Free vibration analysis of laminated composite cylindrical shells by DQM

PII: S1359-8368(96)00052-2

ELSEVIER

Composites Part B 28B (1997) 267-274 © 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/97/$17.00

Free vibration analysis of laminated composite cylindrical shells by DQM C. S h u a and H. Du b

aDepartment of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 bSchool of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263 (Received 3 May 1995; accepted 20 June 1996) In this paper, free vibrations of laminated composite cylindrical shells are investigated by the global method of generalized differential quadrature (GDQ). The GDQ method was developed to improve the differential quadrature (DQ) method for the computation of weighting coefficients. The differential equations of motion are formulated using Love's first approximation classical shell theory. The spatial derivatives in both the governing equations and the boundary conditions are discretized by the GDQ method. The GDQ method is examined by comparing its results with those available in the literature. It is demonstrated that, with the use of the GDQ method, the natural frequencies can be easily and accurately obtained by using a considerably small number of grid points. © 1997 Elsevier Science Limited (Keywords: A. layered structures; B. vibration; global method; generalized differential quadrature; cylindrical shells)

INTRODUCTION The study of laminated composite structures has been of considerable interest over the past few decades. Laminated composite structures can enhance certain characteristics of the structure by proper arrangement of the stacking sequence of the layers. This advantage is usually not found in other types of structure. In addition, laminated structures also offer the advantage of combining the best aspects of the constituent layers to achieve better or suitable properties for the whole structure by proper selection of materials for the layers. Hence, the study of laminated composite structures is very important for practical applications. Currently, there are many computational methods available for the free vibration analysis of laminated composite cylindrical shells. These methods can be classified as analytical methods and numerical methods. Among the analytical methods, the method of closed-form solution is used extensively t 3. The finite element method 4'5 is a numerical method. Generally, the analytical methods require less computational effort than the finite element methods. However, they may not be as convenient as numerical methods in cases of application to complicated problems. The numerical methods discretize the partial differential equations at each interior point and then reduce them into a set of algebraic equations. The eigenvalues of the resultant system of algebraic equations provide vibrational frequencies of the problem. Usually, the finite element method requires a lot of virtual storage and computational effort to obtain accurate numerical solutions.

As we know, the finite element method is a low-order method. To obtain accurate numerical results, one needs to use a small mesh size to minimize the truncated error. As a result, a large number of grid points should be used to keep the small mesh size. Usually, the number of interior grid points equals the dimension of the resultant algebraic equation system, giving the same number of eigenfrequencies. For this case, the computed eigenfrequencies represent numerical solutions. It is well known that the accuracy of numerical solutions depends on the order of the finite element method used. Thus, all the computed eigenfrequencies have the same order of accuracy. Among all the computed eigenfrequencies, only low frequencies are of practical interest. However, since all the computed eigenfrequencies have the same order of accuracy, one still needs to use a large number of grid points to obtain the better accuracy of such low frequencies. As a result, it requires a lot of virtual storage and computational effort. As will be shown in this paper, the global method of generalized differential quadrature (GDQ) offers a promising way to obtain the accurate low eigenfrequencies of laminated composite cylindrical shells by using just a few grid points. The GDQ method, which is a type of numerical method, was developed by Shu e t al. 6 8 to improve the differential quadrature (DQ) method 9 for the computation of weighting coefficients. Both the GDQ and DQ approximate a spatial derivative of a function with respect to a coordinate at a discrete point as a weighted linear sum of all the functional values in the whole computational domain. The advantage of GDQ over DQ is its ease for the computation of weighting coefficients. In DQ, the weighting coefficients

267

Vibration analysis by GDQ: C. Shu and H. Du are usually obtained by solving an ill-conditioned algebraic equation system 9-13. In GDQ, the weighting coefficients of the first-order derivative are computed by a simple algebraic formulation, and the weighting coefficients of the secondand higher-order derivatives are given by a recurrence relationship. For details of GDQ method, the reader is referred to 6. The GDQ method has been successfully applied to simulate some fluid flow problems 6-8 and the structural dynamics of beams and plates 14'15. Currently, there is no work in the literature to show application of the DQ-type method to laminated composite cylindrical shell problems. The present paper will show the detailed implementation of the GDQ method in the free vibration analysis of laminated composite cylindrical shells having various boundary conditions.

found that all methods for the determination of weighting coefficients can be generalized under above two analyses 6. In GDQ, two sets of base polynomials: i.e. Lagrangian interpolation polynomials and x k, k = 1, 2 .... A t - 1, are used to determine the weighting coefficients. As a result, the weighting coefficients of the first-order derivative are computed by a simple algebraic formulation without any restriction on the choice of grid points while the weighting coefficients of the second- and higher-order derivatives are given by a recurrence relationship (for details, see 6). Some basic results of the one-dimensional case are described as follows. For a smooth function fix,t), GDQ discretizes its nth order derivative with respect to x at the grid point xi, as N

f(xn)(xi't) : Z Cik (n)"f(x~,t),

for i = 1,2 ..... N;

k=l

GENERALIZED DIFFERENTIAL QUADRATURE (GDQ)

n = 1,2 ..... N - 1

The GDQ approach was developed by Shu et al. 6-8 to improve the differential quadrature (DQ) technique 9 for the computation of weighting coefficients. DQ approximates the first-order derivative of a function at a grid point as a weighted linear sum of all the functional values in the whole domain. This can be demonstrated by the following example. For the one-dimensional unsteady problem, DQ approximates the first-order derivative of a smooth function fix,t) with respect to x at xi as: y. k=l

(1) "f (xk, t), cik

for i = 1,2, ..., N

Weighting coefficients for the first-order derivative: i,j= 1,2 ..... N, b u t j #= i (3)

C(1)__ A(1)(xi) ij - - ( x i _ x j ) . A ( 1 ) ( x j ) ,

c(i)ii ---- -

~2- cij _0) ,

i=1,2,

"" .,N

(4)

j = 1,j4:i

(1)

where N is the number of grid points, xi is the coordinate of the grid point, and c12)- is the weighting coefficient. Obviously, the key procedure in this approach is to determine the weighting coefficient C~(1) ik • Bellman et al. 9 suggested two methods to carry this out. One is to solve an algebraic equation system, which is ill-conditioned. The other is to use an algebraic formulation but with a condition that the coordinates of grid points should be chosen as the roots of the Legendre polynomial. The former method is usually used since it lets the coordinates of grid points be chosen arbitrarily. However, because of the drawbacks described above, the number of grid points used should be small. To overcome the drawbacks of DQ, GDQ was developed where the weighting coefficients are calculated by a simple algebraic formulation or by a recurrence relationship without any restriction on the choice of grid points. GDQ introduces a generalized way to compute the weighting coefficients of the DQ approximation under the analysis of a high-order polynomial approximation and the analysis of a linear vector space. First, it is supposed that, for a smooth problem, the solution of a partial differential equation in a domain is approximated by a high-order polynomial. Then, it can be shown that this high-order polynomial constitutes a linear vector space. Therefore many properties of a linear vector space, such as linear independence of base polynomials and the existence of several sets of base polynomials, can be used to determine the weighting coefficients in the DQ approximation. It was

268

where N is the number of grid points in the x direction and c(n) ik are the weighting coefficients to be determined as follows.

where

N j~xl)(xi, t)=

(2)

N A(1)(xi)

=

H (xi-xj) j = 1,j:/:i

Weighting coefficients for the second- and higher-order derivatives: _(n- O\ ~(n)__ (n-l) ~0) ~J ~, for i,j= 1,2, .. .,N, c ij - - •" C ii "c ij X i -- XjJ

b u t j :~ i , n = 2,3 ..... N - 1

(5)

N c(n) __ ii - - - -

E

_(n), cij

for i---- 1,2,

"" .,N,

j = 1,j~=i

n = 2 , 3 ..... N - 1

(6)

It is obvious from the above equations that the weighting coefficients of the second- and higher-order derivatives can be determined completely from those of the first-order derivative. It should be indicated that, by using the Lagrangian interpolation polynomials and the general collocation method, Quan and Chang 16'17 independently obtained the same formulation as eqn (3) for computing the weighting coefficients of the first-order derivative.

GOVERNING EQUATIONS Consider a cylindrical shell as shown in Figure 1. The mean radius, thickness, and length are denoted by R, h and L,

Vibration analysis by GDQ: C. Shu and H. Du where

e= k=- {

N = (Nx,No,N~o)T

(12)

M= (Mx, Mo, Mxo)T

(13)

{Ou ll'Ov )(Ov lOu~} T O--£'-R~+w , ~x + RoOj

02W 1 (

02w

OX2' R 2

Or) 2 (

02w

-- - ~ - ' } - "~ ,

(14)

Or)} T

-- ~ - 1 -

(15)

Aij, Bij and Dij are the extensional, coupling and bending stiffnesses. For a shell that is composed of different layers of orthotropic materials, these stiffnesses can be written as:

Figure 1 Geometryof a laminated cylindrical shell

&

h2

hl

3f

Aij= Z O,j(h -k -

h4

,)

(16)

1 N, QiJ(hk-h~-l)

(17)

k=l

w,-;l

k=l

Nt

Dij=

1 ~" 7.k ih3 ~k~=l(,e~ij(, k - h 2 _ , )

(18)

where Nt denotes the number of layers in the laminated shell, ht and hk-i denote the distances from the shell reference surface to the outer and inner_,,surfaces of the kth layer, as indicated in Figure 2, and Q~j are elastic stiffness constants of the kth layer. By using eqn (11), eqns (7)-(9) can be reduced to:

\

02U

(19a)

Lll u -+-L12v -1- LI3W = oh Figure 2 Cross-sectionalview of a laminated cylindrical shell

respectively. The orthogonal coordinate system (x,O,z) is taken to be at the middle surface. The deformations of the shell in the x, 0 and z directions are defined by u, v and w. The equilibrium equations of motion for a thin cylindrical shell can be written as:

ONx 10Nxo c)2u Ox ~-R O0 = Pt Ot2

2

cgX2 1- R

02Mxo 1 oeMo OOOx t- R2 O02

No R=

oh

02V

(19b)

02w L3]u + L32v + L33w----oh Ot2

(19C)

where

(7)

02

A66 02

Lll =All ~x2 + R~ O0-5

ONxo 10No 10Mxo+ 10M 0 02v a~-+ ~ - - t - R~x-x R2 O0 =Or at 2 02Mx

L21 u -}- LzzV q- L23w =

L12 =

(9)

AI2 0 L13 -- R Ox

02w Pt at 2

(AI2+A66) 02

(8)

R

OxO0q--

03 Bll Ox3

(BI2+2B66) 02

OxO0

R2

(B12 -{'-2B66 )

03

R2

OxO02

where Pt =

~h/?td2p dz

(10)

in which P and at are the density and mass per unit area, respectively. The force and moment resultants can be written aslS: [NI = [;

;].[:]

(11)

L21 =

(AI2+A66) o32

L22 =

R A

(B12+2B66) 02

OxO0

R2

OxO0

3B66 2D66"~ 02 66 + ~--qt_ ~ - / CgX2

q_ (~222 q_ 2B22 ~22) c32

-Rr-+

269

Vibration analysis by GDQ: C. Shu and H. Du

/--23 =

-}-

R3/] ~ -

It can be seen from eqns (23) and (24) that there are two boundary conditions for W and one boundary condition for U and V at each end.

~-'-I- R41100 ~

((DI2 + 2 D 6 6 ) ( B 1 2 ~2B66).)03 R2

OX200

F

IMPLEMENTATION OF THE GDQ METHOD AI2 0

03 (B12 q- 2B66) O3 R Ox t-Bll 0~5+ R2 OxO02

L31 --

(-~22 L32= -Jr-

B22"~ 0 + g 3) ~+

In the following, the GDQ method is applied to discretize the derivatives in the governing equations [eqns (21a), (21b) and (21c)] and the boundary conditions [eqns (23) and (24)]. After spatial discretization, eqns (21a), (21b) and (21c) become N N Sill Ui .-Jr- ~ . Jll2t.i, ~ ~(2) rr "Jr- Z ~J121Vi, ktak ~(1)I" k Vk k=l k=l

(~_~232 D22"~ 03 + e 4 ; 003

( -~(B12 ( D 1 2 + 4 D 6 6 ) ~ 03 ~2B66) -IR2 ; OX200 A22

L33

R2

2B12 02 F R

2B22 02

Ox2 q'- R 3 002

2(D12 "F 2D66) 04 R2 Ox2002

04 Dliox-4

N

(25a)

D22 04

k=l

R 4 004

A general relation for the displacement fields of a thin cylindrical shell with general boundary conditions can be written as: u = U(x) cos(n0) cos(o~t)

(20a)

v = V(x) sin(n0) cos(wt)

(20b)

w = W(x) cos(n0) cos(o~t)

(20c)

Substituting eqn (20) into eqn (19), the governing equations can be rewritten as:

k=l

k=l

N + Z o~232Ci,(2),,, k Wk = -- phw2Vi k=l N k=l

N

(21b)

= -- phoo 2 V

$311U (1) -t- $312U(3) -t- $321W -'t'-$322V(2) q- $331W Jr- $332W(2)

phoo2W

k=l

Nx=O,

Mx=0

(21c)

(22)

By using eqn (11), eqn (22) can be simplified as V=0,

Fully clamped end

V=0,

270

W=0,

u(l)=0,

W{2)=0

(23)

(FC) W=0,

U=0,

w(l)=0

(24)

,4,)

S332Ci,k-+- S333Ci,k Wk = - phw2Wi

where cl,~ is the GDQ weighting coefficient of the nth order derivative, and N is the number of grid points. It is noted that eqns (25) are the governing equations for variables U, V and W, respectively. Similarly. the derivatives in the boundary conditions can be discretized by the GDQ method. Using the GDQ approach, the boundary conditions eqns (23) and (24) can be generalized as VI=0,

Wl=0

N Z ~(n0)r T t'l,k ~ k - - v k=l

Freely supported end (FS)

W=0,

(

(25c)

where U (i), V (i), W (i) denote the ith order derivative of U, V and W, and S~jkare the constant coefficients defined in the Appendix. Equations (21) are the governing equations to be solved by the GDQ method. The following two types of boundary conditions are considered in the present study.

V=0,

k=l

-'}-S331Wi-t- Z

$211 U (1) Jr- S221g -q--$222g (2) -q--S231W -{-S232W (2)

(25b)

~o311t.i,k -t- $312ci, k S k -t- $321 V/-~- Z q•322t'i, ~(2)I/ k vk

(21a)

= -- phoo2U

--

N

N

C 17(I) . e SlllUq-Sll2U(2) q-Ol21 v To131 11,11( ,v I ) .-I- ¢'o132I,11( ,v 3 )

+ $333W(4)=

N

E S21lcll u,+s22,v,+ Y S222Ci, kV k -Jr-$231W i

(26a) (26b)

N Z .~(n0+ 1)I,11 ~l,k VVk~---0 k=l

(26C)

VN =O,

(27a)

at x = 0, and WN =O

N Z ~(nl)lr __ t~N,k u k - - O k=l

(27b)

N Z ~(nl + 1)1,iI __/3 t.N, k ~vk -- t~ k=l

(27c)

Vibration analysis by GDQ: C. Shu and H. Du (0)k at x = L, where nO, nl can be taken as either 0 or 1, and c~, and CU, _~0)k are defined as

n0 = 0, nl = 0

fully clamped-fully clamped

n0 = 0, nl = 1

fully clamped-freely supported

number of equations should be equal to the number of unknowns. For the case here, the variables U, V and W at each grid point are taken as unknowns. So, we have N unknowns for U, V and W, respectively. Since eqns (26) and (27) provide two boundary equations for variables U and V, respectively, and four boundary equations for variable W, eqns (25a) and (25b) should be applied at N - 2 interior points, and eqn (25c) should be applied at N - 4 interior points. This can be done by taking the index i as 2,3 ..... N - 1 for eqns (25a) and (25b), and 3,4 ..... N - 2 for eqn (25). By substituting eqns (26a), (27b), (29) and (30) into eqns (25), the following system of eigenvalue equations can be obtained:

n0=l,

freely supported-fully clamped

[A]{X} = f~{X}

c(0)

~1

l,k= l,

c(O)

N, k =

whenk=l 0

(28a)

others

~ 1 whenk=N 1,

0

(28b)

others

By choosing nO, nl, eqns (26) give the following four sets of boundary conditions:

nl=0

nO= l,nl = 1

freely supported-freely supported

According to eqns (26) and (27), the functional values of variable V at two boundary points, x = 0, L, are zero. The boundary condition for variable U is given by eqns (26b) and (27b), which can be coupled to give the solutions Ul and UN as:

(31)

where I X l T = { U 2 , U3..... UN ~, V2, V3..... VN_,, W3, W4 . . . . . W N _ 2 } T

= ohco2, andA is a (3N - 8) × (3N - 8) dimension matrix. Obviously, solving the eigenvalues of matrix [A] provides the natural frequencies co of the thin cylindrical shells.

N-l

1 72AUK1.U ~ U1 = AUN =

(29a) NUMERICAL RESULTS AND DISCUSSION

1

N-I

UN = - - . E AUKN.Uk AUN k=2

(29b)

where AUK 1

(nO) (n l)

(nO) (n 1)

In this section, the global method of GDQ is applied to study the free vibration of cross-ply laminated circular cylindrical shells. For the numerical computation, the coordinates of grid points are chosen as

CI, k "CN,N -- C1,N'CN, k

1

cos[(/

Xi ~

(nO)

(nO) (nl)

It can be seen from eqn (29) that U~ and UN are expressed in terms of Uz, U3..... UN-b According to eqns (26a) and (27a), the functional values of variable W at two boundary points, W1,WN, are zero. Equations (26c) and (27c) provide another two boundary conditions for W, which can be coupled to give two solutions, W2 and Wu-b as: N-2

(30a)

N-2

1

WN-I =AWN'k=

EAWKN.Wk 3

i = 1,2, .. .,N

(32)

where L is the length of a cylindrical shell. In the present study, the numerical results are displayed by the frequency parameter X, defined as

AUN = CN, 1"CI,N -- el, 1 "CN,N

1 E AWK1.Wk W2 = AWN' k=3

1)Tr]L,

2

AUKN = c~n,°)'c~,~ - Cl,(nO)k"CN,(nl'1 (nl)

I/N

(30b)

k = 27rR

co

(33)

where co is the natural frequency and A Jl is an extensional stiffness term defined in eqn (16). Since there is an infinite complexity of the class of crossply laminates, only a regular antisymmetric cross-ply laminated shell is considered in the present study. For this kind of laminated composite, the coefficients in eqns (16)(18) can be simplified as: h A1, =A22 = ~(Q1, + Q22),

Aj2 =- Q12h,

A66 =

Q66h

where

(34) AWK1

c(nO+1)

~(nl + 1)

_(n0+ 1) ~(nl + I)

~(n0+ 1) ~(nl + 1)

,..(nO+ 1) _(nl + 1)

l,k

AWKN ='q,2

"¢'N,N- 1 -- e l , N - l'CN, k "t'U,k

--~l,k

h2 B,,

=

-

=

-

"CN, 2

A W N = ..(nl + l)...(n0+ 1) __ (nO+ 1) _(nl + 1) ~N, 2 ~ I , N - 1 C1,2 "ON,N - 1

According to eqn (30), W2 and Wu-1 are expressed in terms of W 3 , W 4 . . . . . WN_ 2. It is well known that, for a well-posed problem, the

h3 D,| = 022 = ~-~(Q|I + Q22),

B12 = 0,

B66 = 0

1 DI2 •

~ Q 1 2 h3,

1

D66 = -~Q66 h3

271

Vibration analysis by GDQ: C. Shu and H. Du Table 1 Convergence of fundamental frequency parameter ~, for an antisymmetric cross-ply laminated shell with freely supported boundary conditions (h/R = 0.05,/JR = 5, n = 3) Two-layered shell Soldatos 3

0.4129

N N N N N

0.4120 0.4127 0.4127 0.4127 0.4127

= = = = =

5 7 9 13 21

1.0

2.0

10.0

n

0.7083 0.7073 0.7082 0.7082 0.7082 0.7082

Two-layered shell

Infinitely-layered shell

Present

Soldatos3

Present

Soldatos3

2.1064 1.3440 0.9588 0.7494 0.6422 0.6137 1.0733 0.6710 0.4710 0.3773 0.3632 0.4168 0.1944 0.1008 0.0941 0.1523 0.2417 0.3536

2.106 1.344 0.9589 0.7495 0.6423 0.6138 1.073 0.6710 0.4710 0.3774 0.3632 0.4168 0.1944 0.1008 0.0941 0.1523 0.2417 0.3536

2.1140 1.3564 0.9799 0.7975 0.7505 0.8192 1.0762 0.6757 0.4882 0.4395 0.5094 0.6668 0.1949 0.1092 0.1480 0.2661 0.4275 0.6265

2.114 1.356 0.9800 0.7976 0.7506 0.8193 1.076 0.6757 0.4882 0.4395 0.5094 0.6668 0.1949 0.1092 0.1480 0.2661 0.4275 0.6265

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

where h is the thickness of the shell, Nt is the number of layers, Qij are material constants t8. The elementary material parameters of each layer are given as EI_~I= 40.0, E22

Pl2 = 0.25,

G12 = 0.5 E22

(35)

It can be observed from eqn (34) that the extension-bending coupling terms reach their maximum values with two plies ( N l = 2) and become zero with an infinite number of plies (Nt = ~). These two cases will be chosen to study the effects of extension-bending coupling on the frequency parameters. For the case of two-layered shells, we have B ll = B22 = (Qll - Q22)×h2/8, while for the case of infinitelylayered shells, we have B]1 = B22 = 0. The convergence of GDQ results for the two abovementioned cases with freely supported boundary condition is studied, and the numerical results for the case of h/R = 0.05, L/R = 5, n = 3 are listed in Table 1. From this table, it is apparent that the convergence of GDQ results is very good. When N, which is the number of grid points, is larger than 7, the GDQ result is independent of N. This indicates that seven grid points are sufficient to obtain accurate numerical results when the GDQ method is applied. Since accurate GDQ results can be obtained by using such a few grid points, the computational time required is tiny. Actually, for every numerical result shown in Table 1, the CPU time required on the IBM 3081 is less than one second. Table 1 also shows a good agreement between the GDQ -

-

272

h/R

Infinitely-layered shell

Table 2 Comparison of fundamental frequency parameter h for an antisymmetric cross-ply laminated shell with freely supported boundary conditions (h/R = 0.01) /JR

Table 3 Fundamental frequency parameter ), of an antisymmetric crossply laminated cylindrical shell with FS-FS boundary conditions LIR 1

Two-layered shell 0.01 0.6137 (6) 0.02 0.7950(5) 0.03 0.9152 (4) 0.04 1.0385 (4) 0.05 1.1339 (3) Infinitely-layered shell 0.01 0.7505 (5) 0.02 0.9760 (4) 0.03 1.1692 (3) 0.04 1.3125 (3) 0.05 1.4764 (3)

2

5

20

10

0.3632 (5) 0.4600(4) 0.5233 (3) 0.5659 (3) 0.6161 (3)

0.1734 (3) 0.2186(3) 0.2572 (2) 0.2664 (2) 0.2780 (2)

0.0941 (3) 0.1108(2) 0.1260 (2) 0.1446 (2) 0.1653 (2)

0.0410 (2) 0.0624(2) 0.0766 (1) 0.0764 (1) 0.0762 (i)

0.4395 0.5502 0.6404 0.7210 0.7469

0.2085 0.2656 0.2877 0.3161 0.3491

0.1092 0.1380 0.1759 0.1949 0.1950

0.0577 0.0771 0.0771 0.0772 0.0772

(4) (3) (3) (2) (2)

(3) (2) (2) (2) (2)

(2) (2) (2) (1) (1)

(1) (1) (1) (1) (1)

Parameters in brackets indicate the circumferential numbers at which the fundamental frequencies occur

Table 4 Fundamental frequency parameter X of an antisymmetric crossply laminated cylindrical shell with FS-FC boundary conditions h/R

LIR 1

Two-layered shell 0.01 0.6440 (6) 0.02 0.8557 (5) 0.03 1.0148 (4) 0.04 1.1734 (4) 0.05 1.3201 (3) Infinitely-layered shell 0.01 0.8044 (5) 0.02 1.0944 (4) 0.03 1.3632 (3) 0.04 1.5867 (3) 0.05 1.8327 (3)

2

10

5

20

0.3750 0.4764 0.5467 0.5957 0.6518

(5) (4) (3) (3) (3)

0.1858 0.2293 0.2665 0.2764 0.2886

(3) (3) (2) (2) (2)

0.1030 0.1238 0.1376 0.1549 0.1745

(3) (2) (2). (2) (2)

0.0496 0.0683 0.0879 0.0877 0.0875

(2) (2) (1) (1) (1)

0.4545 0.5769 0.6775 0.7736 0.8146

(4) (3) (3) (2) (2)

0.2193 0.2752 0.2984 0.3277 0.3615

(3) (2) (2) (2) (2)

0.1223 0.1487 0.1846 0.2018 0.2021

(2) (2) (2) (1) (1)

0.0641 0.0886 0.0886 0.0886 0.0886

(2) (1) (1) (1) (1)

Parameters in brackets indicate the circumferential numbers at which the fundamental frequencies occur

results and those of Soldatos 3. Soldatos's results are based on the Love's first approximation shell theory and the closed-form solution. The detailed comparison between the GDQ results and those of Soldatos 3 for freely supported boundary condition, h/R = 0.01, various values o f / J R and circumferential wavenumbers is shown in Table 2, where the GDQ results are obtained by using seven grid points. It can be seen from Table 2 that for all the cases, including the two-layered shells and the infinitely-layered shells, the present GDQ results are almost identical to those of Soldatos 3 although just a few grid points are used in the present work. This demonstrates that the GDQ method is a very efficient numerical technique for the free vibration analysis of laminated cylindrical shells owing to its high order of accuracy and smaller requirement of computational time. Having gained confidence in the GDQ method, the fundamental (lowest) frequency parameters X for three sets of boundary conditions [namely, the freely supportedfreely supported (FS-FS), the freely supported-fully clamped (FS-FC) and the fully clamped-fully clamped (FC-FC)]; L/R = 1, 2, 5, 10 and 20; and h/R = 0.01, 0.02, 0.03, 0.04 and 0.05 have been computed and the numerical results are listed in Tables 3-5. The fundamental frequency

Vibration analysis by GDQ: C. Shu and H. Du Table 5 Fundamental frequency parameter k of an antisymmetric crossply laminated cylindrical shell with FC-FC boundary conditions h/R

L/R 1

Two-layered shell 0.01 0.6867(6) 0.02 0.9378 (5) 0.03 1.1469 (4) 0.04 1.3582 (4) 0.05 1.5579(3) Infinitely-layered shell 0.01 0.8762 (5) 0.02 1.2544 (4) 0.03 1.6204 (3) 0.04 1.9575 (3) 0.05 2.3186 (3)

2

10

5

20

0.3943(5) 0.5020 (4) 0.5811 (3) 0.6377 (3) 0.7014(3)

0.2028(3) 0.2440 (3) 0.2842 (2) 0.2945 (2) 0.3071 (2)

0.1131(3) 0.1385 (2) 0.1510 (2) 0.1670 (2) 0.1855(2)

0.0590(2) 0.0754 (2) 0.0968 (2) 0.1003 (l) 0.1001(1)

0.4784 0.6146 0.7286 0.8409 0.9017

0.2342 0.2929 0.3167 0.3465 0.3806

0.1371 0.1614 0.1951 0.2156 0.2162

0.0716 0.1013 0.1013 0.1013 0.1014

(4) (3) (3) (2) (2)

(3) (2) (2) (2) (2)

(2) (2) (2) (1) (1)

(2) (1) (1) (1) (1)

Parameters in brackets indicate the circumferential numbers at which the fundamental frequencies occur

than one second. The fundamental frequency parameters for three sets of boundary conditions--namely, the freely supported-freely supported, the freely supported-fully clamped, the fully clamped-fully clamped--and various shell thicknesses and lengths are computed and tabulated in the paper. From the numerical computation, it can be concluded that the present approach is an efficient method for the free vibration analysis of laminated composite cylindrical shells owing to its high order of accuracy and small requirement of virtual storage and computational effort.

REFERENCES

1

parameters k for an antisymmetric cross-ply laminated circular cylindrical shell with the FS-FS boundary condition are shown in Table 3. The fundamental frequency parameters X for an antisymmetric cross-ply laminated circular cylindrical shell with the FS-FC and FC-FC boundary conditions are listed in Tables 4 and 5, respectively. In Tables 3-5, the GDQ results are obtained by using nine grid points, and the parameters in brackets indicate the circumferential wavenumbers at which the fundamental frequencies occur. It can be seen from these tables that, with the same boundary condition, the fundamental frequency parameter X for an infinitely-layered shell is larger than that for a two-layered shell. As mentioned earlier, the infinitely-layered shell shows no extensionbending coupling while the two-layered shell shows maximum extension-bending coupling. Thus, the numerical results imply that the less the extension-bending coupling, the larger the fundamental frequency parameters. Numerical results in Tables 3 - 5 also reveal that as the shell thickness increases, the fundamental frequency parameter increases accordingly for the short shell, and changes a little for the long shell (L/R = 20). For all the cases, as the length of the shell increases, the fundamental frequency parameter decreases. The boundary condition has some effect on the fundamental frequency parameter. Numerical results in Tables 3 - 5 demonstrate that the FC-FC boundary condition provides the largest fundamental frequency parameter while the FS-FS boundary condition gives the smallest fundamental frequency parameter.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

CONCLUSIONS

17

In this paper, a new approach is proposed to study the free vibration of cross-ply laminated circular cylindrical shells. In the proposed approach, the derivatives in both the goveming equations and the boundary conditions are discretized by the global method of generalized differential quadrature (GDQ). It was found that the convergence of GDQ results is very fast. The GDQ result using only seven grid points agrees very well with that in the literature. And accordingly, the CPU time required on the IBM 3081 is less

18

Reddy, J.N. Exact solutions of moderately thick laminated shells. J. Engng Mech. ASCE, 1984, 110, 794-809. Chaudhuri, R.A. and Abu-Arja, K.R. Closed form solutions for arbitrary laminated anisotropic cylindrical shells (tubes) including shear deformation. AIAA J., 1989, 27, 1597-1605. Soldatos, K.P. A comparison of some shell theories used for the dynamic analysis of cross-ply laminated circular cylindrical panels. J, Sound Vibr., 1984, 97, 305-319. Reddy, J.N. Finite element modelling of layered, anisotropic composite plates and shells: a review of recent research. Shock Vibr. Digest, 1981, 13, 3-12. Kapania, R.K. A review on the analysis of laminated shells. J. Pressure Vessel Tech. ASME, 1989, 111, 88-96. Shu, C. 'Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation', PhD Thesis, University of Glasgow, 1991. Shu, C. and Richards, B.E. Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. Int. J. Num. Meth. Fluids, 1992, 15, 791-798. Shu, C. and Richards, B.E. Parallel simulation of incompressible viscous flows by generalized differential quadrature. Comput. Syst. Engng, 1992, 3, 271-281. Bellman, R., Kashef, B.G. and Casti, J. Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. Z Comput. Phys., 1972, 10, 40-52. Mingle, J.O. The method of differential quadrature for transient nonlinear diffusion. J. Math. Anal. Appl., 1977, 60, 559-569. Civan, F. and Sliepcevich, C.M. Application of differential quadrature to transport processes. J. Math. Anal. Appl., 1983, 93, 711724. Civan, F. and Sliepcevich, C.M. Differential quadrature for multidimensional problems. J. Math. Anal. Appl., 1984, 101, 423-443. Jang, S.K., Bert, C.M. and Striz, A.G. Application of differential quadrature to static analysis of structural components. Int. J. Num. Meth. Engng, 1989, 28, 561-577. Du, H., Lim, M.K. and Lin, R.M. Application of generalized differential quadrature method to structural problems. Int. J. Num. Meth. Engng, 1994, 37, 1881-1896. Du, H., Lim, M.K. and Lin, R.M. Application of differential quadrature to vibration analysis. J. Sound Vibr., 1995, 181, 279-293. Quan, J.R. and Chang, C.T. New insights in solving distributed system equations by the quadrature method--l. Analysis. Computers Chem. Engng, 1989, 13, 779-788. Quan, J.R. and Chang, C.T. New insights in solving distributed system equations by the quadrature method--II. Numerical experiments. Computers Chem. Engng, 1989, 13, 1017-1024. Lam, K.Y. and Loy, C.T. Free vibration of a rotating multi-layered cylindrical shell. Int. J. Solids Struct., 1995, 32, 647-663.

APPENDIX

A66r/2

Sl 11

--

R2

273

Vibration analysis by GDQ: C. Shu and H. Du SI12=AII

$232 = (O12 -+-2066)n

(B12 -I- 2B66)n

R2

(912 + 2966) n S121 :

R2

(A12-l- A66) n {-

(B12 + 2B66)n 2 A12 R2 + S132 =

-

-

$312 : Bll

D22n3 A22n

Bll

$321 --

( B12n 966n Al2n ~ )

8211 = - ~ R 2 + --R-~--k- --~---.1-

R4

$221 --

2B22 n2 R3

A22n2 R2 $331 -

2D66 $222 :A66 -I- ~ - +

D22n3 $231 = -

274

~q---+

3B66 ~-

B22n(l + n2) A22n"~ R3

R2

q- R2 J

(B12 --t--2B66)n R

F

D22 n4 R4 2D12 n2

B22n3 + B22n R3

R2

(D12 + 4D66)n $322 :

022 n2 R4

R

B12n2 + 2B66n2 --b - ~ ) R2

R $311 = -

S131 =

+

2B22 n2 R3 4D66 n2

$332- RT-- - + ~

$333 = - D11

A22 R2 2B12

+

---~