cap_8

Chapter 9

Vibrations of Cylindrical Shells

Tiejun Yang, Wen L. Li and Lu DaiAdditional information is available at the end of the chapter

http://dx.doi.org/10.5772/51816

1. IntroductionBeams, plates and shells are the most commonly-used structural components in industrialapplications. In comparison with beams and plates, shells usually exhibit more complicateddynamic behaviours because the curvature will effectively couple the flexural and in-planedeformations together as manifested in the fact that all three displacement components simultaneously appear in each of the governing differential equations and boundary conditions. Thus it is understandable that the axial constraints can have direct effects on apredominantly radial mode. For instance, it has been shown that the natural frequencies forthe circumferential modes of a simply supported shell can be noticeably modified by theconstraints applied in the axial direction [1]. Vibrations of shells have been extensively studied for several decades, resulting in numerous shell theories or formulations to account forthe various effects associated with deformations or stress components.Expressions for the natural frequencies and modes shapes can be derived for the classicalhomogeneous boundary conditions [2-9]. Wave propagation approach was employed byseveral researchers [10-13] to predict the natural frequencies for finite circular cylindricalshells with different boundary conditions. Because of the complexity and tediousness of the(exact) solution procedures, approximate procedures such as the Rayleigh-Ritz methods orequivalent energy methods have been widely used for solving shell problems [14-18]. In theRayleigh-Ritz methods, the characteristic functions for a similar beam problem are typically used to represent all three displacement components, leading to a characteristic equation in the form of cubic polynomials. Assuming that the circumferential wave length issmaller than the axial wave length, Yu [6] derived a simple formula for calculating the natural frequencies directly from the shell parameters and the frequency parameters for the analogous beam case. Soedel [19] improved and generalized Yus result by eliminating the shortcircumferential wave length restriction. However, since the wavenumbers for axial modalfunction are obtained from beam functions which do not exactly satisfy shell boundary con

2012 Yang et al.; licensee InTech. This is an open access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited.

ditions, it is mathematically difficult to access or ensure the accuracy and convergence ofsuch a solution.The free vibration of shells with elastic supports was studied by Loveday and Rogers [20]using a general analysis procedure originally presented by Warburton [3]. The effect of flexibility in boundary conditions on the natural frequencies of two (lower order) circumferentialmodes was investigated for a range of restraining stiffness values. The vibrations of circularcylindrical shells with non-uniform boundary constraints were studied by Amabili and Garziera [21] using the artificial spring method in which the modes for the corresponding less-restrained problem were used to expand the displacement solutions. The non-uniformspring stiffness distributions were systematically represented by cosine series and theirpresence was accounted for in terms of maximum potential energies stored in the springs.A large number of studies are available in the literature for the vibrations of shells underdifferent boundary conditions or with various complicating features. A comprehensive review of early investigations can be found in Leissas book [22]. Some recent progresses havebeen reviewed by Qatu [23]. Regardless of whether an approximate or an exact solution procedure is employed, the corresponding formulations and implementations usually have tobe modified or customized for different boundary conditions. This shall not be considered atrivial task in view that there exist 136 different combinations even considering the simplest(homogeneous) boundary conditions. Thus, it is useful to develop a solution method thatcan be generally applied to a wide range of boundary conditions with no need of modifyingsolution algorithms and procedures. Mathematically, elastic supports represent a generalform of boundary conditions from which all the classical boundary conditions can be readilyderived by simply setting each of the spring stiffnesses to either zero or infinity. This chapter will be devoted to developing a general analytical method for solving shell problems involving general elastically restrained ends.

2. Basic equations and solution proceduresFigure 1 shows an elastically restrained circular cylindrical shell of radius R, thickness h andlength L. Each of the eight sets of elastic restraints shall be understood as a distributedspring along the circumference. Let u, v, and w denote the displacements in the axial x, circumferential and radial r directions, respectively. The equations of the motions for theshell can be written as

21 122

212 22

21 1 22

(a)

(b)

(c)

N N uhx R tN N vhx R tQ Q N whx R R t

rqrq

rq

+ = + = + - =

(1)

Advances in Vibration Engineering and Structural Dynamics206

where is the mass density of the shell material, and N1, N12, N, Q1 and Q denote the resultant forces acting on the mid-surface.

1k

qx

l

R

r 2k

3k

4k5k

6k

7k

8k

h

Figure 1. A circular cylindrical shell elastically restrained along all edges.

In terms of the shell displacements, the force and moment components can be expressed as

1

2

122 2

1 2 2 22 2

2 2 2 22

123 31 121 3

(a)

(b)(1 ) (c)2

(d)

(e)

(1 ) (f)

u v wN K x R Rv w uN K R R x

u vN K R xw wM D x R

w wM D R xwM D R x

M M w wQ Dx R x

s sqsq

sq

s qsq

s q

q

= + + = + +

- = + = - + = - +

= - - = + = - + 2 2

3 32 122 3 3 2

(g)

(h)R x

M M w wQ DR x R R x

q

q q q

= + = - +

(2)

where

Vibrations of Cylindrical Shellshttp://dx.doi.org/10.5772/51816

207

23 2

2

/ (1 ), (a)/ 12(1 ) (b)

/ / 12 (c)

K EhD Eh

D K h

ss

k

= -= -= =

(3)

E and are respectively the Youngs modulus and Poisson ratio of the material; M1, M2 andM12 are the bending and twisting moments.The boundary conditions for an elastically restrained shell can specified as:at x=0,

12

3

4

0 (a)0 (b)

0 (c)

0 (d)

xx

xx

x

N k uN k v

MQ k wRwM k x

qqq

- =- =+ - =

+ =

(4)

at x=L,

56

7

8

0 (a) 0 (b)

0 (c)

0 (d)

xx

xx

x

N k uN k v

MQ k wRwM k x

qqq

+ =+ =+ + =

- =

(5)

where k1, k2, , k8 are the stiffnesses for the restraining springs. The elastic supports represent a set of general boundary conditions, and all the classical boundary conditions can beconsidered as the special cases when the stiffness for each spring is equal to either zero orinfinity.The above equations are usually referred to as Donnell-Mushtari equations. Flgges theoryis also widely used to describe vibrations of shells. In terms of the shell displacements, thecorresponding force and moment components are written as


21 2

212

22 2 2 2

221 2

(a)1 (1 ) (b)2 2

(c)

1 (1 )2 2

u v w D wN K Kx R R R xu v D v wN K R x R R x R x

v w u D w wN K R R x R R Ru v D u wN K R x R R xR

s sqs s

q qsq q

s sq qq

= + + - - - = + + -

= + + + + - - = + + +

2 21 2 2 2 2

212

2 22 2 2 2 2

221 2

3 3 21 3 2 2

(d)

(e)

(1 ) (f)

(g)

1 1(1 ) (h)2 21

w w u vM D R xx R Rw vM D R x R x

w w wM D R R xw u vM D R x R xR

w w uQ D x R x R

s sq qs q

sqs q q

q

= - + - -

= - - - = - + +

= - - + - = - + -

22 3 2 2

1 1 (i)2 2u v

xx R Rs s

qq - + + -

(6)

A shell problem can be solved either exactly or approximately. An exact solution usually implies that both the governing equations and the boundary conditions are simultaneously satisfied exactly on a point-wise basis. Otherwise, a solution is considered approximate inwhich one or more of the governing equations and boundary conditions are enforced onlyin an approximate sense. Both solution strategies will be used below.

2.1. An approximate solution based on the Rayleigh-Ritz procedureApproximate methods based on energy methods or the Rayleigh-Ritz procedures are widelyused for the vibration analysis of shells with various boundary conditions and/or complicating factors. In such an approach, the displacement functions are usually expressed as

0

0

0

( , ) ( ) cos (a)

( , ) ( ) sin (b)

( , ) ( ) cos (c)

mm u

mm

m vm

mm w

m

u x a x n

v x b x n

w x c x n

q j q

q j q

q j q

=

=

=

=

=

=

(7)

where m(x), =u, v, and w, are the characteristic functions for beams with similar boundary conditions. Although characteristic functions generally exist in the forms of trigonometric and hyperbolic functions, they also include some integration and frequency constantsthat have to be determined from boundary conditions. Consequently, each boundary condi


209

tion basically calls for a special set of modal data. In the literature the modal parameters arewell established for the simplest homogeneous boundary conditions. However, the determination of modal properties for the more complicated elastic boundary supports can become,at least, a tedious task since they have to be re-calculated each time when any of the stiffnessvalues is changed. It should also be noted that calculating the modal properties will typically involve seeking the roots of a nonlinear transcendental equation, which mathematicallyrequires an iterative root searching scheme and careful numerical implementations to ensure no missing data. To overcome this problem, a unified representation of the shell solutions will be adopted here in which the displacements, regardless of boundary conditions,will be invariably expressed as

u(x, )= (m=0

amcosmx + pu(x)) cosn, (m = mL )(a)v(x, )= (

m=0

bmcosmx + pv(x)) sinn , (b)w(x, )= (

m=0

cmcosmx + pw(x)) cosn(c)(8)

where p(x), =u, v, and w, denote three auxiliary polynomials which satisfy

100

2

300

4

500

6

33

( ) ( ,0) , (a)( ) ( ,0) (b)( ) ( , / 2) , (c)( ) ( , / 2) , (d)( ) ( ,0) , (e)( ) ( ,0) , (f)

( )

uxx

ux Lx L

vxx

vx Lx L

wxx

wx Lx L

wx

p x u xx x

p x u xx x

p x v xx x

p x v xx x

p x w xx x

p x w xx x

p xx

b

bp bp b

b

b

==

==

==

==

==

==

= = = = = = = = = = = =

373

003 3

83 3

( ,0) , (g)

( ) ( ,0) (h)x

wx Lx L

w xx

p x w xx x

b

b==

==

= = = =

(9)

It is clear from Eqs. (9) that these auxiliary polynomials are only dependent on the first andthird derivatives i, (i=1,2,,8) of the displacement solutions on the boundaries. In terms ofboundary derivatives, the lowest-order polynomials can be explicitly expressed as [24, 25]


1 1 2 21 3 2 41 5 2 6 3 7 4 8

( ) ( ) ( ) (a)( ) ( ) ( ) (b)( ) ( ) ( ) ( ) ( ) (c)

uvw

p x x xp x x xp x x x x x

z b z bz b z bz b z b z b z b

= += += + + +

(10)

where

2 21

2 22

4 3 2 2 434 2 2 44

(6 2 3 ) / 6( )( ) (3 ) / 6( ) ( ) (15 60 60 8 ) / 360 ( ) (15 30 7 ) / 360

T

Lx L x Lxx x L Lx x x Lx L x L Lx x L x L L

zzzz

- - - = = - - + - - +

z (11)

This alternative form of Fourier series recognizes the fact that the conventional Fourier series for a sufficiently smooth function f(x) defined on a compact interval [0, L] generally failsto converge at the end points. Introducing the auxiliary functions will ensure the cosine series in Eqs. (8) to converge uniformly and polynomially over the interval, including the endpoints. As a matter of fact, the polynomial subtraction techniques have been employed bymathematicians as a means to accelerate the convergence of the Fourier series expansion foran explicitly defined function [26-28].The coefficients i represent the values of the first and third derivatives of the displacementsat the boundaries, and are hence related to the unknown Fourier coefficients for the trigonometric terms. The relationships between the constants and the expansion coefficients can bederived either exactly or approximately.In seeking an approximate solution based on an energy method, the solution is not requiredto explicitly satisfy the force or natural boundary conditions. Accordingly, the derivative parameters i in Eqs. (10) will be here determined from a simplified set of the boundary conditions, that is,

1,5

2,63

3,732

4,82

0 (a)(1 ) 0 (b)2

0 (c)

0 (d)

u k uxv k vx

w k wxw wk xx

m

k

k

=- = = =

mm

m

(12)

where

/ .i ik k K= (13)

By substituting Eqs. (8) and (10) into (12), one will obtain


211

{ }

10

10

10

(a)

(b)

(c)

mu u m

mm

v v mm

T mw w m

m

a

b

c

bbbbb b b b

-= -=

-=

= =

=

H Q

H Q

H Q

1234

5 6 7 8

(14)

where

{ }

{ }

1 1

5 5

T11 5

2 2

6 6

T12 6

3 33 3 3 3

3 37 7 7 7

4

13 6 (a) 16 3 ( 1) (b)

13 2 6 (c) 1

6 3 2 ( 1) (d)

73 6 45 360

76 3 360 45

u

m mu

v

m mv

w

k L k L

k L k L

k kk L k L

k L k L

k kk L k L k L k L

k L k L k L k L

k

m

m

k

k

+

+

+ = + = - -+ = - +

= -

- - +

- - +=

H

Q

H

Q

H

8

(e)3 6

6 3

L LL L

L LkL L

k k k k

k k k k

+ - - +

(15)

and

{ }2 23 7 ( 1) ( 1) Tm m mw m mk k kl kl= - - - -Q (16)

In light of Eqs. (13), Eqs (8) can be reduced to Eqs. (7) with the axial functions being definedas


( ) cos ( ) , ( , , )m T mmx x x u v wa aj l a= + =Qz (17)

where

-1

(a)

0 (b)0

( ) (c)

m mw w

mm

m m

aa

a a a

= =

=

Q QQ

Q

Q H Q

%%

%

(18)

Since the boundary conditions are not exactly satisfied by the displacements such constructed, the Rayleigh-Ritz procedure will be employed to find a weak form of solution. The current solution is noticeably different from the conventional Rayleigh-Ritz solutions in that: a)the shell displacements are expressed in terms of three independent sets of axial functions,rather than a single (set of) beam function(s), b) the basis functions in each displacement expansion constitutes a complete set so that the convergence of the Rayleigh-Ritz solution isguaranteed mathematically, and c) it does not suffer from the well-known numerical instability problem related to the higher order beam functions or polynomials. More importantly,the current method is that it provides a unified solution to a wide variety of boundary conditions.The potential energy consistent with the Donnell-Mushtari theory can be expressed from

2 22

0 02 22 2 2 2 2

2 2 2 2 2 2

2 2 21 2 3

(1 )2(1 )2 2

2(1 )

1 2 [

LK u v w u v w v uV x R R x R R x R

w w w w w RdxdR xx R x R

k u k v k w k

p nnq q q

k n qqq q

- = + + - - + + + + + - - -

+ + + +

( )( )

( ) ( ) ( )

2 24 0

02 22 2 2

5 6 7 80

2 2 2 2

0 0

and the total kinetic energy is calcul

(a)]

1 2 [ ]

1

ated

(b)

from

2

x

x L

L

w x Rd

k u k v k w k w x Rd

T h u t v t w t Rdxd

p

p

p

q

q

r q

=

=

+ + + +

= + +

(19)

By minimizing the Lagrangian L=V-T against all the unknown expansion coefficients, a finalsystem of linear algebraic equations can be derived


213

T 2T T

0ss s sr ss

s rsr r rr rr

qq qq q qq

qw

- =

M 0 0a ab 0 M 0 bc c0 0 M

L L LL L LL L L

(20)

where

{ }{ }{ }

T00 01

T01 02

T00 01

, ,..., ,.. , (a) , ,..., ,... , (b) , ,..., ,... , (c)

mn

mn

mn

a a ab b bc c c

===

a

b

c

(21)

2' ' ' ', ' ' ' ,11 ,00 1 52

' ', ' ' ' ,10 ,01

', ' ' ' ,10

2 ', ' ' ' ,002

(1 ) 2 2 [ (0) (0) ( ) ( )] (a)2(1 )[ ] (b)2

(c)(1[

ss mm mm m m m mmn m n nn uu uu u u u u

s mm mmmn m n nn uv uv

sr mmmn m n nn uw

mmmn m n nn vv

nI I k k L LL LRn nI IR RIR

n IR

q

qq

md j j j jm mdmd

md

-L = + + +-L = -

L =-L = + ' ' ',11 2 6

', ' ' ' ,002

4 2' ' ' ', ' ' ' ,00 ,22 ,00 ,112 4 2

2 ',02 ,22

) 2 2 (0) (0) ( ) ( )] (d)2(e)

1{ [ 2(1 )

(

mm m m m mvv v v v v

r mmmn m n nn vw

mm mm mm mmmn m n nn ww ww ww ww

mmww ww

I k k L LL Ln IR

n nI I I IR R Rn I IR

q

qq

j j j j

d

d k mm

+ +

L =

L = + + + -

- + ' ' '0 3 7' '

4 8

2 2 )] (0) (0) ( ) ( ) (f)(0) (0) ( ) ( )2 2 }

mm m m m mw w w w

m m m mw w w w

k k L LL LL Lk kL x x L x x

j j j jj j j j

+ + + + +

(22)

', ' ' ' ,00

', ' ' ' ,00

', ' ' ' ,00

(a)(b)(c)

ss mmmn m n nn uu

mmmn m n nn vvrr mmmn m n nn ww

hIhIhI

qqd rd rd r

===

MMM

(23)

and

''

, 02 ( , , , ) q mp mLmm

pq p qI L dx u v wx xbaab

ff a b= = (24)

The integrals in Eq. (23) can be calculated analytically; for instance

I,00mm' = 2 L 0

L

mm' dx


''

0' '

' '0

' ' ''

2 (cos ( ) )(cos ( ) ) 2 (cos cos cos ( ) cos ( ) ( ) ( ) )

L T m T mm m

L T m T m mT T mm m m m

m m mm mmm mm

x x x x dxLx x x x x x x x dxL

S S Z

a b

a b a b

a b ab

l l

l l l le d

= + +

= + + +

= + + +

Q Q

Q Q Q Q

z z

z z z z (25)

where

0' '

' '

(1 ) (a)(b)

(c)

m mm m m m

cmm mT m

Sa aab a b

e d= +=

Z =P QQ QX

(26)

and

0

2 ( ) cos Lm T

c w mx x dxL l= P z (27)

{ }1

2 2 4 4

2

2 2

0 4 4 6

4

0 0 0 0 , for 02 (a)1 ( 1) 1 ( 1) , for 0

245

.7 2180 452 ( ) ( )4 31 2945 7560 472531756

T

Tm m

m m m m

L T

mL m

L

symL LL x x dx

L L L

L

l l l l+

== - - -

= =- -

-

X z z4 6 6

(b)

4 127 20 945 302400 4725

L L L

-

(28)

2.2. A strong form of solution based on Flgges equations

As aforementioned, the displacement expressions in terms of beam functions cannot exactlysatisfy the shell boundary conditions; instead they are made to satisfy the boundary conditions in a weak sense via the use of the Rayleigh-Ritz procedure. To overcome this problem,the displacement expressions, Eqs. (8), will now be generalized to

0 0

0 0

0 0

( , ) cos ( ) cos (a)

( , ) cos ( ) sin (b)

( , ) cos ( ) cos (c)

umn m n

n mv

mn m nn m

wmn m n

n m

u x A p x n

v x B p x n

w x C p x n

xxx

q q

q q

q q

lll

= =

= =

= =

= += += +

(29)


215

which represent a 2-D version of the improved Fourier series expansions, Eqs. (8).To demonstrate the flexibility in choosing the auxiliary functionspnu(x), pnv(x)andpnw(x), analternative set is used below:

( ) ( ) (a)( ) ( ) (b)( ) ( ) (c)

u un nv vn nw wn n

p x xp x xp x x

===

L aL aL b

(30)

herenu = an bn , nv = cn dn , nw = en f n gn h n with an, bn,..., gn and hn being the unknown coefficients to be determined; (x)= {1(x) 2(x)}Tand (x)= {1(x) 2(x) 3(x) 4(x)}T and withtheir elements being defined as

21

22

( ) ( 1) (a)

( ) ( 1) (b)

xx x lx xx l l

a

a

= -

= -(31)

and

1

23 3

3 3 33 3

4 3 3

9 3( ) sin( ) sin( ) (a)4 2 12 29 3( ) cos( ) cos( ) (b)4 2 12 2

3( ) sin( ) sin( ) (c)2 233( ) cos( ) cos( ) (d)2 23

l x l xx l ll x l xx l l

l x l xx l ll x l xx l l

p pb p pp pb p p

p pb p pp pb p p

= -

= - -

= -

= - -

(32)

In Eqs. (27), the sums of x-related terms are here understood as the series expansions in x-direction, rather than characteristic functions for a beam with similar boundary condition.This distinction is important in that the boundary conditions and governing differentialequations can now be exactly satisfied on a point-wise basis; that is, the solution can befound in strong form, as described below.Substituting Eqs. (6) and (27) into (4) and (5) will lead to


( )

33

210 0 0

20 0

2 32 2

7 3 43 4 3 (a)

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

74 2 (3 ) (2 )32

n n n

mn mn m mnm m m

n n mn mnm m

n n n n n n

l R l l Ra f hR l Rk nA B R CK R R

kn nc e A BR K

lkn na b c e f glR lR DR R

s p g s gp pp

s s l g

s s g sg

s sp

= = =

= =

- + - + = - - +

- - -+ + = +

- -- - + + + - +

3

33

2 2 330 0

2 3 2 42 3 2

222 20 0

33

43 (c)(1 )

2

1 3 7 n 44 3 3 (d)

3 7 44 3 3

n

m mn mnm m

n n n n

mn m mnm m

n n

l k hDkn A CR DR

kl l l na f h eR l DR Rn nB CR R

R l l R lb el R R

pl s

p n npp p

s sl

pg s g sp p p

= =

= =

- = - +

- + + + + - = + +

- - + - +

( )

250 0 0

60 0

3272 2

(e)cos( ) cos( ) cos( )

1 (1 ) (1 )1 cos( ) cos( ) (f)2 2 2

7 42 4 (3 ) (2 )32

n

mn mn m mnm m m

n n mn mnm m

n n n n n

gk nm A m B R m CK R R

kn nd f m A m BR K

lk l kn na b d e flR lR DR R

s sp p l g p

s s g sg p p

s sp

= = =

= =

= + + +

- - -- + - = - +

- -- - - - - -

73

2 2 730 0

2 3 282 3 2

222 20 0

3 (g)cos( ) (1 )cos( ) cos( )2

1 3 7 n 4 n4 3 3 (h)cos( ) cos( )

n n

m mn mnm m

n n n n

mn m mnm m

g hDm km n A m CR DR

kl l lb e f gR l DR Rn nm B m CR R

pp l s p p

p s spp p

s sp l p

= =

= =

+ - = - - +

+ + - + + = - - +

(33)

Equations (31) represent a set of constraint conditions between the unknown (boundary)constants, an, bn,..., gn and hn, and the Fourier expansion coefficients Amn, Bmn, and Cmn ( m, n =0, 1, 2,... ). The constraint equations (31a-h) can be rewritten more concisely, in matrix form,as

=Ly Sx (34)


217

The elements of the coefficient matrices can be readily derived from Eqs. (31); for example,Eq. (31a) implies

{ }{ }{ }{ } { } { } { } { }{ }{ }{ }

31

36

38

32 33 34 35 37

11

12

13

,

,3

3,

, , , , ,1,

,2 2

2,

(a)7 3 (b)3 44 (c)3

0 (d)/ (e)

/ (f)

nnn n

nnn n

nnn n

n n n n n n n n n nnnmn n

nnmn n

nnmn n

l RR ll l RR

k Kn R

m RR l

ds pg dps g dpp

ds d

s p gd

= = - + = - +

= = = = === -

= - +

L

L

L

L L L L LSS

S (g)

(35)

Other sub-matrices can be similarly obtained from the remaining equations in Eqs. (31).In actual numerical calculations, all the series expansions will have to be truncated to m=Mand n=N. Thus there is a total number of (M+1)(3N+2)+8N+6 unknown expansion coefficients in the displacement functions. Since Eq. (33) represents a set of 8N+6 equations, additional (M+1)(3N+2) equations are needed to be able to solve for all the unknown coefficients.Accordingly, we will turn to the governing differential equations.In Flgges theory, the equations of motion are given as

21 212

212 2 2 122 2

2 2 2 2 21 12 21 2 22 2 2 2

(a)

(b)

(c)

N N uhx R tN N M M vhx R R xR tM M M M N whR x R x Rx R t

rqrq q

rq q q

+ = + + + =

+ + + - =

(36)

Substituting Eqs. (6) and (37) into Eqs. (34) results in


2 222 20 0 0

0 0 023

0 0

(1 )(1 ) (1 )(1 )cos ( )2 2(1 ) sin ( )2

(1 ) sin2

um m mn n

m n nv

m m mn nm n n

m m m m mnm n

wn

n nA xR Rn B n xR

nR CR R

R

xx

x

s g s g

s

s s gg

s

l ll l

l l l l

= = =

= = =

= =

T

- + - +- - + - + + - +

-- + - +

L a

L a

L b2

02

0 0 0

0 0 02 220 0

(a)

(1 )( ) ( ) ( )2cos ( ) 0

(1 ) sin ( )2(1 )(1 3 ) cos2

nu

m mn nm n n

um m mn n

m n n

m m mnm n

vn

nx R x xRh A xK

n A n xRn BRn

xx

x

s gg

w r

s

s g

ll l

l l

=

= = =

= = =

= =

T

- - - + + =

+ + - - +- +

-

b b

L a

L a

L220

220 0

202

0 0 02 2

(1 )(1 3 )( ) ( )2 (b)(3 ) cos2

(3 )( ) ( )2cos ( ) 0

(1 ) s2

n

m m mnm n

wn

nv

m mn nm n n

m m

x xRn CRn nx xR

h B xKRR R

n x

xn

s g

s g

s g

w r

s s g g

l l

ll l

=

= = T=

= = =

- + - -- + - - - + + = -- - +

a a

L b b

L a

0 02

02

20 0

202 2 2 4 2 220

in

(1 )( ) ( ) ( )2(3 ) cos2

(3 )( ) ( )21 ( 1) 2 cos

m mnm n

un

n

m m mnm n

un

n

m m m mnm n

A

x x R xR Rn n BR

n nx xRn R CR

xn

x

n x

s s g gs g

s g

g g g

l

l l

l l l

= = T=

= = T=

= =

- + - - -+ + - + - + ++ + +

L a a a

L a a

02 2 2 220

2

0 0 0

(c)

1 ( 1) ( ) 2 ( ) ( )

cos ( ) 0

wn

nw

m mn nm n n

n x x R xRh C xK

nx

g g g

w r l

T=

= = =

+ + + - + - + =

L b b b

L b

(37)


219

By expanding all non-cosine terms into Fourier cosine series and comparing the like terms,the following matrix equation can be obtained

+ + = 0.2

( )hKr w-Ex Fy Px Qy (38)

where E, F, P and Q are coefficient matrices whose elements are given as:

{ }{ }{ }{ }{ }

11

12

13

21

22

222,

,23

,

,

(1 )(1 ) (a)2(1 ) (b)2

(1 ) (c)2(1 ) (d)2

m mm nnmn m n

mm mm nnmn m n

mm m mm nnmn m n

mm mm nnmn m n

nR

m nlR

m m nRlR lRm n

lR

s gl d ds p c d d

s p s pg l c d ds p c d d

- += - + += -

- = - + - +=

E

E

E

E

E

{ }{ }{ }

23

31

32

222,

22,

2 3,

22,

(1 )(1 3 ) (e)2(3 ) (f)2

(1 ) (g)2(3 )

2

m mm nnmn m n

m mm nnmn m n

mm m m m mm nnmn m n

m mmmn m n

nR

nnR

n RR Rnn

R

n g l d d

n g l d d

nl s lg l c d d

s l g d

- + = - + - = - +

- = - + - - = +

E

E

E

{ }3322

2,

(h)

1 1 2 (i)

nn

m m mm nnmn m nnR RR

d

g l l d d

- = + + + E

(39)


{ }{ }{ }{ }{ }

11

11

12

12

13

21 1, 2

22 1, 2

1,

2,2 2

, 2

(1 )(1 ) (a)2(1 )(1 ) (b)2

(1 ) (c)2(1 ) (d)2

9 9 9(1 )8 1632

a m m nnmn n

b m m nnmn n

c m nnmn n

d m nnmn n

e mn n

nR

nR

nR

nR

R nR Rl

s gy f d

s gy f d

s j ds j ds p sg

-

-

-

-

-

- += -

- += -

+=+=

-= + -

F

F

F

F

F

{ }{ }

13

13

2 22 42

2 2 2 21 32 2,

2

,

9 (1 ) (e)8 16329 9 9(1 ) 9 (1 ) (f)8 16 8 1632 32

2

m m nn

f m m nnmn n

g mn n

R nR Rl

R n R nR R R Rl ll

s p sk g k d

s p s s p sg k g k d

sp

-

-

-- + -

- -= + - + + -

=

F

F

{ }13

2 2 2 2 22 42 2 2 2

2 2 2 2 2 21 3, 2 2 2 2

(1 ) 9 (1 ) (g)8 84 2 4(1 ) 9 (1 )

8 82 4 2 4

m m nn

h mmn n

R l n l R l nR R R R

l R l n l R l nR R R R

s s sg k g k dp p ps s s sg k g kp p p p

-

- -+ - - + -

- -= + - + + -

F

{ }{ }{ }{ }{ }

21

21

22

22

23

1,

2,2

1 1, 2

22 2, 2

, 2

(h)

(1 ) (i)2(1 ) (j)2

(1 )(1 3 ) (k)2(1 )(1 3 ) (l)2

9 9 (34

m nn

a m nnmn n

b m nnmn n

c m m nnmn n

d m m nnmn n

e mn n

nR

nR

nRnR

nlR

d

s j ds j d

s gf y d

s gf y d

pp

-

-

-

-

-

+= -+= -

- += - -

- += - -

= - +

F

F

F

F

F

{ }{ }

23

23

1 32

2 42 2,3 3

13 2 3 2,

) 3 (3 ) (m)32 32129 9 (3 ) 3 (3 ) (n)32 324 12

(3 ) 3(3 )8 3

m m nn

f m m nnmn n

g mmn n

n nl nl lR

nl n nl nl lR R

nl nl nl nR R

s g p s gk k dpp s g p s gk k dp p

s skpp p

-

-

- -- +

- -= + + +

- -= - + - +

F

F

{ }{ }{ }

23

31

31

3

3 32 4, 3 2 3 2

21 0, 2

2,

(o)8(3 ) 3(3 ) (p)8 83

(1 ) 6 (o)2(1 )

2

m nn

h m m nnmn n

a m m nnmn n

b mn n

l

nl nl nl nlR R

n RR R l

nR R

g k dps s gk k dp pp p

s s g gj d d

s s g

-

-

-

- -= + + +

-= - -

-= -

F

F

F

{ }{ }

{ }

32

32

33

2 02

1 1, 2

21 2, 2

22 2 21, 2 2 2 2

6 (r)

(3 ) (s)2(3 ) (t)2

9 1 1 1 94 34 2 4

m m nn

c m m nnmn n

d m m nnmn n

e mmn n

Rl

n nRn n

Rl R n l

RR l l R

gj d d

s gf y ds gf y d

p pg k gp p

-

-

-

-

-= -

-= -

-= + + + - +

F

F

F

{ }33

22 2 232 2

2 22 2 2 2 2 22 42 2 2 2 2 2,

1 9 (u)28 8

9 1 1 1 9 1 9 (v)4 3 24 2 4 8 8

m nn

f m m nnmn n

R nRl l

l R n l R nR RR l l R l l

p p k d

p p p pg k g k dp p

-

-+ +

- -= - + + + - + + +

F

{ } ( ) ( )

{ } ( )33

33

2 22 22 21 32 2 2 2,

222, 2 2

1 11 9 9 (w)4 2 3 4 2

1 14 2

g m m nnmn n

h mn n

n l n ll l R l l Rl R l RR R

n ll l Rl RR

p pg k g k dp p p pp p

pgp pp

-

-

- -= + + - - + + -

-= - + + -

F

F ( ) 2222 42 2 19 9 (x)3 4 2m m nnn ll l Rl RR pk g k dp pp -

- + + -

(40)


221

{ } { } { }{ } { }{ } { }{ }{ }{ }

11 22 33

11 22

11 22

33

33

33

, , ,1, ,2, ,

31,

42,33,

(a)(b)(c)

9 (d)4 39 (e)4 3

mm nnmn m n mn m n mn m na c m nnmn n mn nb d m nnmn n mn n

me m nnmn n

mf m nnmn n

g mn n

l

l

l

d df df d

kk dpkk dp

p

- - - -

-

-

-

= = - = -= = -= = -

= - = - +

=

P P PQ QQ Q

Q

Q

Q

{ }33

313 423,

(f)3(g)3

mm nn

mh m nnmn nl

kk dkk dp

-

- = - +

Q

(41)

The symbols1m, 2m, 3m, 4m, 1m, 2m, 1m, 2m, 1m, 2m, and mi in the above equations aredefined as

1 10 2

2 20

2

30

2 0,sin( ) cos ; (a,b)42 0,(1 4 )

2 0,cos( ) cos ; (c,d)4( 1)2 0,(1 4 )

3sin( ) co2

m m mm

m m mm

mm

mx xl m

mx xl m

xl

p pk l kp

pp k l kp

p k

=

=

=

== = -

== = - -

=

m

m

m

1

32

4 40

2

1 1 10

2 0,3s ; (e,f)12 0,(9 4 )2 0,33cos( ) cos ; (g, h)12( 1)2 0,(9 4 )

( ) cos ;

m m

m m mm

m m mm

mx

m

mx xl m

x x

pl kp

pp k l kp

a f l f

=

=

== - - == = - -

= =

m+

m

m

( )

( )

2 2

4 4

2 22 2 20

4 4

1 1

0,12 (i,j)2 6 6( 1) 0,

0,12( ) cos ; (k, l)2 ( 1) 6 6( 1) 0,

( ) cos

m

m mm m mm

m mm

l ml m mm

l mx x l m mm

x x

pp

a f l f pp

a j l

=

=

= - + -- - == = - + - -

=

( )

( )

10 2 2

2 2 20 2 2

1 1 10

0 0,; (m,n)4 2 ( 1) 0,

0 0,( ) cos ; (o,p)4 1 2( 1) 0,

1 ( ) cos ;

mm

mm m mm

m m mm

m

mmm

x xmm

lx x

jp

a j l jp

a y l y

=

=

== + - = = = + -

- = =

( )

( )

2 2

2 2 20

2 2

0,(q,r)12 1 ( 1) 0,

1 0,( ) cos ; (s, t)12 1 ( 1) 0,

sin cos sin cos ;

m

mm m mm

m i im i i i m m m

i m

m

mlmmlx xmlm

x x x x

p

a y l yp

l c l l c l c

=

= - + - = = = - + -

= = = =

2 2

0 0,1 ( 1) 0, (u,v) 0.2 ( 1) 1 0,( )

i

m i

imi ii mm i

p

p+

= - - = - - -

(42)


All the unmentioned elements in matrices P and Q are identically equal to zero.Equations (32) and (36) can be combined into

= 0,2h

Kr w -

K M x (43)

where K = E + F L -1S andM = P + QL -1S .The final system of equations, Eq. (19) or (41), represents a standard characteristic equationfor a matrix eigen-problem from which all the eigenvalues and eigenvectors can be readilycalculated. It should be mentioned that the elements in each eigenvector are actually the expansion coefficients for the corresponding mode; its physical mode shape can be directlyobtained from Eqs. (7) or (27).In the above discussions, the stiffness distribution for each restraining spring is assumed tobe axisymmetric or uniform along the circumference. However, this restriction is not necessary. For non-uniform elastic boundary restraints, the displacement expansions, Eq. (27),shall be used, and any and all of stiffness constants can be simply understood as varyingwith spatial angle . For simplicity, we can universally expand these functions into standardcosine series and modify Eq. (31) accordingly to reflect this complicating factor.

3. Results and discussionSeveral numerical examples will be given below to verify the two solution strategies described earlier.

3.1. Results about the approximate Rayleigh-Ritz solutionWe first consider a familiar simply-supported cylindrical shell. The simply supported boundary condition, Nx =Mx =v =w =0at each end, can be considered as a special case whenk2,6 =k3,7 = and k1,5 =k4,8 =0 (in actual calculations, infinity is represented by a sufficientlylarge number). To examine the convergence of the solution, Table 1 shows the frequency parameters, =R (1 2) / E , calculated using different numbers of terms in the series expansions. It is seen that the solution converges nicely with only a small number of terms. Inthe following calculations, the expansions in axial direction will be simply truncated toM=15. Given in Table 2 are the frequencies parameters for some lower-order modes. Exactsolution is available for the simply supported case and the results are also shown there forcomparison. An excellent agreement is observed between these two sets of results. Althoughthe simply supported boundary condition represents the simplest case in shell analysis, thisproblem is not trivial in testing the reliability and sophistication of the current solutionmethod. From numerical analysis standpoint, it may actually represent a quite challengingcase because of the extreme stiffness values involved. The non-trivialness can also been seen


223

mathematically from the fact that the simple sine function (in the axial direction) in the exactsolution is actually expanded as a cosine series expansion in the current solution.

Number of terms used in theseries

=R (1 2) / En=0 n=1 n=2 n=3 n=4

M=5M=7M=9M=10

0.4646520.4646490.4646480.464648

0.2573890.2573860.2573850.257385

0.1271320.1271290.1271280.127128

0.1433290.1433270.1433270.143327

0.2348230.2348220.2348220.234822

Table 1. Frequency parameters, =R (1 2) / E , obtained using different numbers of terms in the displacementexpansions.

Mode =R (12) / E

n=0 n=1 n=2 n=3 n=4m=1, CurrentExactm=2, CurrentExactm=3, CurrentExact

0.4646480.4646480.9289070.9289070.9481720.948172

0.2573850.2573850.5741790.5741760.7643750.764355

0.1271280.1271280.3376520.3376490.5329510.532923

0.1433270.1433270.2488130.2488100.3998930.399865

0.2348220.2348220.2856200.2856190.3836880.383667

Table 2. Frequency parameters, =R (1 2) / E , for a simply-supported shell; L=4R, h/R=0.05 and =0.3.

Next, consider a cylindrical shell clamped at each end, that is,u =v =w =w / x =0. Theclamped-clamped boundary condition is a case when the stiffnesses of the restrainingsprings all become infinitely large. The related shell and material parameters are as follows:L=0.502 m, R=0.0635 m, h=0.00163 m, E=2.11011, =0.28, and =7800. Listed in Table 3 aresome of the lowest natural frequencies for this clamped-clamped shell. The reference resultsgiven there are calculated from

6 4 22 1 0 0A A AW - W + W - = (44)

where =R (1 2) / E , and the coefficients A0, A1 and A2 are the functions of the modalindices, shell parameters, and the boundary conditions [27]. Equation (42) can be derivedfrom the Rayleigh-Ritz procedure by adopting the beam characteristic functions as the axialfunctions for all three displacement components. A noticeable difference between these twosets of results may be attributed to the fact that: a) Eq. (42) given in ref. [29] is based on theFlgge shell theory, rather than the Donnell-Mushtari theory, and b) Eq. (42) uses only a sin


gle beam characteristic function in contrast to the three complete sets (of basis functions) inthe current method.

Mode Currentm=1 Eq. (42)Currentm=2 Eq. (42)

n=12345

1886.74934.220982.2651598.552484.78

2035.05971.531990.3391600.902486.49

3854.752039.661454.801769.542572.31

4302.052189.591500.071782.282578.07

Table 3. The natural frequencies in Hz for a clamped-clamped shell; L=0.502 m, R=0.0635 m, h=0.00163 m, E=2.1E+11,=0.28, =7800 kg/m3.

Mode n=2 n=3 n=4 n=5 n=6 n=7Translation:

current(2.130) [22]

FEARotation:Current

(2.132) [22]FEAm=1:

CurrentFEA

0.004130.003100.003100.019070.003430.003430.240750.23810

0.009860.008760.008760.016760.009240.009230.131900.12836

0.017920.016800.016790.020680.017340.017310.083430.07938

0.028300.027170.027140.029950.027740.027690.062920.05893

0.040990.039860.039800.042200.040450.040370.059060.05555

0.055990.054870.054750.057130.055460.055330.066060.06332

Table 4. Frequency parameters, =R (1 2) / E , for a free-free shell; R=0.5 m, L=4R, h=0.002R, and =0.28.

Another classical example involves a completely free shell. Vibration of a free-free shell is ofparticular interest as manifested in the debate between two legendary figures, Rayleigh andLove, about the validity of the inextensional theory of shells. The lower-order modes aretypically related to the rigid-body motions in the axial direction. Theoretically, the Hw matrix given in Eqs. (14) will become non-invertible for a completely free shell. However, thisnumerical irregularity can be easily avoided by letting one of the bending-related springshave a very small stiffness, such as,k^ 4 =106. Table 4 shows a comparison of the frequencyparameters calculated using different techniques. While the results obtained from the current technique agree reasonably well with the other two reference sets, perhaps within thevariance of different shell theories, the frequency parameters for the two lower order modeswith rigid-body rotation (n=2 and 3) are clearly inaccurate which probably indicates that theinability of exactly satisfying the shell boundary conditions by the beam functions tends


225

to have more serious consequence in such a case. Amazingly enough, the inextensional theory works very well in predicting the frequency parameters for the rigid-body modes(those with rigid-body motions in the axial direction). It is also seen that the frequency parameters of the rigid-body modes increases monotonically with the circumferential modalindex n.After it has been adequately illustrated how the classical boundary conditions can be easilyand universally dealt with by simply changing the stiffness values of the restraining springs,we will direct our attention to shells with elastic end restraints. For the purpose of comparison, the problems previously studied in ref. [20] will be considered here. It was observed inthat study that the tangential stiffness had the greatest effect on the natural frequency of thecylinder supported at both ends while the axial boundary stiffness had the greatest influence on the natural frequency of the cylinder supported at one end. It was also determinedthat natural frequencies varied rapidly with the boundary flexibility when the non-dimensionalized stiffness is between 10-2 and 102.The frequency parameters for the clamped-free shell are shown in Table 5 for the reducedaxial stiffness k^ 1L (1 2)=1 (corresponding to ku* =1 in ref. [20]). It is seen that the currentresults are slightly larger than those taken from ref. [20]. The possible reasons include: 1) thedifference in shell theories (the Flgge theory, rather than the Donnell-Mushtari, was usedthere), and 2) different Poisson ratios may have been used in the calculations.

Mode n=0 n=1 n=2 n=3 n=4 n=5

m=1m=2

0.97521.22044

0.5146861.12788

0.32866(0.315*)1.08573

0.3610361.10467

0.532604(0.498)

1.16021

0.7826611.432

Note: the numbers in parentheses are taken from ref. [20]

Table 5. Frequency parameters, =R (1 2) / E , for a clamped-free shell; R=0.00625 m, L=R, h=0.1R, =0.28,andk^1L (1 2) = 1.

Although all eight sets of springs can be independently specified here, for simplicity we willonly consider a simple configuration: a cantilevered shell with an elastic support being attached to its free (right) end in the radial direction. Listed in Table 6 are the four lowest natural frequencies for several different stiffness values. Obviously, the cases for k^ 7 =0 and representthe clamped-free and clamped-simply supported boundary conditions, respectively.The mode shapes for the three intermediate stiffness values are plotted in Figs. 2-4. It is seenthat the modal parameters can be significantly modified by the stiffness of the restrainingsprings. The four modes in Fig. 2 for k^ 7 =0.01 m-1 closely resemble their counterparts in theclamped-free case, even though the natural frequencies have been modified noticeably.While all the first four natural frequencies happen to increase, more or less, with the springstiffness, the modal sequences are not necessarily the same. For example, when the spring


stiffness k^ 7 is increased from 0.01 to 0.1 m-1, the third natural frequency goes from 886.66 to926.17 Hz. However, this frequency drift may not necessarily reflect the direct effect of thestiffness change on the (original) third mode. It is evident from Figs. 2 and 3 that the thirdand fourth modes are actually switched in these two cases: the original third mode now becomes the fourth at 1200.88 Hz. It is also interesting to note that while stiffening the elasticsupport k^ 7 (from 0.01 to 0.1 m-1) has significantly raised the natural frequencies for the firsttwo modes, the fourth mode is adversely affected: its frequency has actually dropped from1023.61 to 926.17 Hz (see Figs. 2 and 3). A similar trend is also observed between the fourthmode for k^ 7=0.1 m-1 and the second mode for k^ 7=1 m-1, as shown in Figs. 3 and 4.

Mode k^ 7 = 0 k^ 7 = 0.01 k^ 7 = 0.1 k^ 7 =1 k^ 7 =1010

1234

404.108487.598865.6031003.38

451.242513.222886.6561023.61

627.345679.082926.1731200.88

729.593935.7451084.91,

1319.99, ,

742.920936.7191269.581333.37

Table 6. Natural frequencies in Hz for a clamped-elastically supported shell; L=0.502 m, R=0.0635 m, h=0.00163 m,E=2.1E+11, =0.28, =7800 kg/m3;k^5 = k^6 = k^8 = 0.

Figure 2. First four modes for the clamped-elastically supported shell; k^7 = 0.01 m-1andk^5 = k^6 = k^8 = 0.


227

Figure 3. First four modes for the clamped-elastically supported shell; k^7 = 0.1 m-1andk^5 = k^6 = k^8 = 0.

Figure 4. First four modes for the clamped-elastically supported shell; k^7 = 1 m-1andk^5 = k^6 = k^8 = 0.


3.2. An exact solution based on the Flgges equationsTo validate the exact solution method, the simply supported shell is considered again. Givenin Table 7 are the calculated natural frequency parameters =R (1 2) / E . The currentresults agree well with the exact solutions based on Flgges theory [30], solutions based onbeam functions [31] and three-dimensional linear elasticity solutions [30].

h/R n =R (12) / E

Ref. [30]a Ref. [31] Ref. [30]b Present

0.05

0 0.0929586 0.0929682 0.0929296 0.09295901 0.0161065 0.0161029 0.0161063 0.01610642 0.0393038 0.0392710 0.0392332 0.03930353 0.1098527 0.1098113 0.1094770 0.10984684 0.2103446 0.2102770 0.2090080 0.2103419

0.002

0 0.0929296 0.0929298 0.0929296 0.09292991 0.0161011 0.0161011 0.0161011 0.01610232 0.0054532 0.0054530 0.0054524 0.00545473 0.0050419 0.0050415 0.0050372 0.00504274 0.0085341 0.0085338 0.0085341 0.0085344

a Exact solutions based on Flgges theory.b Three-dimensional linear elasticity solutions.

Table 7. Comparison of values of the natural frequency parameter =R (1 2) / E for a circular cylindrical shellwith simply supported boundary conditions, m = 1, R/l = 0.05, = 0.3.

nm = 1 m = 2

FEM present difference (%) FEM present difference (%)0 3229.8 3230.3 0.015% 5131.4 5131.1 0.006%1 2478.6 2479.3 0.028% 4830.4 4830.6 0.004%2 269.20 269.30 0.037% 276.62 278.58 0.704%3 761.25 761.01 0.032% 770.99 771.62 0.082%4 1459.2 1458.6 0.041% 1469.6 1469.3 0.020%5 2359.4 2358.6 0.034% 2369.9 2369.0 0.038%

Table 8. Comparison of values of the natural frequency for a circular cylindrical shell with free-free boundaryconditions, L=0.502 m, R=0.0635 m, h=0.00163 m, =0.28, E=2.1E+11 N/m3, =7800 kg/m3.


229

The current solution method is also compared with the finite element model (ANSYS) forshells under free-free boundary condition. In the FEM model, the shell surface is divided into 8000 elements with 8080 nodes. The calculated natural frequencies are compared in Tables 8. An excellent agreement is observed between these two solution methods.In most techniques, such as the wave approach, the beam functions for the analogous boundary conditions are often used to determine the axial modal wavenumbers. While such anapproach is exact for a simply supported shell, and perhaps acceptable for slender thinshells, it may become problematic for shorter shells due to the increased coupling of the radial and two in-plane displacements. To illustrate this point, we consider relatively shorterand thicker shell (l=8R and R =39h). The calculated natural frequencies are compared in Table 9 for a clamped-clamped shell. It is seen that while the current and FEM results are ingood agreement, the frequencies obtained from the wave approach (based on the use ofbeam functions) are significantly higher, especially for the lower order modes.

nm = 1 m = 2

FEM Ref. [32] present FEM Ref. [32] Present0 3229.8 4845.5 3230.3 5146.0 8075.8 5139.81 1882.8 2350.2 1880.9 3850.7 4775.6 3848.92 899.59 985.48 898.18 2017.8 2303.4 2014.13 896.97 919.01 896.56 1390.9 1479.2 1388.94 1501.9 1517.45 1501.6 1676.4 1714.0 1676.05 2386.1 2402.05 2386.0 2472.5 2501.8 2472.6

Table 9. Comparison of the natural frequencies for a circular cylindrical shell with clamped-clamped boundaryconditions, L=0.502 m, R=0.0635 m, h=0.00163 m, =0.28, E=2.1E+11 N/m3, =7800 kg/m3.

The exact solution method can be readily applied to shells with elastic boundary supports.Since the above examples are considered adequate in illustrating the reliability and accuracyof the current method, we will not elaborate further by presenting the results for elasticallyrestrained shells. Instead, we will simply point out that the solution method based on Eqs.(27) is also valid for non-uniform or varying boundary restraint along the circumferential direction, which represents a significant advancement over many existing techniques.

4. ConclusionAn improved Fourier series solution method is described for vibration analysis of cylindrical shells with general elastic supports. This method can be easily and universally applied toa wide variety of boundary conditions including all the 136 classical homogeneous boundary conditions. The displacement functions are invariantly expressed as series expansionsin terms of the complete set of trigonometric functions, which can mathematically ensure


the accuracy and convergence of the present solution. From practical point of view, thechange of boundary conditions here is as simple as varying a typical shell or material parameter (e.g., thickness or mass density), and does not involve any solution algorithm andprocedure modifications to adapt to different boundary conditions. In addition, the proposed method does not require pre-determining any secondary data such as modal parameters for an analogous beam, or modifying the implementation algorithms to avoid thenumerical instabilities resulting from computer round-off errors. It should be mentionedthat the current method can be readily extended to shells with arbitrary non-uniform elasticrestraints. The accuracy and reliability of the current solutions have been demonstratedthrough numerical examples involving various boundary conditions.

AcknowledgmentsThe authors gratefully acknowledge the financial support from the National Natural ScienceFoundation of China (No. 50979018).

Author detailsTiejun Yang1, Wen L. Li2 and Lu Dai1

1 College of Power and Energy Engineering, Harbin Engineering University, Harbin, PRChina2 Department of Mechanical Engineering, Wayne State University, Detroit, USA

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Vibrations of Cylindrical Shells1. Introduction2. Basic equations and solution procedures2.1. An approximate solution based on the Rayleigh-Ritz procedure2.2. A strong form of solution based on Flgges equations

3. Results and discussion3.1. Results about the approximate Rayleigh-Ritz solution3.2. An exact solution based on the Flgges equations

4. ConclusionAcknowledgmentsAuthor detailsReferences