Natural mode analysis of hollow and annular elliptical cylindrical cavities

Natural mode analysis of hollow and annular elliptical cylindrical cavities

Journal of Sound and Vibration (1995) 183(2), 327–351 NATURAL MODE ANALYSIS OF HOLLOW AND ANNULAR ELLIPTICAL CYLINDRICAL CAVITIES K. H  J. K ...

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Journal of Sound and Vibration (1995) 183(2), 327–351

NATURAL MODE ANALYSIS OF HOLLOW AND ANNULAR ELLIPTICAL CYLINDRICAL CAVITIES K. H  J. K Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 45221-0072, U.S.A. (Received 8 November 1993, and in final form 1 April 1994) Analytical solutions for natural frequencies and mode shapes of elliptical cylindrical acoustic cavities were obtained by solving the homogeneous wave equation in elliptical cylindrical co-ordinates. Both hollow cylinder and annular cylinder cases were considered. The characteristic equation was obtained in terms of Mathieu functions. Natural frequencies and mode shapes were calculated for cavities of the size used as a typical hermetic compressor shell or a small automotive muffler. The results were compared with those of circular cylindrical cavities which have the same cross-sectional area and volume. The influence of the eccentricity on the general characteristics of cylindrically shaped acoustic devices is discussed.

1. INTRODUCTION

In-depth understanding of the natural modes of gas pulsations in an acoustic cavity is important because it provides fundamental information for the design work related to the cavity. Although numerical methods such as the finite element method [1] or the boundary element method [2–4] can be used to obtain the natural modes of a cavity of any general shape, their results usually fail to provide good physical insights to the problem, as do analytical solutions. Acoustic cavities that resemble hollow or annular elliptic cylinders are often encountered in industrial machinery such as automobile mufflers and hermetic compressors. While analytical solutions are readily found for cavities of hollow and annular circular cylinder shape [5, 6], or spherical shape [7], they are not found for the cavities of elliptical cylindrical shape, probably because of the associated mathematical complexities. Vibration problems of elliptical plates are somewhat related [8–10]. This paper presents an analytical solution method to obtain natural frequencies and mode shapes of pressure pulsations in both hollow and annular elliptical cylinders. The homogeneous wave equation formulated in elliptical cylindrical co-ordinates is solved, which gives the general solution in terms of Mathieu and modified Mathieu functions. Applying proper boundary conditions, natural frequencies and mode shapes are obtained for both hollow cross-section and annular cross-section cavities. Natural frequencies evaluated for various eccentricities are given in graphical and tabular forms. Some cross-sectional views of typical mode shapes are also shown. By taking extreme cases, it is shown that the solutions converge to the circular cases as they should. A circular cylindrical cavity may be used in practical applications as an alternative to the elliptical cylindrical cavity if they have the same axial length and cross-sectional area. One common question has been how much the geometry, the eccentricity in this case, would influence the acoustic characteristics of the cavities. To answer this question, natural frequencies and mode shapes of the elliptic cavities are compared with those of the 327 0022–460X/95/220327 + 25 $08.00/0

7 1995 Academic Press Limited

.   . 

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equivalent cylindrical cavities. It is shown that an elliptical shaped cavity has similar characteristics in its natural modes except that it has two different frequencies and mode shapes associated with one mode number. In a wider range of eccentricity than one might think, the natural frequencies of the elliptical cavities, especially in annular cases, are found to be fairly close to the frequencies of the equivalent circular cavity. Also it is shown that the mode shapes are basically distorted shapes from those of the circular cavity. Most of the mathematical techniques related with Mathieu functions used in this work are based on the work done by McLachlan [11], Morse and Feshbach [12] and Stratton et al. [13]. The notations of Mathieu functions used in this work follow those adopted by Morse and Feshbach [12]. 2. GENERAL SOLUTION OF THE WAVE EQUATION IN ELLIPTIC CO-ORDINATES

The linear, homogeneous wave equation in a three-dimensional cavity is given as [12] 9 2P(x, y, z, t) −

01 1 c

2

1 2P(x, y, z, t) = 0, 1t 2

(1)

where P is the acoustic pressure and c is the speed of sound in the acoustic medium. Because of the geometry of the cavity in this work, elliptical cylindrical co-ordinates should be used, as shown in Figure 1. The co-ordinates are related to the Cartesian co-ordinate system by the following equation [11]: x = h cosh j cos h,

y = h sinh j sin h,

z = z,

(2–4)

where h is the semi-interfocal distance of the ellipse. The eccentricity of the ellipse formed by the boundary defined by j = j1 is e = sec hj1 = z1 − (b/a)2, where a and b are the half-lengths of major and minor axes, respectively. Parameters a, e and h are related by h = ae. Now we consider the elliptical–cylindrical cavity whose boundaries are represented by j = j1 , z = 0 and l. All the boundaries are considered rigid walls where no normal velocities exist. The Laplacian operator in equation (1) is represented as [11] 92 =

0

1

2 12 12 12 . 2+ 2 + h (cosh 2j − cos 2h) 1j 1h 1z 2 2

Figure 1. Elliptical co-ordinates.

(5)

  

329

Therefore, equation (1) becomes

0

1

2 1 2P 1 2P 1 2P 1 1 2 P − = 0. 2 + 2 + h (cosh 2j − cos 2h) 1j 1h 1z 2 c 2 1t 2 2

(6)

Assuming harmonic response, using the method of separation of variables, P(j, h, z, t) = F(j, h)Z(z) exp(ivt + c),

(7)

where i = z−1, v is the frequency and c is an arbitrary phase angle to be decided by the excitation. Then, equation (6) becomes

6

0

1

01 7

2Z(z) 1 2F 1 2F d2Z(z) v +F + FZ exp(ivt + c) = 0. 2 + 2 h (cosh 2j − cos 2h) 1j 1h dz 2 c 2

2

(8)

Equation (8) can be rearranged as

0

1 01

2 1 2F 1 2F v d2Z(z) + 2 + FZ = −F h 2(cosh 2j − cos 2h) 1j 2 1h c dz 2 2

(9)

Dividing by FZ, we have two separated equations as follows: d2Z(z) + R2Z(z) = 0. dz 2 1 2F 1 2F + 2+ 1j 2 1h

$0 1 v c

%

2

− R2

h2 (cosh 2j − cos 2h)F = 0, 2

(10)

(11)

where R is a separation constant. Applying the boundary conditions dZ(0)/dz = dZ(l)/dz = 0 in equation (10), it is found that R = np/l and, Z(z) = C1 cos (npz/l),

(12)

where n = 0, 1, 2, . . . and C1 is an arbitrary constant. Furthermore, if we let F(j, h) = U(j)V(h) and substitute R = np/l in equation (11), the equation becomes U0V + UV0 + [(v/c)2 − (np/l)2](h 2/2)(cosh 2j − cos 2h)UV = 0

(13)

U0V + UV0 + h 2[(v/c)2 − (np/l)2](cosh2 j − cos2 h)UV = 0,

(14)

or where U0 = d2U/dj 2

and

V0 = d2V/dh 2.

Letting K 2 = h 2[(v/c)2 − (np/l)2],

(15)

U0/U + K 2 cosh2 j = −V0/V + K 2 cos2 h = s,

(16)

equation (14) is separated as where s is an arbitrary separation constant. Finally, we have two more separated equations as follows d2V d 2U 2 2 − (s − K 2 cosh2 j)U = 0. (17, 18) 2 + (s − K cos h)V = 0, dh dj 2

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330

If we replace j in equation (18) by ij, the equation becomes the same form as equation (17). Equations (17) and (18) are so-called Mathieu and modified Mathieu equations, respectively. The solutions of the Mathieu equations are given in terms of Mathieu functions. The separation constant s is called the characteristic number or eigenvalue of the Mathieu function. Solutions of equation (17) are in the form of [11, 12] V(h) = Cm Sem (K, cos h) + Dm Fem (K, cos h),

s = sm ,

m = 0, 1, 2, 3, . . . ,

(19)

, cos h) + D m Fom (K , cos h), m Som (K V(h) = C

s = s¯ m ,

m = 1, 2, 3, . . . ,

(20)

, cos h) are the even and odd Mathieu functions of the first where Sem (K, cos h) and Som (K , cos h) are the even and odd kind of order m, respectively. Fem (K, cos h) and Fom (K , Mathieu functions of the second kind. The values sm and s¯ m are related to K and K respectively, in the form of a power series [11]. Notice that equations (19) and (20) represent two different sets of general solutions, corresponding to two different characteristic numbers sm and s¯ m . The first set of general solution is a linear combination of the periodic function Sem (K, cos h) and the non-periodic function Fem (K, cos h). Sem (K, cos h) has period p or 2p in h, depending upon whether m is even or odd. Similarly, the second set of general solutions is a combination , cos h) and the non-periodic function Fom (K , cos h). of the periodic function Som (K , cos h) has the period p or 2p in h depending upon whether m is even or odd [11]. Som (K The general solutions of equation (18) are [11, 12] U(j) = Am Jem (K, cosh j) + Bm Nem (K, cosh j),

s = sm ,

m = 0, 1, 2, 3, . . . ,

(21)

m Jom (K , cosh j) + B m Nom (K , cosh j), U(j) = A

s = s¯ m ,

m = 1, 2, 3, . . . ,

(22)

, cosh j) are the modified Mathieu functions of the first where Jem (K, cosh j) and Jom (K  kind. Nem (K, cosh j) and Nom (K, cosh j) are the modified Mathieu functions of the second , cosh j) have period of pi or 2pi in j, depending kind. Both Jem (K, cosh j) and Jom (K upon whether m is even or odd. On the other hand, both Nem (K, cosh j) and , cosh j) are non-periodic in j [11]. Nom (K Some general properties of Mathieu functions can be summarized as follows [11]. (1) Functions Se2n , Se2n + 1 , So2n + 1 and So2n + 2 have n real zeros in the range of 0 Q h Q p/2. (2) Jem (K, cosh j) is proportional to Sem (K, cosh j), which means that one is a constant , cosh j) and Som (K , cosh j) have the same relationship to each multiple of the other. Jom (K other. (3) Se2n (K, cos h) is symmetric about both the major and minor axes of the ellipse. Se2n + 1 (K, cos h) is symmetric about the major axis but antisymmetric about the minor axis. , cos h) is antisymmetric about the major axis but symmetric about the minor axis. So2n + 1 (K  So2n (K, cos h) is antisymmetric about both axes. The general solution for F(j, h) = U(j)V(h) is the product of any two functions which are the solutions of equations (17) and (18), respectively, for the same values of s and K. Because s may have any value, the number of solutions are unlimited. The solution should be written as F(j, h) = U(j)V(h) = [Cm Sem (K, cos h) + Dm Fem (K, cos h)][Am Jem (K, cosh j) + Bm Nem (K, cosh j)],

s = sm ,

m = 0, 1, 2, 3, . . . ,

(23a)

m Som (K , cos h) + D m Fom (K , cos h)][A m Jom (K , cosh j) + B m Nom (K , cosh j)], F(j, h) = [C s = s¯ m ,

m = 1, 2, 3, . . . .

(23b)

  

331

Substituting equations (12) and (23) in equation (7), the general solution of the wave equation in an elliptical cylindrical co-ordinate is obtained. In the following, we will consider the cases for hollow and annular elliptical cylindrical cavities separately. 3. NATURAL MODES OF THE HOLLOW ELLIPTICAL CYLINDER

3.1.   If we consider a hollow-closed elliptical cylinder, then F(j, h) must satisfy the following physical conditions. (1) For a closed cylinder, F(j, h) must satisfy the condition of continuity in the h direction. Since Fem and Fom are not periodic in h, they cannot be solutions of the boundary value problem that we consider here. Therefore, we have V(h) = Cm Sem (K, cos h), , cos h), m Som (K V(h) = C

s = sm ,

m = 0, 1, 2, 3, . . . ,

(24)

s = s¯ m ,

m = 1, 2, 3, . . . .

(25)

(2) At points (0, −h) and (0, h) on the interfocal line, the solution must have (a) continuity in displacement, i.e., F(0, h) = F(0, −h), (26) and (b) continuity in gradient, i.e., 1 1 [F(j, h)]j:0 = − [F(j, −h)]j:0 . 1j 1j

(27)

Equations (26) and (27) have to be satisfied only when the interfocal line is included in the domain of consideration. For the annular elliptical cylinder, the conditions in equations (26) and (27) are not necessary. After considering the physical conditions in equations (24) and (25), equation (23) reduces to F(j, h) = Cm Sem (K, cos h)[Am Jem (K, cosh j) + Bm Nem (K, cosh j)], s = sm , m = 0, 1, 2, 3, . . . , m Som (K , cos h)[A m Jom (K , cosh j) + B m Nom (K , cosh j)], F(j, h) = C

(28)

s = s¯ m , m = 1, 2, 3, . . . . (29) , cosh h)Jom (K , cosh j) satisfy the physical Both Sem (K, cos h)Jem (K, cosh j) and Som (K conditions in equations (26) and (27). On the other hand, the functions Sem (K, cos h) , cos h)Nom (K , cosh j) satisfy only equation (27). So we find that Nem (K, cosh j) and Som (K the permissible form of the solution consists of all the product solutions in equations (28) and (29). This gives a

a

m=0

m=1

, cos h)Jom (K , cosh j) F(j, h) = s am Sem (K, cos h)Jem (K, cosh j) + s bm Som (K where am = Cm Am ,

m Bm . bm = C

(30)

(31)

Finally we have a

a

P(j, h, z, t) = s s amn Sem (K, cos h)Jem (K, cosh j) cos m=0 n=0 a

a

npz exp(ivmn t + cmn ) l

, cos h)Jom (K , cosh j) cos + s s bmn Som (K m=1 n=0

npz exp(iv¯ mn t + c  mn ). l

(32)

This is the complete solution of the wave equation for the hollow elliptical cylinder cavity.

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332

3.2.   At the rigid boundary of the elliptical cylinder wall, the normal component of the fluid particle velocity should be zero: 1P (j, h, z, t)=j = j 1 = 0. 1j

(33)

Substituting equation (32) into the boundary condition and considering that amn and bmn are arbitrary constants, we have Je'm (K, cosh j1 ) = 0,

, cosh j1 ) = 0, m = 1, 2, 3, . . . . Jo'm (K

m = 0, 1, 2, 3, . . . ;

(34, 35) Equations (34) and (35), known as the period or pulsatance equation [11], are the  of which are used to calculate the frequencies characteristic equations the roots K and K mr of the modes. For each m there are an infinite number of zeros. The roots Kmr and K of equations (34) and (35), respectively, provide the rth natural frequencies of the modes corresponding to m. mr found, the equations Jem (Kmr , cosh j) = 0 and For the values Kmr and K mr , cosh j) = 0 define two different series of confocal nodal ellipses which define the Jom (K r − 1 nodal lines on the elliptical cross-section. Similarly, the functions Sem (K, cos h) and , cos h) also have zeros in h. P(j, h, z, t) vanishes also if h satisfies the equations Som (K mr , cos h) = 0. The roots h of the equations define a series of Sem (Kmr , cos h) = 0 or Som (K mr , cos h) each have m zeros confocal nodal hyperbolas such that Sem (Kmr , cos h) and Som (K in 0 E h E p. Thus, for a given m each function gives rise to m nodal hyperbolas [1]. As the result of the above discussion, we have two different pressure pulsation modes for each m, n, r combination, as follows: P(m,n,r)1 (j, h, z) = Sem (Kmr , cos h)Jem (Kmr , cosh j) cos (npz/l), m, n = 0, 1, 2, 3, . . . , r = 1, 2, 3, . . . ,

(36)

mr , cos h)Jom (K mr , cosh j) cosh (npz/l), P(m,n,r)2 (j, h, z) = Som (K m, r = 1, 2, 3, . . . , n = 0, 1, 2, 3, . . . .

(37)

Since any of the modes in equations (36) and (37) can become a solution of the wave equation, the general solution becomes a

a

P(j, h, z, t) = s s m=0 n=0

$

a

s amnr Sem (Kmr , cos h)Jem (Kmr , cosh j) cos (npz/l)

r=1

a

a

× exp(ivmnr t + cmnr ) + s s m=1 n=0

$

%

%

a

mr , cos h) s bmnr Som (K

r=1

mr , cosh j) cos (npz/l) exp(iv¯ mnr t + c × Jom (K  mnr ),

(38)

where amnr , bmnr , cmnr and c  mnr are arbitrary constants to be determined from the initial conditions, if necessary.

  

333

3.3.       If we fix the half major axis length a and let the eccentricity e approach zero, then the elliptical cylinder becomes a circular cylinder of radius r = a = b. In such a case, h:u, h cosh j:r. After we omit the proportional constant, we have [11, 12] sm :s¯ m :m 2,

Sem (K, cos h):cos mu,

(39, 40)

, cosh j) Jem (K :Jm (K cosh j), , cosh j) Jom (K

7

, cos h):sin mu, Som (K

(41, 42)

where Jm (K cosh j) is the Bessel function of the first kind of the order m. Also, from equation (15), K = hz(v/c)2 − (np/l)2, and h cosh j = r when e:0. This gives Jm (K cosh j) = Jm (kr),

k = z(v/c)2 − (np/l)2

The boundary conditions in equations (34) and (35) become a single equation: J'm (ka) = 0,

m = 0, 1, 2, 3, . . . .

(43)

Equation (43) is the characteristic equation of the cylindrical cavity. The natural frequencies are obtained from vmnr = cz(kmr )2 + (np/l)2, (44) where kmr is the rth root of equation (43). The two natural modes become: P(m,n,r)1 (r, u, z) = Jm (kmr r) cos (mu) cos (npz/l),

(45)

P(m,n,r)2 (r, u, z) = Jm (kmr r) sin (mu) cos (npz/l).

(46)

Two natural modes in the above equations are actually the same with a phase angle difference of 90°/m from each other. Similar to the elliptical cylinder case, the general response becomes P(r, u, z, t) =

6

a

a

s s m=0 n=0

a

$

a

+ s s m=1 n=0

a

s amnr cos (mu)Jm (Kmr r) cos (npz/l)

r=1

$

a

%

s bmnr sin (mu)Jm (Kmr r) cos (npz/l)

r=1

× exp(ivmnr t + Cmnr ).

%7 (47)

The major difference between the elliptical and circular cylinder is that for the circular cylinder there is only one natural frequency corresponding to the given wavenumber (m, n, r), because both pressure modes P(m,n,r)1 and P(m,n,r)2 have the same frequency, whereas, the frequencies corresponding to P(m,n,r)1 and P(m,n,r)2 are different for the elliptical cylinder as they are from two different characteristic equations (equations (34) and (35)). In mnr is always greater than elliptical cavities, the natural frequency v¯ mnr corresponding to K vmnr corresponding to Kmnr . The reason for this can be best explained by looking at the mode shape of the case m = r = 1, shown in Figure 2(a). For this case, v¯ corresponds to the mode where the wave mainly travels across the major axis, whereas v corresponds to the mode where the wave travels across the minor axis. Less time would be required for the wave to travel across the major axis, therefore v¯ is greater than v. This is somewhat similar to the relation between the rectangular and the square cavities. The two pressure modes P(m,n,r)1 and P(m,n,r)2 for a given wavenumber in the elliptical cylindrical cavities look different, as seen in Figure 2. Because we can consider that the

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334

circular cylinder also has two modes for one set of mode numbers, but with a 90°/m degree phase angle between them, we can treat the circular cylinder case as a degenerated special case of the corresponding elliptical cylinder. 3.4.      In order to solve the boundary conditions (34) and (35), mathematical expressions for , cosh j) should be known first. The Mathieu and modified Jem (K, cosh j) and Jom (K Mathieu function can be expressed in terms of trigonometric functions and Bessel functions as follows [12]: a

Sem (K, cos h) = s Bne (K, m) cos (nh),

a

, cos h) = s Bn0 (K , m) sin (nh), Som (K

n=0

n=1

(48, 49)

Fig. 2 (Part 1) See Caption on page 11

  

Fig. 2 (Part 2) See caption on page 11

335

336

.   .  a

e Je2m (K, cosh j) = zp/2 s (−1)n − mB2n (K, m)J2n (K cosh j),

(50)

n=0 a

e Je2m + 1 (K, cosh j) = zp/2 s (−1)n − mB2n + 1 (K, m)J2n + 1 (K cosh j),

(51)

n=0 a

0 , cosh j) = zp/2(tanh j) s (−1)n − m(2n)B2n , m)J2n (K  cosh j), Jo2m (K (K

(52)

n=1 a

0 , cosh j) = zp/2(tanh j) s (−1)n − m(2n + 1)B2n , m)J2n + 1 (K  cosh j), Jo2m + 1 (K + 1 (K

(53)

n=0

, m) are dependent upon the parameter K and where the coefficients Bne (K, m)e and Bn00 (K the order m. The coefficients Bn and Bn for some ranges of the parameters are listed in a tabular form in the work by Stratton et al. [13] and in [14]. From equations (48)–(53), mr in equations (34) and (35) and utilizing the tables, we can find the zeros of Kmr and K for the boundary defined by j = j1 . The values in the table should be interpolated as necessary. In Figures 3 and 4 it is shown how the zeros of equations (34) and (35) are distributed for the case corresponding to e = 0·6 and j = 1·1.

Fig. 2 (Part 3) See caption on page 11

  

337

Fig. 2 (Part 4) Figure 2. The cross-section view mode shapes of (a) an elliptical cylinder and (b) a circular cylinder for various m, n, r combinations.

Figure 3. Zeros of Je'm (K, cosh (1·1)) with m = 0, 1, 2, 3, 4.

, cosh (1·1)) with m = 1, 2, Figure 4. Zeros of Jo'm (K 3, 4.

.   . 

338

T 1 (a) The comparison of frequencies between circular and elliptical cylinders (based on the same major axis length) Circular cylinder e = 0 Elliptical cylinder e = 0·2 (h = 0, radius r = 0·15) (h = 0·03, j = 2·29) ZXXXXCXXXXV ZXXXXXXXCXXXXXXXV m, r vc v v¯ v¯ /v 0, 1 0, 2 0, 3 0, 4 0, 5 0, 6 0, 7 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 1, 7 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

1 279·4 4 353·5 7 725·7 11 122·4 14 526 17 932·7 21 341·5 2 373·7 5 929·6 9 358·3 12 777 16 192 19 606·7 23 020·1 3 555·1 7 394·3 10 902·3 14 360·1 17 799·3 21 229·5 4 738·2 8 798·1 12 389 15 892 19 360·7 22 876·9 5 914·8 10 161·6 13 831·7 17 384·1 20 886·3 24 361

1 279·4 4 409·7 7 819·3 11 309·1 14 767·7 18 170·4 21 662 2 376·3 6 003·8 9 447·9 12 888·6 14 309·2 19 776·9 23 184·3 3 592 7 437·6 11 000·2 14 492·2 17 874·3 21 419·6 4 799·7 8 892·1 12 462·6 16 077·2 19 528 23 015 5 987·5 10 302·8 13 974·7 17 573·3 21 134·9 24 569·2

0 0 0 0 0 0 0 2 407·8 6 067·5 9 540·4 12 972·3 16 480 19 968·8 23 317·2 3 593·6 7 439·3 11 030·9 14 547·8 17 958·5 21 250 4 799·8 8 892·9 12 466·1 16 087·2 19 552·8 23 070·1 5 987·5 10 302·9 13 974·9 17 574·9 21 139·7 24 579·7

0 0 0 0 0 0 0 1·013 1·011 1·010 1·007 1·010 1·006 1·006 1·000 1·000 1·003 1·004 1·005 1·005 1·000 1·000 1·000 1·001 1·001 1·002 1·000 1·000 1·000 1·000 1·000 1·000

(b) The comparison of frequencies between elliptical cylinders with different eccentricities (based on the same major axis length) e = 0·4 (h = 0·06, j = 1·5668) e = 0·6 (h = 0·09, j = 1·1) e = 0·8 (h = 0·12, j = 0·6931) ZXXXXXXCXXXXXXV ZXXXXXCXXXXXV ZXXXXXCXXXXXV m, r v v¯ v¯ /v v v¯ v¯ /v v v¯ v¯ /v 0, 1 0, 2 1, 1 2, 1 3, 1 4, 1

1279·41 4559·93 2382·06 3674·94 4924·04 6155·28

0 0 2522·16 3700·89 4977·08 6155·70

0 0 1·059 1·007 1·011 1·000

1279·41 5003·73 2388·29 3783·62 5159·35 6490·91

0 0 2784·24 3952·47 5216·32 6508·62

0 0 1·166 1·045 1·011 1·003

1279·41 6341·49 2398·62 3893·98 5426·02 6961·10

0 0 3484·07 4639·30 5914·80 7268·72

0 0 1·453 1·191 1·090 1·044

  

339

Figure 5. Convergence of the natural modes of the hollow elliptical cavity to the circular cavity modes shown in terms of the non-dimensional frequency difference, when the major axis length a is fixed. Non-dimensional c c c c frequencies are defined as W(m, r) = [(vmr1 − vmr1 )/vmr1 ] × 100%, W (m, r) = [(v¯ mr1 − vmr1 )/vmr1 ] × 100%; the superscript c indicates the circular cylindrical cavity.

The natural frequencies can be found from equation (15) by utilizing corresponding values of the zeros found: vmnr = Cz(Kmr /h)2 + (np/l)2,

v¯ mnr = Cz(K mr /h)2 + (np/l)2.

(54, 55)

A numerical calculation was carried out for the specific case of a = 0·15 m, l = 0·4 m, C = 162·9 m/s, which represents the condition in a typical shell cavity of a small hermetic refrigeration compressor. Some numerical values of v and v¯ for n = 1 and various eccentricities e are listed in Table 1. The results are compared to the circular cylinder (e = 0) whose radius is the same as the major axis of the elliptical cavity. Figure 5 is derived from the same information, but in terms of the non-dimensional difference of the natural frequencies compared with those of the circular cylinder. It shows that the frequencies approach those of the circular cylinder cavities when the eccentricity e approaches zero, as it should be, which validates the actual calculation procedure used in this work. When m = 0, only one pressure mode exists: P(0,n,r)1 = J0 (K0r cosh j) cos (npz/l) with frequencies v0nr . It will be more convenient to define P(0,n,r)2 = 0 with v¯ 0nr = 0 for other purposes such as forced response analysis. The modes with m = 0 correspond to modes with pressure variation only in the radial (j) direction. The nodal lines are only the ellipses corresponding to the constant values of j in these modes. One of the interesting case is when m = 0, r = 1; which corresponds to a mode with constant pressure in the cross-section. Therefore P(0,n,1)1 = cos(npz/l). The frequency of this mode depends only on the length and the volume of the cylinder, and is therefore the same as that of the circular cylinder. More meaningful conclusions can be made by comparing cavities of the same volume and cross-sectional area. In practice this means comparing the characteristics of two interchangeable acoustic devices because they will occupy approximately the same space

340

.   . 

Figure 6. Natural frequencies of hollow elliptical cavities compared with the equal volume circular cavity in terms of the non-dimensional frequency difference defined as in Figure 5.

in actual machinery. Under this condition, the natural frequencies of the elliptical cylinder of various eccentricities e are compared with the circular cylinder of radius r = 0·15 m. Other parameters are taken as l = 0·4 m, C = 162·9 m/s and n = 1 for both cavities, as before. The results are shown in Tables 2(a) and (b). Again, in Figure 6 the same information is shown, but in terms of the non-dimensionalized differences of the natural frequencies. The curves in Figure 6 indicate the errors in the estimation of the natural frequencies of elliptical cavities by approximating them as equivalent circular cavities. In Figure 6 it is shown that for m = 0 and r = 1 the natural frequency of the elliptical cavity remains the same as that of the circular cavity for the entire range of e. For a mode corresponding to m = 1 and m = 2, the frequency of the circular cavity v c is somewhere between two frequencies of the elliptical cavity, i.e., v Q v cv¯ . This is also expected from the geometry, as discussed before. It is interesting that for higher m values, both frequencies of the elliptical cylinder become lower than those of the circular cylinders, i.e., v Q v¯ Q v c, which is not quite obvious from the geometry. It is considered to be caused by the fact that m represents the wavenumber in the circumferential direction, and that the longer circumferential length of the elliptical cavity becomes a dominant factor in higher m modes. From Figure 6, it is also seen that the two frequencies corresponding to a higher m number become close to each other and to the frequency of the circular cavity. Furthermore, the figure shows that for an eccentricity up to 0·6, the frequencies v and v¯ are within only 8% of the frequency of the equivalent circular cylinder. The eccentricity, used to illustrate the ellipses in Figure 1, is approximately 0·6. This suggests that the frequencies of the equivalent circular cavity would serve as fairly good approximations for the elliptic cavity in a fairly wide range.

  

341

T 2 (a) The comparison of frequencies between circular and elliptical cylinders (based on the same volume of the cavities) Circular cylinder e = 0 (h = 0, radius r = 0·15) ZXXXXXCXXXXXV m, r vc 0, 1 0, 2 0, 3 0, 4 0, 5 0, 6 0, 7 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 1, 7 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

1 279·414 4 353·515 7 725·689 11 122·378 14 525·963 17 932·674 21 341·481 2 373·747 5 929·634 9 358·348 12 776·997 16 192·438 19 606·660 23 020·145 3 555·103 7 394·303 10 902·258 14 360·061 17 799·333 21 229·464 4 738·221 8 798·056 12 388·929 15 891·909 19 360·690 22 876·923 5 914·835 10 161·623 13 831·706 17 384·061 20 886·299 24 360·974

Elliptical cylinder e = 0·2 (h = 0·0303077, j = 2·29) ZXXXXXXXXXCXXXXXXXXXV v v¯ v¯ /v 1 279·4 4 368·7 7 742 11 195·7 14 618·9 17 986·9 21 442·8 2 359·2 5 916 9 353·8 12 759 16 144·6 19 576·9 22 949·6 3 548·3 7 364·3† 10 890 14 346·2 17 733·8 21 202·9 4 720·9 8 747·6 12 354 15 856·3 19 330·6 22 782 5 910 10 148·9 13 822·8 17 354·3 20 871·7 24 320·4

0 0 0 0 0 0 0 2 390·3 5 976·1 9 445·3 12 841·9 16 313·6 19 766·9 23 081·2 3 561·7 7 366† 10 920·5 14 401·2 17 807·8 21 302·2 4 721·5 8 747·6 12 357·4 15 861·3 19 355·2 22 836·6 5 910 10 148·9 13 823·2 17 355·9 20 876·1 24 330·8

0 0 0 0 0 0 0 1·01 1·01 1·01 1·01 1·01 1·01 1·01 1·00 1·01 1·00 1·00 1·00 1·01 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00

†Some errors due to interpolation of tables may have occurred in this row

(b) The comparison of frequencies between elliptical cylinders with different eccentricities (based on the same volume of the cavities) e = 0·4 e = 0·6 e = 0·8 (h = 0·062673, j = 1·5668) (h = 0·10062, j = 1·1) (h = 0·155, j = 0·6931) ZXXXXXXCXXXXXXV ZXXXXXCXXXXXVZXXXXXCXXXXXV m, r v v¯ v¯ /v v v¯ v¯ /v v v¯ v¯ /v 0, 1 0, 2 1, 1 2, 1 3, 1 4, 1

1279·4 4381·1 2310·2 3537·6 4707·6 5904·3

0 0 2442·7 3562·3 4711·6 5904·7

0 0 1·06 1·01 1·00 1·00

1279·4 4511·9 2211·5 3432·2 4650 5833·8

0 0 2555·2 3581·2 4700·6 5849·5

0 0 1·16 1·04 1·01 1·00

1279·4 4978·3 2026·5 3122·9 4280·2 5452·4

0 0 2817·5 3683·6 4652·5 5688·2

0 0 1·39 1·18 1·09 1·04

.   . 

342

The cross-sectional views of some natural modes with various m, n, r combinations are shown in Figure 2. These mode shapes are basically distorted versions of those of the circular cross-section which are shown in Figure 2. The pressure mode P(m, r) has m confocal nodal hyperbolas in 0 E h Q p; and r − 1 confocal nodal ellipses within the cross-section. P(m, r)2 always has h = 0° and h = 180° as its nodal lines. Also, from the basic properties of Sem and Som , it is known that the mode shape of P(m, r)1 is symmetrical about both major and minor axes if m is even, and is symmetrical about the major axis but antisymmetric about the minor axis if m is odd. Similarly, P(m, r)2 is antisymmetric about both the axes if m is even but symmetrical about the minor axis and antisymmetric about the major axis if m is odd. The number of nodal hyperbolas is decided by m, and . However, if K or K  is large, the hyperbolas will is independent of the value of K and K tend to cluster about h = 90°, except the one of Sem along the major axis [11]. 4. NATURAL MODES OF ELLIPTICAL ANNULAR CYLINDRICAL CAVITIES

4.1.   An annular cylindrical cavity the cross-section of which consists of two confocal ellipses of j = j1 (with a = a1 = he1 , b = a1 z1 − e12 ) and j2 (with a = a2 = he2 , b = a2 z1 − e22 ) is shown in Figure 7. The same semi-interfocal length was used only for computational simplicity. The general solution in equation (23) should still satisfy the physical condition (1) in section 3.1. Therefore, we have V(h) = Cm Sem (K, cos h), s = sm , m = 0, 1, 2, 3, . . . ,   V(h) = Cm Som (K, cos h), s = s¯ m , m = 1, 2, 3, . . . . For the annular cylinder, the solution does not have to satisfy the physical condition (2) in section 3.1 because the interfocal line is not included in the domain under consideration. Thus the general solution of the annular elliptical cylinder becomes: a

f(j, h) = U(j)V(h) = s Cm [Am Jem (K, cosh j) + Bm Nem (K, cosh j)]Sem (K, cos h) m=0 a

m [A m Jom (K , cosh j) + B m Nom (K , cosh j)]Som (K , cos h). + s C

(56)

m=1

The full expression of the pressure mode of the annular cylinder becomes a

a

P(j, h, z, t) = s s Cmn [Amn Jem (K, cosh j) + Bmn Nem (K, cosh j)]Sem (K, cos h) m=0 n=0

a

a

mn [A mn Jom (K , cosh j) × cos (npz/l) exp(ivmn t + cmn ) + s s C m=1 n=0

mn Nom (K , cosh j)]Som (K , cos h) cos (npz/l) exp(iv¯ mn t + c mn ). +B

Figure 7. The cross-section of an annular elliptical cylinder.

(57)

  

343

4.2.   For an annular elliptical cylinder with rigid boundary conditions, the boundary conditions are defined as 1P 1P (j , h, z, t) = 0. (j , h, z, t) = 1j 2 1j 1

(58a, b)

These conditions are represented as

$

%6 7 6 7

(59)

$

%6 7 6 7

(60)

Je'm (K, cosh j1 ) Ne'm (K, cosh j1 ) Je'm (K, cosh j2 ) Ne'm (K, cosh j2 )

Am 0 = , 0 Bm

and , cosh j1 ) No'm (K , cosh j1 ) Jo'm (K , cosh j2 ) No'm (K , cosh j2 ) Jo'm (K

m A 0 m = 0 . B

For non-trivial solutions to exist, the determinants of equations (59) and (60) should become zero, which gives two independent characteristic equations Je'm (K, cosh j1 )Ne'm (K, cosh j2 ) − Je'm (K, cosh j2 )Ne'm (K, cosh j1 ) = 0,

(61)

, cosh j1 )No'm (K , cosh j2 ) − Jo'm (K , cosh j2 )No'm (K , cosh j1 ) = 0. Jo'm (K

(62)

 are determined by solving equations (61) and (62) respectively. The eigenvalues K and K mr , respectively, where r = 1, 2, 3, . . . . For Let the rth zero of the equations be Kmr and K each (m, r), we have Umr (j) = Amr [Jem (Kmr , cosh j) + mmr Nem (Kmr , cosh j)] mr [Jom (K mr , cosh j) + m¯ mr Nom (K mr , cosh j)], =A

(63) (64)

where mmr = Bmr /Amr = −Je'm (Kmr , cosh j1 )/Ne'm (Kmr , cosh j1 ) = −Je'm (Kmr , cosh j2 )/Ne'm (Kmr , cosh j2 ), mr /A mr = −Jo'm (K mr , cosh j1 )/No'm (K mr , cosh j1 ) m¯ mr = B mr , cosh j2 )/No'm (K mr , cosh j2 ). = −Jo'm (K

(65) (66) (67) (68)

Finally, the general solution of the homogeneous wave equation is obtained as a

a

a

P(j, h, z, t) = s s s gmnr [Jem (Kmr , cosh j) + mmr Nem (Kmr , cosh j)]Sem (Kmr , cos h) n=0 m=0 r=1

a

a

a

mr , cosh j) × cos (npz/l) exp(ivmnr t + cmnr ) + s s s g¯ mnr [Jom (K n=0 m=1 r=1

mr , cosh j)]Som (Kmr , cos h) cos (npz/l) exp(ivmnr t + c mnr ), (69) + m¯ mr Nom (K where gmnr and g¯ mnr are arbitrary constants. The frequencies vmnr and v¯ mnr are obtained from equations (54) and (55), respectively, as before. Again, for every m, n, r combinations, we have two sets of pressure modes as follows: P(mnr)1 (j, h, z) = [Jem (Kmr , cosh j) + mmr Nem (Kmr , cosh j)]Sem (Kmr , cos h) cos (npz/l), m, n = 0, 1, 2, 3, . . . , r = 1, 2, 3, . . . ,

(70)

344 .   .  mr , cosh j) + m¯ mr Nom (K mr , cosh j)]Som (K mr , cos h) cos (npz/l), P(mnr)2 (j, h, z) = [Jom (K m, r = 1, 2, 3, . . . , n = 0, 1, 2, 3, . . . .

(71)

4.3.       If the eccentricities of both confocal ellipses become zero, we have two concentric circles of radius r = r1 and r = r2 , which forms a circular annular cylinder. In addition to the results in equations (39)–(42), as before, we have [11, 12] Nem (K, cosh j) :Nm (K cosh j), , cosh j) Nom (K

(72)

where Nm (K cosh j) is the Neumann function of order m. By utilizing equations (39)–(42) and equation (72), and noting that K cosh j = kr when e:0, both of the characteristic equations in equations (61) and (62) degenerate to a single equation: J'm (kr1 )N'm (kr2 ) − J'm (kr2 )N'm (kr1 ) = 0,

(73)

where k = (v/c) − (np/l) . The two sets of pressure modes become 2

2

2

P(mnr)1 (r, u, z) = [Jm (kmr r) + mmr Nm (kmr r)] cos (mu) cos (npz/l),

(74)

P(mnr)2 (r, u, z) = [Jm (kmr r) + m¯ mr Nm (kmr r)] sin (mu) cos (npz/l),

(75)

mmr = m¯ mr = −J'm (kmr r2 )/N'm (kmr r2 )

(76)

with All these results agree with those derived by Kim and Soedel [6] for circular annular cylindrical cavities. 4.4.    To solve the characteristic equations (61) and (62), we need general expressions for Nem and Nom , which are given as [12] Ne2m (K, cosh j) =

Ne2m + 1 (K, cosh j) =

1 B0e (K, m)[ 1 B (K, m) e 1

X

X

p a e s (−1)n − mB2n (K, m)Nn (12 K ej)Jn (12 K e−j), 2 n=0

p a 1 e −j s (−1)n − mB2n )Nn + 1 (12 K ej) + 1 (K, m)[Jn (2 K e 2 n=0

+ Jn + 1 (12 K e−j)Nn (12 K ej)] , cosh j) = No2m + 1 (K

1 , m) B (K 0 1

X

1 , m) B (K 0 2

X

(78)

p a 0 , m) s (−1)n − m(2n)B2n + 1 (K 2 n=0

 e−j)Nn + 1 (12 K  ej) − Jn + 1 (12 K  e−j)Nn (12 K  ej)], × [Jn (12 K No2m (K, cosh j) =

(77)

(79)

p a 0 , m)[Jn − 1 (12 K  e−j)Nn + 1 (12 K  e j) s (−1)n − m(2n)B2n (K 2 n=0

 e−j)Nn − 1 (12 K  ej)]. − Jn + 1 (12 K

(80)

Utilizing these equations, together with equations (50)–(53), zeros of equations (61) and (62), which provide natural frequencies, can be found. Again, the values of the functions in the tables of references [13] and [14] were interpolated during the root-finding procedure.

  

345

Figure 8. Zeros of equation (61) with e1 = 0·48 and e2 = 0·3.

In Figures 8 and 9 it is shown how the zeros of equations (61) and (62) are distributed. Some results of v and v¯ for a specific case of l = 0·185 m, c = 162·9 m/s and n = 1, with various eccentricities e1 and e2 , are shown in Table 3 and Figure 10. The eccentricities of the ellipses were chosen so that two ellipses have the same enclosed area as the inner and outer circles, respectively. The same value of the interfocal distance h was used for both inner and outer ellipses as mentioned. The radii of the circular cavity were r1 = 0·05 m, r2 = 0·08 m and the length l was again taken as 0·185 m. As it was before, this allows to compare the characteristics of two types of acoustic device of approximately the same space requirement. Again, practical examples can be taken from a hermetic shell of a

Figure 9. Zeros of equation (62) with e1 = 0·48 and e2 = 0·3.

.   . 

346

T 3 (a) The comparison of frequencies between annular cylinders of circular and elliptical cross-sections (based on the same volume of the cavities) Annular circular e1 = e2 = 0, h = 0, r1 = 0·05, r2 = 0·08 ZXXXCXXXV m, r vc 0, 1 0, 2 0, 3 1, 1 1, 2 1, 3 2, 1 2, 2 2, 3 3, 1 3, 2 3, 3 4, 1 4, 2 4, 3

2 766·300 17 418·42 34 296·31 3 746·74 17 621·90 34 401·85 5 751·26 18 207·75 34 694·13 8 025·92 19 756·24 35 173·18 10 368·32 20 420·42 35 843·09

Annular elliptical cylinders ZXXXXXXXXXXXXXCXXXXXXXXXXXXXV e1 = 0·198, e2 = 0·1245, e1 = 0·5486, e2 = 0·3621, h = 0·01 h = 0·03 ZXXXXXXCXXXXXXV ZXXXXXXCXXXXXXV v v¯ v/v¯ v v¯ v/v¯ 2 766·3 17 319·3 34 175·3 3 750·8 17 806·1 34 565·9 5 750·8 18 265·7 34 824·9 8 024·7 19 668·2 35 287·1 10 359·5 20 511·3 35 878·4

5 0 0 3 712·6 17 677·2 34 344·3 5 743·6 18 254·5 34 749 8 024·4 19 668·2 35 283 10 358·7 20 492·2 35 878·2

a a a 1·010 1·017 1·006 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00

2 766·3 17 068·5 33 162·1 3 789·9 17 863 34 195 5 748·1 18 616·4 35 141·3 8 008·6 19 367·7 35 927·9 10 343·8 20 522·8 36 516·7

0 0 0 3 699·7 17 084·2 33 162·2 5 735·6 18 136·2 34 197 8 007·4 19 227·7 35 163·1 10 343·7 20 501·8 36 067·6

a a a 1·024 1·046 1·031 1·00 1·03 1·03 1·00 1·01 1·02 1·00 1·00 1·012

(b) The comparison of frequencies between annular elliptical cylinders with different eccentricities (based on the same volume of the cavities)

m, r

e1 = 0·7862, e2 = 0·5672, h = 0·05 e1 = 0·9075, e2 = 0·7258, h = 0·07 ZXXXXXXXXCXXXXXXXXV ZXXXXXXXCXXXXXXXV v v¯ v/v¯ v v¯ v/v¯

0, 1 0, 2 1, 1 1, 2 2, 1 2, 2 3, 1 3, 2 4, 1 4, 2

2 766·3 16 366·7 3 834·38 17 866·8 5 719·4 19 231·3 7 907·9 20 445·3 10 202·1 21 474·7

m, r 0, 1 0, 2 1, 1 2, 1 2, 1 2, 2

0 0 3 600·3 16 365·7 5 626·6 17 845·3 7 883 19 310 10 195·6 20 780·2

a a 1·065 1·092 1·016 1·078 1·00 1·06 1·00 1·033

2 766·3 16 080·8 3 837·7 18 531·6 5 616 19 299·2 7 667·4 20 738·1 9 837·54 22 594·4

a a 1·11 1·15 1·05 1·09 1·02 1·07 1·00 1·08

0 0 3 457·6 16 079·7 5 361·6 17 673·6 7 356·7 19 304·5 9 775·4 20 963·4

e1 = 0·9589, e2 = 0·8348, h = 0·09 ZXXXXXXXXXXXXCXXXXXXXXXXXXV v v¯ v/v¯ 2 766·3 16 579·9 3 764·4 5 418·7 7 298·5 9 283·1

0 0 3 309·3 4 982·9 6 979·4 9 066·2

a a 1·14 1·09 1·05 1·02

  

347

compressor or an automotive muffler where either circular or elliptic geometry may be used. As the hollow cavity case, Figure 10 represents the non-dimensional frequency difference of the annular elliptical cavity compared to the annular circular cavity. The figure shows the error in the estimation of the frequency of the elliptical cavity caused by approximating it as a cricular one. From the top graph in Figure 10, it seems that the frequencies of the elliptical cavity are generally lower than those of the circular cavity for m = 1. The difference becomes larger as h (therefore, also the eccentricities e1 and e2 ) increases. It is concluded that the increased effective circumferential length of the elliptical cavities caused this. For other cases shown (m e 2, r e 2), since the acoustic wave travels in both circumferential and radial directions, the frequencies are not determined predominantly by the circumferential dimension. Increasing the eccentricities e1 and e2 increases the frequencies v and decreases v¯ . For m = 0, which corresponds to the modes with the pressure variation only in radial direction, the frequencies of the elliptical annular cavity become very close to the frequencies of the circular cases, as it was in the hollow cavities. For the modes corresponding to m = 0, the influence of the shape of the cavity seems to be very small.

Figure 10. Natural frequencies of annular elliptical cavities compared with the equal volume annular circular cavity in terms of the non-dimensional frequency difference as defined in Figure 5. (a) and (b) are different sets of modes.

348

.   . 

The cross-sectional views of various mode shapes of the annular elliptical cylinder for several m, n, r combinations are given in Figure 11. Compared with the hollow cylinder cases, less difference is found in mode shapes between the elliptical cavity and circular cavity. Also, the difference in natural frequencies at each mode number is found to be smaller compared with the hollow cavity cases. For example, when the eccentricities for the inner ellipse and outer ellipse are 0·7862 and 0·5672, and h = 0·05 (see Table 3), the natural frequencies are different only by about 7% from those of the annular circular cylinder cavity in Figure 10. Therefore, using the natural modes of the annular circluar cylinder of the same volume and cross-sectional area to approximate the elliptical annular cylinder will give reasonable results. 5. CONCLUDING REMARKS

Natural modes and natural frequencies of the acoustic cavities of hollow and annular elliptic cylindrical shapes have been obtained analytically in this work. To the extent of the authors’ literature survey, no analytical work has been found for elliptical cylinder shaped cavities. The desire to compare the acoustic characteristics of the elliptical cylindrical cavity and the equivalent circular cylindrical cavity was the motivation for this work. The effect of eccentricity of the cylindrically shaped cavity on its acoustic performance has been often questioned because circular or elliptical cylindrical enclosures are frequently found in industrial applications.

Fig. 11 (Part 1) See caption on page 24

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The natural frequencies were calculated and tabulated for a lower mode range. It was shown that the solution, both in its analytical form and in terms of numerical values of frequencies, converges to the circular cylinder case when the eccentricity approaches zero. Several mode shapes were shown which are distorted patterns of the corresponding mode shapes of the circular cylinderical cavity. The mode shapes of an annular elliptical cylinder were found to be even closer in their form to those of the equivalent annular circular cylinder. Unique aspects of the acoustic characteristics of elliptical cavities were pointed out, such as the existence of two distinct modes and natural frequencies associated with the same mode number. Natural frequencies of the elliptical cylindrical cavities were compared with the frequencies of the equivalent circular cylindrical cavities. If the eccentricity is moderate, as encountered in most of the practical applications, the natural frequencies of an elliptical shaped cavity were shown to be fairly close to those of the equivalent circular cylindrical cavity. This suggests that the latter, which can be analyzed much more easily, would serve as a good approximation for the former. Some results from the qualitative observations in this study, for example the general trend of the natural frequency change owing to the eccentricity and the mode number, may be used to increase the effective range of approximation.

Fig. 11 (Part 2) See caption on page 24

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Fig. 11 (Part 3) Figure 11. The cross-sectional view of mode shapes of an annular elliptical cylinder for various m, n, r combinations.

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