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TRANSVERSE VIBRATIONS OF A RECTANGULAR MEMBRANE WITH DISCONTINUOUSLY VARYING DENSITY
Journal of Sound and Vibration (1999) 222(2), 341±344 Article No. jsvi.1999.2021, available online at http://www.idealibrary.com on
TRANSVERSE VIBRATIONS OF A RECTANGULAR MEMBRANE WITH DISCONTINUOUSLY VARYING DENSITY
M. E. PRONSATO,y P. A. A. LAURAz
AND
A. JUANy
y Physics Department, Universidad Nacional del Sur and z Engineering Department and Institute of Applied Mechanics (CONICET), Universidad Nacional del Sur, 8000 BahõÂa Blanca, Argentina (Received 28 October 1998) 1. INTRODUCTION
The dynamic behavior of structural systems with continuously or discontinuously varying material properties is of considerable interest in several areas of applied science and technology [1±3]. The present note deals with the study of a vibrating rectangular membrane with a discontinuously varying density distribution (Figure 1). The problem is solved by expanding the displacement amplitude in terms of a double Fourier Series which constitutes the exact solution in the case of an homogeneous membrane. The Rayleigh±Ritz method is then used to generate the determinantal equation. 2. APPROXIMATE ANALYTICAL SOLUTION
In the case of normal modes the problem is governed by the differential system r 1a; b r2 W o2 W 0, W @D2 0, T with
r x; y
r1 8 x; y 2 D1 , : x, y 2 D2 ÿ D1 r2 8
2
From the point of view of ®nding an approximate solution it is convenient to formulate the problem in terms of the functional o2 2 2 2 2 Wx Wy d x d yÿ W d x d y W d x d y , r g JW T 2 D2 D1 D2 ÿD1 2 where g r1 =r2 , 0022±460X/99/170341+4 $30.00/0
D1 D2 : # 1999 Academic Press
342
LETTERS TO THE EDITOR y
b
v
u a
Figure 1. Non-homogeneous u u=a =b:
rectangular
membrane
considered
in
x
the
present
study
Introducing the dimensionless variables x x =a; y y=b, expression (3) becomes o2 r 2 a 2 W 2x l2 W 2y dx dy ÿ lJW T C 2 g W 2 dx dy W 2 dx dy , 4 C1
C2 ÿC1
where l = a/b. The displacement amplitude will now be approximated by a truncated Fourier series of the form W Wa
N X M X n1 m1
bnm sin
npx mpy sin : a b
5
Substituting equation (5) in equation (4) and requiring @J Wa 0, @bnm
6
one obtains a linear system of equations in the bnm's. The non-trivial solution of the system leads, ®nally, to the determinantal equation in the eigenvalues of the problem.
343
LETTERS TO THE EDITOR
TABLE 1 p Values of the fundamental frequency coef®cient O1 r2 =T o1 a as a function of l, g and u = v (Figure 1) l=1 z}|{ W2 W8 W1
l = 1 5 z}|{ W1 W2 W8
g
u
l=2 z}|{ W1 W2 W8
0 1 3 10
0 2
4776 3897 2899
4765 475667 6089 3877 385974 4967 2819 274548 3695
6078 606862 7552 495 492989 6161 3619 353542 4583
7541 752943 6142 611907 4501 440164
0 1 3 10
0 3
5235 3488 2279
5182 516617 6673 3456 344392 4447 2211 218316 2906
6624 660432 8277 4417 440214 5516 2839 280628 3604
8224 819986 5483 546554 3532 349159
0 1 3 10
0 4
596 3151 1904
5809 578831 7598 3123 311758 4017 1862 185357 2427
7458 743036 9424 399 398372 4982 2387 237616 301
9274 923879 4953 49455 2966 295344
3. NUMERICAL RESULTS
The present investigation deals with the determination of the lower natural frequencies of symmetric normal modes of the system depicted in Figure 1. For this con®guration u u=a v v=b the inner, concentric portion possesses the same aspect ratio as the outer boundary of the membrane. The frequency coef®cients were determined using the terms: n = 1, m = 1, 3, 5, 7; n = 3, m = 3, 5, 7; n = 5, m=5. pTable 1 depicts values of the fundamental frequency coef®cient O1 r2 =T o1 a for several combinations of values of l, g and u = v when one, two and eight terms of the approximating function are employed. Table 2 shows the second frequency coef®cient corresponding to a symmetric mode when one and eight terms of the truncated double Fourier series are used. From the analysis of TABLE 2 p Values of the frequency coef®cient O2 = r2 =T o2 a in the case of symmetric modes (Figure 1) l=1 z}|{ W8 W2
l = 1 5 z}|{ W2 W8
l=2 z}|{ W2 W8
0 2
1062 919 8464
104668 903024 805261
15471 13379 12252
151337 130143 113749
20411 17645 16126
197972 170086 147275
01 3 10
0 3
11403 8993 8198
109616 880481 715777
16576 13078 11861
155947 126817 101834
21851 17242 15608
201661 165803 132428
01 3 10
0 4
12422 8729 7216
11579 837533 554573
17979 12696 10462
161315 120911 79857
23664 16739 13777
206305 158351 104495
g
u
01 3 10
344
LETTERS TO THE EDITOR
TABLE 3 Comparison of values of O1 in the case of a square non-homogeneous membrane g
u=v
0 1 3 10
0 3
Present study
O1
5166 3444 2183
Reference [1] 517 354 234
Tables 1 and 2 one concludes that the rate of convergence appears to be satisfactory. Finally, Table 3 depicts a comparison of fundamental frequency coef®cients in the case of a square, non-homogeneous membrane between the results obtained in the present investigation and those determined in reference [1], where a single polynomial expression was used to represent, the fundamental mode shape. The agreement is excellent in the case of moderate values of g leading to the conclusion that a simple, polynomial co-ordinate function yields very good accuracy in the case of a rather complex elastodynamics problem. ACKNOWLEDGMENTS
The present study has been sponsored by SecretarõÂ a General de Ciencia y TecnologõÂ a of Universidad Nacional del Sur and by CONICET Research and Development Program. Miss M. E. Pronsato has been supported by a CONICET Fellowship. The authors express their gratitude to Professor D. V. Bambill for her valuable cooperation in the preparation of the computer program. REFERENCES 1. P. A. A. LAURA and R. H. GUTIERREZ 1984 Applied Acoustics 17, 405±411. A simple approximate method for the solution of the Helmholtz equation in the case of rectangular, non-homogeneous domains. 2. J. P. SPENCE and C. O. HORGAN 1983 Journal of Sound and Vibration 87, 71±81. Bounds on natural frequencies of composite circular membranes: integral equation methods. 3. R. H. GUTIERREZ, P. A. A. LAURA, D. V. BAMBILL, V. A. JEDERLINIC and D. HODGES 1998 Journal of Sound and Vibration 212, 611±622. Axisymmetric vibrations of solid circular and annular membranes with continuously varying density.
TRANSVERSE VIBRATIONS OF A RECTANGULAR MEMBRANE WITH DISCONTINUOUSLY VARYING DENSITY
M. E. PRONSATO,y P. A. A. LAURAz
AND
A. JUANy
y Physics Department, Universidad Nacional del Sur and z Engineering Department and Institute of Applied Mechanics (CONICET), Universidad Nacional del Sur, 8000 BahõÂa Blanca, Argentina (Received 28 October 1998) 1. INTRODUCTION
The dynamic behavior of structural systems with continuously or discontinuously varying material properties is of considerable interest in several areas of applied science and technology [1±3]. The present note deals with the study of a vibrating rectangular membrane with a discontinuously varying density distribution (Figure 1). The problem is solved by expanding the displacement amplitude in terms of a double Fourier Series which constitutes the exact solution in the case of an homogeneous membrane. The Rayleigh±Ritz method is then used to generate the determinantal equation. 2. APPROXIMATE ANALYTICAL SOLUTION
In the case of normal modes the problem is governed by the differential system r 1a; b r2 W o2 W 0, W @D2 0, T with
r x; y
r1 8 x; y 2 D1 , : x, y 2 D2 ÿ D1 r2 8
2
From the point of view of ®nding an approximate solution it is convenient to formulate the problem in terms of the functional o2 2 2 2 2 Wx Wy d x d yÿ W d x d y W d x d y , r g JW T 2 D2 D1 D2 ÿD1 2 where g r1 =r2 , 0022±460X/99/170341+4 $30.00/0
D1 D2 : # 1999 Academic Press
342
LETTERS TO THE EDITOR y
b
v
u a
Figure 1. Non-homogeneous u u=a =b:
rectangular
membrane
considered
in
x
the
present
study
Introducing the dimensionless variables x x =a; y y=b, expression (3) becomes o2 r 2 a 2 W 2x l2 W 2y dx dy ÿ lJW T C 2 g W 2 dx dy W 2 dx dy , 4 C1
C2 ÿC1
where l = a/b. The displacement amplitude will now be approximated by a truncated Fourier series of the form W Wa
N X M X n1 m1
bnm sin
npx mpy sin : a b
5
Substituting equation (5) in equation (4) and requiring @J Wa 0, @bnm
6
one obtains a linear system of equations in the bnm's. The non-trivial solution of the system leads, ®nally, to the determinantal equation in the eigenvalues of the problem.
343
LETTERS TO THE EDITOR
TABLE 1 p Values of the fundamental frequency coef®cient O1 r2 =T o1 a as a function of l, g and u = v (Figure 1) l=1 z}|{ W2 W8 W1
l = 1 5 z}|{ W1 W2 W8
g
u
l=2 z}|{ W1 W2 W8
0 1 3 10
0 2
4776 3897 2899
4765 475667 6089 3877 385974 4967 2819 274548 3695
6078 606862 7552 495 492989 6161 3619 353542 4583
7541 752943 6142 611907 4501 440164
0 1 3 10
0 3
5235 3488 2279
5182 516617 6673 3456 344392 4447 2211 218316 2906
6624 660432 8277 4417 440214 5516 2839 280628 3604
8224 819986 5483 546554 3532 349159
0 1 3 10
0 4
596 3151 1904
5809 578831 7598 3123 311758 4017 1862 185357 2427
7458 743036 9424 399 398372 4982 2387 237616 301
9274 923879 4953 49455 2966 295344
3. NUMERICAL RESULTS
The present investigation deals with the determination of the lower natural frequencies of symmetric normal modes of the system depicted in Figure 1. For this con®guration u u=a v v=b the inner, concentric portion possesses the same aspect ratio as the outer boundary of the membrane. The frequency coef®cients were determined using the terms: n = 1, m = 1, 3, 5, 7; n = 3, m = 3, 5, 7; n = 5, m=5. pTable 1 depicts values of the fundamental frequency coef®cient O1 r2 =T o1 a for several combinations of values of l, g and u = v when one, two and eight terms of the approximating function are employed. Table 2 shows the second frequency coef®cient corresponding to a symmetric mode when one and eight terms of the truncated double Fourier series are used. From the analysis of TABLE 2 p Values of the frequency coef®cient O2 = r2 =T o2 a in the case of symmetric modes (Figure 1) l=1 z}|{ W8 W2
l = 1 5 z}|{ W2 W8
l=2 z}|{ W2 W8
0 2
1062 919 8464
104668 903024 805261
15471 13379 12252
151337 130143 113749
20411 17645 16126
197972 170086 147275
01 3 10
0 3
11403 8993 8198
109616 880481 715777
16576 13078 11861
155947 126817 101834
21851 17242 15608
201661 165803 132428
01 3 10
0 4
12422 8729 7216
11579 837533 554573
17979 12696 10462
161315 120911 79857
23664 16739 13777
206305 158351 104495
g
u
01 3 10
344
LETTERS TO THE EDITOR
TABLE 3 Comparison of values of O1 in the case of a square non-homogeneous membrane g
u=v
0 1 3 10
0 3
Present study
O1
5166 3444 2183
Reference [1] 517 354 234
Tables 1 and 2 one concludes that the rate of convergence appears to be satisfactory. Finally, Table 3 depicts a comparison of fundamental frequency coef®cients in the case of a square, non-homogeneous membrane between the results obtained in the present investigation and those determined in reference [1], where a single polynomial expression was used to represent, the fundamental mode shape. The agreement is excellent in the case of moderate values of g leading to the conclusion that a simple, polynomial co-ordinate function yields very good accuracy in the case of a rather complex elastodynamics problem. ACKNOWLEDGMENTS
The present study has been sponsored by SecretarõÂ a General de Ciencia y TecnologõÂ a of Universidad Nacional del Sur and by CONICET Research and Development Program. Miss M. E. Pronsato has been supported by a CONICET Fellowship. The authors express their gratitude to Professor D. V. Bambill for her valuable cooperation in the preparation of the computer program. REFERENCES 1. P. A. A. LAURA and R. H. GUTIERREZ 1984 Applied Acoustics 17, 405±411. A simple approximate method for the solution of the Helmholtz equation in the case of rectangular, non-homogeneous domains. 2. J. P. SPENCE and C. O. HORGAN 1983 Journal of Sound and Vibration 87, 71±81. Bounds on natural frequencies of composite circular membranes: integral equation methods. 3. R. H. GUTIERREZ, P. A. A. LAURA, D. V. BAMBILL, V. A. JEDERLINIC and D. HODGES 1998 Journal of Sound and Vibration 212, 611±622. Axisymmetric vibrations of solid circular and annular membranes with continuously varying density.