Vibration of an arbitrarily shaped membrane with point supports

Journal of Sound and Vibration

VIBRATION

(1979) 65(l),

l-9

OF AN ARBITRARILY

SHAPED MEMBRANE

WITH POINT SUPPORTS K. NAGAYA Department of Mechanical Engineering, Yamagata University, Yonezawa, Japan (Received 14 October 1978)

A method for solving vibration problems of an arbitrarily shaped membrane on point supports is presented. The frequency equation is derived by the use of the exact solution of the equation of motion which includes terms representing the reaction forces of the point supports. Numerical calculations are carried out for an elliptical membrane with two point supports, and the non-dimensional natural frequencies are shown for various aspect ratios.

1. INTRODUCTION

The vibration problems of membranes of arbitrary shape can be discussed by using the results obtained in certain acoustical waveguide problems. Methods for problems of waveguides of general cross section shapes include the following: finite differences [l, 21; point matching [3-51; Rayleigh-Ritz [6]; Galerkin method [7]; finite element [8]. These general methods have many advantages for solving problems of membranes of general shape. However, since in all these methods approximate solutions of the equation of motion and approximate boundary conditions are used, they cannot be expected to yield good results in cases of higher mode vibration and in such cases a large size digital computer is required for the numerical calculations. More rigorous analyses have been given for these problems. Laura, Chi, Meinke, Lange, Rurger and Romanelli [9-121 have discussed some complicated problems by means of the conformal mapping method, and Mazumdar [13] gave a method for solving membrane vibration problems by using the method of constant-deflection contours. However these methods have been applied to the case of the first mode only in most of the studies. Recently the author described another analytical method [14]. In this paper this method is expanded to deal with vibration problems of arbitrarily shaped membranes with some point supports. The author’s previous method has some advantages for solving the higher mode vibration and dynamic response problems of arbitrarily shaped membranes. In that method, however, the boundary of the membrane was restricted to one made up of curves. In this paper an improved method is developed in which no restrictions on the shape of the membrane are required. In the analysis the reaction forces of the point supports are regarded as unknown external forces, and the results can therefore be obtained from the equation of motion in a form involving the reaction forces. The exact solution of the equation of motion is obtained, and the outer boundary condition on the membrane is satisfied by means of the Fourier expansion method. As an example, the vibration of an elliptical membrane with two point supports is discussed. 1

0022&460X/79/130001 +09 rSO2.00/0

0 1979Academic Press Inc. (London)

Limlted

2

K. NAGAYA

2. GENERAL

FORMULATIONS

An arbitrarily shaped uniform membrane with point supports is shown in Figure 1, where the origin o of Cartesian co-ordinates x and y and polar co-ordinates I, 0 is chosen at an arbitrary point inside the boundary r, and the z-axis is normal to the membrane surface. If the displacement of the membrane in the z-direction is denoted by w, the equation of motion of the membrane, in the co-ordinates I, 8, is ~[d~~/ar~

+(1/t-)

aw/ar +(i/r2) a2W/ae21 - pa2W/at2 = 4

(1)

where T is the tension of the membrane, p is the mass density per unit area, t is the time and q is the unknown force corresponding to the reaction force of the point supports. The displacement and the force for free vibration problems can be written as follows: w(r, 8, t) = W(r, 6) sin cot,

q = Q(r, 0) sin wt.

(2)

Upon substituting equations (2) into equation (l), the solution for a uniform membrane is found to be IV,@

= (l/T) i j=l

&,CAj,J,(crr)+Fj,(r)l3,,

f n=O

(3)

where Fj,(r) = (n/2) rQj.(t3KW

J,b3-

s0

J,W)Y,(~5)15 dt,

Qjn(d= (l/n) R Qk, 4 Qjjn de, s -Z a2 = po2/T,

&g = l/2,

qn =

cos

no,

s,=l

forn>l,

@22,= sin ntJ,

(4)

o is the frequency (radians per second), Aj, is the constant of integration, J,(ar) and Y,(m) are Bessel functions of the first and second kinds of order n, respectively, and Fj,(r) is the particular solution of equation (1). When the membrane is supported on I points, the applied force Q is denoted by

Qhe) = - i Ri6(r-bi)6(e-Oi)

(5)

i=l

Figure

1. Geometry

of an arbitrarily

shaped membrane

with point supports

VIBRATING

MEMBRANE

OF ARBITRARY

3

SHAPE

where Ri is the reaction force at the support i, s(r-t~~) and S(f3- Oi) are Dirac delta functions, and bi and Oi are the polar co-ordinates at the support point i. From equations (4) and (5), one has

Qj,(r) = - i (Rji/‘n)6(r-bj)$jn,

(6)

i=l

where $1, = cosn@,,

I+& = sinnOi.

(7)

Substituting equation (6) into equations (4) yields Fj,(r) = -(l/2)

i

Rji{Y,(crr)J,(crbi)-J,(ccr)Y,(nbi)}bi~:,u(r-b,),

i=l

(8)

where u (r - bi) = 1 for r > bi, u(r - bi) = 0 for r < bi is the unit step function. When the membrane is symmetric about the x-axis, the motion of the membrane may be separated into two types of vibrations: symmetric and antisymmetric. In that case the results for the symmetric and antisymmetric modes are obtained by taking j = 1 and j = 2, respectively. If one puts W = C Wj and substitutes equation (8) into equation (3), the displacement function becomes

wj= (l/T) C E,CAjnJ&r) n

-(l/2) C biRji Mar) J,WJ - J, W Y,(abJ}u(r - bi)$:,I @jn, i=l

(9) where the summation C, is taken as n = 0,1,2, . . .. co for symmetric modes and is taken as n = 1,2,3 ,..., GOfor antisymmetric modes. The boundary condition for a membrane with a fixed edge is

(10)

(w)r = 0.

The boundary condition (equation (10)) around the whole range of the outer edge cannot be satisfied directly, because equation (10) is not expressed as a trigonometric series in the co-ordinate 13.To satisfy the boundary condition, the Fourier expansion of equation (10) is performed. Then the displacement along the outer edge of the membrane, expanded into a Fourier series, becomes t$ = (l/T)Z~..(aj.s!,-i~~biRji~~)~j~ m ”

= 0

for

j = 1,2,

(11)

where

qm = cos me,

Qz2, = sin m8,

e. = l/2,

E, = 1 for m > 1,

Si,,, = (2471) ’J,(ar,) GjjnGjjmde, s0 ci

= (E,/z) “{Y,(arr)J,(ab,)-J,(ar,)Y,(ab,)) s0

~,!&@~,@~,,,dt?.

(12)

K. NAGAYA

4

The summation C,,, is taken in the same way as previously, and r,- is the value of the coordinate r at the boundary, which is expressed as a function of 8 in the case of arbitrarily shaped membranes. When the terms n, m are truncated to N + 1, equation (11) becomes, in the matrix form,

A’

A 10 A 11

(13)

I

I

a,,

i

biRli~l$,,

i=l

n’

A 21 A2

A 22

(14)

I

A 2N ~ biR,iC~~ i=l n’

where

(15)

The constant Aj, is obtained from equation (13) as A,, = i bi Rji E;,,,

(16)

i=l

where (17) and I&,,l = (-1)“+“1A’,,[ is the cofactor of the determinant /A’[, where IA&,/is the minor which is given by removing the mth row and nth column in the determinant IAjl. The displacement function W is therefore obtained from equations (9) and (16) as 6

= (l/T) i i=l

biRjiCE,[Ej,J”(crr)-(1/2){Y”(crr)J,(abi)-J,(ar)Y,(crbi)}u(r-bi)IG_j.]3,. n (18)

The boundary conditions at the supported points are

wLls,e=e, = 0

for

s=12. > 3 .., I.

(19)

VIBRATING

MEMBRANE

OF ARBITRARY

5

SHAPE

Substituting equation (18) into equation (19) gives

(l/T) C biRji 1 enCfJ’ Jn ;nCabs) - (l/2) {Y,Cubs) Jn(abJ i=l

”

-

Jntabs)Yn(abJ1U(bs-

bi)Icljnl $Tn= 0

for

s=

1,2 ,..., I.

(20)

Therefore the frequency equation of an arbitrarily shaped membrane with point support is

where

- (l/2){Yntabs)Jntabi) Rii = C 8, CR:, JnCabs) ’ Jn(abs)Yn(~bi)I u(bs-biIIl/jnlrl/&.

(24

It is difficult to obtain the analytical values of SL,,,and c,,, in equations (12), and hence the integrations in the equations are performed numerically. Problems of membranes with corners can also be discussed by using this analysis; in such cases, the outer boundary should be considered separately on each side of the corner. The Fourier coefficients Sj,, and T”$ are then obtained by the addition of those for the separately considered boundaries. 2.1.

EXAMPLE

1:

ELLIPTICAL

MEMBRANES

WITH

POINT

SUPPORTS

As an example of an arbitrarily shaped membrane, an elliptical membrane is selected. If the origin o is taken in the center of the membrane, one has the following relations: x = rr cos 8,

x2/a2 + y2/c2 = 1,

y = rr sin

e.

The co-ordinate rr at the outer edge is found to be r,/a = [ l/{cos2 8 +(a2/c2) sin2 S>]1’2.

(23)

By substituting equation (23) into equations (12), the Fourier coefficients S’,, and T’j can be calculated, and the natural frequencies of the membrane then obtained from eq:tion (21). 2.2.

EXAMPLE

2:

CIRCULAR

MEMBRANES

WITH

POINT

SUPPORTS

The exact result can be derived from this analysis for the vibration of a circular membrane with point supports. For a circular membrane, with radius R (= a = c), the integration in equations (12) can be performed analytically, and the Fourier coefficients calculated as Sj,, = E, J,(aR) ci

for

n = m,

sim = 0

= (~,/2){Y,(aR)J,(ab,)-J,(aR)Y,(c&)} Ti.i = 0 “lfl

for

+;,

n # m.

for

n # m,

for

n = m,

(24)

K. NAGAYA

6

The constant A, is therefore obtained for each nodal diameter n as Aj, = (l/2) f: bi Rji{Yn(ER)J,(Mbi)- J,(ctR)Y,(&i)} $j,/J”(aR).

(25)

i=l

From equation (25), one has Ein = (~/‘~){Y~(uR)J,(u~~)-J,(MR)Y,(u~~)} $;,/J,(aR).

(26)

Substituing equation (26) into equation (21) one obtains RL = C (&n/2)[{Yn(aR) Jn(abi)- Jn(RR)Yntab,)}J, (ab,)/Jn CUR)

” {Y,(ab,)J,(abi)-J,(ab,)Y,(abi)}u(b,-bi)]rl/jnIC/g,.

(27)

The natural frequencies are calculated from equation (21) directly.

3. NUMERICAL

EXAMPLES

Numerical calculations have been carried out for an elliptical membrane with two point supports as shown in Figure 2. The supported points lie on the x axis, and their distances from the origin are the same (bi = b). In this case one has the following relations: b = b, = b,,

0,

= 0,

0,

= 71,

I = 2.

(28)

k------20Figure 2. An elliptical membrane

with two point supports

on the x-axis.

Therefore Bji is, for the elliptical membrane, Bii = 1 E, I$, J, (ab) I&. n

(29)

The following expression is similarly obtained for the circular membrane: R!i = c (42) C{YA@R) J, rb)- J, WV Y, W)I J, W)/J,G41 It/j.$;, n

(30)

The frequency equation is B’,,B’,,

- B{,B’,,

= 0.

(31)

VIBRATING

MEMBRANE OF ARBITRARY TABLE

7

SHAPE

1

Non-dimensional natural frequencies aR of symmetric modes for a circular membrane with two point supports equidistant from the center (b = bI = b,)

W 7

Mode First Second Third Fourth

,

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.0

3.002 3.897 5.520 6.278

3.159 4.074 5.150 6.234

3.190 4.321 5.240 5.871

3.096 4.584 5.471 5.653

2.934 4.703 5.484 5.940

2.765 4.596 5.421 6.234

2.634 4.371 5.384 6.159

2,528 4.146 5.340 5.896

2.404 3.832 5.135 5.520

The terms n, m are taken up to 5 - 7 according to the convergence of the Fourier series in the numerical calculation. Table 1 shows the non-dimensional natural frequencies aR( = oR fl) of the circular membrane with two point supports for the first through the fourth modes in the case of symmetric vibrations. Table 2 shows the fundamental nondimensional natural frequencies cla(=wa m) of symmetric modes for the elliptical membrane with two point supports. It can be observed that the fundamental frequency decreases with the supported length b/a. When the inner supports lie on the outer edge (b/R = 1, b/a = l), the frequency should coincide with that of the membrane without inner supports. The frequencies for such special cases are obtained from the expression 1.&l= 0, and are also shown in the tables. The result obtained in that case coincides with that of the exact analysis [15] for the first mode in the range shown in the tables. Figure 3 shows the relation between the frequency aa of symmetric modes and supported length b/a for the case of the elliptical membrane for various modes. In the figure the solid line shows the value for a/c = 2 and the dashed line that for a/c = 1.4. Since the supported points lie on the nodal line in the case of antisymmetric vibrations, the frequencies for antisymmetric modes have no dependence on the supports.

TABLE

2

Fundamental non-dimensional natural frequencies aa of symmetric modes for an elliptical membrane with two point supports equidistant from the center and on the x-axis (b = b, = b2)

ale

ro.l

1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0

3.002 3.338 3.673 4.001 4.328 4.665 5.498 6.333

0.2

0.3

0.4

0.5

0.6

0.7

0.8

3.159 3.518 3.879 4.242 4.604 4.963 5.854 6.745

3.190 3.552 3.911 4.267 4.615 4.957 5.774 6.546

3.096 3.416 3.729 4.033 4.334 4.624 5.329 6.023

2.934 3.214 3.489 3.766 4.039 4.313 4.992 5.658

2.765 3.027 3.288 3.555 3.821 ‘4.091 4.764 5.445

2.634 2.879 3.138 3.400 3.671 3.945 4.635 4.366

2.528 2.773 3.033 3.304 3.582 3.869 4.607 5.353

1.0 > 2.404 2.655 2.921 3.199 3.485 3.777 4.526 5.289

K. NAGAYA 9

1

/

I

/

I

(

I

I

a-

.-__

--

3-

----Is’_=2.cJ2,

2 2c 2b 11 0 0.1

20

PJr I

I

0.2

0.3

I

I

0.4

0.5

I

I

I

0.6

0.7

0.8

I 0.9

I.0

b/o

Figure 3. Non-dimensional natural frequencies aa of symmetric modes versus the supported elliptical membrane with two point supports equidistant from the center (b = b, = b2).

length h/a for an

4. CONCLUSIONS

In this paper an analytical method for solving vibration problems of an arbitrarily shaped membrane with point supports has been presented. The frequency equation was obtained by using the exact solution of the equation of motion including terms representing the reaction forces of the point supports. The boundary condition around the edge was satisfied by means of the Fourier expansion method. Numerical calculations were carried out for an elliptical membrane with two point supports, and the natural frequencies were given for various aspect ratios. The convergence of the Fourier series was good, and results of reasonable accuracy were obtained readily for the elliptical membrane. Therefore the method developed in this paper has some advantages compared with other general approximate methods.

ACKNOWLEDGMENT The author wishes to thank heartily Professor Y. Hirano, Yamagata University, for his kind guidance in the present work.

REFERENCES J. B. DAVIES and C. A. MUILWYK 1966 Proceedings of the Institution of Electrical Engineers 113, 277-284. Numerical solution of uniform hollow waveguides with boundaries of arbitrary shape. 2. C. W. STEELE 1968 Journal ofComputational Physics 3,48-l 53. Numerical computation of electric and magnetic field in a uniform waveguide of arbitrary cross-section. 3. H. Y. YEE 1965 Proceedings of the Institution of Electrical and Electronic Engineers 54, 64. On determination of cutoff frequencies of waveguides with arbitrary cross-section. 1.

VIBRATING MEMBRANEOF ARBITRARY SHAPE

9

4. H. Y. YEE and N. F. AUDEH 1966 Institution ofElectricaland Electronic Engineers Transactions on Microwave Theory and Techniques MTT-14,487-493. Cut-off frequencies of eccentric waveguides. 5. R. H. T. BATES 1969 Institution of Electrical and Electronic Engineers Transactions on Microwave Theory and Techniques MTT-17, 294301. The theory of point-matching method for perfectly conducting waveguides and transmission lines. 6. R. M. BULLEY and J. B. DAVIES 1969 Institution of Electrical and Electronic Engirleers Transactions on Microwave Theory and Techniques MTT-17,440-446. Computation of approximate polynomial solutions to TE modes in an arbitrary shaped waveguide. 7. P. A. A. LAURA 1966 Proceedings of the Institution of Electrical and Electronic Engineers 54, 1495-1497. A simple method for the calculation of cut-off frequencies ofwaveguides with arbitrary cross-section. 8. P. L. ARLETT, A. K. BAHRANI and 0. C. ZIENKIEWICZ 1968 Proceedings of the Institution of Electrical Engineers 115. 1762-l 766. Application of finite elements to the solution of Helmholtz’s equation. 9. M. CHI and P. A. A. LAURA 1964 Institution of Electrical and Electronic Engineers Transactions on Microwave Theory and Techniques MTT-12, 248-249. Approximate method of determining the cut-off frequencies of waveguides of arbitrary cross-section. 10. H. H. MEINKE, K. P. LANGE and J. F. RURGER 1963 Proceedings ofthe Institution of Electricaland Electronic Engineers51, 14361433. TE and TM waves in waveguides of very general cross-section. Il. P. A. A. LAURA, E. ROMANELLIand M. J. MAURIZI 1972 Journal of Sound and Vibration 20, cross-section by the method of 27-38. On the analysis of waveguides of doubly-connected conformal mapping. 12. P. A. A. LAURA 1912 Institution of Electrical and Electronic Engineers Transactions on Microwave Theory and Techniques MTT-20, 292. The solution of Helmholtz equation in elliptical coordinates. 13. J. MAZUMDAR1973 Journal ofSoundand Vibration 27,47-57. Transverse vibration ofmembranes of arbitrary shape by the method of constant-deflection contours. 14. K. NAGAYA 1978 American Society of Mechanical Engineers Journal of Applied Mechanics 45, 153-l 58. Vibrations and dynamic response of membranes with arbitrary shape. 15. K. SATO 1976 Journal of Physical Society of Japan 40, 1199-1206. Forced vibration analysis of a composite elliptical membrane consisting of conformal elliptical parts.