Free vibration analysis of joined spherical-cylindrical shells by matched Fourier-Chebyshev expansions

Author’s Accepted Manuscript Free vibration analysis of joined sphericalcylindrical shells by matched Fourier-Chebyshev expansions Jinhee Lee www.elsevier.com/locate/ijmecsci

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S0020-7403(16)31189-4 http://dx.doi.org/10.1016/j.ijmecsci.2016.12.025 MS3537

To appear in: International Journal of Mechanical Sciences Received date: 20 May 2016 Revised date: 9 November 2016 Accepted date: 29 December 2016 Cite this article as: Jinhee Lee, Free vibration analysis of joined sphericalcylindrical shells by matched Fourier-Chebyshev expansions, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2016.12.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Free vibration analysis of joined spherical-cylindrical shells by matched Fourier-Chebyshev expansions

Jinhee Lee

Department of Mechanical and Design Engineering, Hongik University 2639 Sejong-ro, Sejong 30016, Korea

Abstract A free vibration analysis of joined spherical-cylindrical shell structures is presented. The effects of transverse shear and rotary inertia are taken into account. The deflections and the rotations of spherical and cylindrical shells are represented by the expansions of Chebyshev polynomials in the colatitudinal and axial directions and Fourier functions in the circumferential direction. When the open ends of two hemispheres face each other the opposed circumferential directions should be taken into account in the governing equations and in the continuity conditions. The equations of motion are collocated to yield the system of equations that correspond to the circumferential wave number. To satisfy the continuity conditions the expansions are matched at the junctions of the substructures. The number of collocation points is chosen to be less than the number of expansion terms, and the set of algebraic equations is condensed so that the number of expansions matches the number of degrees of freedom of the problem. Numerical examples are provided for full sphere, bell structure and hermetic capsule.

Keywords: collocation; free vibration; joined shells; matched Fourier-Chebyshev expansion; sphericalcylindrical shell

1. Introduction Shell is one of the most essential elements in the civil, mechanical and aeronautical engineering practices and its dynamic characteristics had been extensively investigated and numerous studies were reported as can be found in the lists of references of the publications on shell vibrations [1-4]. The research on the vibration analysis of joined spherical-cylindrical shell structures, however, had been rather scarce. Galletly and Mistry [5] studied the dynamic characteristics of cylindrical shells clamped at one end and closed at the other by different types of shells including hemispheres and ellipsoids. Their study was based on the kinematic relations of Novozhilov, Flügge and Reissner and applied variational finite difference method and finite element method. Tavakoli and Singh [6] proposed a substructure synthesis method for the free vibration analysis of spherical shells and hermetic capsules. Structural elements were formulated from Love’s shell theory, and a system of eight coupled first order differential equations was solved for each shell substructure using Padé method, and the substructures were then joined by matching the displacement and force boundary variables. However, in their analyses of full sphere and hermetic capsule the range of spherical shells under consideration was limited between the angle 0=0.1° and 1= 179.9° and allowed pinholes with free boundary conditions at the zenith and the nadir. Wong and Sze [7] assumed a membrane approximation for the spherical and cylindrical shells and applied a matched asymptotic expansion method to study the axisymmetric vibration of hermetic capsules. They found that the continuity of axial stress resultants conflicted with the continuity of the normal component of the displacement across the junction joining the hemispherical and cylindrical shells which resulted in finite jumps in the normal displacement. Shang [8] developed an analytical solution for the axisymmetric and torsionless free vibration of a hermetic capsule based on the Naghdi-Reissner shell theory. He employed Legendre function and trigonometric functions as the basis functions of the hemispherical and the cylindrical shells, respectively. The conditions of the junction between the spherical and cylindrical shells were given by the continuity of deformation and the equilibrium relations. Buchanan [9] presented finite element solutions for a hermetic capsule, where three-dimensional models based on the elasticity theory were developed. Natural frequencies of torsional mode of a hermetic capsule were found that had not been previously reported. Lee et al. [10] analyzed the free vibration of a joined shell structure where a cylindrical shell was joined by a hemispherical shell at one end. The hemispherical shell and the cylindrical shell were assumed to have free boundary conditions and the simply supported boundary conditions, respectively, at

2

the junction and the effects of the spherical shell were assumed to be smaller than the cylindrical shell to the free vibration of joined structure. Redekop [11] presented a free vibration analysis of torus-cylindrical shell assembly. The equations based on the Sanders-Budiansky shell theory were applied to toroidal and cylindrical shells and the solutions were obtained using a differential quadrature method. Ma et al. [12] presented a study on the vibration analysis of joined conical-cylindrical shells based on the Reissner’s shell theory. In their study each of displacement components was expressed by the modified Fourier series composed of a standard Fourier series and closed-form auxiliary functions and artificial spring technique was adopted to simulate the boundary conditions and the continuity conditions. The expansion coefficients were determined by using Rayleigh-Ritz method. Kang [13] conducted a free vibration analysis of a joined hemispherical-cylindrical shell structure with a top opening, which was not based on a shell theory but on the three-dimensional elasticity theory. He formulated the potential and kinetic energies from the assumed displacements in the radial, circumferential and axial directions, and applied Ritz method to solve the eigenvalue problem. Qu et al. [14] presented a free vibration of joined conical-cylindrical-spherical shells with ring stiffeners by using modified variational method. Reissner-Naghdi’s shell theory was assumed and rotations and displacements of each shell segment were expanded by means of Fourier series and orthogonal polynomials which included Chebyshev polynomials of first- and second-kind, and Legendre polynomial of first kind. Wu et al. [15, 16] analyzed the vibration characteristics of joined sphericalcylindrical-spherical and cylindrical-spherical shells, respectively, by domain decomposition method. The analyses were based on Reissner-Naghdi’s shell theory and Fourier series. Chebyshev polynomials were employed as the admissible functions for the displacements. The solutions were obtained by using modified variational principle and least-squares weighted residual method. The interface potentials between the two adjacent shell segments were introduced into the energy functional of the joined shell. In the present study a free vibration analysis of joined spherical-cylindrical shell structures is developed, which is an extension of the axisymmetric vibration analysis of hermetic capsule [17] where the numbers of expansions of each shell element was larger than the number of intended degrees of freedom. The continuity conditions were considered as side constraints, and the set of algebraic equations was condensed so that the number of degrees of freedom matched the total number of expansion coefficients. The idea was also utilized in the free vibration analysis of stepped beams [18] where both the boundary conditions and the continuity conditions were considered as side constraints.

3

2. Elemental shell formulations 2.1 Hemispherical shell The equations of motion of hemispherical shells with the effects of transverse shear and rotary inertia taken into account were derived by Soedel [3] and employed in the previous study [19] as

N  N sin     N cos   Q sin    2 R sin   hU   

(1a)

N  N sin      N cos   Q sin    2 R sin   hV   

(1b)

Q  Q sin       N  N  sin    2 R sin   hW   

(1c)

M   h3 M  sin     M cos   Q R sin    2 R sin      12

(1d)

M   h3 M  sin     M cos   Q R sin    2 R sin      12

(1e)

for harmonic motions at natural frequency



in radian/second. U, V and W are the displacements in the

colatitudinal, circumferential and normal directions, respectively, as shown in Fig. 1.



W R



V U 



Fig. 1 Displacements and rotations in the spherical coordinate system

 and  are the rotations in the colatitudinal and circumferential directions. R, h and  are the radius, the wall thickness and the density of the spherical shell. The stress resultants N, N, N, Q, Q, M, Mand M are defined as

 1 U W  1 V U W  N  C        R   R R sin    R tan  R   

(2a)

 1 V  1 U W   U W N  C        R sin    R tan  R  R  R   

(2b)

N  N 

1   1 V V 1 U  C    2 R   R tan  R sin    

4

(2c)

 1   1    M  D      R   R sin    R tan    

(2d)

 1      M  D     R sin    R tan  R   

(2e)

M   M 

1   1   1   D    2 R   R tan  R sin    

(2f)

 U 1 W  Q   Gh      R R   

(2g)

 V 1 W  Q   Gh      R R sin    

(2h)

C and D are defined as C=Eh/(1-2) and D=Eh3/12(1-2). E, G,  and  represent Young’s modulus, the shear modulus, Poisson’s ratio and the shear correction factor, respectively. When the stress resultants of Eqs. (2a-h) are substituted into the equations of motion (1a-e) we have

 2U U 1   2U  1  Gh  1   2V  cos       sin  U     2sin   2  tan 2  C  2   2

sin 

3  V   Gh  W  Gh h   1     R sin     2 R 2 sin  U,  sin  2 tan    C   C C

(3a)

1    2U 3  U 1    2V V     sin  2  cos   2  2 tan   2     

  Gh  1  2V 1   1   1   sin  V  2 2 sin   C   2  tan  

(3b)

 Gh  W  Gh h   1    R sin     2 R 2 sin  V,  C   C C  

 Gh   U V   1    cos  U    sin  C     2 1    sin  W 

 Gh

  Gh   2W W 1  2W   sin   cos      C   sin   2   2 

 Gh  C

   h 2 2 R  sin   cos    W,    R sin      C  

 W R sin   U  D  

 2  1   2   sin   cos     2sin   2  2 

 1  Gh 2  1   2  3    h3 2 2     R sin        R sin  ,  2 D 2  2 tan   12 D  tan  

W  1    2  3   1    2   R  sin  V     sin  2  cos    D   2  2 tan   2     

(3c)

(3d)

 Gh 

1  2  1    sin   2  2

 1   Gh 2   h3 2 2  1  R sin      R sin  .    2 D 12 D  tan   

Assume a function f(,) which is periodic in the circumferential direction

5

(3e)

f ,  



 f   cos n

n 0

(4)

n

and follow the term fn()cosn along a meridian over the pole. The polar projections of Fig. 2(a) shows that the component fn() does not change sign as the pole is crossed when n is even, while Fig. 2(b) indicates that fn() must always change sign as the pole is crossed when n is odd. Fig. 2(a) and Fig. 2(b) imply that fn() is an odd function when n is an odd number and that fn() is an even function when n is an even number.

fn() H

L

 -1 H

0

1

L

(a) n = 2

H L

L

H

H

fn()

 -1

0

1

L

(b) n = 3

Fig. 2 Polar projections in the vicinity of the pole showing the positive (H) and negative (L) regions for a term fn() cos n

The Chebyshev polynomials of the first kind are defined recursively as

T0  x   1,   T1  x   x,   Tk  x   2 xTk 1  x   Tk  2  x 

 1  x  1

(5)

 k  2

which makes the terms T0(x), T2(x), T4(x) ∙∙∙ T2m(x) be even functions of x, while T1(x), T3(x), T5(x) ∙∙∙

6

T2m-1(x) are odd functions of x. The colatitude  ranges from 0 to /2, and it is convenient to use the normalized form

2





 0,1 .

(6)

The displacements and the rotations are then approximated as follows: K 1

U  , n    akS Fk   cos n

(7a)

k 1

K 1

V  , n    bkS Fk   sin n

(7b)

k 1

K 1

W  , n    ckS Fk   cos n

(7c)

k 1

K 1

  , n    d kS Fk   cos n

(7d)

k 1

K 1

  , n    ekS Fk   sin n

(7e)

k 1

where aSk, bSk, cSk, dSk and eSk are the expansion coefficients. Superscript S in the expansion coefficients of Eqs. (7a-e) denotes the spherical shell. K is the number of collocation points in the colatitudinal direction. Fk() is the basis function selected to be an odd function for odd n, and an even function for even n. Such a basis function can be realized by use of Chebyshev polynomials as follows:

T    Fk     2 k  2  T2 k 1  

 n  0, 2, 4,   n  1, 3, 5, 

(8)

which makes the approximations of Eqs. (7a-e) Fourier-Chebyshev expansions. Expansions of Eqs. (7a-e) are substituted into Eqs. (3a-e), which are then collocated at the GaussLobatto collocation points

i  cos

  2i  1

 i  1, 2, , K 

4K

(9)

to yield the collocated equations as follows K 1

a k 1

S k

2  4sin i  2 cos i 1  Gh   1   n Fki   Fk i         sin i Fk i   2 2 2  C      2sin i tan i

K 1  3   n  Gh  2sin i  1    n  K 1    bkS  Fk i   Fk i     ckS 1    Fk i    2 tan  C   k 1 i   k 1  K 1

 Gh

k 1

C

  d kS

K 1

R sin i Fk i    2  akS k 1

R 2 sin i  h Fk i  C

7

(10a)

K 1  1    n  K 1  3   n  Gh    akS  Fk i   Fk i     ckS n 1     Fk i  2 tan i C      k 1 k 1 K 1  2 1   sin i 1   cos i    bkS  Fki   Fk i  2   k 1 

(10b)

  n   Gh  1   1  2   1   sin i Fk i    2 2  tan   C   sin i  2

K 1

 Gh

k 1

C

  ekS

K 1

R sin i Fk i    2  bkS k 1

R 2 sin i  h Fk i  C

K 1  Gh   2sin i  Gh   K 1    akS 1    Fk i   cos i Fk i     bkS n 1      Fk i  C   C     k 1 k 1 K 1  4 Gh sin i    Gh n 2  2 Gh cos i     ckS  F   F    2 1    sin i  Fk i        k i k i 2 C   k 1  C   C sin i  K 1

  d kS k 1

(10c)

 Gh  2sin i  Gh  R Fk i   cos i Fk i     ekS n RFk i  C  C   k 1 K 1

K 1

  2  ckS k 1

K 1

a k 1

S k

 Gh D

R 2 sin i  h Fk i  C K 1

R sin i Fk i    ckS k 1

2 Gh R sin i Fk i  D 

K 1  4sin i   1   n 2 2 cos i 1  Gh 2      d kS  F   F      R  sin i Fk i    k  i k  i 2 2 2  D k 1  2sin i tan     

(10d)

K 1 K 1  1    n   3   n R 2 sin i  h3   ekS  Fk i   Fk i     2  d kS Fk i  2 tan i 12 D k 1 k 1   

K 1

b k 1

S k

 Gh D

K 1

 Gh

k 1

D

R sin i Fk i    ckS

RnFk i 

K 1  K 1  2 1   sin i  3   n 1   cos i   1    n  d kS  Fk i   Fk i     ekS  Fki   Fk i  2 2 tan i     k 1  k 1 

(10e)

2 3 2 K 1    Gh 2  1   1  n 2 S R sin i  h  2   1  R sin  F     e Fk i ,        i k i k 2  tan 2 i D 12 D k 1  sin i   

 i  1,

2,  , K 

with i  i 2 . The notation ( ' ) stands for the differentiation with respect to . Eqs. (10a)-(10e) can be rearranged in the matrix form



 A1  n      A2  n      2 B1  n     B2  n    The vectors in Eq. (11) are defined as

8



(11)

   a1S

a2S  aKS b1S b2S  bKS c1S c2S  cKS d1S d 2S  d KS e1S e2S  eKS  , T

   aKS 1 bKS 1 cKS 1 d KS 1 eKS 1

T

(12)

.

Matrices [A1(n)], [A2(n)], [B1(n)] and [B2(n)] correspond to the circumferential wave number n.  in Eq. (11) denotes the positive circumferential direction even though it does not appear explicitly in matrices [A1(n)], [A2(n)], [B1(n)] and [B2(n)].



W V

U



R



x

L

Fig. 3 Displacements and rotations in cylindrical coordinate system 2.2 Cylindrical shell The equations of motion of a cylindrical shell as shown in Fig. 3 were derived by Soedel [3] as N x 1 N x    2  hU x R 

(13a)

N x 1 N Q     2  hV x R  R

(13b)

Qx 1 Q N     2  hW x R  R

(13c)

M x 1 M  x  h3   Qx   2  x R  12

(13d)

M x 1 M  h3 (13e)   Q   2  x R  12 with the effects of transverse shear and rotary inertia taken into account. U, V and W are the displacements in the axial, circumferential and normal directions, respectively, as shown in Fig. 3.  and  are the rotations in the axial and circumferential directions. The stress resultants Nx, N, Nx, Qx, Q, Mx, Mand Mx are defined as

U  1 V W   Nx  C       R  R    x

9

(14a)

U   1 V W N  C     R   R x   N x  N x 

(14b)

1   V 1 U  C   2  x R  

(14c)

     Mx  D    x R  

(14d)

   1  M  D    R   x  

(14e)

M x  M x 

1    1   D   2  x R  

(14f)

W   Qx   Gh     x  

(14g)

V 1 W   Q   Gh      R R   

(14h)

When the stress resultants of Eqs. (14a-h) are substituted into the equations of motion (13a-e) we have

  2U 1   2U 1   2V  W C 2    2 R x R x 2 R 2  2  x

 2     hU 

(15a)

 1   2U 1   2V 1  2V 1 W  1 W  V C   2  2    Gh   2  2 2 2 2 x R  R   R  R R  2 R x

 2     hU 

(15b)

 1 V  2W 1  2W  1     U 1 V W  2 C   2  2    Gh   2  2  2       hW (15c) R  2 x R    R x R  R   R  x   2  1   2  1    2   W  D 2      Gh    2 2 2 R  x   x  x 2 R     

3  2 h     12 

 1    2  1   2  1  2   V 1 W  D   2   Gh     2 2  2 x R R  R     2 R x

3  2 h      12 

(15d)

(15e)

assuming simple harmonic motions. The axial distance x ranges from –L/2 to L/2, where L is the length of the cylinder, and it is normalized as

  2 x L   1,1

(16)

The displacements and the rotation of the cylindrical shell are approximated using the Fourier-Chebyshev expansions as: P2

U  , n    aCpTp 1   cos n

(17a)

p 1

P2

V  , n    bCp Tp 1   sin n

(17b)

p 1

P2

W  , n    cCp Tp 1   cos n p 1

10

(17c)

P2

  , n    d Cp Tp 1   cos n

(17d)

p 1

P2

  , n    eCp Tp 1   sin n

(17e)

p 1

where aCp, bCp, cCp, dCp and eCp are the expansion coefficients. Superscript C in the expansion coefficients (17a-e) denotes the cylindrical shell. P is the number of collocation points in the axial direction. Expansions of (17a-e) are substituted into Eqs. (15a-e), which are then collocated at the Gauss-Lobatto collocation points   2i  1 (18) i   cos  i  1, 2, , P  2P to result in the collocated equations as follows P2

a p 1

C p

n 2 1   C  4C **  P  2 C n 1   C * T  Tp 1 i     2 Tp 1 i      bp p  1 i RL 2R2  L  p 1 P2 P2 2 C *   cCp Tp 1 i    2  a Cp  hTp 1 i  RL p 1 p 1

P2

n 1    C

p 1

RL

  a Cp P2

 c p 1

C p

P2 n 2 C   Gh  2 1   C **  Tp*1 i    bpC  Tp 1 i   Tp 1 i   2 2 L R p 1  

n  C   Gh  R2

P2

Tp 1 i    e p 1

P2

  a Cp p 1

C p

 Gh R

Tp 1 i   

P2

2

b p 1

C p

P2 n  C   Gh  2 C * Tp 1 i    bpC Tp 1 i  RL R2 p 1

K 2

  eCp p 1

P2 p 1

 e p 1

P2

b p 1

C p

 Gh R

C p

n 1   D

P2

Tp 1 i    cCp p 1

(19c)

P2 n Gh Tp 1 i    2  c Cp  hTp 1 i  R p 1

P2    n 2 1   D  2 Gh *  4D  Tp 1 i    d pC  2 Tp**1 i      Gh  Tp 1 i   2   L 2R p 1     L  K 2

(19b)

 hTp 1 i 

P2  4 Gh  P  2 C 2 Gh *  C n 2 Gh    cCp  2 Tp**1 i    2  Tp 1 i   Tp 1 i     d p 2 L R  p 1 R  L  p 1

  cCp

(19a)

R

P2

T

* p 1

i     d 2

p 1

C p

 h3 12

(19d)

Tp 1 i 

P2 n 1    D * n Gh Tp 1 i    d pC Tp 1 i  R RL p 1

3 P2  n2 D   2 1   D **  2 C h   eCp  T     Gh T     e Tp 1 i          p  1 i p  1 i p 2 12 L2 p 1 p 1  R    K 2

 i  1, 2,

(19e)

, P

The notation (*) in Eqs. (19a-e) stands for the differentiation with respect to  . Eqs. (19a-e) can be rearranged in the matrix form



 X1  n      X2  n      2 Y1  n     Y2  n   

11



(20)

The vectors in Eq. (20) are defined as

   a1C

a2C  aPC b1C b2C  bPC c1C c2C  cPC d1C d 2C  d PC e1C e2C  ePC  , T

   aPC1 aPC 2

bPC1 bPC 2 cPC1 cPC 2 d PC1 d PC 2 ePC1 ePC 2 

T

(21)

Matrices [X1(n)], [X2(n)], [Y1(n)] and [Y2(n)] correspond to the circumferential wave number n.

(W)1

1





(V)1 (U)1 

1



1 = -2 2



(V)2 

(U)2 (W)2

Fig. 4 Displacements and rotations in the subdomains 1 and 2 of a full sphere

3. Joined shell structures 3.1 Full sphere A full sphere can be constructed by assembling two hemispheres as shown in Fig. 4. Superscripts 1 and 2 denote each subdomain. It is worthwhile to note that the circumferential directions of subdomains are in the opposed direction, i.e.,   1  2 as shown in Fig. 4 and the displacements and the rotations of subdomain 2 are then approximated as follows:

U 

2

K 1

   akS 

2

k 1

V 

2

K 1

   bkS 

2

2

2

K 1

   ckS  K 1

2

   d kS 

2

k 1



2

K 1

   ekS  k 1

Fk   cos n

(22a)

K 1

Fk   sin  n     bkS 

Fk   sin n

(22b)

Fk   cos n

(22c)

Fk   cos n

(22d)

Fk   sin n

(22e)

2

k 1

k 1



2

k 1

k 1

W 

K 1

Fk   cos  n     akS 

2

K 1

Fk   cos  n     ckS 

2

k 1

K 1

Fk   cos  n     d kS 

2

k 1

K 1

Fk   sin  n     ekS  k 1

12

2

Eq. (11) is applied to each hemisphere and the equations of motion are written as 1 1  A1  n  0      A 2  n  0            A1  n    2   0 A 2  n    2   0    

(23)

1 1  B  n  0     B 2  n  0       2   1        0 B1  n    2   0 B 2  n    2       

Submatrices [A1(-n)], [A2(-n)], [B1(-n)] and [B2(-n)] in Eq. (23) are formed using the expansions of Eqs. (22a-e) instead of Eqs. (7a-e). The total number of equations in Eq. (23) is 10K whereas the total number of expansion coefficients is 10K+10. The remaining ten equations are obtained from the continuity conditions between the hemispherical shells. The continuity conditions of displacements and rotations and stress resultants at 1= 1∩2 are dictated by

U  1   U  1 , V  1   V  1 , W  1  W  1 , 1

2

1

2

1

2

   1      1 ,    1      1 ,  N  n   1   N  n   1 , 1

2

 N  n   

1

  N  n   1 ,

1

 M  n   

1

 1

Q  n  

2

 1

1

2



1

2

(24a-j)

   Q  n   1 , 2

 1

  M   n   1 ,  M   n   1   M   n   1 2

1

2

Using the expansions of Eqs. (7a-e) and Eqs. (22a-e) and the relationships of

Tk 1 1  1, Tk1 1   k  1

2

(25)

the continuity conditions (24a-j) can be written as

 a  K 1

S k

k 1

1

 c  K 1 k 1

S 1 k

  k  1  a  K 1

2

k 1

  akS 

  ckS 

S 1 k

K 1

S 1 k

k 1

K 1 k 1

S 1 k

2

  akS 

2

  0,

2

 n  d  k 1

S 1 k

S 1 k

k 1

K 1

2

S 1 k

  nb 

S 1 k

  akS 

2

2

S 1 k

  d kS 

  bkS 

2

S 1 k

2

  ckS 

2

2

2

0

  0 , e  K 1 k 1

S 1 k

  bkS 

2

S 1 k

  ekS 

   k 1 e 

  ekS 

S 1 k

2

13

S 1 k

S 1 k

  ekS 

  ckS 

2

2

2

  d kS 

2

  0 (26c-e)

  0

  0

  R d 

  ne 

2

(26a, b)

  1  c 

S 1 k

2

  d kS 

  d kS 

  bkS 

   k 1 b 

   k 1 c 

k 1

K 1

K 1

  0 ,  d 

  k  1  d  K 1

 b  k 1

  akS 

 n  a    a 

2

(26f)

(26g) 2

  0

(26h)

  0

(26i)

  0

(26j)

The continuity conditions of (26a-j) can be rearranged together in a matrix form 1 1           Z1   2    Z2   2   0          

(27)

The sizes of matrices [Z1] and [Z2] are 10×10K and 10×10, respectively, and the vector {(aSK+1)1 (bSK+1)1 (cSK+1)1 (dSK+1)1 (eSK+1)1 (aSK+1)2 (bSK+1)2 (cSK+1)2 (dSK+1)2 (eSK+1)}T can be computed by 1 1   1            Z 2   Z1    2  2           

(28)

which is substituted into Eq. (23) to produce    A1  n     1  0   A 2  n  0  1      Z   Z1    2  A1  n    0 A 2  n  2  0       B1  n     1  0  B 2  n  0  1 2      Z   Z1    2   B1  n    0 B 2  n   2  0    

(29)

To investigate the accuracy and the convergence property of the present method the natural frequencies of complete spherical shells are computed for different K and the results are given in the dimensionless form  R  E in Table 1. The material properties are E=206.85 GPa, =0.3 and =7287 kg/m3, and the thickness-to-radius ratios h/R of the spherical shells are 0.01 and 0.05, respectively. In Table 1 the computational results of Chao and Chern [20] based on the Ritz method are given for comparison. Chao and Chern [20] found that their Ritz solution coincided with the exact solution. Both the work of Chao and Chern [20] and the present study are based on the Reissner shell theory, and Table 1 shows that the natural frequencies are slightly higher for the thicker spherical shell. The natural frequencies converge to four significant figures for K=30, and the converged natural frequencies are practically identical to those based on the Ritz method, which demonstrates the rapid convergence and good accuracy of the present method. The numbers of collocation points K=40 and P=40 are employed in the additional computations.

3.2 Bell structure A bell-like structure can be constructed by assembling a hemispherical shell and a cylindrical shell as shown in Fig. 5. Eq. (11) and Eq. (20) are applied to the hemispherical and the cylindrical section, respectively. The equations of motion are written together as

14

1 1  A1  n  0      A 2  n  0           X1  n    3   0 X 2  n    3   0    

(30)

1 1  B  n  0     B 2  n  0       2   1        0 Y1  n    3   0 Y2  n    3       

Table 1 Convergence test: nondimensional natural frequencies  R  E of full spherical shells (E=206.85 GPa, = 7287 kg/m3, = 0.3, =5/6) Ritz method

present method h/R

0.01

0.05

m

[20]

K=5

K=10

K=20

K=30

K=40

1

0.72509

0.73432

0.73484

0.73485

0.73485

0.73486

2

0.83627

0.86901

0.87043

0.87046

0.87046

0.87047

3

0.87995

0.92173

0.92451

0.92453

0.92453

0.92454

4

0.92881

0.94844

0.95269

0.95270

0.95269

0.95269

1

0.73407

0.73646

0.73644

0.73644

0.73644

0.73661

2

0.87840

0.88101

0.88080

0.88078

0.88078

0.88110

3

0.96481

0.95907

0.95806

0.95798

0.95797

0.95841

4

1.07840

1.03490

1.03180

1.03160

1.03160

1.03230

 

(V) 

(W)

(W) (U) 

(V) (U)



2



3

Fig. 5 Displacements and rotations in the subdomains 1 and 3 of a bell

15

Superscripts 1 and 3 denote the hemispherical and cylindrical subdomains. The continuity conditions of displacements, rotations and stress resultants at 2= 1∩3 are

U  1  U  1 , V  1  V  1 , W  1  W  1 , 1

3

1

3

1

   1     1 ,

3

1

3

   1     1 ,  N  1   N x  1 ,  N  1   N x  1 , 1

M 

1

3

1

  1

1

3

3 3   M x   ,  M   1   M x   , 1 1

1





3

Q 

1

  1

(31a-j)

3   Qx   1



The total number of equations in Eq. (30) is 5K+5P whereas the total number of expansion coefficients is 5K+5P+15. The continuity conditions of Eqs. (31a-j) provide ten additional equations. The remaining five equations are obtained from the boundary conditions of the cylindrical shell. For free boundary conditions

 N x  1  0,  N x  1  0,  M x  1  0,  M x  1  0, Qx  1  0 3

3

3

3

3

(32a-e)

The continuity conditions (31a-j) can be written as K 1

P2

a 

   aCp 

 b 

K 1

   bCp 

 c 

K 1

   cCp 

d 

K 1

   d pC 

K 1

   eCp 

k 1

k 1

k 1

k 1

S 1 k

S 1 k

S 1 k

S 1 k

 e  k 1

S 1 k

3

 1

3

3

p 1

0

(33a)

 1

p 1

0

(33b)

 1

p 1

0

(33c)

0

(33d)

0

(33e)

p 1

P2 p 1

P2 p 1

P2

3

 1

p 1

p 1

P2

3

 1

p 1

p 1

 2  k  12 S 1 1 1   ak   n  bkS   1     ckS       k 1    K 1

 2 R  1 p  p  12 C 3 3 3  p 1 p 1    a p   n  1  bCp     1  cCp    0  L p 1   

(33f)

P2

2 p 2   P2  2  k  1 2 R  1  p  1 3  p 1    S 1 S 1  C 3  n a  b   n  1 a  bCp    0     k     p   k    L k 1      p 1   

(33g)

 2  k  12 S 1  P2   2 R  1 p  p  12 C 3  3  p 1  S 1  d  n  e  d p   n  1  eCp    0           k k L k 1   p 1       

(33h)

K 1

K 1

16

2 p 2  2  k  1 S 1  2 R  1  p  1 C 3  p 1   P2    S 1 C 3 ek     n  1  d p   ep    0   n  d k     L k 1      p 1    K 1

(33i)

2 p 2   P2  2  k  1 3  p 1   2 R  1  p  1 C 3  S 1 S 1 S 1   a  c  R d  c p   R  1  d Cp    0 (33j)      k    k  k    L k 1   p 1       K 1

using the expansions of Eqs. (7a-e), Eqs. (17a-e) and the relationships

Tp 1  1   1

p 1

, Tp*1  1   1

p

 p  1

2

(34)

The boundary conditions (32a-e) are expressed by 2  3 3 3   2 R  p  1  aCp   n  bpC    cCp    0    L p 1     P2

(35a)

2  2 R  p  1 3    C 3 bCp    0 ,  n  a p    L p 1    

2  3 3   2 R  p  1  d Cp   n  eCp    0    L p 1    

(35b, c)

2  2 R  p  1 C 3   C 3  n d     ep    0 ,   p L p 1    

2  3   2  p  1 C 3  c p    d Cp    0    L p 1    

(35d, e)

P2

P2

P2

P2

The continuity conditions of (33a-j) and the boundary conditions (35a-e) can be rewritten together in a matrix form 1 1           Z3   3    Z4   3   0          

(36)

The sizes of matrices [Z3] and [Z4] are 15×(5K+5P) and 15×15, respectively, and the vector {(aSK+1)1 (bSK+1)1 (cSK+1)1 (dSK+1)1 (eSK+1)1 (aCP+1) (aCP+2) (bCP+1) (bCP+2) (cCP+1) (cCP+2) (dCP+1) (dCP+2) (eCP+1)(eCP+2)}T can be computed by 1 1   1          3     Z 4   Z3   3               

which is substituted into Eq. (30) to produce

17

(37)

   A1  n     1  0   A 2  n  0  1  Z Z           3  3  X1  n    0 X 2  n  4  0       B1  n     1  0  B 2  n  0  1 2      Z   Z3    3   Y1  n    0 Y2  n  4  0    

(38)

The solution of Eq. (38) yields the estimate for the natural frequencies and corresponding expansion coefficients. The lowest two natural frequencies of a bell structure, of which the geometric constants and the material properties are R=0.175 m, L=0.7 m, h=0.0022 m, E=210 GPa, =0.3, =5/6 and =7850 kg/m3, are computed for different zonal number n using Eq. (38) and are given in Table 2. In Table 2 the computational results of FEM (ANSYS) along with the experimental measurements of Lee et al. [10] are given for comparison. FEM results are based on a model that consists of 2542 eight-node structural shell elements. The solutions of the present method are in good agreements with FEM and the experimental measurements.

Table 2 Natural frequencies of a bell structure with free boundary conditions in Hz (R=0.175 m, L=0.7 m, h=0.0022 m, E=210 GPa, = 7850 kg/m3, = 0.3, =5/6, K=40, P=40, FEM: 2542 eight-node structural shell elements) n

present method

FEM (ANSYS)

0

1961 2727

1959 2726

1

1780 2678

1781 2678

2

49.35 684.2

48.93 684.1

51.9

3

137.2 407.9

137.0 407.7

146.1 406.9

4

261.6 363.5

261.4 363.6

278.1 378.3

18

experiment [10]

 

(W)

(V)

  (U) (W)

(U) 

 (V) (U)

(W)

(V) 







2

3

Fig. 6 Displacements and rotations in the subdomains 1, 2 and 3 of a hermetic capsule

3.3 Hermetic capsule Hermetic capsule can be constructed by assembling two hemispheres and a cylinder as shown in Fig. 6. Eq. (11) and Eq. (20) are applied to the hemispherical and cylindrical sections, respectively. The equations of motion are written together as 1 1 0 0 0 0  A1  n       A 2  n      3     3    X1  n  0 X 2  n  0  0       0      0 0 A1  n    2   0 0 A 2  n   2      1 1   0 0 0 0     B 2  n        B1  n  3     3      2   0 Y1  n  0 Y2  n  0       0              0 0 B1  n     2 0 B 2  n     2    0       

(39)

The total number of equations in Eq. (39) is 10K+5P whereas the total number of expansion coefficients is 10K+5P+20. The continuity conditions of Eq. (31a-j) provide ten additional equations. The remaining ten equations are obtained from the continuity conditions at 3= 3∩2:

U  1   U  1 , V  1   V  1 , W  1  W  1 , 3

2

3

2

3

2

   1      1 ,    1      1 ,  N x  n   1   N  n   1 , 3

2

 N   n    M  n  

1

3

x

1

  N  n   1 ,

2

3

2

2

3

x

3

 Q  n  

3

x

1

   Q  n   1 , 2

  M   n   1 ,  M x  n   1   M   n   1 2

3

The continuity conditions (40a-j) can be written as

19

2

(40a-j)

P2

a  p 1

C 3 p

K 1

   akS 

2

 0,

k 1

P2

 b  p 1

P2

d  p 1

C 3 p

C 3 p

K 1

K 1

   bkS 

   d kS 

2

0,

k 1

2

 0,

k 1

P2

 c 

C 3 p

p 1

P2

 e  p 1

C 3 p

K 1

K 1

   ckS 

   ekS 

2

0

(41a-c)

k 1

2

0

(41d-e)

k 1

 2 R  p  12 C 3 3 3   a p    n  bCp     c Cp      L p 1    P2

 2  k  12 S 2 2 2      ak    n  bkS   1     ckS    0   k 1   

(41f)

K 1

2 2   K 1  2 R  p  1 2  k  1 2     C 3 C 3  S 2  n a  b  n a  bkS    0     p    k   p  L  p 1   k 1      

(41g)

2 2   K 1  2  k  1 2 2   2 R  p  1 C 3   C 3  S 2 c p   R  d p       ak   ckS   R  d kS    0     L  p 1      k 1   

(41h)

P2

P2

2 2   K 1  2 2   2 R  p  1  2  k  1  C 3 C 3  d   n e   d kS   n  ekS    0        p  p  L  p 1   k 1      

(41i)

2 2  2 R  p  1 C 3  2  k  1 S 2    K 1    C 3 S 2 e p     n  d k   ek    0   n  d p    L  p 1      k 1   

(41j)

P2

P2

The continuity conditions of Eq. (33a-j) as well as Eq. (41a-j) can be written together in a matrix form

 1   1        Z5    3    Z6   3   0        2  2      

(42)

The sizes of matrices [Z5] and [Z6] are 20×(10K+5P) and 20×20, respectively, and Eq. (39) can be rewritten as 1   A1  n      0 0 0 0   A 2  n       3  1    X1  n  0  0 X n  0 Z Z       2 6 5  0        2   0     0 A  n  0 0 A  n      1 2         1  B1  n      0 0 0 0  B 2  n       3  1     2   0 Y1  n  0  0 Y n  0 Z Z        2 5        6  2   0     0 B1  n    0 0 B 2  n       

(43)

The natural frequencies of a hermetic capsule are computed and three lowest natural frequencies for

20

different circumferential wave number n using Eq. (43) are given in Table 3. The geometric constants and the material properties are R=0.1143 m, L=0.343 m, h=0.00203 m, E=207 GPa, =0.3, =5/6 and =7800 kg/m3, respectively. In Table 3 the computational results of FEM (ANSYS) along with the solutions of Tavakoli and Singh [6], Shang [8], Buchanan [9] and Wu et al. [15] are given for comparison. FEM results are based on a model that consists of 1949 eight-node structural shell elements. The results of Shang [8] and Buchanan [9] were based on the axisymmetric assumptions. The solutions of the present method are in excellent agreements with FEM and existing solutions. It is readily seen in Table 3 that the fundamental natural frequency occurs for n=3 and that the natural frequencies for n=2, 3, and 4 are lower than those of axisymmetric modes. Also a parametric study is carried out to investigate the influence of the length-to-radius ratio L/R on the natural frequencies of the hermetic shell. Fig. 7 shows the changes of the nondimensional natural frequencies    R  1  2  E as L/R varies from 2.25 to 4.5. The thickness-to-radius ratio h/R is 0.01. Fig. 7 supplements the work of Wu et al. [15], where L/R ranged from 5 to 40. In Fig. 7(a) the decrease of the lowest natural frequencies is more obvious for n=0 and n=1 than the other cases. In Fig. 7(b) the constant decreases of the second lowest natural frequencies are observed in all six cases.

Table 3 Natural frequencies of hermetic capsule in Hz (R=0.1143 m, L=0.343 m, h=0.00203 m, E=207 GPa, = 7800 kg/m3, = 0.3, =5/6, K=40, P=40) n

0

1

2

3

4

5

m

present method

FEM (ANSYS)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

3219 4005 5536 2740 4200 5109 854 2471 3946 602 1727 2996 674 1351 2343 955 1374 2061

3205 4003 5534 2740 4199 5087 854 2471 3947 601 1728 3001 675 1350 2349 964 1330 2045

space state method [6]

axisymmetric analytical [8]

axisymmetric FEM [9]

4011 5539 2740 4198 5086 854 2471 3946 601 1732 3000

4002.8 5533.5

4012 5542

21

variational method[15] 3205 4014 5562 2750 4210 5099 873 2500 3975

Fig. 7 Nondimensional natural frequencies    R  1  2  E (E=207 GPa, =7800 kg/m3, = 0.3, h/R= 0.01, K=40, P=40)

4. CONCLUSIONS A Fourier-Chebyshev expansion method is applied to the joined spherical-cylindrical shell structures to study their dynamic characteristics. The equations of motion of the joined shell structures take the effects of transverse shear and rotary inertia into account. The displacements and the rotations of the spherical and cylindrical shells are represented by Chebyshev polynomials in the colatitudinal and axial directions and Fourier functions in the circumferential direction. As for the hemispherical shells odd functions and even functions of Chebyshev polynomials are used alternatively as basis functions according to the circumferential wave number to incorporate the symmetry and the antisymmetry of the solutions. When two hemispheres are involved in a joined shell structure and their open ends face each other they have opposed circumferential directions and it is important to take extra care to reflect 1=-2 in the governing equations and in the continuity conditions. The equations of motion are collocated to yield the system of equations that correspond to the circumferential wave number. To handle the continuity conditions Fourier-Chebyshev expansions are matched at the junctions of the adjoining shells. The number of collocation points is chosen to be less than the number of expansion terms. The continuity conditions and boundary conditions are used as the side constraints of the expansions, and the set of algebraic equations is condensed so that the number of expansions matches the number of degrees of

22

freedom of the problem. Numerical examples are provided for full sphere, bell structure and hermetic capsule. The results are in good agreements with those of existing literatures, which suggests that the present method may be utilized in the analyses of shell structures where different types of shells are joined together.

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Acknowledgements This work was supported by 2016 Hongik University Research fund.

REFERENCES [1] Leissa AW. Vibration of Shells. Acoust Soc Am; 1993. ISBN-13: 978-1563962936. [2] Blevins RD. Formulas for Natural Frequencies and Mode Shapes. Krieger Pub; 2001. ISBN-13: 9781575241845. [3] Soedel W. Vibrations of Shells and Plates CRC Press; 2004. ISBN-13: 978-0824756291. [4] Kraus H. Thin Elastic Shells. Wiley and Sons; 1967. ISBN-13:978-0471507208. [5] Galletly GD, Mistry J. The free vibrations of cylindrical shells with various end closures. Nucl Eng Des 1974;30(2):249-68. http://doi.org/10.1016/0029-5493(74)90170-8. [6] Tavakoli MS, Singh R. Eigensolutions of joined/hermetic shell structures using the state space method. J Sound Vib 1989;130(1):97-123. http://doi.org/10.1016/0022-460X(89)90522-1. [7] Wong SK, Sze KY. Application of matched asymptotic expansions to the free vibration of a hermetic shell. J Sound Vib 1998;209(4):593-607. http://doi.org/10.1006/jsvi.1997.1200. [8] Shang X. Exact solution for free vibration of a hermetic capsule. Mech Res Commun 2001;28(3):2838. http://doi.org/10.1016/S0093-6413(01)00175-6. [9] Buchanan GR. An analysis of the free vibration of the free vibration of a hermetic capsule. J Sound Vib 2003;259(2):490-6. http://doi.org/10.1006/jsvi.2002.5171. [10] Lee YS, Yang MS, Kim HS, Kim JH. A study of the free vibration of the joined cylindrical-spherical shell structures. Comput Struct 2002; 80:2405-14. http://doi.org/10.1016/S0045-7949(02)00243-2. [11] Redekop D. Vibration analysis of a torus-cylinder shell assembly. J Sound Vib 2004;277(4-5):919-30. http://doi.org/10.1016/j.jsv.2003.09.034. [12] Ma X, Jin G, Xiong Y, Liu Z. Free and forced vibration analysis of coupled conical-cylindrical shells with arbitrary boundary conditions. Int J Mech Sci 2014; 88:122-137. http://doi.org/10.1016/j.ijmecsci.2014.08.002. [13] Kang JH. Free vibrations of combined hemispherical-cylindrical shells of revolution with a top opening. Int J Struct Stab Dy 2014;14(1):1-14. http://doi.org/10.1142/S0219455413500235. [14] Qu Y, Wu S, Chen Y, Hua H. Vibration analysis of ring-stiffened conical-cylindrical-spherical shells based on a modified variational approach. Int J Mech Sci 2013; 69:72-84.

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http://doi.org/10.1016/j.ijmecsci.2013.01.026. [15] Wu S, Qu Y, Hua H. Vibration characteristics of a spherical-cylindrical-spherical shell by a domain decomposition method. Mech Res Commun 2013;49:17-26. http://doi.org/10.1016/j.mechrescom.2013.01.002. [16] Wu S, Qu Y, Hua H. Vibrations of joined cylindrical-spherical shell with elastic-support boundary conditions. J Mech Sci Technol 2013; 27(5):1265-1272. http://doi.org/10.1007/s12206-013-0207-7. [17] Lee J. Free vibration analysis of a hermetic capsule by pseudospectral method. J Mech Sci Technol 2012; 26(4):1011-5. http://doi.org/10.1007/s12206-012-0216-y. [18] Lee J. Application of Chebyshev-tau method to the free vibration analysis of stepped beams. Int J Mech Sci 2015;101-2:411-20. http://doi.org/10.1016/j.ijmecsci.2015.08.012. [19] Lee J. Free vibration analysis of spherical caps by the pseudospectral method. J Mech Sci Technol 2009; 23:221-8. http://doi.org/10.1007/s12206-008-0906-7. [20] Chao CC, Chern YC. Axisymmetric free vibration of orthotropic complete spherical shells. J Compos Mater 1988; 22:1116-1130. http://doi.org/10.1177/002199838802201203.

Highlights ∙ Free vibration analysis of joined spherical-cylindrical shells is presented. ∙ Deflections and rotations are represented by Fourier-Chebyshev expansions. ∙ Expansions are matched at the junctions to satisfy continuity conditions. ∙ Collocation method is applied to produce the system of equations. ∙ Numerical examples include full sphere, bell-structure and hermetic capsule.

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