a hydrostatic state of in-plane stress S.
2. APPROXIMATE ANALYTICAL SOLUTION
The analytic function
z=f(~)=[al(l+~l)]C~+'q~"+'),
~ = r e i~
(1)
maps the unit circle in the ~:-plane into the simply connected corrugated domain shown in Figure 1. The physical domain is circumscribed by a circle of radius a and it possesses n axes o f symmetry. It is also assumed that r/<~ 1In (if O > 1In the boundary curve develops loops in the z-plane and this is clearly unacceptable from a physical viewpoint). As shown in reference [1], if r = to<< l the first term of the Complex polynomial shown in expression (1) prevails and a circle of radius ro is mapped, approximately, on to a circle of radius go = [ a / ( 1 + o)]ro
(2)
in the z-plane. Accordingly, expression (1) provides a valid approximate mapping function for transforming an annulus in the ~:-plane on to the doubly connected domain shown in Figure 1, where Ro<< a. 622 0022-460x/88/030622+04$03.00/0 O 1988 Academic Press Limited
LETTERS TO TIlE
623
EDITOR
In the case o f n o r m a l m o d e s o f v i b r a t i o n o n e m a k e s
w(r, O, t) = W(r, O) e +~'-- Wo(r) e +~',
(3)
where ~t~ (r) = Ct( r ~"+ ch r + an) + C2( r ~+t + a2 r 2 + a j r), a n d where the a , ' s a r e d e t e r m i n e d b y substituting each c o - o r d i n a t e function in the g o v e r n i n g b o u n d a r y c o n d i t i o n s in the ~:-plane,
w(i)
= W ( r n ) = 0.
(4)
C o n s t r u c t i n g n o w the f u n c t i o n a l J [ W] for the m e c h a n i c a l system u n d e r c o n s i d e r a t i o n ,
J[ W] = U~,~, - T,~,,
(5)
where
U,,o,=~jo
,LkOr
Tin., = ~p~o2
]
\r~-]
J
in+i/ I I,o IV 2 f ' ( ~ )
d r dO,
(6a)
dO
(6b)
substituting then e x p r e s s i o n (4) into e q u a t i o n (5) a n d r e q u i r i n g that
OJ[ tV~I/,gG = 0,
(7)
o n e d e t e r m i n e s a l i n e a r system o f h o m o g e n e o u s e q u a t i o n s in the C,'s. F r o m the n o n - t r i v i a l i t y c o n d i t i o n one d e t e r m i n e s an a p p r o x i m a t e f r e q u e n c y e q u a t i o n c o r r e s p o n d i n g to q u a s i - a x i s y m m e t r i e m o d e s . Since each n a t u r a l f r e q u e n c y coefficient ~ , = x / p / S w,a is a f u n c t i o n o f the o p t i m i z a t i o n p a r a m e t e r y, b y r e q u i r i n g 0.O, --=0 0),
(8)
o n e is a b l e to m i n i m i z e each eigenvalue o f the p r o b l e m u n d e r study. In o r d e r to assess the a c c u r a c y o f the p r o c e d u r e it was first a p p l i e d to the case o f c i r c u l a r a n n u l a r m e m b r a n e s a n d the results were then c o m p a r e d with exact e i g e n v a l u e s given in the literature; see T a b l e 1. F o r Ro/a = 0.1 the f u n d a m e n t a l e i g e n v a l u e d e t e r m i n e d TABLE 1
Comparison o f values o f the first two natural freqiwnO' coefficients corresponding to axisymmetric modes o f vibration o f a circular, annular membrane Present study 2-term solution
2-term solution
Exact [ 1]
[5]
0-1
-Or = 3-315 .02 = 7.010
3"32 --
3"3139 6"8576
3-3156 --
0"2
Dn = 3-819 902 = 7.998
3"83 --
3"8159 7"7855
3"8224 --
0.3
.or = 4.412 .02 = 9.190
4.43 --
4-4184t 9.2869t
4.2641 --
0-4
.Or = 5.184 .02 = 10.755
5-21 --
5.1830 10.4432
---
Ro/a
9t Eigenvalues determined by using Galerkin's method and polynomial approximations svithout optimization [ I].
624
LE'ITERS TO TIlE EDITOR
b y using the o p t i m i z e d R a y l e i g h - R i t z a p p r o a c h is in excellent a g r e e m e n t with the exact result a n d the e i g e n v a l u e given in r e f e r e n c e [5]. On the o t h e r h a n d , for Ro/a = 0-2, 0.3 a n d 0.4 the results given in reference 15] are not in g o o d a g r e e m e n t with o t h e r values a v a i l a b l e in the literature, while the a c c u r a c y a c h i e v e d using the p r e s e n t a p p r o a c h is excellent. O n e also observes that the s e c o n d e i g e n v a l u e is, in g e n e r a l , less t h a n 2% h i g h e r t h a n the exact result. This fact s h o w s the p o w e r o f the o p t i m i z e d R a y l e i g h a p p r o a c h , e x t e n d e d in reference [6] to the o p t i m i z a t i o n o f h i g h e r eigenvalues. 3.
FINITE ELEMENTS ALGORITHMIC
PROCEDURE
The a p p r o a c h p r e s e n t e d in reference [3] is b a s e d , essentially, on the m e t h o d o l o g y d e v e l o p e d in reference [2]. It consists o f i n c l u d i n g an u n k n o w n e x p o n e n t i a l p a r a m e t e r k in the s h a p e functions. T h e stiffness a n d mass matrices then b e c o m e , functions o f k a n d the e i g e n v a l u e is t h e n m i n i m i z e d with respect to k. C o n s i d e r a b l e e c o n o m y in m e m o r y a n d c o m p u t e r time is a c h i e v e d for a given d e g r e e o f a c c u r a c y b y f o l l o w i n g this p r o c e d u r e [3]. 4.
DISCUSSION OFTItE
NUMERICAL
RESULTS
T a b l e 2 d e p i c t s e i g e n v a l u e s o f c o r r u g a t e d m e m b r a n e s with a c o n c e n t r i c fixed c i r c u l a r h o l e for several values o f n a n d "O. T h e a g r e e m e n t b e t w e e n a n a l y t i c a l p r e d i c t i o n s a n d the results o b t a i n e d by m e a n s o f the k - o p t i m i z e d finite e l e m e n t a l g o r i t h m i c p r o c e d u r e is TABLE 2
Conlparison of values of Oi(i = 1, 2) of corntgated, doubly connected metnbranes 2-term solution
Ro/a n=4, r/= '
0.1
-Q,= 3.906 -O2 = 8.043
3.91 .
.
.
.
4.576 9.311
4.59 .
.
4.65 .
.
5"402 10"935
5.42
3"725 7.75
3-73
4"348 8"913
4.36 .
.
4"51 .
.
5" 113 10"38
5" 13 .
.
5-26 .
.
3"553 7.480
3.56 .
.
.
.
4"121 8"570
4"14 .
.
4"27 .
.
4"814 9"925
4-84 .
.
0.2 0-3 n = 7, =~
0- I 0-2 0"3
n = 12, r/= ~
Finite elem. solutiont 25 elements 64 elements
l-term solution
0.1 0"2 0.3
I" Quadrangular elements [3].
--
--
.
.
.
.
.
--
3-98
--
4.4023
5"30
5"0224
--
3"8330
--
4.2454
5.16
4"8642
m
3.6792
--
4"0798
4.88
4.6823
.
3"91 .
[5]
.
--
-.
.
LE'I'rERS TO TIlE EDITOR
625
quite good. On the other hand considerable discrepancies with the fundamental eigenvalues given in reference [5] are found. It is important to point out that present results may 'be of some importance when considering acoustic and electromagnetic waveguides of annular cross section where a corrugated outer wall may be convenient from an engineering viewpoint (for instance if an increase in heat dissipation rate is desired). ACKNOWLEDGMENTS
The present investigation has been sponsored by C O N I C E T Research and Development program (PID 3009400). The authors are indebted to research engineer B. H. Valerga de Greco for her generous and valuable co-operation.
Department of Engineering, Universidad Naeional del Sur, 8000 Bahia Blanca, Argentina hlstitute of Applied Afechanics,
V. H. CORTINEZ
P . A . A . LAURA
8111 Puerto Belgrano Naval Base,
Argemina ENACE, S.A., Av. Alem 712--1001 Buenos Aires, Argentina
H . C . SANZl A. BERGMANN
(Received 23 September 1987) REFERENCES
1. P. A. A. LAURA, E. ROMANELLI and M. J. MAURIZI 1972 Journal of Sound and Vibration 20, 27-38. On the analysis of waveguides of doubly connected cross section by the method of conformal mapping. 2. P.A.A. LAURA, J. C. UTJES and G. SANCtlEZ SARMIENTO 1986 Journal of Sound and Vibration 111,219-228. Non-linear optimization of the shape functions when applying the finite element method to vibration problems. 3. J. C. UTJES, G. SANCHEZ SARMIENTO, P. A. A. LAURA and H. C. SANZl 1987 Institute of Applied Mechanics lhtblication No 87-6, l'hterto Belgrano Naval Base. Non-linear optimization of the shape function when solving the 2-D Helmholtz equation by means of the finite element method. 4. LORD RAYLEIGtl 1894 Theory of Sound, Volume 1, London: MacMillan second edition. Reprinted New York: Dover. 5. J. MAZUMDARand D. HILL 1987 Applied Acoustics 21, 22-37. A note on the determination of cut-ott frequencies of hollow waveguides by a contour line-conformal mapping technique. 6. P. A. A. LAURA and V. H. CORTINEZ 1986 American Institute of Chemical Engineers Journal 32, 1025-1026. Optimization of eigenvalues when using the Galerkin method.
FREE VIBRATIONS OF A CORRUGATED MEMBRANE WITH A FIXED, INNER CIRCULAR BOUNDARY 1. I NTRODUCTION
The present study deals with the analysis o f transverse vibrations o f t h e doubly connected membrane with fixed boundaries shown in Figure 1. The fundamental frequency coefficient o f such a system is determined by two independent methodologies: an approximate analytical approach based on the approximate conformal mapping transformation of the given domain on to a circular annulus in such a manner that the boundary conditions can be satisfied identically [1]; a finite element approach in which the shape functions contain a k optimization exponential parameter which allows for minimization of the desired eigenvalue [2, 3]. Since the analytical methodology is based upon the construction of co-ordinate functions in the transformed (~:) plane and they do contain a y exponential parameter which also allows for minimization o f the frequency coefficient it is concluded that both approaches, the analytical and the computational method, are based on Rayleigh's minimization criterion [4].
~-plone
Figure 1. Doubly connected annular membrane o f outer corrugated boundary subjected to a hydrostatic state of in-plane stress S.
2. APPROXIMATE ANALYTICAL SOLUTION
The analytic function
z=f(~)=[al(l+~l)]C~+'q~"+'),
~ = r e i~
(1)
maps the unit circle in the ~:-plane into the simply connected corrugated domain shown in Figure 1. The physical domain is circumscribed by a circle of radius a and it possesses n axes o f symmetry. It is also assumed that r/<~ 1In (if O > 1In the boundary curve develops loops in the z-plane and this is clearly unacceptable from a physical viewpoint). As shown in reference [1], if r = to<< l the first term of the Complex polynomial shown in expression (1) prevails and a circle of radius ro is mapped, approximately, on to a circle of radius go = [ a / ( 1 + o)]ro
(2)
in the z-plane. Accordingly, expression (1) provides a valid approximate mapping function for transforming an annulus in the ~:-plane on to the doubly connected domain shown in Figure 1, where Ro<< a. 622 0022-460x/88/030622+04$03.00/0 O 1988 Academic Press Limited
LETTERS TO TIlE
623
EDITOR
In the case o f n o r m a l m o d e s o f v i b r a t i o n o n e m a k e s
w(r, O, t) = W(r, O) e +~'-- Wo(r) e +~',
(3)
where ~t~ (r) = Ct( r ~"+ ch r + an) + C2( r ~+t + a2 r 2 + a j r), a n d where the a , ' s a r e d e t e r m i n e d b y substituting each c o - o r d i n a t e function in the g o v e r n i n g b o u n d a r y c o n d i t i o n s in the ~:-plane,
w(i)
= W ( r n ) = 0.
(4)
C o n s t r u c t i n g n o w the f u n c t i o n a l J [ W] for the m e c h a n i c a l system u n d e r c o n s i d e r a t i o n ,
J[ W] = U~,~, - T,~,,
(5)
where
U,,o,=~jo
,LkOr
Tin., = ~p~o2
]
\r~-]
J
in+i/ I I,o IV 2 f ' ( ~ )
d r dO,
(6a)
dO
(6b)
substituting then e x p r e s s i o n (4) into e q u a t i o n (5) a n d r e q u i r i n g that
OJ[ tV~I/,gG = 0,
(7)
o n e d e t e r m i n e s a l i n e a r system o f h o m o g e n e o u s e q u a t i o n s in the C,'s. F r o m the n o n - t r i v i a l i t y c o n d i t i o n one d e t e r m i n e s an a p p r o x i m a t e f r e q u e n c y e q u a t i o n c o r r e s p o n d i n g to q u a s i - a x i s y m m e t r i e m o d e s . Since each n a t u r a l f r e q u e n c y coefficient ~ , = x / p / S w,a is a f u n c t i o n o f the o p t i m i z a t i o n p a r a m e t e r y, b y r e q u i r i n g 0.O, --=0 0),
(8)
o n e is a b l e to m i n i m i z e each eigenvalue o f the p r o b l e m u n d e r study. In o r d e r to assess the a c c u r a c y o f the p r o c e d u r e it was first a p p l i e d to the case o f c i r c u l a r a n n u l a r m e m b r a n e s a n d the results were then c o m p a r e d with exact e i g e n v a l u e s given in the literature; see T a b l e 1. F o r Ro/a = 0.1 the f u n d a m e n t a l e i g e n v a l u e d e t e r m i n e d TABLE 1
Comparison o f values o f the first two natural freqiwnO' coefficients corresponding to axisymmetric modes o f vibration o f a circular, annular membrane Present study 2-term solution
2-term solution
Exact [ 1]
[5]
0-1
-Or = 3-315 .02 = 7.010
3"32 --
3"3139 6"8576
3-3156 --
0"2
Dn = 3-819 902 = 7.998
3"83 --
3"8159 7"7855
3"8224 --
0.3
.or = 4.412 .02 = 9.190
4.43 --
4-4184t 9.2869t
4.2641 --
0-4
.Or = 5.184 .02 = 10.755
5-21 --
5.1830 10.4432
---
Ro/a
9t Eigenvalues determined by using Galerkin's method and polynomial approximations svithout optimization [ I].
624
LE'ITERS TO TIlE EDITOR
b y using the o p t i m i z e d R a y l e i g h - R i t z a p p r o a c h is in excellent a g r e e m e n t with the exact result a n d the e i g e n v a l u e given in r e f e r e n c e [5]. On the o t h e r h a n d , for Ro/a = 0-2, 0.3 a n d 0.4 the results given in reference 15] are not in g o o d a g r e e m e n t with o t h e r values a v a i l a b l e in the literature, while the a c c u r a c y a c h i e v e d using the p r e s e n t a p p r o a c h is excellent. O n e also observes that the s e c o n d e i g e n v a l u e is, in g e n e r a l , less t h a n 2% h i g h e r t h a n the exact result. This fact s h o w s the p o w e r o f the o p t i m i z e d R a y l e i g h a p p r o a c h , e x t e n d e d in reference [6] to the o p t i m i z a t i o n o f h i g h e r eigenvalues. 3.
FINITE ELEMENTS ALGORITHMIC
PROCEDURE
The a p p r o a c h p r e s e n t e d in reference [3] is b a s e d , essentially, on the m e t h o d o l o g y d e v e l o p e d in reference [2]. It consists o f i n c l u d i n g an u n k n o w n e x p o n e n t i a l p a r a m e t e r k in the s h a p e functions. T h e stiffness a n d mass matrices then b e c o m e , functions o f k a n d the e i g e n v a l u e is t h e n m i n i m i z e d with respect to k. C o n s i d e r a b l e e c o n o m y in m e m o r y a n d c o m p u t e r time is a c h i e v e d for a given d e g r e e o f a c c u r a c y b y f o l l o w i n g this p r o c e d u r e [3]. 4.
DISCUSSION OFTItE
NUMERICAL
RESULTS
T a b l e 2 d e p i c t s e i g e n v a l u e s o f c o r r u g a t e d m e m b r a n e s with a c o n c e n t r i c fixed c i r c u l a r h o l e for several values o f n a n d "O. T h e a g r e e m e n t b e t w e e n a n a l y t i c a l p r e d i c t i o n s a n d the results o b t a i n e d by m e a n s o f the k - o p t i m i z e d finite e l e m e n t a l g o r i t h m i c p r o c e d u r e is TABLE 2
Conlparison of values of Oi(i = 1, 2) of corntgated, doubly connected metnbranes 2-term solution
Ro/a n=4, r/= '
0.1
-Q,= 3.906 -O2 = 8.043
3.91 .
.
.
.
4.576 9.311
4.59 .
.
4.65 .
.
5"402 10"935
5.42
3"725 7.75
3-73
4"348 8"913
4.36 .
.
4"51 .
.
5" 113 10"38
5" 13 .
.
5-26 .
.
3"553 7.480
3.56 .
.
.
.
4"121 8"570
4"14 .
.
4"27 .
.
4"814 9"925
4-84 .
.
0.2 0-3 n = 7, =~
0- I 0-2 0"3
n = 12, r/= ~
Finite elem. solutiont 25 elements 64 elements
l-term solution
0.1 0"2 0.3
I" Quadrangular elements [3].
--
--
.
.
.
.
.
--
3-98
--
4.4023
5"30
5"0224
--
3"8330
--
4.2454
5.16
4"8642
m
3.6792
--
4"0798
4.88
4.6823
.
3"91 .
[5]
.
--
-.
.
LE'I'rERS TO TIlE EDITOR
625
quite good. On the other hand considerable discrepancies with the fundamental eigenvalues given in reference [5] are found. It is important to point out that present results may 'be of some importance when considering acoustic and electromagnetic waveguides of annular cross section where a corrugated outer wall may be convenient from an engineering viewpoint (for instance if an increase in heat dissipation rate is desired). ACKNOWLEDGMENTS
The present investigation has been sponsored by C O N I C E T Research and Development program (PID 3009400). The authors are indebted to research engineer B. H. Valerga de Greco for her generous and valuable co-operation.
Department of Engineering, Universidad Naeional del Sur, 8000 Bahia Blanca, Argentina hlstitute of Applied Afechanics,
V. H. CORTINEZ
P . A . A . LAURA
8111 Puerto Belgrano Naval Base,
Argemina ENACE, S.A., Av. Alem 712--1001 Buenos Aires, Argentina
H . C . SANZl A. BERGMANN
(Received 23 September 1987) REFERENCES
1. P. A. A. LAURA, E. ROMANELLI and M. J. MAURIZI 1972 Journal of Sound and Vibration 20, 27-38. On the analysis of waveguides of doubly connected cross section by the method of conformal mapping. 2. P.A.A. LAURA, J. C. UTJES and G. SANCtlEZ SARMIENTO 1986 Journal of Sound and Vibration 111,219-228. Non-linear optimization of the shape functions when applying the finite element method to vibration problems. 3. J. C. UTJES, G. SANCHEZ SARMIENTO, P. A. A. LAURA and H. C. SANZl 1987 Institute of Applied Mechanics lhtblication No 87-6, l'hterto Belgrano Naval Base. Non-linear optimization of the shape function when solving the 2-D Helmholtz equation by means of the finite element method. 4. LORD RAYLEIGtl 1894 Theory of Sound, Volume 1, London: MacMillan second edition. Reprinted New York: Dover. 5. J. MAZUMDARand D. HILL 1987 Applied Acoustics 21, 22-37. A note on the determination of cut-ott frequencies of hollow waveguides by a contour line-conformal mapping technique. 6. P. A. A. LAURA and V. H. CORTINEZ 1986 American Institute of Chemical Engineers Journal 32, 1025-1026. Optimization of eigenvalues when using the Galerkin method.