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ELSEVIER Applied Acoustics, Vol. 48, No. 4, pp. 301-309, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0003-682X/96/%15.00+0.00 PII:SOOO3-682X(96)00005-9 Approximate Determination of Vibration Frequencies of Doubly Connected Membranes of Complicated Boundary Shape V. H. Cortinez, P. A. A. Laura Institute of Applied Mechanics (CONICET-SENID-ACCE) and Department of Engineering, Universidad National de1 Sur, 8000-Bahia Blanca, Argentina & H. C. Sanzi ENACE SA, Atucha II-National Atomic Energy Commission, Buenos Aires, Argentina (Received 1 June 1995; revised version received 17 October 1995; accepted 15 December 1995) ABSTRACT This paper deals with the determination of upper bounds of vibration frequencies of doubly connected membranes fixed at both boundaries by means of a conformal mapping approach. If the exact Laurent series which maps the given domain onto a circular annulus in the transformed domain is not known, it is shown that for certain cases of practical interest one can still obtain an approximate value of the frequency, not necessarily being an upper bound. The numerical values obtained in the present investigation are in good agreement with those deter- mined by means of (a) an extended Pnueli’s method and (b) a$nite element code. Copyright 0 1996 Elsevier Science Ltd Keywords: Upper bounds, Laurent series, conformal mapping, Pnueli’s method, finite element. INTRODUCTION The study of the dynamic behavior of membranes is of interest in several fields of applied science and technology. i-lo The problem is governed by the two-dimensional wave equation when several simplifying assumptions are introduced. 301

Approximate determination of vibration frequencies of doubly connected membranes of complicated boundary shape

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Page 1: Approximate determination of vibration frequencies of doubly connected membranes of complicated boundary shape

ELSEVIER

Applied Acoustics, Vol. 48, No. 4, pp. 301-309, 1996 Copyright 0 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0003-682X/96/%15.00+0.00

PII:SOOO3-682X(96)00005-9

Approximate Determination of Vibration Frequencies of Doubly Connected Membranes of Complicated

Boundary Shape

V. H. Cortinez, P. A. A. Laura

Institute of Applied Mechanics (CONICET-SENID-ACCE) and Department of Engineering, Universidad National de1 Sur, 8000-Bahia Blanca, Argentina

&

H. C. Sanzi

ENACE SA, Atucha II-National Atomic Energy Commission, Buenos Aires, Argentina

(Received 1 June 1995; revised version received 17 October 1995; accepted 15 December 1995)

ABSTRACT

This paper deals with the determination of upper bounds of vibration frequencies of doubly connected membranes fixed at both boundaries by means of a conformal mapping approach. If the exact Laurent series which maps the given domain onto a circular

annulus in the transformed domain is not known, it is shown that for certain cases of practical interest one can still obtain an approximate value of the frequency, not necessarily being an upper bound. The numerical values obtained in the present investigation are in good agreement with those deter- mined by means of (a) an extended Pnueli’s method and (b) a$nite element code. Copyright 0 1996 Elsevier Science Ltd

Keywords: Upper bounds, Laurent series, conformal mapping, Pnueli’s method, finite element.

INTRODUCTION

The study of the dynamic behavior of membranes is of interest in several fields of applied science and technology. i-lo The problem is governed by the two-dimensional wave equation when several simplifying assumptions are introduced.

301

Page 2: Approximate determination of vibration frequencies of doubly connected membranes of complicated boundary shape

302 b’. H. C‘ortiner. P. ‘4. .A. Luuru. H. C’. Sanzi

In the case of rectangular, circular and elliptical simply connected domains, exact solutions are well known in terms of standard mathematical functions. In the case of doubly connected domains exact solutions are pos- sible in a limited number of cases, namely: the circular annular shape and the elliptical confocal domain. For other configurations approximate methods must be used. The present paper deals with the approximate determination of the fundamental frequency of transverse vibration of doubly connected membranes of complicated boundary shape by the following approaches:

(1)

(2)

(3)

an extension of Szego’s theorem’ which allows for the calculation of an upper bound if the exact mapping function which maps the given domain onto an annulus of unit outer radius and internal radius rO; a direct application of the formulation presented by Laura et al.” which, following the previous methodology, allows for the approx- imate determination of the eigenvalue under investigation although it is not necessarily an upper bound; an extension of Pnuelli’s approach;’

The determined eigenvalues are in very good agreement with those determined by means of a finite element algorithmic procedure.‘”

It is important to point out that the results are also applicable when calculating the lowest cut-off frequency in a soft-walled acoustic waveguide or electromagnetic wave propagation in a waveguide of doubly connected cross-section in the case of TM modes.

DETERMINATION OF AN UPPER BOUND

Since configurations of ‘s’ axes of symmetry are of primary interest in the present investigation it must be recalled that the mapping function can be expressed in general as3 [Fig. 1 (A)]:

: = .,- + i,r =.j(C) Z E ajc ,=_-“L 1 +.i.y

(1)

where < = reic). Approximating the fundamental displacement amplitude W(x,v) by means

of W,(x,y) one expresses Rayleigh’s quotient by:

(2)

where A is the area of the membrane.

Page 3: Approximate determination of vibration frequencies of doubly connected membranes of complicated boundary shape

Approximate determination ?f vibration frequencies ?f membranes 303

Z-plane

(A] Regular Poligonal Shape with a Concentric Circular Fmforation and its Mapping in the q- Plane

(61 ~~=__d: n 5 l/s (10) (l+n)

ICI “A = 0.9224 L (9)

(D) ff;, = $ L 191 IE) “;, = 1.132 L (91

(L = a : original square ) (L=opl

Fig. 1. Conformal mapping of doubly connected regions and some of the configurations studied in the present investigation.

Expressing eqn (2) in terms of the (r,O) variables results in:”

and neglecting the 0 dependence in the t-plane one can assume

Wl = WI(r)

(3)

(4)

Page 4: Approximate determination of vibration frequencies of doubly connected membranes of complicated boundary shape

304 V. H. Cortinc, P. A. A. Laura, H. C. Sanzi

Accordingly, integrating eqn (3) with respect to 0, one obtains:

c’ (d WI /drfr dr

where

(5a)

(5b)

In view of the fact that all the terms appearing in eqn (5b) are positive one can write

ai < G(r) (6)

J’ I .I

a;Wfrdr < G(r) W:r dr Tli . T,)

and the expression

(7)

(8)

satisfies the inequality

A;! > 1; (9)

Since Xi is an upper bound of the exact eigenvalue, it turns out that X: is also an upper bound, according to eqn (9).

If W,(I) is made equal to the exact eigenfunction for the case of an annular membrane of unit outer radius and inner radius equal to YO, one concludes that the ratio

s ’ (dWi/dr)2rdr )‘<I

1’ W:r dr

)‘o

is equal to the exact fundamental eigenvalue

%.

(10)

for such a structural element.

Page 5: Approximate determination of vibration frequencies of doubly connected membranes of complicated boundary shape

Approximate determination of vibration frequencies of membranes 305

Accordingly

A.2 - n:1 s 4

and then Qll

AlI < --&

(11)

(12)

In the case of a simply connected membrane the inequality reads:

2.4048 AlI <-

a0 (13)

which is from Szego.7 It seems reasonable, therefore, to use the approximation

A,, ” R,, QO

(14)

for higher modes. Finding the exact mapping function and ~0 in the case of a doubly

connected region is an extremely difficult task. However, approximate mapping and values of r. can be obtained in a straightforward fashion3 if one knows the analytic function which maps the simply connected region for configurations such as those shown in Fig. 1. Since

(15)

for values of r. < 1 one can approximate

R. 2: c&r0 (16)

Clearly one can approximate the parameters appearing in eqn (12) by those obtained from eqn (16) and, as it will be shown, good engineering accuracy will result but one cannot guarantee, necessarily, that an upper bound will also result.

GENERALIZATION OF PNUELI’S EXPRESSION

Pnueli8 has demonstrated that one can approximate the eigenvalues, X,,, of a simply connected membrane by the proper values Q,,, corresponding to an equivalent circular membrane:

r = A/n

where A is the area of the membrane.

(17)

Page 6: Approximate determination of vibration frequencies of doubly connected membranes of complicated boundary shape

306 V. H. Cortinez, P. A. A. Laura, H. C. Satm

TABLE 1 Frequency coefficients corresponding to circular. annular membrane fixed at both boundaries

(unit outer radius and inner radius equal to rO)

r0 Q/i n I2 .~

0.1 3.3139 6.8576 0.2 3.8 159 7.7855 0.3 4.418 9.286 0.4 5.1830 10.4432

One can, pragmatically, extend Pnueli’s finding to the case of doubly connected membranes and approximate their eigenvalues X,, by the Q,,s corresponding to circular annular membranes of inner radius r = A/X and outer radius Y = A/n, where Ai is the area bounded by the inner boundary of the membrane and A is the total planform area of the membrane.

NUMERICAL RESULTS

Table 1 depicts values of exact frequency coefficients of circular annular membranes and which are used in the present study for the determination of approximate eigenvalues of doubly connected membranes of complicated boundary shape.

Table 2 shows a comparison of frequency coefficients for square penta- gonal and hexagonal outer configurations and circular inner boundaries between the results previously obtained3 and those determined using eqn (14) with approximate parameters, and good engineering agreement is observed.

Figures 2 and 3 depict comparison of fundamental eigenvalues obtained using the approximate approaches explained in this paper and those deter- mined using a finite element algorithmic procedure previously described in the literature” for the cases of outer square and regular pentagonal config- urations with circular concentric holes. Good agreement is observed between the predictions of the finite element method and the values obtained using eqns (12) and (16). On the other hand, Pnueli’s approach yields results which are, in general, extremely low but the agreement with the eigenvalues obtained by means of the finite element method improves as Ro/a, acquires higher values, say Rota, > 0.40.

Table 3 depicts a comparison of eigenvalues for the case of a corrugated membrane with a concentric circular boundary [Fig. l(B)]. It is again concluded that the agreement with previous resultslo is very reasonable.

Page 7: Approximate determination of vibration frequencies of doubly connected membranes of complicated boundary shape

Approximate determination of vibration frequencies of membranes 307

TABLE 2 Frequency coefficients corresponding to the first two quasi-axisymmetric modes in the case of

outer boundaries of regular polygonal shape

s Present studya

r. = Ro/a’o xa, Aa,, eqn (4)

4a’s = l.O787a,

W. = 1.0526a,

6~‘~ = l.O376a,

0.1 3.07 6.3 3.06+ 6.56i 0.2 3.53 7.2 3.50 7.34 0.3 4.08 8.6 4.04 8.35 0.1 3.15 6.5 3.15 6.81 0.2 3.61 7.4 3.61 7.64 0.3 4.19 8.7 4.17 8.74 0.1 3.20 6.6 3.20 6.95 0.2 3.67 7.5 3.67 7.79 0.3 4.25 8.8 4.24 8.88

“These values have been determined using eqns (14) and (16).tXlIa,; fX12a,.

4

3

2.5

+

/ /

0

0

-/, ,

0

0 0

+ 0 ,

. ,

.

0.1 0.2 0.3 0.4 0.5 '%/a~

Fig. 2. Fundamental eigenvalue of a square membrane with a concentric circular perforation: comparison of results.

Table 4 shows eigenvalues corresponding to quasi-axisymmetric config- urations in Fig. l(Ck(E). I n view of the good engineering accuracy of the eigenvalues exhibited in Table 2 and Table 3 one may reasonably expect that the accuracy of the values contained in Table 4 is acceptable from a practical viewpoint.

Page 8: Approximate determination of vibration frequencies of doubly connected membranes of complicated boundary shape

308 V. H. Cortinez, P. A. A. Laura, H. C. Sanzi

/’ /

/ /

4_ /

/

_/ .

/’ /

/ /

/ ,

3

01 0.2 0.3 0.4 w+

- ExpressIons (12)cnd(16) Flnlte Element M&hod

-_‘- ExtendedPnwW’sMelhod

Fig. 3. Fundamental eigenvalue of a pentagonal membrane with a concentric circular perforation: comparison of results.

TABLE 3 Frequency coefficients corresponding to the first two quasi-axisymmetric modes in the case of a corrugated membrane with a fixed. concentric circular boundary (Z = a/l + n(< + nF+‘)),

eqn (15)

s Present study”

f-0 = &i a Xa

4(9 = l/6; o’e = 0.857a) 0.1 3.95 8.6 0.2 4.66 9.8 0.3 5.51 11.6

7(?/= I i’9: Q’,] = 0.9a) 0.1 8.76 8.1 0.2 4.3 69.3 0.3 5.16 10.8

“These values have been determined using eqns (14) and (16).+X, tu; 1X,,a.

Xa, eqn (IS)

3.90+ 8.04: 8.57 9.31 5.40 10.93 3.72 7.75 4.34 8.91 5.1 I 10.3

ACKNOWLEDGEMENTS

The present study has been sponsored by CONICET Research and Devel- opment Program (PID 1992-1994) and by Secretaria General de Ciencia Y Tecnologia of Universidad National de1 Sur (Program 1994-1995; Director: Professor R. E. Rossi).

Page 9: Approximate determination of vibration frequencies of doubly connected membranes of complicated boundary shape

Approximate determination of vibration frequencies of membranes 309

TABLE 4 First, two-frequency coefficients corresponding to two quasi-axisymmetric modes of config-

uration shown in Fig. l(CHE)

ro = Ro/a’o XL Fig. I(C) Fig. I(D) Fig. I(E)

0.1 3.59 3.17 2.92” 7.42 6.57 6.05b

0.2 4.13 3.65 3.37 8.43 7.46 6.87

0.3 4.78 4.23 9.39 8.32

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

REFERENCES

Eastep, F. E., Estimation of fundamental frequencies of doubly connected membranes. Journal of Sound and Vibration, 37 (1974) 399410. Irie, T., Yamada, H. & Ashida, K., Vibrations of cross shaped I-shaped and L-shaped membranes and plates. Journal of the Acoustical Society of America, 72 (1982) 46&465. Laura, P. A. A.Romanelli, E. & Maurizi, M. J., On the analysis of waveguides of doubly connected cross-section by the method of conformal mapping. Jour- nal of Sound and Vibration, 20 (1972) 27-38. Nagaya, K., Vibration and dynamic response of membranes with arbitrary shape. ASME Journal of Applied Mechanics, 45 (1987) 153-l 58. Kuttler, J. R. & Sigilito, V. G., Frequencies of limacons and cardiods that have applications to waveguides and mitral valves. Journal of Sound and Vibration, 84 (1982) 603-605. Mazumdar, J., A review of approximate methods for determining the vibra- tional modes of membranes. The Shock and Vibration Digest, 16 (1984) 5-15. Szego, G., Conformal mapping related to torsional rigidity, principal frequency and electrostatic capacity. In: Proceedings of a Symposium. National Bureau of Standards, June 1949.. Pnueli, D., Lower bound to then eigenvalue of the Helmholtz equation over two dimensional regions of arbitrary shape. Journal of Applied Mechanics, 36 (1969) 630-631. Schinzinger, R. & Laura, P. A. A., Conformal Mapping: Methods and Applica- tions. Elsevier, Amsterdam, 199 1. Cortinez, V. H., Laura, P. A. A. Sanzi, H. C. & Bergmann, A., Free vibrations of a corrugated membrane with a fixed, inner, circular boundary. Journal of Sound and Vibration, 120 (1988) 622-625.