1122 Li et al.

/ J Zhejiang Univ SCI 2005 6A(10):1122-1127

Journal of Zhejiang University SCIENCE

ISSN 1009-3095
http://www.zju.edu.cn/jzus
E-mail: jzus@zju.edu.cn

Study on applicability of modal analysis of thin finite length

cylindrical shells using wave propagation approach*

LI Bing-ru (李冰茹)†1, WANG Xuan-yin (王宣银)†‡1, GE Hui-liang (葛辉良)2, DING Yuan-ming (丁渊明)1
(1State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China)
2
( State Key Laboratory of Oceanic Acoustics, Hangzhou Applied Acoustics Research Institute, Hangzhou 310012, China)
†
E-mail: libingru2000@yahoo.com.cn; xywang@zju.edu.cn
Received Nov. 24, 2004; revision accepted Mar. 31, 2005

Abstract: Donnell’s thin shell theory and basic equations based on the wave propagation method discussed in detail here, is used
to investigate the natural frequencies of thin finite length circular cylindrical shells under various boundary conditions. Mode
shapes are drawn to explain the circumferential mode number n and axial mode number m, and the natural frequencies are cal-
culated numerically and compared with those of FEM (finite element method) to confirm the reliability of the analytical solution.
The effects of relevant parameters on natural frequencies are discussed thoroughly. It is shown that for long thin shells the method
is simple, accurate and effective.

Keywords: Wave propagation, Natural frequency, Mode shape, Cylindrical shell

doi:10.1631/jzus.2005.A1122 Document code: A CLC number: TB532

INTRODUCTION modes of circular cylindrical shells in which trans-

verse deflections dominate; Chung (1981) used
Vibrations of cylindrical shells are of consider- Stokes’ transformation technique to obtain the natural
able importance as they are extensively used in in- frequencies of circular cylindrical shells with differ-
dustry, flight structures and marine crafts. The natural ent boundary conditions; Chakravorty and
frequencies and mode shapes are important sources of Bandyopadhyay (1995), Bouabdallah and Batoz
information for understanding and controlling the (1996), and Guo et al.(2002) used a finite element
vibration of these structures, so many papers on the method (FEM) to obtain the natural frequencies of the
prediction of the natural frequencies of cylindrical cylindrical shells; Callahan and Baruh (1999) pre-
shells have been published over the past years. sented a systematic procedure using the computa-
Many shell theories have been developed and tional power of existing commercial software pack-
various solution methods have been proposed. ages for obtaining the closed-form eigensolution for
Sharma (1974) investigated the natural frequencies of thin circular cylindrical shell vibrations; Zhang et
fixed-free circular cylindrical shells, and gave a de- al.(2001a) used wave propagation method to evaluate
tailed analysis for the case of the axial mode being the natural frequencies of finite cylindrical shells.
approximated by characteristic beam functions with This paper focuses mainly on Zhang et
appropriate end conditions; Soedel (1980) used a al.(2001a)’s method using an interesting technique
formula to calculate the natural frequencies and combining an exact frequency wavenumber charac-
teristics formula with appropriate beam functions in
the axial direction to give relatively more accurate
‡
Corresponding author
*
Project (No. 51446020203JW0401) supported by the State Key
predictions of circular cylindrical shells’ natural fre-
Laboratory of Oceanic Acoustics Foundation, China quencies. Although they highlighted the advantage of
 Li et al. / J Zhejiang Univ SCI 2005 6A(10):1122-1127 1123

the method, its applicability is still to be explored. In θ, z) are not independent of each other.
this paper, the effects of relevant parameters on
natural frequencies are thoroughly discussed. It is Applied wave propagation approach in cylindrical
shown that for the long-thin finite cylindrical shells shells
the method is more simple and effective than other In the wave propagation approach, the solution
methods. of Eq.(1) can be expressed in the form of traveling
wave form as:

THEORETICAL ANALYSIS u = U m cos[n(φ − α )]e j(ω t − k z z )

v = Vm sin[ n(φ − α )]e j(ω t − k z z ) (2)
Equation of cylindrical shells motion j(ω t − k z z )
The shell under consideration is shown in Fig.1. w = Wm cos[n(φ − α )]e

where u, v and w are the displacement components in

L
the axial, tangential and radial directions, respectively;
w
the coefficients Um, Vm and Wm in the equations are
u
the displacement amplitudes; α is an arbitrary angle,
z φ
v to account for the fact that there is no preferential
direction of the mode shape in the circumferential
direction; n is circumferential mode parameter (where
2n=the number of cross points in the radial dis-
placement shape); m is axial mode parameter (where
Fig.1 A circular cylindrical shell with relevant parameters m=the number of cross points in the radial displace-
ment shape along any axial generatrix); the meanings
For this analysis we will use the equations of of m and n are illustrated in Fig.2; ω is the angular
motion derived by Junger and Feit (1986). The equa- natural frequency for (m, n) vibration mode; kz is the
tions of motion for cylindrical shells can be written wavenumber in the axial direction.
as: For infinite length cylindrical shells, all the vi-
bration displacements are symmetric about φ, so the
∂ 2u 1 − µ ∂ 2u 1 + µ ∂ 2 v µ ∂w wavenumber in the circumferential direction can be
+ + + written as:
∂z 2 2a 2 ∂φ 2 2a ∂z∂φ a ∂z
1 − µ 2 ∂ 2u kφ=n/a, n∈N
= ρ 2
E ∂t
1 + µ ∂ 2 u 1 − µ ∂ 2 v 1 ∂ 2 v 1 ∂w Substituting Eq.(2) into Eq.(1) yields:
+ + +
2a ∂z∂φ 2 ∂z 2 a 2 ∂φ 2 a 2 ∂φ
 L11 L12 L13  U m  0 
1 − µ 2 ∂ 2v L    
= ρ 2
 21 L22 L23  Vm  = 0  (3)
E ∂t
 L31 L32 L33  Wm  0 
µ ∂u 1 ∂v w  ∂4 w ∂4 w 1 ∂4 w 
+ 2 + 2 + β 2  a2 4 + 2 2 2 + 2 4 
a ∂z a ∂φ a  ∂z ∂z ∂φ a ∂φ  The items of this matrix can be expressed as:
1− µ ∂ w
2 2
=− ρ 2 1− µ 2 
E ∂t 
L11 = K  k z 2 + kφ  − ρ hω 2 ,
(1)  2 
K (1 + µ ) Kµ
where β2=h2/(12a2); a is the radius of the cylinder; ρ L12 = L21 = k z kφ , L13 = L31 = kz ,
2 a
is the density of the material; h is the thickness of the
shells; E is Young’s modulus; and µ is Poisson’s ratio. 1− µ 2 
L22 = K  k z + kφ 2  − ρ hω 2 ,
The vibration displacements in the three directions (r,  2 
1124 Li et al. / J Zhejiang Univ SCI 2005 6A(10):1122-1127

Expanding Eq.(4) yields the following polyno-

mial for the natural frequencies:

ω 6 + a1ω 4 + a2ω 2 + a3 = 0 (5)

where
n=0 m=0
1
a1 = − [c1 + c2 + c3 ],
ρh
1 c1c3 + c2 c3 + c1c2 
a2 =  ,
( ρ h) 2  − L12 2 − L232 − L132 
1 c1 L23 + c2 L13 + c3 L12 
2 2 2
n=1 m=1 a3 =  ,
( ρ h)3  −c1c2 c3 − 2 L12 L13 L23 
c1 = L11 + ρ hω 2 , c2 = L22 + ρ hω 2 , c3 = L33 + ρ hω 2 .

In this paper, the software ANSYS was used to

perform the finite element analysis.
n=2 m=2 Fig.2 and Fig.3 were calculated by ANSYS.
Fig.2 presents circumferential nodal pattern and axial
nodal pattern respectively. The results can be used to
explain the parameter n and parameter m. Fig.3 shows
some typical combined mode shapes.

n=3 m=3

n=4 m=4 (a) (b)

(a) (b)

Fig.2 Illustration of parameter n and m

(a) Circumferential nodal pattern; (b) Axial nodal pattern

K K
L23 = L32 = kφ , − ρ hω 2 .
L33 = Dk 4 +
a a2
Eh Eh3 (c) (d)
where k = k z 2 + kφ 2 , K = , D= .
1− µ 2
12(1 − µ 2 ) Fig.3 Some typical combined mode shapes
(a) n=1, m=1; (b) n=0, m=2; (c) n=1, m=2; (d) n=3, m=1
To obtain the non-trivial solution of Eq.(3), the
determinant of the characteristic matrix in Eq.(3) From the figures, the typical modes can be de-
must be zero: scribed as follows:
When n=0, the circumferential nodal pattern is a
L11 L12 L13
circle, indicating that this mode is an extensional
L21 L22 L23 = 0 (4) mode referred to as breathing type mode.
L31 L32 L33 The mode is a pure radial mode when m=0. Here,
 Li et al. / J Zhejiang Univ SCI 2005 6A(10):1122-1127 1125

the cylinder retains a constant cross-sectional shape

along its length.
When m, n are both equal to one, the mode is a
circumferential mode. When n=1 and m≠1, the mode

Ω
is an axial bending mode and the mode is radial mo-
tion with shearing mode when m=1 and n≠1.

Beam functions
In this paper, the wave propagation method was
used in conjunction with beam functions. The natural (a)
frequencies of finite length cylindrical shell with
different boundary conditions can be obtained.
Zhang et al.(2001b)’s wavenumbers for different
boundary conditions of beams are listed in Table 1.
Substitution of beam functions into motion Eq.(5),

Ω
yields different approximate mode shapes and natural
frequencies.

Table 1 Wavenumbers for different boundary conditions

Boundary conditions Wave numbers
Clamped-free kL=(2m−1)π/2 (b)
Free-simply supported kL=(4m+1)π/4
Simply supported-simply supported kL=mπ
Clamped-simply supported kL=(4m+1)π/4
Clamped-clamped kL=(2m+1)π/2
Sliding-simply supported kL=(2m−1)π/2
Ω

Free-free kL=(2m+1)π/2

RESULTS AND DISCUSSION

(c)
To check the validity of the present analysis, the
results were compared with those calculated by FEA.
The non-dimensional frequency parameter Ω
was used to make the conclusions more widely ap-
plicable. Here define Ω=ω/ωr, where
Ω

1 E
ωr = .
a ρ (1 − µ 2 )

Relationship between the natural frequencies and

the circumferential mode number n with different (d)
radius-thickness ratio a/h
In the computation of Fig.4, the material used is Fig.4 Variation of the non-dimensional frequencies with
the parameter n for different parameter m
aluminum with mass density ρ of 2700 kg/m3, the (a) a/h=10; (b) a/h=20; (c) a/h=25; (d) a/h=30
Poisson ratio µ is equal to 0.33, and Young’s modulus
E=7.1×1010 N/m3. The boundary condition consid-
ered is clamped-clamped. ter Ω versus circumferential wave number n with
Fig.4 shows non-dimensional frequency parame- different radius-thickness ratio a/h. The solid curves
1126 Li et al. / J Zhejiang Univ SCI 2005 6A(10):1122-1127

correspond to the results obtained by using the present From this section, we know that the present
method and curves marked “∆”, “□” and “○” corre- method can be used to evaluate the natural frequen-
spond to those from FEM. cies of thin cylindrical shells.
These shells all have the same thickness h=0.01
and the same length L=1. Relationship between the natural frequencies and
From these plots, the following observations can the axial mode number m with different
be made: length-thickness ratio L/a
1. We can find that, for the same thickness h, the The results are plotted in Fig.5 to study the effect
smaller the shell radius-to-thickness ratio a/h is, the of the axial length on the modes of the radius.
larger is the difference between the results from FEM
and those from the present method. Figs.4a, 4b and 4c
show the results from FEM are lower than those from
the present method. This can be attributed to the fact
that the effects of shear deflection and rotary inertia of
the shell (which would reduce the natural frequencies)

Ω
should not be neglected for small a/h (Soedel, 1982).
As the above effects are not taken into account by
Eq.(5), it may be expected that Eq.(5) can be applied
to thin finite cylindrical shells.
2. The relative error decreases with decreasing
axial mode number m. It means that the results are m
more exact when the shells are longer. This indicates Fig.5 Variation of the non-dimensional frequencies with
that, for the long-thin shell, the effect of the boundary the parameter m for different length-thickness ratio L/a
while n=2
conditions are small, the wave propagation in the
cylindrical shell trends to the form of an approaching
wave. It was found that, when L/a becomes larger, the
3. When the ratio a/h=30, the results from the natural frequencies of shells become smaller for the
method are more accurate than the results when ratio same m, because larger L/a ratio leads to smaller shell
a/h<30. And we also can find that higher order modes rigidity. The curves’ variations level off with in-
led to more accurate results. All these phenomena creasing L/a ratio, that is to say, the natural frequency
indicate that although the coupling of the vibration is more sensitive to the geometric sizes when the
between the axial and circumferential direction was cylindrical shell is short. It indicates that the method
neglected in this method, the effects of such coupling is more effective for long cylindrical shells.
were less important for long-thin shells and higher
order modes.
4. The lowest frequency does not occur at the CONCLUSION
lowest values of n, and for different values of m, the
lowest frequency occurs at different mode. For ex- Donnell’s shell theory and wave propagation
ample, at a/h=30 and m=1, the lowest frequency oc- method were applied to analyze the free vibration
curs at the mode when n=3, this phenomenon can be characteristics of long-thin finite circle cylindrical
explained by considering the strain energy of the shells. The curves of the relation between the pa-
middle surface under both bending and stretching rameter n and the shell radius-to-thickness ratio a/h,
(Kraus, 1967). as well as between the parameter m and the shell
5. For the larger circumferential mode number n, length-to-radius ratio L/a are numerically presented
the curves change dramatically, in other words, the and some important conclusions can be obtained from
natural frequency is sensitive to the geometric sizes them.
when the ratio a/h is small. It indicates that the The results from the present paper compared
method is more effective for thin cylindrical shells. with the solutions obtained from FEM showed that
 Li et al. / J Zhejiang Univ SCI 2005 6A(10):1122-1127 1127

the method is effective for long thin cylindrical shells. Guo, D., Zheng, Z.C., Chu, F.L., 2002. Vibration analysis of
As far as the applications are concerned, the results spinning cylindrical shells by finite element method. In-
ternational Journal of Solids and Structures, 39:725-739.
obtained can commendably satisfy the criterion of
Junger, M.C., Feit, D., 1986. Sound, Structures, and Their
precision. The method can be extended to long thin Interaction, Second Edition. The MIT Press.
ring-stiffened cylindrical shells and some long-thin Kraus, H., 1967. Thin Elastic Shells. John Wiley, New York.
shell structures with complicated boundary condi- Sharma, C.B., 1974. Calculation of natural frequencies of
tions. fixed-free circular cylindrical shells. Journal of Sound
and Vibration, 35(1):55-76.
Soedel, W., 1980. A new frequency formula for closed circular
References
cylindrical shells for a large variety of boundary condi-
Bouabdallah, M.S., Batoz, J.L., 1996. Formulation and
tions. Journal of Sound and Vibration, 70(3):309-317.
evaluation of a finite element model for the linear analysis
Soedel, W., 1982. On the vibration of shells with Ti-
of stiffened composite cylindrical panels. Finite Elements
moshenko-Mindlin type shear deflections and rotatory
in Analysis and Design, 21:265-289.
inertia. Journal of Sound and Vibration, 83(1):67-79.
Callahan, J., Baruh, H., 1999. A closed-form solution proce-
Zhang, X.M., Liu, G.R., Lam, K.Y., 2001a. Vibration analysis
dure for circular cylindrical shell vibrations. International
of thin cylindrical shells using wave propagation ap-
Journal of Solids and Structures, 36:2973-3013.
proach. Journal of Sound and Vibration, 239(3):397-403.
Chakravorty, D., Bandyopadhyay, J.N., 1995. On the free
Zhang, X.M., Liu, G.R., Lam, K.Y., 2001b. Frequency analy-
vibration of shallow shells. Journal of Sound and Vibra-
sis of cylindrical panels using a wave propagation ap-
tion, 185(4):673-684.
proach. Applied Acoustics, 62:527-543.
Chung, H., 1981. Free vibration analysis of circular cylindrical
shells. Journal of Sound and Vibration, 74(3):331-350.

Welcome visiting our journal website: http://www.zju.edu.cn/jzus

Welcome contributions & subscription from all over the world
The editor would welcome your view or comments on any item in the
journal, or related matters
Please write to: Helen Zhang, Managing Editor of JZUS
E-mail: jzus@zju.edu.cn Tel/Fax: 86-571-87952276