Prediction of natural frequencies of finite length circular cylindrical shells

Applied Acoustics 59 (2000) 385±400 www.elsevier.com/locate/apacoust

Prediction of natural frequencies of ®nite length circular cylindrical shells C. Wang, J.C.S. Lai* Acoustics and Vibration Unit, University College, The University of New South Wales, Australian Defence Force Academy, Canberra, ACT 2600, Australia Received 12 January 1999; received in revised form 3 June 1999; accepted 27 July 1999

Abstract Analysis of the vibration characteristics of ®nite circular cylindrical shells is more complex than for beams and plates. This is because the coupling of vibration of shells between the three directions can no longer be neglected. A literature review of the subject reveals that in traditional analysis, assumptions are made to simplify the equations of motion so that they can be solved directly with the appropriate boundary conditions. Consequently, the results obtained could be in error under certain conditions. Based on the Love's equations and an in®nite length model, a novel wave approach is introduced to predict the natural frequencies of ®nite length circular cylindrical shells with di€erent boundary conditions without simplifying the equations of motion. Results obtained compare favourably with those obtained using the ®nite element method. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Natural frequency; Finite length; Circular cylindrical shell

1. Introduction Like beams and plates, cylindrical shells are structures being widely used in industries. Consequently, the vibration and acoustic noise radiated from cylindrical shells are of special concern to vibration and acoustics engineers. However, the vibration and acoustic behaviour of cylindrical shells is not as easily understood as that of beams and plates. This is mainly because the equations of motion governing cylindrical shells are more complex than those of beams and plates. Indeed, according to previous research [1], beams and plates are actually two special cases of * Corresponding author. +61-2-6268-8272; fax: +61-2-6268-8276. E-mail address: [email protected] (J.C.S. Lai). 0003-682X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(99)00039-0

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shells with in®nite curvature. Therefore, the curvature e€ects may have some signi®cant consequences for the vibration and acoustic performance of shells. The vibration theory of cylindrical shells has been studied extensively for more than 100 years. As the boundary conditions signi®cantly a€ect the mode shapes and the natural frequencies, the prediction of natural frequencies and mode shapes for di€erent boundary conditions has been the main thrust in the study of ®nite length cylindrical shells [2]. However, because the equations of motion of cylindrical shells together with relevant boundary conditions are more complex than those of beams and plates, it is very dicult to obtain an analytical solution. So in the past, numerous papers have been published to discuss how to establish and solve the equations e€ectively by invoking various assumptions. According to Leissa [2], who summarised and compared about 12 of these methods, all these methods invoked assumptions that simplify the equations of motion so that they can be solved analytically with di€erent boundary conditions. Normally the relationship between stress, strain and displacement was ®rst simpli®ed to yield a set of equations of motion. Then, further simpli®cations were made when solving the equations of motion with di€erent boundary conditions. Obviously, as the approximations were made directly to the equations of motion, the relationships derived for the natural frequencies and wavenumbers, the modal wavenumbers and the mode shapes were all approximate solutions which only apply under speci®c conditions. For example, Donnell's equation [3] provides good results only for very thin and long circular cylindrical shells, and the equation developed by Lord Rayleigh [4], is only valid for high order modes of circular cylindrical shells, etc. In recent years, some researchers have used the ®nite element method to analyse the vibration behaviour of ®nite length cylindrical shells [5,6]. However, it is tedious to explore the in¯uence of all the parameters on the vibration behaviour by FEM. For engineering applications and acoustic analysis, an approximate analytical solution may provide more insight into the vibration behaviour of ®nite length cylindrical shells provided that its accuracy satis®es certain predetermined criteria. In this paper, therefore, based on Love's equations, the basic vibration properties of cylindrical shells with circular cross section are ®rst discussed by using an in®nite length model. Then, a wave approach is introduced to study the vibration behaviour of ®nite length circular cylindrical shells. Finally, an approximate method for calculating the natural frequencies of ®nite length circular cylindrical shells with different boundary conditions without simplifying the exact equations of motion is given. These approximate solutions are compared with numerical results obtained using a ®nite element code ANSYS and with experimental data. 2. Theoretical analysis 2.1. Love's equations and an in®nite length model Vibration analysis of cylindrical shells was ®rst made in the nineteenth century [1,2]. Among the theories developed in the past, the approach by Love [7] has been

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proven to be the most e€ective and successful in linear theories. As many other theories are of Love's type [1], the set of equations developed by Love, namely the Love's equations, is generally used to study the vibration of continuous structures of which the thickness is much less than the length or radius. By assuming (a) the thickness of the shell is much less than its radius and (b) the shear de¯ection is small so that the in¯uence of rotatory inertia could be neglected, the equations of motion of thin cylindrical shells, as given by Love [7] are:  2  @ u~ z 1 ‡  @2 u~   @u~ r 1 ÿ  @2 u~ z @2 u~ z ‡ ‡ :
‡
…1† ‡ qz ˆ h 2 K 2 2 2 @z @t 2a @@z a @z 2a @     K…1 ÿ † D…1 ÿ † @2 u~  K…1 ‡ † @2 u~ z K D @2 u~  ‡ ‡
‡ ‡

@z2 @z@ 2 2a2 2a a2 a4 @2 K @u~ r D @3 u~ r D @3 u~ r @2 u~  ÿ 2
2 ÿ 4
3 ‡ q ˆ h 2 ‡ 2
@t a @ a @z @ a @

…2†

@4 u~ r 2D @4 u~ r D @4 u~ r D @3 u~  D @3 u~  ÿ 2
2 2 ÿ 4
4 ‡ 2
2 ‡ 4
3 4 a @z @ a @ a @z @ a @ @z K @u~  K @u~ z K @2 u~ r ÿ ÿ 2
ur ‡ qr ˆ h 2
ÿ 2
@t a @ a @z a

…3†

ÿD


where u~ z ; u~  ; u~ r are the vibration displacements in the axial …z†, circumferential …†, and radial …r† directions, respectively, qz ; q ; qr are the external pressures in the three Eh Eh3 ; E is Young's modulus,  is Poisson's ratio, directions, K ˆ ; D ˆ 12…1 ÿ 2 † 1 ÿ 2 a is the radius of the cylinder,  is the density of the material and h is the thickness of the shell, as shown in Fig. 1. From this set of equations, it can be seen that for cylindrical shells, the vibration displacements in the three directions …r; ; z† are not independent of each other.

Fig. 1. A circular cylindrical shell.

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For an in®nite length circular cylindrical shell which is closed in the circumferential direction and in®nite in the axial direction, all the vibration displacements should satisfy circular symmetry: u~ i …; z; t† ˆ u~ i … ‡ 2; z; t†

i ˆ z; ; r

…4†

From Eq. (4), the wavenumber in the circumferential direction for the three vibrations can be determined as n n2N …5† k ˆ a where N is the natural number set. Eq. (5) indicates that the wavenumber in the circumferential direction has to be a series of discrete numbers. Therefore, in®nite length cylindrical shells have vibration modes in the circumferential direction …†. In the axial direction …z†, there exist three kinds of waves, longitudinal wave u~z , torsional wave u~  and ¯exural wave u~ r . Therefore, the solution of Love's equations for an in®nite length circular cylindrical shell can be assumed as, 8 1 P > > unz cos…n ÿ r †e j…!tÿkzz z‡=2† u~ z ˆ > > > nˆ0 > < 1 P u~  ˆ un  sin…n ÿ r †ej…!tÿkz z† …6† > nˆ0 > > 1 > P > > : u~ r ˆ unr cos…n ÿ r †ej…!tÿkrz z† nˆ0

where kzz ; kz , and krz are the wavenumbers in the axial direction for the three vibrations. Here, u~ z is not a sine function in the  direction because physically, for n ˆ 0 mode, the longitudinal wave should exist and should not have a nodal line. Moreover, as shown by Soedel [1], there is a phase di€erence of =2 between u~z and u~  in the z direction, and this di€erence has been included in u~z in Eq. (6). One may also assume these waves propagating in the negative directions but it can be shown that the ®nal results are the same. By substituting Eq. (6) into Eqs. (1)±(3), we obtain   1ÿ 2 K…1 ‡ † 2 2
k ÿ ! h
unz eÿjkzz z ‡
kz k un eÿjkz z K kzz ‡ 2 2 K krz unr eÿjkrz z ˆ 0 ‡ a K…1 ‡ †
kzz k unz eÿjkzz z ‡ 2
k2

2



ÿ ! h
un e

    K…1 ÿ † D…1 ÿ † 2 D ‡
kz ‡ K ‡ 2 2 2a2 a ÿjkz z



 K D D
k ‡
k2rz k ‡
k3
unr eÿjkrz z ˆ 0 ‡ a a a

…7†

…8†

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  K K D 2 D 3 ÿjkzz z
kzz unz e
k ‡
kz k ‡
k
un eÿjkz z ‡ a a a a   K 4 2 2 4 2 ‡ Dkrz ‡ 2Dkrz k ‡ Dkrz ‡ 2 ÿ ! h
unr eÿjkrz z ˆ 0 a

389

…9†

For these three equations to be always valid for all z and , kzz ˆ kz ˆ krz ˆ kz

…10†

and k2 ˆ k2z ‡ k2 Also, for non-trivial solution, the determinant of this set of equations must be zero, i.e.   1ÿ 2 2 K k2z ‡
k  ÿ ! h 2 K…1 ‡ †
kz k 2 K
kz a

K…1 ‡ †
kz k ; 2    D 1‡ 2 K ‡ 2 k2 ÿ kz ÿ !2 h a 2   K D
k ‡
k2 k a a

K kz a   K D 2
k ‡
k k ˆ 0 a a K Dk4 ‡ 2 ÿ !2 h a …11†

Eq. (10) indicates that when three waves exist in the structure, their wavenumbers in the axial direction must be the same. Rewriting      1ÿ 2 D 1‡ 2 K 2 2
k ; k22 ˆ K ‡ 2 k ÿ kz ; k33 ˆ Dk4 ‡ 2 k11 ˆ K kz ‡ 2 a 2 a   K…1 ‡ † K K D 2
kz k ; k13 ˆ k31 ˆ kz ; k23 ˆ k32 ˆ
k ‡
k k k12 ˆ k21 ˆ 2 a a a then Eq. (11) becomes, !6 ‡ a1 !4 ‡ a2 !2 ‡ a3 ˆ 0 where a1 ˆ ÿ

1 …k11 ‡ k22 ‡ k33 †; h

a2 ˆ


1 ÿ k11 k33 ‡ k22 k33 ‡ k11 k22 ÿ jk23 j2 ÿjk13 j2 ÿjk12 j2 2 …h†

a3 ˆ


1 ÿ k11 k223 ‡ k22 k213 ‡ k33 k212 ‡ 2k12 k23 k13 ÿ k11 k22 k33 3 …h†

…12†

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From Eq. (12), three solutions can be obtained [1] q 2 2  a1 a1 ÿ 3a2 cos ÿ !21 ˆ ÿ 3 3 3 q 2 2  ‡ 2 a1 ÿ a ÿ 3a2 cos !22 ˆ ÿ 3 3 1 3 q 2 2  ‡ 4 a1 ÿ !23 ˆ ÿ a ÿ 3a2 cos 3 3 1 3

…13† …14† …15†

where  ˆ cosÿ1

27a3 ‡ 2a31 ÿ 9a1 a2 q 2 …a21 ÿ 3a2 †3

Eqs. (13)±(15) are actually the three relationships between frequencies and wavenumbers. They govern the characteristics of wave propagation in the structure. By substituting any of the three solutions back to Eqs. (7)±(9), the amplitudes of the three waves can be determined, in which normally there is only one type of wave dominating over the other two waves. Therefore, each of the three solutions actually corresponds to a combination of three waves. Since the wavenumbers of the waves in the axial direction must be the same sa shown in Eq. (10), their propagation speeds should be the same, except their amplitudes and phase angles. For acoustic analysis, one is usually interested in the combination in which the ¯exural wave dominates the response because it normally contributes signi®cantly to acoustic noise. Unlike ¯at plates, it can be seen that for a given kz ; k , there are always three frequencies corresponding to three waves. Normally, the lowest frequency represents the ¯exural wave [1], and the other two are in-plane waves. 2.2. Finite length cylindrical shells For ®nite length circular cylindrical shells, Eqs. (1)±(3) still apply because these three equations are set up based on a cylindrical neutral element. However unlike in®nite length shells, there must be a series of standing waves with di€erent wavenumbers caused by the wave re¯ection at the boundaries and the superposition of two relevant waves propagating in opposite directions. Thus, the solution of basic equations does not have the form of an approaching wave, and must be written in the form of standing waves. 8 1 P 1 P > > ~ ˆ umnz m …z† cos…n ÿ r †ej!t u > z > > mˆ0nˆ0 > > > < 1 P 1 P u~  ˆ umn m …z† sin…n ÿ r †ej!t …16† > mˆ0 nˆ0 > > > > 1 P 1 > P > > umnr m …z† cos…n ÿ r †ej!t : u~ r ˆ mˆ0nˆ0

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where m …z†; m …z†; m …z† are the amplitude distributions of the three displacements along the z direction, namely, the mode shapes of the longitudinal, torsional and ¯exural vibration in the z direction. Subscripts m and n denote the mode number in the z and the  directions, respectively. According to the vibration theory [1], m …z†; m …z†; m …z† and the wavenumbers are determined by the boundary conditions. So normally, Eq. (16) should be substituted back into Eqs. (1)±(3), and by using the boundary conditions at both ends of the shell, am …z†; m …z†; m …z† and the three relationships between natural frequencies !imn and the wavenumbers …kzm ; kn † can be obtained. However, since Eq. (16) is obtained by linear superposition of Eq. (6), Eqs. (13)±(15), which apply to Eq. (6), should still be valid for Eq. (16), independent of the boundary conditions. This can be easily understood if one realises that Eqs. (13)±(15) actually de®ne the speed of three waves in the structure, which depends only on the material properties, such as Young's modulus, density and thickness, and is independent of the length and boundary conditions. Hence, if the wavenumbers for di€erent boundary conditions could be determined, the natural frequencies of ®nite length cylindrical shells can be predicted. For circular cylindrical shells which are closed in the circumferential direction, the wave number k is always determined by Eq. (5). Therefore, the accuracy of predicting the natural frequencies is primarily determined by that in determining the wave numbers kzm . For ®nite length cylindrical shells, the modes of three kinds of vibration could exist simultaneously. For a certain …kzm ; kn † there are three natural frequencies corresponding to the three dominant vibrations. Generally, the lowest one represents the ¯exural vibration. When a set of natural frequencies and …kzm ; kn † satisfy any one of the three relations, all three vibrations exist but only one dominates the performance. 2.3. Application of the beam functions In order to calculate the natural frequencies, it is only necessary to determine the wavenumber kz in the axial direction. Unfortunately as kz strongly depends on the boundary conditions, one has to solve the equations of motion with appropriate boundary conditions in order to obtain kz . However, if one is only interested in ¯exural vibration, one may use the beam function to determine the modal wavenumbers and mode shapes of cylindrical shells in the axial direction by assuming the ¯exural mode shapes of cylindrical shells in the axial direction m …z† to be of the same form as that of a transversely vibrating beam of the same boundary conditions. According to previous studies [1,2], beam functions have already been widely used to obtain approximate solutions for ®nite length cylindrical shells. Normally beam functions are substituted back into motion equations, then approximate solutions of natural frequencies, modal wavenumbers and mode shapes for di€erent boundary conditions could be obtained. However, note that in Eq. (13) which is normally associated with the dominant ¯exural vibration, the only unknowns that need to be determined for the prediction of natural frequencies are the modal wave numbers. The results of a vibrating beam thus can be used directly in Eq. (13) which

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is exact for circular cylindrical shells, and the approximation only needs to be made to modal wavenumbers. For example, the modal wavenumbers in the axial direction for a shell simplysupported at both ends can be obtained by using the beam function as kzm ˆ

m ; l

m ˆ 1; 2; 3; . . .

…17†

For circular cylindrical shells clamped at both ends, the wavenumber can be approximately derived from a beam as   1 m‡  2 ; m ˆ 1; 2; 3; . . . …18† kzm ˆ l When both ends of a cylindrical shell are free, the modal wavenumbers obtained by using the beam function are given by   1 mÿ  2 ; m ˆ 2; 3; 4; . . . …19† kzm ˆ l It can be seen that in this case, the beam function only provides a solution for m greater than 2. The modes m ˆ 0 and m ˆ 1 are actually the two inextensional modes of the circular cylindrical shells [1]. For a free-free beam, these two vibration modes correspond to the rigid body translation and rotation modes. Since for the beam vibration, they are trivial modes having zero frequencies, it is impossible to predict the frequencies of these two modes of cylindrical shells by using the beam function. However, by using inextensional approximation, Lord Rayleigh [4] and Love [9] discussed these two cases and obtained the natural frequencies and the corresponding mode shapes in the axial directions [2]. !20n ˆ

!21n

E h2 n2 …n2 ÿ 1†2
 12a4 n2 ‡ 1

E h2 n2 …n2 ÿ 1†2 1 ‡
ˆ
 12a4 n2 ‡ 1 1‡

…20† 24…1 ÿ †a2 n2 l 2 12a2 n2 …n2 ‡ 1†l2

…21†

According to Eqs. (20) and (21), it can be seen that as l=a increases, !1n approaches !0n . Therefore, for long or small radius cylindrical shells, one can just calculate !0n for these two modes without much error. An alternative way to calculate !0n , is to substitute kz0 ˆ 0 into Eq. (13).

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3. Discussions 3.1. The scope of application of Eq. (13) Since Eq. (13) is the exact solution for thin circular cylindrical shells by neglecting the shear and rotatory inertia, it is important to determine the limits for applying this equation. Soedel [10] derived the motion equations of shells including the e€ects of the shear de¯ection and rotatory inertia, and showed that Timoshenko's thick beam equation [11] is a special case of his general thick shell equations. By applying the solution obtained by Timoshenko [11] for thick beams to thick plates and cylindrical shells, Soedel showed that the inclusion of shear de¯ection makes the structure behave as if it were less sti€ and the inclusion of rotatory inertia increases the mass e€ect. Both e€ects tend to decrease the calculated natural frequencies. Also, Soedel [10] found that if kz h  1, shear and rotatory inertia may be neglected. This condition is equivalent to, a …22† kz a  h Therefore, it may be expected that Eq. (13) could apply to practical cylindrical shells for which Eq. (22) is satis®ed. 3.2. Comparison of wavenumber-frequency relationship with ®nite element model One of the most widely used approximate solutions for circular cylindrical shells is due to Donnell [3]. In this method, the in-plane de¯ections are neglected, which seems to work for very shallow shells and bending-dominated modes. Although this equation was ®rst derived for simply supported cylindrical shells, Soedel [12] obtained the same form of equation as shown in Eq. (23) for all boundary conditions by using an approximate technique. !2mn ˆ

2 K…1 ÿ 2 † D  2 k4zm
kzm ‡ k2n ‡
 2 h ha2 k2zm ‡ k2n

…23†

n where kn ˆ and kzm take di€erent forms for di€erent boundary conditions. In a order to examine the results obtained by Eq. (13), comparisons are made with the Table 1 Various con®gurations of cylindrical shells for ANSYS calculations Length (m)

l=0.2

l=0.5

l=1

Thickness

a/h=5

a/h=20

a/h=5

a/h=20

a/h=5

a/h=20

Number of nodes Number of elements

4680 3000

2480 2400

9180 6000

4080 4000

18 180 12 000

6480 6400

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Fig. 2. Comparisons of the results by Eqs. (13) and (23) and ANSYS for a=h ˆ 5.

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Fig. 3. Comparisons of the results by Eqs. (13) and (23) and ANSYS for a=h ˆ 20.

395

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approximate results provided by Eq. (23), and with ®nite element models for the cylindrical shells with two radius/thickness ratios (a=h ˆ 5 and 20), three di€erent lengths (l ˆ 0:2, 0.5 and 1 m) and three di€erent boundary conditions (namely, simply supported, free-free and clamped-clamped ends). A ®nite element code ANSYS [13] version 5.2 was used, and six corresponding models were created. The numbers of the nodes and the elements for each model are listed in Table 1. The element types used were SHELL63 for a=h ˆ 20, and SOLID45 for a=h ˆ 5. By using these models, the natural frequencies for each case can be calculated. All calculations were performed using a SUN SPARC20 workstation. Note that for the axial wavenumber of each mode, only Eq. (17) for simply-supported condition is exact. In order to obtain the correct axial wavenumber for each mode for free-free and clamped-clamped conditions, the displacements at each node along the axial direction are curve ®tted by using the corresponding beam functions [2]. Results obtained by Eqs. (13) and (23) and ANSYS for circumferential modes n ˆ 1, 2 and 3 are shown in Fig. 2(a)±(c), respectively, for a=h ˆ 5, while those for a=h ˆ 20 are shown in Fig. 3(a)±(c). Note that for convenience, a non-dimensional frequency parameter is used.

2 ˆ

!2 a2 !2 ˆ 2 E !r

…24†

Fig. 4. Comparisons of the natural frequencies of a clamped±clamped circular cylindrical shell by Eqs. (13) and (23) and ANSYS.

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397

s 1 E . For a simply-supported shell, it where !r is the ring angular frequency, !r ˆ a  can be seen from Figs. 2(a)±3(c) that for large a=h…ˆ 20†, the agreement between the results obtained by Eq. (13) with those of ANSYS is excellent. However, for small a=h…ˆ 5† as shown in Figs. 2(a)±3(c), ANSYS results are slightly lower than those obtained by Eq. (13) at high kz a. This is because for small a=h, shear de¯ection and rotatory inertia of the shell (which would reduce the natural frequencies as discussed above) are not taken into account by Eq. (13). For a circular cylindrical shell either clamped or free at both ends, it can be seen from Figs. 2 and 3 that most of the points for free-free and clamped-clamped conditions are very close to the corresponding curve except for a few points which correspond to the modes of the shells with short length and of low axial mode order. This could be because ANSYS mode shapes were curve®tted based on the beam functions which are less accurate for lower order modes due to the end e€ects. Nevertheless, Figs. 2 and 3 show that results obtained by Eq. (13) are much more accurate than those of Eq. (23) and suggest that Eq. (13) should be valid for di€erent boundary conditions.

Fig. 5. Comparisons of the natural frequencies of a free-free circular cylindrical shell by Eqs. (13) and (23) and ANSYS.

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Table 2 Experimental and theoretical natural frequencies for a clamped±clamped shell l/a=8.13, a/h=19.1, h=2.54 mm Mode number (m)

Source

Mode number (n) 2

3

4

5

6

7

1

Experiment [14] Koval and Cranch's equation Yu's equation Morley's equation Equation (23) Equation (13)

1240 1569 1756 1611 1729 1386

2150 2326 2460 2206 2541 2159

3970 4092 4246 3982 4334 3942

6320 6419 6615 6351 6704 6308

9230 9251 9522 9257 9611 9212

12 12 12 12 12 12

600 566 959 695 048 647

2

Experiment [14] Koval and Cranch's equation Yu's equation Morley's equation Equation (23) Equation (13)

2440 3605 4036 3975 3417 2938

2560 2782 2943 2734 3060 2672

4160 4185 4342 4085 4572 4171

6475 6444 6641 6376 6885 6481

9380 9259 9530 9266 9776 9372

12 12 12 12 13 12

750 569 962 698 209 803

3

Experiment [14] Koval and Cranch's equation Yu's equation Morley's equation Equation (23) Equation (13)

± 6883 7706 7674 5460 4806

3380 3918 4144 3998 4019 3613

4540 4474 4642 4402 5016 4606

6720 6524 6723 6463 7186 6773

9540 9286 9558 9295 10037 9624

12 12 12 12 13 13

900 580 974 709 456 043

4

Experiment [14] Koval and Cranch's equation Yu's equation Morley's equation Equation (23) Equation (13)

± 11310 12662 12643 7422 6670

4480 5759 6092 5994 5266 4828

5130 5087 5278 5068 5688 5268

7100 6708 6913 6660 7627 7204

9890 9349 9623 9362 10401 9979

13 12 12 12 13 13

220 605 999 736 792 372

5

Experiment [14] Koval and Cranch's equation Yu's equation Morley's equation Equation (23) Equation (13)

8020 16862 18879 18866 9135 8382

5740 8233 8709 8641 6625 6157

5910 6104 6333 6160 6553 6125

7710 7051 7267 7027 8213 7782

10310 9471 9748 9490 10876 10445

13 12 13 12 14 13

570 653 050 787 223 794

3.3. Accuracy of natural frequency predicted using beam functions In order to examine the accuracy of Eq. (13) combined with beam functions for predicting the natural frequencies, the results obtained for clamped-clamped and free-free boundary conditions are compared with a FEM model in Figs. 4 and 5, respectively. For short and thick shells [Fig. 4(a)], the di€erences between results obtained by Eq. (13) and FEM are substantial. This is because the coupling between the circumferential and axial modes is ignored by using the beam function. The e€ects of this coupling are less important for long thin shells and for higher order modes such as shown in Fig. 4(d).

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For short shells free at both ends [Fig. 5(a) and (c)], the agreement is considerably better than that for shells clamped at both ends. This is because the coupling e€ects between the circumferential and axial modes under free-free condition are less signi®cant due to lack of constraints at both ends. For long thin shells [Fig. 5(b) and (d)], the agreement between results obtained using beam functions and FEM is excellent. Love's inextensional modes are also shown in Fig. 5(a) and (c). Although Eq. (13) cannot predict these kind of modes because of the diculty in determining the axial wavenumbers, it can be seen that the frequencies of Love's inextensional modes are getting close to those of Rayleigh's inextensional modes as the length/ thickness ratio increases. For all curves shown in Figs. 4 and 5, the discrepancies between Eq. (23) and FEM are much more substantial than for Eq. (13), especially for low circumferential wavenumbers. The accuracy of our proposed method of prediction using beam functions can be further examined by comparisons with four other methods and the experimental data [14] for a clamped±clamped steel cylindrical shell as shown in Table 2. The four other methods of prediction are Eq. (23), Koval and Cranch's equation [14], Yu's [8] and Morley's equations [15]. Table 2 clearly shows that the predictions obtained by Eq. (13) consistently outperform all the other approximate methods. In particular, among all the methods given in Table 2, only Eq. (13) is capable to predict low circumferential modes to within 5% of the experimental value except for modes (1,2) and (2,2). 4. Conclusion The basic vibration behaviour of circular cylindrical shells has been examined using Love's equations. It has been shown, through the use of in®nite length cylindrical shell model and the wave approach, that the relationships between the natural frequencies and wavenumbers apply also to ®nite length cylindrical shells, independent of the boundary conditions. This is because these relationships de®ne the speed of the three waves present in the structures. Hence, in the prediction of natural frequencies for ®nite length circular cylindrical shells, instead of the traditional approach of simplifying the Love's equations and solving the simpli®ed equations directly with the boundary conditions at both ends, the boundary conditions are used in conjunction with the beam functions to determine the wavenumbers. Results obtained by this approach compare very favourably with those obtained using a ®nite element code. Furthermore, they outperform several other predictive methods when compared with experimental data. Acknowledgements This project is being supported by the Australian Research Council under the large grant scheme. The ®rst author (C. Wang) acknowledges receipt of an Overseas Postgraduate Research Scholarship for the pursuit of this study.

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