ARTICLE IN PRESS

JOURNAL OF
SOUND AND
VIBRATION
Journal of Sound and Vibration 294 (2006) 927–943
www.elsevier.com/locate/jsvi

The natural vibration of a conical shell with an annular end plate

Sen. Liang, H.L. Chen
Institute of Vibration and Noise Control, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China
Received 23 November 2004; received in revised form 27 June 2005; accepted 17 December 2005
Available online 9 March 2006

Abstract

In this paper, natural frequencies and mode shapes for a conical shell with an annular end plate or a round end plate are
investigated in detail by combining the vibration theory with the transfer matrix method. The governing equations of
vibration for this structure are expressed in terms of a matrix differential equation, and a novel recurrence formula method
that can close in on analytical solution of transfer matrix is presented. Once the transfer matrix of single component has
been determined, and the product of each component matrix and the joining matrix can obtain entire structure matrix. The
frequency equations and mode shape functions are represented in terms of the elements of the structural transfer matrices.
The 3D finite element numerical simulations have verified the present formulas of natural frequencies and mode shapes.
The conclusions show that the transfer matrix method can accurately obtain natural vibration characteristics of the conical
shell with an annular end plate.
r 2006 Elsevier Ltd. All rights reserved.

1. Introduction

A conical shell with an annular end plate is a kind of useful structure in the aeronautical, aerospace and
civil industries. Nowadays, the shell structure becomes larger and thinner and its vibration problem becomes
more and more important than before. Therefore, the wide engineering applications of the shell structures
have attracted considerably researchers’ interest and many methods for investigating their dynamical
characteristics have been promoted. Presently, many researchers concentrate their investigations on dynamic
characteristics of a conical shell, a thin-walled round plate and a combination of cylindrical shell and
round plates [1–12], but there are few research reports relative to this system of the conical shell with an
annular end plate.
Transfer matrix method has been utilized in engineering applications for many years. Holzer originally
employed the transfer matrix approach for an approximate solution of the differential equation governing the
torsional vibration of rod in 1921 and the method is generally known as Holzer’s method. Myklestad
presented an approach quite analogous to Holzer’s for the treatment of beam [13]. The relative equations were
rearranged and simplified by Thomson to permit a systematic tabular computation and to expand this method
to more general problems [14]. One of the earliest applications of transfer matrix method was the steady-state
description of four terminal electrical networks, in which the method is commonly designated as a ‘‘four-pole
Corresponding author. Tel.: +86 29 82663186; fax: +86 29 82660487.
E-mail address: liangsen98@mailst.xjtu.edu.cn (S. Liang).

0022-460X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jsv.2005.12.033
 ARTICLE IN PRESS
928 S. Liang, H.L. Chen / Journal of Sound and Vibration 294 (2006) 927–943

parameters’’. Rubin et al. [15–17] systematically applied four-pole parameters approach to acoustical,
mechanical, and electromechanical vibrations. Murthy, Kalnins, Irie et al. [18–26] extended the applications of
transfer matrix approach to a rotor blade, plate, symmetric shell and stiffened ring through a completely
general treatment.
In this paper, natural frequencies and mode shapes of a conical shell with an annular end plate or a
round plate are investigated in detail by combining the vibration theory with the transfer matrix method.
The governing equations of vibration for a conical shell and an annular end plate are expressed in
terms of matrix differential equations, and a new recurrence formula method that can close in on analytical
solution of transfer matrix is presented. Once the transfer matrix of single component has been determined,
the product of each component matrix and the joining matrix can form the matrix of entire structure. The
frequency equations and mode shape functions are represented in terms of the matrix elements. 3D finite
element analysis has validated the present formulas of natural frequencies and mode shapes. The conclusions
show the transfer matrix obtained by present formulas can accurately reveal natural vibration characteristics
of the conical shell with an annular end plate.

2. Theoretical analysis

In order to investigate the dynamic characteristics of the combined shell, the coefficient matrices of a conical
shell and an annular end plate should be derived, respectively. Then the elements of transfer matrix can be
obtained by employing recurrence formula method. The frequency equations and mode shape functions are
represented in terms of the elements of the structural transfer matrices.

2.1. Coefficient matrix of the conical shell

Fig. 1 shows a combined structure of the conical shell with an annular end plate. a denotes the semi-vertex
angle of the truncated conical shell, h expresses the thickness of conical shell and annular plate, and H
is distance form the middle surface of annular plate to the large edge of conical shell. The radius of large edge
for the conical shell is R, and outer radius and inner radius of the annular plate are R1 and R2. The surface
co-ordinates (B; y; z) are taken in the middle surface as shown in Fig. 1. Based on the flÜgge theory [27],
the equations of flexural vibration for a thin-walled conical shell can be written in a differential equation of
the coefficient matrix A(B) as
dX ðBÞ=dB ¼ AðBÞ:X ðBÞ, (1)


here state vector X ðBÞ ¼ ū v w f M B V B N B S By is denoted by the dimensionless variables. u, v and w are
the displacements in circumferential, meridional and normal directions, respectively; f, N B , M B , V B and S By are
the bending slope, the membrane force, the bending moment, the Kevlin–Kirchhoff shearing force and shear

Fig. 1. The conical shell with an annular end plate.

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S. Liang, H.L. Chen / Journal of Sound and Vibration 294 (2006) 927–943 929

resultant, respectively. They are defined as follows:

1
fu v wg ¼ fu v wg; f̄ ¼ f; N B ¼ N B R2 =D; M B ¼ M B R=D,
R


  
V B S By ¼ V s SBy R2 =D; D ¼ Eh3 = 12ð1
n2 Þ , (2)
where D is flexural rigidity expressed by Young’s modulus E, Poisson’s ratio v and thickness h of the shell. In
order to simplify the formulas, the following dimensionless parameters are introduced:
 
B ¼ d=R; B1 ¼ d 1 =R; B2 ¼ d 2 =R; B3 ¼ R1 =R; B4 ¼ R2 =R . (3)
The frequency parameter is described as
p4 ¼ rhR4 o2 =D, (4)
here r is the mass density, and o is the frequency in the rad/s. d is the distance form vertex to arbitrary point in
the middle surface of the conical shell. The non-zero elements of the coefficient matrix A(B) in Eq. (1) can be
derived as follows (a detailed derivation is shown in Appendix A):
1 n nh2 nh2
A11 ¼ ; A12 ¼ ; A13 ¼ 2
; A14 ¼
2
,
B B sin a 6B3 R tga sin a 6B2 R tga sin a

h2 nn n n h2
A18 ¼ ; A21 ¼
; A22 ¼
; A23 ¼
; A27 ¼ ,
6R2 ð1
nÞ B sin a B Btga 12R2

nn2 n ð3 þ nÞð1
nÞn2

A34 ¼ 1; A43 ¼ ; A44 ¼
; A45 ¼ 1; A53 ¼
,
B2 sin2 a B B3 sin2 a
   
2n2 ð1
nÞ 1
n 12 1
n2 nR2
A54 ¼ 1þnþ 2 ; A55 ¼
; A56 ¼ 1; A61 ¼
,
sin a B2 B h2 B2 tga sin a
   

12 1
n2 R2 4 12 1
n2 R2 ð1 þ nÞn2 ð1
nÞn2
A62 ¼
; A63 ¼ p

2þ ,
h2 B2 tga h2 B2 tg2 a sin a B4 sin2 a

ð3 þ nÞð1
nÞn2 nn2 1 n

A64 ¼ ; A65 ¼ ; A66 ¼
; A67 ¼
,
B3 sin2 a B2 sin2 a B Btga
   
12 1
n2 R2 4 12 1
n2 R2
A71 ¼ ; A72 ¼
p þ ,
h2 B2 sin a h2 B 2

12ð1 þ nÞR2 n2 1
n ð1
nÞn2 1
n

A73 ¼ 2

2
; A74 ¼ ; A77 ¼
,
h B sin a 2 tga
2 B B3 sin2 atga B
   
n 4 12 1
n2 n2 R2 12 1
n2 nR2
A78 ¼
; A81 ¼
p þ ; A82 ¼ ,
B sin a B2 h2 sin2 a B2 h2 sin2 a


12ð1 þ nÞR2 1 ð1 þ nÞn2 ð1
nÞn ð2 þ nÞð1
nÞn
A83 ¼ 2
þ 2þ 2 2 tga sin a
; A84 ¼
,
h n 2
n sin a B B3 tga sin a

nn nn 2
A85 ¼
; A87 ¼ ; A88 ¼
. (5)
B2 tga sin a B sin a B
here n is the circumferential wavenumber.
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2.2. Coefficient matrix of the annular plate

For an annular end plate, the governing

  equations
 can be derived as a special case of a conical shell by
taking the limiting values 1=B sin a ! 1=B ; 1=Btga ! 0. The matrix equation has the same expression as
Eq. (1). In this case, the non-zero elements of the coefficient matrix A(B) become:

1 n h2 nn n h2
A11 ¼ ; A12 ¼ ; A18 ¼ ; A21 ¼
; A22 ¼
; A27 ¼ ; A34 ¼ 1,
B B 6R2 ð1
nÞ B B 12R2

nn2 n ð3 þ nÞð1
nÞn2 ð 1
nÞ

A43 ¼ ; A44 ¼
; A45 ¼ 1; A53 ¼
; A54 ¼ ð1 þ n þ 2n2 Þ ,
B2 B B3 B2

1
n   ð1
nÞn2 ð3 þ nÞð1
nÞn2

A55 ¼
; A56 ¼ 1; A63 ¼ p4
2 þ ð1 þ nÞn2 ; A64 ¼ ,
B B4 B3

   
nn2 1 12 1
n2 R2 4 12 1
n2 R2 1
n
A65 ¼ 2 ; A66 ¼
; A71 ¼ ; A72 ¼
p þ ; A77 ¼
,
B B h2 B 2 h2 B 2 B

   
n 4 12 1
n2 n2 R2 12 1
n2 nR2 nn 2
U 78 ¼
; A81 ¼
p þ ; A82 ¼ ; A87 ¼ ; A88 ¼
. (6)
B B 2 h2 B 2 h2 B B

2.3. The joining matrix

In order to investigate the dynamic characteristics of the combined structure, the conical shell and the
annular end plate should be connected together. Therefore, the following continuity conditions and
equilibrium relations must be satisfied:



X ðB1 þ 0Þ c ¼ ½Bp!c X ðB3
0Þ p , (7)
here the subscripts c and p denote conical shell and annular plate, respectively. The joining matrix B can be
given by
2 3
1 0 0 0 0 0 0 0
6 0 cos b
sin b 0 0 0 0 07
6 7
6 7
6 0 sin b cos b 0 0 0 0 07
6 7
60 0 0 1 0 0 0 07
6 7
½Bp!c ¼ 6 7, (8)
60 0 0 0 1 0 0 07
6 7
60 0 0 0 0 cos b
sin b 07
6 7
6 7
40 0 0 0 0 sin b cos b 05
0 0 0 0 0 0 0 1
 
here b ¼ p=2
a .

2.4. The transfer matrix

It is difficult to obtain the analytical solution of the coupled equations set (1) for a conical shell
or an annular end plate, so the transfer matrix approach is adopted here. In Refs. [18,19], the derivation
is only involved in zero-initial value (B0 ¼ 0). In this paper, the initial value has been extended to more
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S. Liang, H.L. Chen / Journal of Sound and Vibration 294 (2006) 927–943 931

general problems (non-zero initial value). Based on the definition of transfer matrix, the following
equation can be written as



X ðBÞ ¼ ½T ðBÞ X ðB0 Þ , (9)
here B0 denotes initial value. Differentiating Eq. (9) with respect to B yields
d
 d

X ðBÞ ¼ ½T ðBÞ X ðB0 Þ . (10)
dB dB
According to Eq. (9), the initial value of state vector can be obtained obviously



X ðB0 Þ ¼ ½T ðBÞ
1 X ðBÞ . (11)
Substituting Eq. (11) into Eq. (10), the following relation is given:
d
 d

X ðBÞ ¼ ½T ðBÞ½T ðBÞ
1 X ðBÞ . (12)
dB dB
By comparing Eq. (1) with Eq. (12), the following equation is derived:
 
d

½AðBÞ
½TðBÞ½T ðBÞ
1 X ðBÞ ¼ f0g. (13)
dB
For all values of B, the state vector X cannot be zero. The following equation can be easily written:
d
½AðBÞ ¼ ½T ðBÞ½T ðBÞ
1 . (14)
dB
Then, post-multiplying T(B) on both sides of Eq. (14) yields
d
½TðBÞ ¼ ½AðBÞ½TðBÞ. (15)
dB
Therefore, the transfer matrix is obtained by the solution of Eq. (15). If B equals to B0 in Eq. (9), the initial
condition will be
½TðB0 Þ ¼ ½1. (16)
Eq. (16) provides the sufficient initial conditions in order to solve the differential equations set (15). Thus,
the transfer matrix of entire structure can be obtained
X ðBÞB¼B2 ¼ T c Bp!c T p X ðBÞB¼B4 . (17)

2.5. Solution of the transfer matrix

Presently, the transfer matrix TðBÞ can be obtained by many ways. Kalnins [28] developed a multisegment,
direct, numerical integration approach, Cohen [29] presented an iteration method using approximate
eigenfunctions, Tottenham and Shimizu [30] used a matrix progression method, Sankar [31] showed an extend
transfer matrix method, and Irie et al. [22–26] employed Runge–Kutta–Gill method. Most of them are
approximate methods, and this paper demonstrates that the recurrence formula method that can close in on
analytical solution solves transfer matrix.

2.5.1. Recurrence formula

On the assumption that AðBÞ is continuous in the range [a, b], a and bX0, and TðBÞ is the only solution of
Eq. (15) under the initial condition (16), the series fT k ðBÞg can be constructed firstly
T 0 ðBÞ ¼ I.
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932 S. Liang, H.L. Chen / Journal of Sound and Vibration 294 (2006) 927–943

Z B
T 1 ðBÞ ¼ I þ AðsÞ:T 0 ðsÞ ds,
B1

..
.
Z B
T k ðBÞ ¼ I þ AðsÞ:T k
1 ðsÞ ds. (18)
B1

The difference between T kþ1 and T k is

Z B
T kþ1 ðBÞ
T k ðBÞ ¼ AðsÞ½T k ðsÞ
T k
1 ðsÞ ds. (19)
B1

Taking norm on both sides of Eq. (19) in normed linear space, the following equation is given
Z Z B
B
T kþ1 ðBÞ
T k ðBÞ ¼ AðsÞ½T k ðsÞ
T k
1 ðsÞ ds AðsÞ T k ðsÞ
T k
1 ðsÞ ds
p
B1 B1
Z B
pM T k ðsÞ
T k
1 ðsÞ ds
B
Z1 Z B
B
pM 2
ds T k
1 ðsÞ
T k ðsÞ ds
B1 B1
..
.
Z Z Z B
B B
pM n ds ds::: T 1 ðsÞ
T 0 ðsÞ ds, ð20Þ
B1 B1 B1

where
Z B Z B
T 1 ðBÞ
T 0 ðBÞ ¼ AðsÞ:T 0 ðsÞ ds ¼ AðsÞ ds. (21)
B1 B1

Thus
Z
B
T 1 ðBÞ
T 0 ðBÞ p AðsÞ ds pM:t, (22)
B1

here M ¼ maxB1 pBpb AðBÞ , and t is the length of integral region. By substituting Eq. (22) into Eq. (20), the
following equation can be obtained
kþ1
T kþ1 ðBÞ
T k ðBÞ pM kþ1 t . (23)
ðk þ 1Þ!
Summing this Eq. (23) on both sides, the relationship is
X1 X 1
ðMtÞkþ1
T kþ1 ðBÞ
T k ðBÞ p  eMt . (24)
k¼0 k¼0
ðk þ 1Þ!
P
1
Obviously, the series ðT k
1
T k Þ is consistent convergence and suppose its convergent result is TðBÞ
I.
k¼0
P
k
By reason of that ðT kþ1
T k Þ ¼ T kþ1
I; the following equation can be derived:
k¼0

lim T kþ1 ðBÞ ¼ TðBÞ.

k!1
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S. Liang, H.L. Chen / Journal of Sound and Vibration 294 (2006) 927–943 933

Taking the limits for recurrence Eq. (18) yields

Z B
TðBÞ ¼ I þ AðsÞ:TðSÞ ds: (25)
B1

Differentiating Eq. (25) with respect to B gives

dTðBÞ
¼ AðBÞTðBÞ.
dB
The result shows that the T k ðBÞ in Eq. (18) will be the solution of matrix Eq. (15) when k becomes large
enough.

2.5.2. The only solution of transfer matrix TðBÞ

On the assumption that TðBÞ and Y ðBÞ are both the solutions of Eq. (15), the following equations can be
written:
Z B Z B
TðBÞ ¼ I þ AðsÞ:TðsÞ ds and Y ðBÞ ¼ I þ AðsÞ:Y ðsÞ ds. (26)
B1 B1

The difference between TðBÞ and Y ðBÞ is

Z B
TðBÞ
Y ðBÞ ¼ AðsÞ½TðsÞ
Y ðsÞ ds. (27)
B1

Taking the norm on both sides of Eq. (27) gives

Z B


kT ðBÞ
Y ðBÞ p AðsÞ TðsÞ
Y ðsÞ ds. (28)
B1

Table 1
Eigenvalue equations of the conical shell with an annular end plate

Fp
Fc F p
Sc F p
Cc
     
 T 51 T 52 T 53 T 54   T 11 T 12 T 13 T 14   T 11 T 12 T 13 T 14 
     
 T 61 T 62 T 63 T 64   T 31 T 32 T 33 T 34   T 21 T 22 T 23 T 24 
  
 ¼0  ¼0  ¼0
 T 71 T 72 T 73 T 74   T 51 T 52 T 53 T 54   T 31 T 32 T 33 T 34 
     
 T 81 T 82 T 83 T 84   T 71 T 72 T 73 T 74   T 41 T 42 T 43 T 44 

Sp
F c Sp
Sc Sp
C c
     
 T 52 T 54 T 56 T 58   T 12 T 14 T 16 T 18   T 12 T 14 T 16 T 18 
     
 T 62 T 64 T 66 T 68   T 32 T 34 T 36 T 38   T 22 T 24 T 26 T 28 
  
 ¼0  ¼0  ¼0
 T 72 T 74 T 76 T 78   T 52 T 54 T 56 T 58   T 32 T 34 T 36 T 38 
     
 T 82 T 84 T 86 T 88   T 72 T 74 T 76 T 78   T 42 T 44 T 46 T 48 

Cp
F c C p
Sc Cp
Cc
     
 T 55 T 56 T 57 T 58   T 15 T 16 T 17 T 18   T 15 T 16 T 17 T 18 
     
 T 65 T 66 T 67 T 68   T 35 T 36 T 37 T 38   T 25 T 26 T 27 T 28 
  
 ¼0  ¼0  ¼0
 T 75 T 76 T 77 T 78   T 55 T 56 T 57 T 58   T 35 T 36 T 37 T 38 
     
 T 85 T 86 T 87 T 88   T 75 T 76 T 77 T 78   T 45 T 46 T 47 T 48 
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Let H ¼ maxB1 pBpb TðBÞ
Y ðBÞ , derive as aforementioned

kþ1
TðBÞ
Y ðBÞ pH: ðMtÞ . (29)
ðk þ 1Þ!

Table 2
Eigenvalue employed FEM and present method

m¼0 m¼1 m¼2

n¼0 FEM 3.4170 6.7660 10.145

Transfer matrix 3.4170 6.7660 10.155
Error/% 0 0 0.00
n¼1 FEM 4.9232 8.3550 11.760
Transfer matrix 4.9282 8.3639 11.772
Error/% 0.10 0.11 0.10
n¼2 FEM 6.3191 9.8670 13.320
Transfer matrix 6.3263 9.8810 13.342
Error/% 0.11 0.14 0.16

Fig. 2. Determinant value vs. frequency parameter p for n ¼ 0, 1, 2: (a) determinant value Dt vs. frequency parameter p for n ¼ 0; (b)
determinant value Dt vs. frequency parameter p for n ¼ 1; (c) determinant value Dt vs. frequency parameter p for n ¼ 2.
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Furthermore,
ðMtÞkþ1
lim ¼ 0. (30)
k!1 ðk þ 1Þ!

Substituting Eq. (30) into Eq. (29) yields

TðBÞ
Y ðBÞ ¼ 0.

Therefore, TðBÞ is the one and only solution of Eq. (15).

2.6. The eigenvalue equation

The elements of transfer matrix TðBÞ can be determined by employing recurrence formula (18). The
eigenvalue equations (or frequency equations) corresponding to the different boundary conditions can be
obtained. Generally, both the inner edge of annular plate and the large edge of conical shell may be one of
following three restriction conditions, i.e. free (F), simply supported (S) and clamped (C) boundary conditions:
At a free edge, M̄ B ¼ V̄ B ¼ N̄ B ¼ S̄ By ¼ 0;
at a clamped edge, ū ¼ v̄ ¼ w̄ ¼ f̄ ¼ 0;
at a simply supported edge, ū ¼ w̄ ¼ M̄ B ¼ N̄ B ¼ 0.

Substituting the boundary conditions of annular plate and conical shell into Eq. (9), the eigenvalue equation
can be derived. Table 1 shows the eigenvalue equations of the structure under all nine combinations. The
natural frequencies of the system are determined by calculating the eigenvalues of these equations in Table 1.

Fig. 3. Eigenvalue and mode shape analyzed by FEM and present method: (a) mode shape relative to eigenvalue p ¼ 3:4170 by FEM; (b)
mode shape of round plate corresponding to p00 ¼ 3:4170 by present method; (c) mode shape of conical shell corresponding to p00 ¼
3:4170 by present method.
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2.7. The mode shape function

Once the eigenvalue is determined, mode shape can be derived. Here, taken the boundary condition F p
C c
for example, the transfer matrix of entire system can be written as
8 9 8 9
>
> ū >> >
> ū >>
> v̄ >
> > > v̄ >
> >
>
> >
> >
> >
>
>
> >
> >
> >
>
>
> w̄ >>
> >
> w̄ >>
>
> > >
> >
>
>
> >
> >
> >
>
< f̄ =  < f̄ =
¼ T ij ði; j ¼ 1; 2; 3; 4; 5; 6; 7; 8Þ. (31)
>
> M̄ > > M̄ >
> B>
> >
> > B>
>
> >
>
>
> V̄ B >
> >
> V̄ B >
>
>
> >
> >
> >
>
>
> >
> >
> >
>
>
> N̄ >
B > >
> N̄ B >
>
>
> >
> >
> >
>
: S̄By ; : S̄By ;
B¼B2 B¼B4

By substituting these boundary conditions into Eq. (31), extracting first, second, and third rows of this
equation, and assigning arbitrarily f̄ðB4 Þ ¼ 1, the following equations set yields:
2 38 9 8 9
T 11 T 12 T 13 > < ū >
= < T 14 >
> =
6T 7
4 21 T 22 T 23 5 v̄ þ T 24 ¼ 0. (32)
>
: >
; >
: >
;
T 31 T 32 T 33 w̄ B¼B T 34
4

Fig. 4. Eigenvalue and mode shape analyzed by FEM and present method: (a) mode shape relative to eigenvalue p ¼ 6:7660 by FEM; (b)
mode shape of round plate corresponding to p01 ¼ 6:7660 by present method; (c) mode shape of conical shell corresponding to p01 ¼
6:7660 by present method.
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Solution of the displacement vector for the inner edge of annular plate can be written as
8 9 2 3 8 9 8 9
< ū >
> = T 11 T 12 T 13
1 >
< T 14 >
= >
b
< 1>=
6T T 22 7
T 23 5 T 24 ¼ b2 .
v̄ ¼
4 21 (33)
: >
> ; >
:T > ; >
:b >;
w̄ B¼B T 31 T 32 T 33 34 3
4

The displacement vector at any point in the middle surface of annular plate can be obtained as
8 9 2 3 8 9 8 9
T 11 T 12 T 13 b
< ū >
> = >
< 1> = > < T 14 >
=
6 7 b2 þ T 24
v̄ ¼ 4 T 21 T 22 T 23 5 . (34)
: >
> ; > > > >
w̄ B pBpB T 31 T 32 T 33 B pBpB : b3 ; : T 34 ;B pBpB
4 3 4 3 4 3

With the same method, the displacement vector at any point in the middle surface of the conical shell can be
derived as
8 9 2 3 8 9 8 9
T 11 T 12 T 13 b
< ū >
> = >
< 1> = > < T 14 >
=
6 7 b2 þ T 24
v̄ ¼ 4 T 21 T 22 T 23 5 . (35)
: >
> ; >
: >
; >
: >
;
w̄ B pBpB T 31 T 32 T 33 B pBpB b3 T 34 B pBpB
1 2 1 2 1 2

Fig. 5. Eigenvalue and mode shape analyzed by FEM and present method: (a) mode shape corresponding to eigenvalue p ¼ 10:145 by
FEM; (b) mode shape of round plate corresponding to p02 ¼ 10:155 by present method; (c) mode shape of conical shell corresponding to
p02 ¼ 10:155 by present method.
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Fig. 6. Eigenvalue and mode shape analyzed by FEM and present method: (a) mode shape corresponding to eigenvalue p ¼ 4:9232 by
FEM; (b) mode shape of round plate corresponding to p10 ¼ 4:9282 by present method; (c) mode shape of conical shell corresponding to
p10 ¼ 4:9282 by present method.

Thus, the mode shape (displacement vector) can be calculated from the Eqs. (34) and (35). Using the same
method, the other elements in state vector will also be obtained easily.

3. Numerical analysis

A conical shell with a round end plate (made from aluminum) is taken for example, geometric parameters of
the combined shell are R ¼ 200 mm, h ¼ 2 mm, R1 ¼ 180 mm, R2 ¼ 0 and H ¼ 96 mm, and material
parameters are E ¼ 68.97 GPa and m ¼ 0:3. The boundary conditions are F p
C c . When B4 ¼ 0, the annular
plate will become a round plate. Therefore, the natural frequencies and their mode shapes can be obtained by
taking an extremely small value for B4 , in this paper B4 ¼ 0:01. In order to verify the formulas presented by this
paper, finite element method (FEM) is also adopted here. The analytical software is ANSYS 7.0, and the
element type employed here is shell 63. By using Lanczos method, the natural vibration characteristics can be
acquired easily.
The nature frequencies of round plates and truncated conical shells can be calculated with the present
formulas, respectively. By comparing these results with those in Refs. [23,32], k (this paper k ¼ 9) in
recurrence Eq. (18) can be determined. Once the transfer matrix of single component has been obtained, the
product of each component matrix and the joining matrix can form the entire structure matrix. The solution of
Eq. (18) can utilize the software Mathematica. Some curves of the determinant value Dt with frequency
parameter p are provided partially. Figs. 2a–c show the curves for n ¼ 0, 1 and 2, respectively. Eigenvalue (or
natural frequency parameter) is expressed by pnm. Here m is the axial mode number and n denotes the
 ARTICLE IN PRESS
S. Liang, H.L. Chen / Journal of Sound and Vibration 294 (2006) 927–943 939

Fig. 7. Eigenvalue and mode shape analyzed by FEM and present method: (a) mode shape corresponding to eigenvalue p ¼ 8:3550 by
FEM; (b) mode shape of round plate corresponding to p11 ¼ 8:3639 by present method; (c) mode shape of conical shell corresponding to
p12 ¼ 8:3639 by present method.

circumferential wave number as above mentioned. The mode shapes corresponding to eigenvalues can be
calculated by Eq. (34) and (35). Figs. 3a–8a are some mode shapes obtained by FEM, and Figs. 3b–8b and
Figs. 3c–8c are some mode shapes acquired by present formulas. A complete mode shape of the combined
shell is comprised of mode shapes of an end plate and a conical shell. In other words, under the same
eigenvalue, combining panel b with panel c forms a complete mode shape of the entire structure. In order to
compare mode shapes obtained by present formulas with those by 3D FEM, panels b and c denote the
dimensionless mode shapes of a meridian, ū, v̄ and w̄ must be transformed into u, v and w by Eq. (A.7a). n ¼ 0,
i.e. circumferential wavenumber is 0, and mode shapes is circular symmetry. n ¼ 1, i.e. circumferential
wavenumber is 1, and mode shape is axial symmetry. To check the accuracy of eigenvalue, comparison is also
made in Table 2.
Table 2 and Figs. 2–8 indicate that the results analyzed by present approach and FEM are in good
agreement with each other, which demonstrates that the method using present study is valid and transfer
matrix obtained by recurrence formula method can accurately calculate natural frequency and mode shape.

4. Conclusions

With the vibration theory and transfer matrix method combined, natural frequencies and mode shapes of
the conical shell with an annular end plate are investigated in detail. The governing equations of vibration for
this system are expressed in terms of matrix differential equations, and a novel recurrence formula method
that can close in on analytical solution of transfer matrix is presented. Once the transfer matrix of single
component has been determined, the product of each component matrix and the joining matrix can form the
 ARTICLE IN PRESS
940 S. Liang, H.L. Chen / Journal of Sound and Vibration 294 (2006) 927–943

Fig. 8. Eigenvalue and mode shape analyzed by FEM and present method: (a) mode shape corresponding to eigenvalue p ¼ 11:760 by
FEM; (b) mode shape of round plate corresponding to p12 ¼ 11:772 by present method; (c) mode shape of conical shell corresponding to
p12 ¼ 11:772 by present method.

matrix of entire structure, and the frequency equations and mode shape functions are represented in terms of
the elements of the structural matrices. The 3D finite element numerical simulation has validated the present
formulas of natural frequencies and mode shapes. The conclusions show the transfer matrix obtained by
present method can accurately reveal dynamic characteristic of the conical shell with an annular end plate.

Acknowledgments

The works was supported by the research Grant no. 10076012 from the National Natural Science
Foundation, and the research Grant no. 20010698011 from Doctoral Science Foundation of Ministry of
Education of the People’s Republic of China.

Appendix A

According to the flÜgge theory, the governing equation of flexural vibration for the conical shell are written
as
 
1 q BN By 1 qN y Q
þ
y þ rho2 u ¼ 0,
B qB B sin a qy Btga
 
1 q BN B 1 qN yB N y
þ
þ rho2 v ¼ 0,
B qB B sin a qy B
 ARTICLE IN PRESS
S. Liang, H.L. Chen / Journal of Sound and Vibration 294 (2006) 927–943 941

 
Ny 1 qQy 1 q BQB



þ rho2 w ¼ 0. ðA:1Þ
Btga B sin a qy B qB
The Kevlin–Kirchhoff shearing force and shear resultant is
1 qM By M By
V B ¼ QB þ ; S By ¼ N By
. (A.2)
B sin a qy Btga
The components of the shearing force are written as
 
1 q BM B 1 qM yB M y
QB ¼ þ
,
B qB B sin a qy B
 
1 q BM By 1 qM y M yB
Qy ¼ þ þ . ðA:3Þ
B qB B sin a qy B
The components of membrane force are given by
  
12D qv n 1 qu w
NB ¼ 2 þ þvþ ,
h qB B sin a qy tga
  
12D qv 1 1 qu w
Ny ¼ 2 n þ þvþ ,
h qB B sin a qy tga
  
6ð1
nÞD qu 1 1 qv
N By ¼ þ
u . ðA:4Þ
h2 qB B sin a qy
The bending moment can be written as
  
qf n 1 q2 w
MB ¼ D þ þf ,
qB B B sin2 a qy2
  
qf 1 1 q2 w
My ¼ D n þ þ f ,
qB B B sin2 a qy2
 
ð1
nÞD qf 1 qw
M By ¼
. ðA:5Þ
B sin a qy B qy
The slope of the displacement w can be expressed by
qw
f¼ . (A.6)
qB
For the steady vibration of shell, let us take



u v w ¼ R u sin ny v cos ny w cos ny ; (A.7a)

f ¼ f cos ny, (A.7b)

n o Dn o
MB My M By ¼ M s cos ny M y cos ny M By sin ny , (A.7c)
R
n o Dn o
NB Ny N By ¼ N s cos ny N y cos ny N By sin ny , (A.7d)
R2
n o Dn o
QB Qy ¼ QB cos ny Qy sin ny , (A.7e)
R2
n o Dn o
VB SBy ¼ V B cos ny S By sin ny . (A.7f)
R2
 ARTICLE IN PRESS
942 S. Liang, H.L. Chen / Journal of Sound and Vibration 294 (2006) 927–943

By eliminating M y M By QB Qy N y N By from the equations (A.1–A.7), the matrix differential equation

yields as follows:
8 9 8 9
>
> ū >> >
> ū >>
>
> >
> >
> >
>
>
> v̄ >
> >
> v̄ >
>
>
> >
> >
> >
>
> w̄ >
> > >
> w̄ >
>
>
> >
> >
> >
>
>
> >
> >
> >
>
d < f̄ =   f̄ =
<
¼ Aij ði; j ¼ 1; 2; 3; 4; 5; 6; 7; 8Þ. (A.8)
dB >
> M̄ B >
> >
> M̄ B >
>
>
> >
> >
> >
>
>
> > > >
> V̄ B >
> >
> > V̄ B >
>
> >
>
>
> > > >
> N̄ B >
> >
> > N̄ B >
>
> >
>
>
> > > >
: S̄ > ; : S̄ >
> ;
By By

References

[1] K.Y. Lam, H. Li, T.Y. Ng, C.F. Chua, Generalized differential Quadrature method for the free vibration of truncated conical panels,
Journal of sound and Vibration 251 (2002) 329–348.
[2] X.X. Hu, T. Sakiyama, H. Masuda, C. Morita, Vibration analysis of rotating twisted and open conical shells, International Journal of
Solids and Structures 39 (2002) 6121–6134.
[3] X.X. Hu, T. Sakiyama, H. Masuda, C. Morita, Vibration analysis of twisted conical shells with tapered thickness, International
Journal of Engineering Science 40 (2002) 1579–1598.
[4] A.M.I. Sweedan, A.A. EI Damatty, Experimental and analytical evaluation of the dynamic characteristics of conical shells, Thin-
Walled Structures 40 (2002) 465–486.
[5] T. Irie, G. Yamada, Y. Muramoto, Natural frequencies of in-plane vibration of annular plates, Journal of Sound and Vibration 97
(1984) 171–175.
[6] T. Irie, G. Yamada, S. Aomura, Free vibration of a mindlin annular plate of varying thickness, Journal of Sound and Vibration 66
(1979) 187–197.
[7] T. Irie, G. Yamada, Y. Kaneko, Vibration and stability of non-uniform annular plate subjected to a follower force, Journal of Sound
and Vibration 73 (1980) 261–269.
[8] A. Selmane, Natural frequencies of transverse vibrations of non-uniform circular and annular plates, Journal of Sound and Vibration
220 (1999) 225–249.
[9] H.N. Arafat, A.H. Nayfeh, W. Faris, Natural frequencies of heated annular and circular plates, International Journal of Solids and
Structures 41 (2004) 3031–3051.
[10] L. Cheng, Y.Y. Li, L.H. Yam, Vibration analysis of annular-like plates, Journal of Sound and Vibration 262 (2003) 1153–1170.
[11] A. Joseph Stanley, N. Ganesan, Frequency response of shell-plate combinations, Computers and Structures 59 (1995)
1083–1094.
[12] D.T. Huang, W. Soedel, On the free vibrations of multiple plates welded to a cylindrical shell with special attention to mode pairs,
Journal of Sound and Vibration 166 (1993) 315–339.
[13] N.O. Myklestad, New method of calculating natural modes of coupled bending-torsion vibration of beams, Transactions of the
American Society of Mechanical Engineers 67 (1945) 61–67.
[14] W.T. Thomson, Matrix solution of vibration of non-uniform beams, Journal of Applied Mechanics 17 (1950) 337–339.
[15] J.C. Snowdon, Mechanical four-pole parameters and their application, Journal of Sound and Vibration 15 (1971) 307–323.
[16] S. Rubin, Review of mechanical immittance and transmission matrix concepts, Journal of the Acoustical Society of America 41 (1967)
1171–1179.
[17] E.C. Pestel, F.A. Leckie, Matrix methods in Elastomechanics, McGraw-Hill Book Company, New York, 1963.
[18] V.R. Murthy, N.C. Nigam, Dynamic characteristics of stiffened rings by transfer matrix approach, Journal of Sound and Vibration 39
(1975) 237–245.
[19] V.R. Murthy, Dynamic characteristics of rotor blades, Journal of Sound and Vibration 49 (1976) 483–500.
[20] T. Irie, G. Yamada, Y. Kaneko, Free vibration of non-circular cylindrical shell with longitudinal interior partitions, Journal of Sound
and vibration 96 (1984) 133–142.
[21] A. Kalnins, Free vibration of rotationally symmetric shells, The Journal of the Acoustical Society of America 36 (1964) 1335–1365.
[22] T. Irie, G. Yamada, Y. Kaneko, Free vibration of a conical shell with variable thickness, Journal of Sound and vibration 82 (1982)
83–94.
[23] T. Irie, G. Yamada, K. Tanaka, Natural frequencies of truncated conical shells, Journal of Sound and Vibration 92 (1984) 447–453.
[24] T. Irie, G. Yamada, K. Tanaka, Free vibration of a thin-walled beam-shell of arc cross-section, Journal of sound and vibration 94
(1984) 563–572.
[25] T. Irie, G. Yamada, Y. Muramoto, Free vibration of joined conical-cylindrical shells, Journal of Sound and Vibration 95 (1984) 31–39.
[26] G. Yamada, T. Irie, T. Tamiya, Free vibration of a circular cylindrical double-shell system closed by end plates, Journal of Sound and
vibration 108 (1986) 297–304.
 ARTICLE IN PRESS
S. Liang, H.L. Chen / Journal of Sound and Vibration 294 (2006) 927–943 943

[27] W. flÜgge, Stresses in Shells, Springer, New York, 1973.

[28] A. Kalnins, Free vibration of rotationally symmetric shells, Journal of the Acoustical Society of America 36 (1964) 1365–1964.
[29] G.A. Cohen, Computer analysis of asymmetric free vibrations of ring-stiffened orthotropic shell of revolution, American Institute of
Aeronautics and Astronautics Journal 3 (1965) 2305–2312.
[30] H. Tottenham, K. Shimizu, Analysis of the free vibration of cantilever cylindrical thin elastic shells by the matric progression method,
International Journal of Mechanical Science 14 (1972) 293–310.
[31] S. Sankar, Extend transfer matrix method for free vibration of shells of revolution, Shock and Vibration Bulletin 47 (1977) 121–133.
[32] Ni. Zhenhua, Vibration mechanics, Xi’an Jiaotong University, Xi’an, 1990.