a simple approximate method for the solution of the helmholtz equation in the case of rectangular,...

Applied Acoustics 17 (1984) 405-411

A Simple Approximate Method for the Solution of the Helmholtz Equation in the Case of Rectangular, Non-

homogeneous Domains

P. A. A. L a u r a and R. H. Gut ierrez

Institute of Applied Mechanics, 8111-Puerto Belgrano Naval Base (Argentina)

(Received: 31 October, 1983)

S U M M A R Y

The dynamic behavior of non-homogeneous continuous media is of interest in many fields, from geophysics to everyday mechanical engineering systems and especially in bioengineering.

Discontinuous types of non-homogeneities are considered in this paper and it is shown that the Ritz method is convenient for this complicated type of structural element.

I N T R O D U C T I O N

The dynamic behavior of mechanical systems with discontinuously varying material properties is of considerable interest in several areas of applied science and technology. Advances in this direction have been reported in several excellent papers. 1-4

The present paper deals with the study of a vibrating rectangular membrane with a discontinuously varying density distribution (Fig. 1). The problem is solved by expanding the displacement amplitude in terms of a complete set of co-ordinate functions which identically satisfy the governing boundary conditions. The Ritz method is then used to generate the determinantal equation.

405 Applied Acoustics 0003-682X/84/$03.00 © Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain

406 P. A. A. Laura, R. H. Gutierrez

-tl,

Fig. 1.

Dz (~2)

J !

I - b 1

Mechanical system under study.

(D 1 ¢= D z)

APPROXIMATE ANALYTICAL METHOD

In the case of normal modes the problem is governed by the differential system:

P V 2 W + ~ 2 ~ W = 0 (l(a))

W(~D2) ---- 0 (l(b)) where:

~'PlV (2, f )eD~

p(Y, )~) = [ p 2 V ( y , f ) e D 2 _ D ,

The solution to this problem can be obtained in a very convenient manner using the Ritz method. The governing functional is the well known expression:

J[W] = (W2~ + W ~ 2 ) d 2 d f - ~ - P2 2

1 where:

=--,Pl DI ~ 02 P2

Approximation for solution of Helmholtz eqn in non-homogeneous regions 407

Int roducing the dimensionless obtains, f rom eqn. (2)"

where

variables x=Y/a 1, y=f/b 1, one

2JtW]= f f c (W2 + 22W2)dxdy °92p2 a2 2 T

x[T f fc W2dxdy+ f fc2_c W2dxdyl (3)

bl

The displacement ampli tude, W, will now be approximated by a t runcated series of the form:

= (1 - x2)(1 - y 2 ) y A,r,x,ym wo (4) n~m

Substituting eqn. (4) into eqn. (3) and applying the minimizat ion condit ion:

OJ[W.] _ 0 (5) (~A.m

one obtains a secular de terminant where the roots are the natural frequencies of the system.

- a

b~

D 3 (~3)

-b I

(D 1 E D 3 ~ D 2 E D 3 )

a 1

Fig. 2. Rectangular membrane with several non-homogeneities.


Y

p

7

0

-7

'-bl

D 1

D~

X &l

i .

b

i_ (a)

Fig. 3.

- a ,

-b 1

(b)

D~

a 1

Rectangular membrane with: (a) Concentric rectangular non-homogeneity. (b) Concentric circular non-homogeneity.


Clearly, the general procedure previously described is applicable to membranes with several non-homogeneities (see Fig. 2).

In the case of the systems shown in Fig. 3(a) and (b) one takes:

N

W a = (1 - x2)(1 _y2 ) ) ' , A.x2,y2, (6) / d n = O

due to symmetry considerations. The analysis is straightforward but too lengthy to be included here. In the case of the system shown in Fig. 3(a) it is convenient to introduce

the parameters u =~/a~ and v = v/b1, while, for Fig. 3(b), one uses U = r / a 1 .

NUMERICAL RESULTS

Fundamental frequency coe f f i c i en t s ~ o 9 1 1 a , (a = 2 a l ) are obtained for the mechanical systems shown in Fig. 3 for several combinations of the parameters 2 = a/b, 7 = Pl/P2 and u = f f / a 1 = v = f / b 1.

In the case of the rectangular membrane with a concentric circular spot, Fig. 3(b), it was found to be of considerable advantage to numerically

TABLE ! Fundamental Frequency Coefficient

~ m l ~ a in the Case of a Rectangular Membrane with a Concentric Rectangular Non-

homogeneity (u = v)

7 u 2 = 1 )~=1"5 ).=2

0" 1 4"76 6"06 7-52 3 0-2 3.94 5"03 6-24

10 2"97 3.79 4"70

0.1 5'17 6.59 8.17 3 0"3 3.54 4.52 5-60

10 2-34 2-98 3"69

0.1 5.81 7'40 9.18 3 0.4 3'20 4.08 5.16

10 1'94 2.48 3"07


evaluate the last two integrals appearing in the governing functional equation (eqn. (3)).

Table 1 presents values of the fundamental frequency coefficient in the case of the non-homogeneous membrane shown in Fig. 3(a) whilst Table 2 shows values of frequency coefficients for the rectangular membrane with the concentric, circular non-homogeneous spot for the case where u = r / a 1 .

TABLE 2 Fundamental Frequency Coefficient

(px/~z/T)to~a in the Case of a Rectangular Membrane with a Concentric Circular Non-

homogeneity (u = F/a l)

7 u 2 = 1 2=1.5 ) t=2

0.1 4.69 6.13 7-80 3 0.2 4'04 4.94 5"93

10 3.16 3.63 4.16

0.1 5.01 6.76 8.81 3 0-3 3.67 4.40 5"22

10 2.52 2-83 3'24

0.1 5'49 7.72 10.27 3 0.4 3.34 3.69 4.73

10 2"09 2.36 2.76

It is important to point out that the approach yields eigenvalues which differ by less than 0.5 percent from the exact results when 7 = 1 (homogeneous case).

CONCLUSIONS

The approach presented in this paper is quite simple and straightforward. Certainly, no claim of originality is made by the authors but it is hoped that analysts and research engineers working on the analysis of vibrating non-homogeneous systems (especially in the case of bioengineering investigations) will find the methodology useful.


R E F E R E N C E S

1. C. O. Horgan and S. Nemat-Nasser, Bounds on eigenvalues of Sturn-Liouville problems with discontinuous coefficients, Zeitschrift flit angewandte Mathematik und Physik, 30 (1979), pp. 77-86.

2. C. O. Horgan, K. W. Lang and S. Nemat-Nasser, Harmonic waves in layered composites: New bounds on eigenfrequencies, Journal of Applied Mechanics, 45 (1978), pp. 829-33.

3. T. Sakata and Y. Sakata, Vibrations of a taut string with stepped mass density, Journal of Sound and Vibration, 71 (1980), pp. 315-17.

4. J. P. Spence and C. O. Horgan, Bounds on natural frequencies of composite circular membranes: Integral equation methods, Journal of Sound and Vibration, 87(1) (1983), pp. 71-81.

a simple approximate method for the solution of the helmholtz equation in the case of rectangular,...

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