three-dimensional vibration analysis of isotropic and orthotropic conical shells with elastic...

Author's Accepted Manuscript

Three-dimensional vibration analysis of iso-tropic and orthotropic conical shells withelastic boundary restraints

Guoyong Jin, Zhu Su, Tiangui Ye, Xingzhao Jia

PII: S0020-7403(14)00308-7DOI: http://dx.doi.org/10.1016/j.ijmecsci.2014.09.005Reference: MS2817

To appear in: International Journal of Mechanical Sciences

Received date: 16 March 2014Revised date: 19 July 2014Accepted date: 10 September 2014

Cite this article as: Guoyong Jin, Zhu Su, Tiangui Ye, Xingzhao Jia, Three-dimensional vibration analysis of isotropic and orthotropic conical shells withelastic boundary restraints, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2014.09.005

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http://dx.doi.org/10.1016/j.ijmecsci.2014.09.005







Three-dimensional vibration analysis of isotropic and orthotropic

conical shells with elastic boundary restraints

Guoyong Jin�, Zhu Su�, Tiangui Ye, Xingzhao Jia College of Power and Energy Engineering, Harbin Engineering University,

Harbin, 150001, P. R. China Abstract

In this paper, a three-dimensional (3-D) solution method is presented for the free vibration

of isotropic and orthotropic conical shells with elastic boundary restraints. The formulation is

derived by means of the Rayleigh-Ritz procedure based on the three-dimensional elasticity

theory. Displacement components of the conical shells are represented by Fourier series in the

circumferential direction and a double Fourier cosine series supplemented with several

auxiliary functions in thickness and meridional directions. The supplementary functions in the

form of the product of a polynomial function and a single cosine series are introduced to ensure

and accelerate the convergence of the series representations. To validate the present method,

the convergence behavior is demonstrated, and several comparisons of the numerical results

with those published in literature and obtained by ANASYS are performed. Numerous new

results for the isotropic and orthotropic conical shells with elastic boundary conditions are

presented. The effects of the geometrical parameters, orthotropic properties and boundary

conditions on the natural frequencies of conical shells are illustrated.

Keywords: isotropic and orthotropic; conical shell; three-dimensional elasticity theory; free

vibration; elastic boundary restraints

1. Introduction

� Corresponding author, Tel: +86 451-82569458 Fax: +86 451-82518264 E-mail address: [email protected]

� Corresponding author, Tel: +86 451-82569458 Fax: +86 451-82518264 E-mail address: [email protected]

Shell structures are extensively used in various engineering applications, particularly in

aerospace, marine and structural engineering. In order to ensure a reliable design, a detailed

understanding of their dynamic characteristics must first be determined. In the past several

decades, various shell theories and different computational methods have been proposed and

developed by researchers in order to predict the vibration behavior of the shells. The shell

theories mainly include three-dimensional (3-D) elasticity theory and two-dimensional (2-D)

theories which reduce the dimensions of the shell problem from three to two by making certain

hypotheses regarding the stress and strain fields along the thickness of the shell and can be

classified into three main categories: classical shell theory (CST), first-order shear

deformation theory (FSDT), and higher-order shear deformation theory (HSDT). A number of

computational methods have been proposed and developed, such as Rayleigh-Ritz method

[1-3], Haar wavelet method [4], differential quadrature method [5, 6], finite element method

[7], and meshless methods [8-15]. More detailed descriptions on this subject may be found in

several monographs respectively by Leissa [16], Qatu [17], Reddy [18], Carrera et al [19], and

Leiss and Qatu [20].

As one of the common shell structures, conical shells play a significant role in many

industrial fields. However, compared with the studies of the cylindrical shells, the literature

about conical shells is limited. Most investigations [21-41] were carried out based on 2-D

theories. Saunders et al [21] applied Rayleigh-Ritz method to compute the frequencies for free

vibration of isotropic thin conical shells with free or simply supported boundary condition.

Garnet and Kemper [22] also employed this method to study the lowest axisymmetric modes of

truncated conical shells, and the transverse shear deformation and rotatory inertia effects are

accounted for. The axisymmetric modes and natural frequencies of isotropic thin conical shells

were obtained by Goldberg et al [23] using numerical intergration method and classical shell

theory (CST). Tong [26] developed the power series expansion approach to analyze free

vibration of isotropic and orthotropic conical shells. Free vibration of isotropic conical shells

was studied by Shu [27] using the global method of generalized differential quadrature (GDQ)

on the basis of classical shell theory (CST) and two types of boundary conditions were

considered including simply supported and clamped edges. Free vibration analysis of thin

isotropic conical shells under different classical boundary conditions was carried out by Liew

et al [28] using the element-free kp-Ritz method based on CST, and the kernel particle (kp)

functions were introduced to approximate the two-dimensional displacement field. Sofiyec et

al [29] studied the vibration and stability of non-homogeneous orthotropic conical shells

subjected to hydrostatic pressure using the Garlerkin method in the context of CST, in which

the material properties of conical shells vary continuously in the thickness direction. The free

and forced vibration of isotropic conical shells were atudied by Li et al [31] by the means of

Hamilton’s principle in conjunction with Rayleigh-Ritz method based on the first-order shear

deformation theory (FSDT). Some researchers [32-38] have analyzed the free vibration of the

rotating conical shells by different methods. Liew et al [39] and Lim and Liew [40, 41]

investigated the free vibration of shallow conical shell using pb-2 Ritz method on the basis of

shallow shell theories. There is limited amount of literature [42-45] concerning the vibration

analysis of conical shells based on 3-D elasticity theory. Leissa and Kang [42] and Kang and

Leissa [43] presented a three-dimensional method of analysis for determining the free vibration

frequencies and mode shapes of thick shells with variable thickness.

From the review of the literature, the available 3-D elasticity solutions for the vibration

problems of isotropic and orthotropic conical shells are relatively scarce, and most of the

previous studies regarding the conical shells are confined to the classical boundary conditions.

However, elastic boundaries are often encountered in engineering applications, and there is a

considerable lack of corresponding research regarding elastic boundary conditions.

In this paper, a three-dimensional (3D) solution method is presented for the free vibration

of isotropic and orthotropic conical shells with elastic boundary restraints, which can be

considered as an extension of the authors’ previous works [46-48]. The formulation is derived

by means of the Rayleigh-Ritz procedure based on the three-dimensional elasticity theory.

Displacement components of the conical shells are represented by Fourier series in the

circumferential direction and a double Fourier cosine series supplemented with auxiliary

functions in meridional and normal directions. The supplementary functions in the form of the

product of a polynomial function and a single cosine series are introduced to ensure and

accelerate the convergence of the series representations. To validate the present method, the

convergence behavior is demonstrated, and comparisons with available results in the literature

are performed. Numerous new results for the isotropic and orthotropic conical shells with

elastic boundary conditions are presented. The effects of the geometrical parameters,

orthotropic properties and boundary conditions on free vibration of conical shells are

illustrated.

2. Theoretical formulations

2.1 Description of model

Let us consider a conical shell with slant height L, thickness H, semi-vertex angle �. The

curvilinear coordinate system composed of coordinates r, �, and s is introduced as shown in

Fig. 1, which can be derived from the Cartesian coordinate system (x, y, and z). The

coordinates r and s along the thickness and meridional directions are measured from the small

edge and inner surface of the conical shell. The conical shell domain is bounded by 0 � r � H, 0

� � � 2�, 0 � s � L. R1 and R2 are the radii of the inner surface of conical shell at the small and

large ends. The displacements of the conical shell are denoted by u, v, and w in r, � and s

directions, respectively. The radius of curvature in the circumferential direction at any point P

is given by

1( , ) tan / cosR r s s R r� ��

and the distance of P from the axis of revolution is expressed as

1( , ) ( , ) cos sin cosR r s R r s s R r� � ��

The relations of the Cartesian coordinates (i.e. x, y and z) and the curvilinear coordinates (i.e.

r, � and s) are given as

( , ) cos cos ( , ) cos ,( , ) cos sin ( , )sin ,cos sin

x R r s R r sy R r s R r sz s r

� � ��

� �

� �

� ��

2.2 Energy functional

According to 3-D elasticity theory, the linear relations between strains and displacement

of an elastic body are given as follows

1 1 2 2

1 1 2 1 3 2 2 3 2 1

3 3 32

3 3 1 3 2 3 2 2 3

3 1 2 1

1 3 3 1 1 2 2 1

1 1 1 1 1 1,

1 1 1 , ( ) ( ),

( ) ( ), ( ) ( )

rr

ss s

rs r

H H H Hu vv w w uH r H H H H s H H H s H H r

H H HHw v wu wH s H H r H H H s H H HH H H Hw u v uH r H H s H H r H H H

��

�

�

� ��

� � �

�

� � � � � �

� � � � �

� � � �

(1)

where H1, H2 and H3 are the Lamè coefficients. For the conical shell, the Lamè coefficients

are given as

2 2 2 1/2 2 2 2 2 2 1/21

2 2 2 1/2 2 2 2 2 1/22

2 2 2 1/2 2 2 2 2 2 1/23

[( ) ( ) ( ) ] (cos cos cos cos sin ) 1

[( ) ( ) ( ) ] ( ( , ) sin ( , ) cos ) ( , )

[( ) ( ) ( ) ] (sin cos sin cos cos )

x y zHr r rx y zH R r s R r s R r s

x y zHs s s

� � � � �

� ��

� � � � �

� � � � � � �

� � � � � �

� � � � � � �

1

(2)

Substituting Eq. (2) into Eq. (1), the strain-displacement relations of a conical shell can be

depicted as

1 sin cos, , ,

1 sin 1 cos, ,

rr ss

s rs r

u v ww ur R R R s

v w w u v uv vs R R r s r R R

��

� �

� ��

� � � �

� � � � �

� � � � � � � �

(3)

Based on Hooke’s law, the relations of the stresses and strains can be expressed in the

form as

11 12 13

12 22 23

13 23 33

5544 66, ,

rr rr ss

rr ss

ss rr ss

rs rss s r r

c c cc c cc c c

c c c

��

��

��

� � � �

� � � ��

� � �

� � ��

� � �

(4)

where ( , 1, 2, 6)ijc i j � � are the elastic stiffness coefficients. For the orthotropic,

three-dimensional conical shell, the elastic stiffness coefficients can be expressed as:

11 12

22 23

33 31

44 55 66

(1 ), ( )(1 ), ( )(1 ), ( )

, ,

r s s r rs s

rs sr s s r rs

s r r r sr s r

s rs r

c E c Ec E c Ec E c E

c G c G c G

� � � � �

� � �

� � � �

� �

� � � � � �

� � � �

� � � ��

� � �

(5)

where 1/ (1 )r r s s sr rs r s sr r rs s� � � � � � � �� . According to the Maxwell

reciprocity relations:

, ,r r r rs s sr r s s sE E E E E E� � � � � � � � � (6)

The isotropic, three-dimensional conical shell can be considered as a special case in which:

11 22 33

12 13 21 23 31 32

44 55 66

2c c c Gc c c c c cc c c G

��

� � � �

� � � � � ��

(7)

where ,(1 )(1 2 ) 2(1 )

E EG �

� ��

and E and � represent Young’s modulus and

Poisson’s ratio.

The strain energy U of the conical shell can be expressed in the integral form as

2

0 0 0

1 ( )21 ( ) ( , )2

rr rr ss ss s s sr sr r rV

L H

rr rr ss ss s s sr sr r r

U dV

R r s drd ds

��

�

��

� � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � �

� � � � � �

��

� � � (8)

Substituting Eqs. (3) and (4) into Eq. (8), the strain energy U can be written as

2 211 22 13 31

212 21 33

223 32 55

244 66

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

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

1 sin cos2 ( )( )( ) ( )

1 sin( ) (

u v u wc c w u c cr R R R r s

v u wc c w u cR R R r sU

v w w uc c w u cR R R s r s

v w vc v cs R R r

� ��

� ��

� ��

��

� � � � �

� � � � � �

� � � � � �

� � � �

2

0 0 0

2

( , )

1 cos )

L H

R r s drd ds

u vR R

�

�

��

� ��

� �

� � � (9)

The kinetic energy T of the conical shell is given as

� �

� �

2 2 2

22 2 2

0 0 0

2

( , )2

VL H

T u v w dV

u v w R r w drd dz�

�

� �

� � �

� � �

��

� � �

� � �

� � � (10)

where � are the mass density per volume, and the over dots denote time derivatives.

In this work, the edges of the conical shells are restrained by springs to simulate the

given or typical boundary conditions. The boundary conditions of thick shell can be defined

as [20]:

1sr sr� �� or 1u u�

1s s� �� or 1v v�

1ss ss� �� or 1w w�

where �sr1, �s�1,and �ss1 represent shear and normal stresses at ends with s=constant. u1, v1, w1

are the displacement functions in at ends, respectively. Therefore, three sets of independent

linear springs (ku, kv, and kw) in r, � and s directions are introduced, and the boundary

conditions for the ends can be expressed as:

0 0 0, ,u sr v s w ssk u k v k w�� at s=0

, ,uL sr vL s wL ssk u k v k w�� at s=L

where ku0, kv0, kw0, kuL, kvL and kwL are the stiffnesses of the spring. The general boundary

conditions can be obtained by assuming spring stiffness equal to proper value. For example,

the free boundary condition corresponds to the case in which the spring stiffness is set equal

to zero. On the contrary, for the clamped edge, the stiffness is infinite. For computational

purposes, infinity is represented by a very large spring value, i.e. 1×1015 Nm-3. The potential

energy L of the elastic reactions of the springs is calculated as

22 2 2 2 2 2

0 0 0 00 0

1 ( ,0)( ) ( , )( )2

H

u v w s uL vL wL s LL R r k u k v k w R r L k u k v k w drd�

�� (11)

The total energy functional � is thus given as

T U L� � � � (12)

2.3 Admissible displacement functions

Considering the circumferential symmetry of the conical shells in about the coordinate �,

the 3-D problem of the conical shell can be transformed to 2-D analysis by using the Fourier

series in circumferential direction. In this study, a 2-D modified Fourier cosine series which is

constructed as the linear superposition of a double Fourier cosine series and several

supplementary functions are employed in the thickness and meridional directions. The

supplementary functions are used to remove the potential discontinuities with the

displacements and their derivatives. The displacements of the conical shell can be expressed

in the following forms [46-48]:

2 2

0 0 0 1 0 1 0

2 2

1 0 0 1 0 1 0

( , , , ) ( , , )

( cos cos cos cos )cos

( cos cos cos cos )sin

j t

Q QN M M

mnq m q lmn ls m lnq lr qn m q l m l q

Q QN M M


u r s t U r s e

A r s a r a s n

A r s a r a s n

��

� � � � � � �

� � � � � � �

� � � � � � �

� � � � � � �

�

��

��

� ��

� ��

�

�

i te �

� ! " # $

(13.a)

2 2

1 0 0 1 0 1 0

2 2

0 0 0 1 0 1 0

( , , , ) ( , , )


( cos cos cos cos ) cos

j t

Q QN M M


Q QN M M


v r s t V r s e

B r s b r b s n

B r s b r b s n

��

� � � � � � �

� � � � � � �

� � � � � � �

� � � � � � �

�

��

��

� ��

� ��

�

�i te �

� ! " # $

(13.b)

2 2

0 0 0 1 0 1 0

2 2

1 0 0 1 0 1 0

( , , , ) ( , , )

( cos cos cos cos )cos


j t

Q QN M M


Q QN M M


w r s t W r s e

C r s c r c s n

C r s c r c s n

��

� � � � � � �

� � � � � � �

� � � � � � �

� � � � � � �

�

��

��

� ��

� ��

�

�

i te �

� ! " # $

(13.c)

where � denotes the natural frequency of the conical shell, t is time variable, the nonnegative

integer n represents the circumferential wave number, 1j � � , /m m H� �� and

/q q L� �� . M and Q are the Fourier series truncated numbers, and N is the maximum wave

number. mnqA , lmna , lnqa� , mnqA , lmna , lnqa� , mnqB , lmnb , lnqb� , mnqB , lmnb , lnqb� , mnqC , lmnc ,

lnqc� , mnqC , lmnc and lnqc� are Fourier coefficients to be determined. The closed-form

functions lr� and ls� are defined separately over [0, H] and [0, L] and the supplementary

functions are introduced to remove any potential discontinuities of the original displacements

and their relevant derivatives, and accelerate the convergence of the series representations.

Based on the 3-D elasticity theory, the displacements of the conical shells have at least

continuous two-order derivatives at any point in the shell space. Therefore, the closed-form

functions lr� and ls� are defined as in the following form:

22

1 2

22

1 2

( ) ( 1) , ( ) ( 1)

( ) ( 1) , ( ) ( 1)

r r

s s

r r rr r rH H Hs s ss s sL L L

� �

� �

� � � �

� � � � (14)

It is easy to verify that

1 1 1 1

2 2 2 2

1 1 1 1

2 2 2 2

(0) ( ) ( ) 0, (0) 1(0) ( ) (0) 0, ( ) 1(0) ( ) ( ) 0, (0) 1(0) ( ) (0) 0, ( ) 1

r r r r

r r r r

s s s s

s s s s

H HH H

L LL L

� � � ��

% %� � � �% %� � � �

% %� � � �% %� � � �

(15)

It can be proven that derivatives of the expansions given in Eq. (13) can be obtained simply

through term-by-term differentiation.

2.4 solution procedure

The eigenvalue problem is formulated by minimizing the total energy functional � with

respect to the Fourier coefficients. Substituting Eqs. (9) (11) and (13) into Eq. (12), and

performing the Rayleigh-Ritz operation, a set of linear algebraic equation against the

unknown coefficients can be obtained as

uu uv uw u uu u

uv vv vw v vv v

uw vw ww w ww w

&

& &

� ��

! " ! "� �� # $ # $� � � �

� �K K K a M 0 0 aK K K b 0 M 0 b 0K K K c 0 0 M c

(16)

where

000 00 00

010 10 10

[ , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , ]u mnq MNQ l lnm lNM l lnq lNQ

mnq MNQ l lnm lNM l lnq lNQ

A A A a a a a a a

A A A a a a a a a &

�a � � ��

(17)

010 10 10

000 00 00

[ , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , ]

u mnq MNQ l lnm lNM l lnq lNQ


B B B b b b b b b

B B B b b b b b b &

�b � � ��

(18)

000 00 00

000 00 00

[ , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , ]u mnq MNQ l lnm lNM l lnq lNQ


C C C c c c c c c

C C C c c c c c c &

�c � � ��

(19)

FG Fg Fg FG Fg Fgij ij ij ij ij ij

fG fg fg fG fg fgij ij ij ij ij ij


FG Fg Fg FG Fg Fgij ij ij ij ij ij


fG fg fg fG fij ij ij ij ij

ij �

K K K K K KK K K K K KK K K K K K

KK K K K K KK K K K K K

K K K K K

��

��

� � � � � ��

��

��

� � � � �� g fgij

� �� K

��

(20)

FG Fg Fgij ij ijfG fg fgij ij ij

fG fg fgij ij ij

FG Fg Fgij ij ij

fG fg fgij ij ij

fG fg fgij ij ij

ij

� ��

�

M M M 0 0 0M M M 0 0 0M M M 0 0 0

M 0 0 0 M M M0 0 0 M M M

0 0 0 M M M

�

�

� � ��

�

�

� � ��

(21)

in which , , ,i j u v w� and the superscripts of the elements indicate the location of

corresponding Fourier coefficients. In order to clarify the calculations of the sub-matrices

ijM and ijK , the detailed expressions for elements in typical matrices uuM and uuK are

given in Appendix A.

All eigenvalues and eigenvectors can be calculated from the Eq. (16). It is mentioned

that the elements of eigenvectors are the Fourier coefficients, and the mode shapes can be

easily obtained by substituting the eigenvectors into Eq. (13).

3. Computed results and discussion

In this section, several examples on free vibration analysis of isotropic and orthotropic

conical shells with different geometric and material parameters are presented to demonstrate

the convergence, accuracy and reliability of the present method. Different boundary

conditions including classical restraints and elastic supports are considered. The free

boundary (F) implies that the two ends of conical shells is stress-free in all coordinate

directions (i.e. 0sr sss�� ), and the clamped boundary is defined as that all

displacement components at the end face are restrained (i.e. 0u v w� � � ). The

simply-supported boundary conditions have a variety of interpretations, and in present works

three types of simply-supported are considered: simply-supported I (S1) is completely

supported in the � and s directions with full slip along the r direction (i.e.

0, 0srv w �� ); simply-supported II (S2) is only free in � directions (i.e.

0, 0su w �� ); simply-supported III (S3) is the standard shear diaphragm case (i.e.

0, 0ssu v �� ); Three types of elastic boundary conditions (i.e. E1, E2, E3) are also

studied in this section: E1 is only elastically restrained in r direction (i.e. 0, 0v w u '� � );

E2 is defined to be elastic in � directions (i.e. 0, 0u w v '� � ); E3 is considered to be

elastic in s direction (i.e. 0, 0u v w '� � ); The corresponding spring stiffnesses of the

boundary condition are given as

1 15 2 15

3 15 1 15 10

2 15 10 3 15 10

15F : 0, C :

S : 1 10 , 0, S : 1 10 , 0

S : 1 10 , 0, E : 1 10 , 1 10

E : 1 10 , 1 10 , E : 1 10 , 1 10

1 10u v w u v w

v w u u w v

u v w v w u

u w v u v w

k k kk k k k k kk k k k k kk k k k k k

k k k� � � � � �

� � ( � � � ( �

� � ( � � � ( � (

� � ( � ( � � ( � (

(

(22)

3.1 convergence study

Since the accuracy of the method depends on the truncated numbers M and Q, the

convergence of the present method must be checked. Table 1 shows the first ten

non-dimensional frequency parameters /L G� �) � of isotropic conical shells subjected

to C-F boundary with different thickness-to-radius ratios H/R1. The conical shells are made

from material with the following properties: E = 168 GPa, � = 5700 kg/m3 and � = 0.3. The

geometric parameters of the conical shells are taken to be H/R1 = 0.1, 0.2 and 0.5, Lcos� = 2

m, R1=1 m and �=30°. It is observed that the current method has stable monotonic

convergence characteristics. The differences between solutions form 13×13 and 14×14 are

very small, and the maximum discrepancy is 0.02%. Thus, in the following examples the

truncated numbers will be uniformly selected as 13×13.

3.2 isotropic and orthotropic conical shells with various boundary conditions

In order to confirm the accuracy of the current method, studies on free vibration of the

conical shells with different boundary conditions are carried out, and the present results are

compared with available solutions in the literature or obtained by ANASYS. Table 2 shows

the non-dimensional frequency parameters /L G� �) � of isotropic conical shells with

different thickness-to-height H/L. The geometric parameters of the conical shells are taken to

be R1/L=0.25, �=30°, H/L=0.25 and 1. The conical shells are made from Zirconia. Young’s

modulus, mass density and Poisson’s ratio for the zirconia are E=168GPa, �=5700kg/m3, and

�=0.3. The completely free and clamped boundary conditions are considered. The

comparisons of the present results with those 3-D solutions reported by Buchanan and Wong

[44] using finite element method are presented. It is seen that very good agreement of the

results is obtained. Table 3 presents the fundamental frequencies of the conical shells with

different semi-vertex angles ( i.e. � = 30° , 45° and 60°). The conical shells also are made

from zirconia with the following geometrical data: R1=1m, Lcos�=2m and H/R1=0.1, 0.2, 0.5

and 1.0. Four kinds of boundary conditions (i.e. F-F, F-C, C-F and C-C) are considered. The

results are compared with those 3-D solutions by using ANSYS with SOLID 45 elements.

The comparisons show very good agreement.

The comparisons of the present results and those published solutions in Tables 2 and 3

indicate that the current analysis is accurate and reliable. Numerous new results of the

isotropic and orthotropic conical shells with different boundary conditions including classical

edges and elastic restraints will been given. Table 4 shows the fundamental frequencies of

isotropic conical shells with various boundary conditions (i.e. S1-S1, S2-S2, S3-S3, C-S2, C-S3,

E1-E1, E2-E2 and E3-E3). The conical shells are made from aluminum with material properties

as: E=70GPa, �=2707 kg/m3 and �=0.3. Different thickness-to-radius ratios (i.e. H/R1 = 0.1,

0.2, 0.5, 1.0) and semi-vertex angles (i.e. � = 15°, 30°, 40°, 60°, 75°) are considered. The

fundamental frequencies of the orthotropic conical shells with various boundary conditions

are presented in Tables 5-7. The orthotropic material properties of the conical shells are given

as: E1= 137.9 GPa, E2= 8.963 GPa, E3= 8.963 GPa, G12= 7.101 GPa, G23= 6.205 GPa, G13=

7.101 GPa, �12= 0.3, �23= 0.49, �13= 0.3, and �= 1605 kg/m3. Three types of fiber orientations

are considered. Type I is defined as that subscripts 1, 2, 3 denote r, �, and s directions; Type II

is considered to be that subscripts 1, 2, 3 denote �, r, and s directions; Type III is considered

to be that subscripts 1, 2, 3 denote s, �, and r directions. The geometrical parameters of the

conical shells are taken to be R1=1m, Lcos�=2m, �= 15°, 30°, 45°, 60°, 75° and H/R1=0.1, 0.2,

0.5, 1.0. Five types of classical edges (i.e. C-C, S1-S1, S2-S2, S3-S3, and C-S2) and three types

of elastic restraints (i.e. E1-E1, E2-E2, E3-E3) are studied. It is observed from tables that the

boundary conditions play a significant role on the natural frequencies of the isotropic and

orthotropic conical shells. It is evident that the fundamental frequency is quite sensitive to the

change of the semi-vertex angle � and thickness-to-radius ratios H/R1. The mode shapes of

the conical shells are illustrated in Figs 2-5.

3.3 Parametric studies

In this section, the effects of the semi-vertex angle � and the thickness-to-radius ratio

H/R1 on the natural frequencies of conical shells are studies. Fig 6 shows the variation of

fundamental natural frequency of isotropic conical shells with different semi-vertex angle �.

It is obvious that the fundamental frequencies of the conical shells decrease as the

semi-vertex angle increases except in the case of the conical shells with S3-S3 boundary

condition. For the isotropic conical shell with S3-S3 boundary condition the fundamental

frequency of the conical shell first increases then decreases with an increase in semi-vertex

angle �. The effect of the semi-vertex angle � on the natural frequencies of orthotropic

conical shells is illustrated in Fig. 7. The variation trends of fundamental frequencies of

orthotropic conical with different fiber orientations are similar. Fig. 8 shows the effect of the

thickness-to-radius ratio H/R1 on the fundamental natural frequencies of isotropic conical

shells with different boundary conditions. The fundamental natural frequencies of isotropic

conical shells with C-C and S3-S3 boundary conditions increase as the thickness-to-radius

ratio H/R1 increases. The effect of the thickness-to-radius ratio H/R1 on the fundamental

natural frequencies of orthotropic conical shells with different boundary conditions is

demonstrated in Fig. 9. For the conical shells with type II materials, the fundamental

frequencies are increase as the thickness-to-radius H/R1 increases. For the conical shells with

type III material subjected to S1-S1, S3-S3 and E2-E2 boundary conditions , the fundamental

frequencies are first increase and then decline as the thickness-to-radius H/R1 increases.

4. Conclusions

In this paper, a three-dimensional (3D) solution method is presented for the free vibration

of isotropic and orthotropic conical shells with elastic boundary restraints. The formulation is

derived by means of the Rayleigh-Ritz procedure based on the three-dimensional elasticity

theory. Displacement components of the conical shells are represented by Fourier series in the

circumferential direction and a double Fourier cosine series and several supplementary

functions in meridional and normal directions. The supplementary functions in the form of the

product of a polynomial function and a single cosine series are introduced to ensure and

accelerate the convergence of the series representations. To validate the present method, the

convergence behavior is demonstrated, and comparisons with available results in the literature

are performed. Numerous new results for the isotropic and orthotropic conical shells with

elastic boundary conditions are presented, which can serve as the benchmark solution for

other computational techniques in the future research. The effects of the geometrical

parameters, orthotropic properties and boundary conditions on free vibration of conical shells

are illustrated.

Acknowledgment

The authors would like to thank the reviewers for their Constructive comments. The

authors gratefully acknowledge the financial support from the National Natural Science

Foundation of China (Nos. 51175098 and 51279035).

Appendix A. Detailed expressions for the stiffness matrix and mass matrix

The superscripts of the elements in the mass matrix ijM and stiffness matrix ijK are

given as follows.

1 1

1 1

1 1

( 1) 1; ( 1) 1

( 1) ( 1) 1; ( 1) ( 1) 1

( 1) ( 1) 1; ( 1) ( 1) 1

F F M q m G G M q mf f M l m g g M l m

f f Q l q g g Q l q

� � � ( � � � � � ( � �

� � � ( � � � � � � ( � � �

� � � ( � � � � � � ( � � ��

The detailed expressions of elements in matrices uuM are:

1 10 0

( , ) cos cos cos cosL H

FGuu oo m m q qC R r s r r s sdrds� � � � �� M (A.1)

1 10 0

( , ) cos cos cos ( )L H

Fguu oo m m q l sC R r s r r s s drds� � � � �� M (A.2)

1 10 0

( , ) cos ( ) cos cosL H

Fguu oo m l r q qC R r s r r s sdrds� � � � �� M (A.3)

1 10 0

( , ) cos cos ( ) cosL H

fGuu oo m m ls qC R r s r r s sdrds� � � � �� M (A.4)

1 10 0

( , ) cos cos ( ) ( )L H

fguu oo m m ls l sC R r s r r s s drds� � � � �� M (A.5)

1 10 0

( , ) cos ( ) ( ) cosL H

fguu oo m l r ls qC R r s r r s sdrds� � � � �� M (A.6)

1 10 0

( , ) ( ) cos cos cosL H

fGuu oo lr m q qC R r s r r s sdrds� � � � �� M (A.7)

1 10 0

( , ) ( ) cos cos ( )L H

fguu oo lr m q l sC R r s r r s s drds� � � � �� M (A.8)

1 10 0

( , ) ( ) ( ) cos cosL H

fguu oo lr l r q qC R r s r r s sdrds� � � � �� M (A.9)

1 10 0

( , ) cos cos cos cosL H

FGuu oo m m q qS R r s r r s sdrds� � � � �� M �� (A.10)

1 10 0

( , ) cos cos cos ( )L H

Fguu oo m m q l sS R r s r r s s drds� � � � �� M �� (A.11)

1 10 0

( , ) cos ( ) cos cosL H

Fguu oo m l r q qS R r s r r s sdrds� � � � �� M �� (A.12)

1 10 0

( , ) cos cos ( ) cosL H

fGuu oo m m ls qS R r s r r s sdrds� � � � �� M �� (A.13)

1 10 0

( , ) cos cos ( ) ( )L H

fguu oo m m ls l sS R r s r r s s drds� � � � �� M �� (A.14)

1 10 0

( , ) cos ( ) ( ) cosL H

fguu oo m l r ls qS R r s r r s sdrds� � � � �� M �� (A.15)

1 10 0

( , ) ( ) cos cos cosL H

fGuu oo lr m q qS R r s r r s sdrds� � � � �� M�� (A.16)

1 10 0

( , ) ( ) cos cos ( )L H

fguu oo lr m q l sS R r s r r s s drds� � � � �� M�� (A.17)

1 10 0

( , ) ( ) ( ) cos cosL H

fguu oo lr l r q qS R r s r r s sdrds� � � � �� M�� (A.18)

The detailed expressions of elements in matrices uuK are

1 1 1

1 1

1 1

1 1 1

11 00

66 11

222 00

55 00

00

( , )sin sin cos cos

/ ( , ) cos cos cos cos

cos / ( , ) cos cos cos cos

( , ) cos cos sin sin

[ ( ,0) ( , )

m m m m q q

m m q q

m m q qFGuu

q q m m q q

uo

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

C R r k R r L k

� � � � � �

� � � �

� � � � �

� � � � � �

�

�

��

�

�

K

1

1

1 1 1 1

0 0

12 00

]cos cos ( 1)

cos ( sin cos cos sin )cos cos

L H

q quL m m

m m m m m m q q

drds

r r

c C r r r r s s

� �

� � � � � � � � �

�

� ��

��

� � (A.19)

1 1 1

1 1

1 1

1 1

11 00

66 11

222 00

'55 00

12 00

( , )sin sin cos ( )

/ ( , ) cos cos cos ( )

cos / ( , ) cos cos cos ( )

( , ) cos cos sin ( )

cos ( sin cos

m m m m q l s

m m q l s

Fguu m m q l s

q m m q l s

m m

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C r

� � � � � �

� � � �

� � � � �

� � � � �

� � � �

�

�

� �

�

K

1 1 1 1

0 0

cos sin )cos ( )

L H

m m m m q l s

drds

r r r s s� � � � �

� ��

� � (A.20)

1 1

1 1

1 1

1 1 1

11 00

66 11

222 00

55 00

00

( , )sin ( )cos cos

/ ( , ) cos ( )cos cos

cos / ( , ) cos ( ) cos cos

( , ) cos ( )sin sin

[ ( ,0) ( , )

m m l r q q

m l r q q

m l r q qFguu

q q m l r q q

uo uL

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

C R r k R r L k

� � � � �

� � � �

� � � � �

� � � � � �

� �

�

��

�

�

K

1

1

1 1 1

0 0

12 00

]cos ( )( 1)

cos ( sin ( ) cos ( ))cos cos

L H

q qm l r

m m l r m l r q q

drds

r r

c C r r r r s s

� �

� � � � � � � �

�

� ��

��

� � (A.21)

1 1 1

1 1

1 1

1 1 1

11 00

66 11

222 00

55 00

12 00

( , ) sin sin ( )cos

/ ( , ) cos cos ( )cos

cos / ( , ) cos cos ( ) cos

( , ) cos cos ( )sin

cos ( sin cos

m m m m ls q

m m ls q

fGuu m m ls q

q m m ls q

m m

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s s

c C R r s r r s s

c C r

� � � � � �

� � � �

� � � � �

� � � � �

� � �

�

�

� �

�

K

1 1 1 1

0 0

cos sin ) ( )cos

L H

m m m m ls q

drds

r r r s s� � � � � �

� ��

� � (A.22)

1 1 1

1 1

1 1

1 1

11 00

66 11

222 00

' '55 00

12 00

( , )sin sin ( ) ( )

/ ( , ) cos cos ( ) ( )

cos / ( , ) cos cos ( ) ( )

( , ) cos cos ( ) ( )

cos ( sin cos

m m m m ls l s

m m ls l s

fguu m m ls l s

m m ls l s

m m m

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C r

� � � � � �

� � � �

� � � � �

� � � �

� � � �

�

�

� �

�

K

1 1 1 1

0 0

cos sin ) ( ) ( )

L H

m m m ls l s

drds

r r r s s� � � � �

� ��

� � (A.23)

1 1

1 1

1 1

1 1 1

11 00

66 11

222 00

'55 00

00

( , ) sin ( ) ( ) cos

/ ( , ) cos ( ) ( ) cos

cos / ( , ) cos ( ) ( ) cos

( , ) cos ( ) ( )sin

[ ( ,0) ( , ) ]

m m l r ls q

m l r ls q

m l r ls qfguu

q m l r ls q

uo uL

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

C R r k R r L k

� � � � �

� � � �

� � � � �

� � � � �

� �

�

��

�

�

K

1

1

1 1 1

0 0

12 00

cos ( )( 1)

cos ( sin ( ) cos ( )) ( ) cos

L H

q qm l r

m m l r m l r ls q

drds

r r

c C r r r r s s

� �

� � � � � � � �

�

� ��

� � (A.24)

1 1 1

1 1

1 1

1 1 1

'11 00

66 11

222 00

55 00

00

( , ) ( )sin cos cos

/ ( , ) ( ) cos cos cos

cos / ( , ) ( ) cos cos cos

( , ) ( ) cos sin sin

[ ( ,0) ( , )

m lr m q q

lr m q q

lr m q qfGuu

q q lr m q q

uo

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

C R r k R r L k

� � � � �

� � � �

� � � � �

� � � � � �

� �

�

��

�

�

K

1

1

1 1 1 1

0 0

'12 00

] ( ) cos ( 1)

cos ( ( )cos ( )sin )cos cos

L H

q quL lr m

lr m m lr m q q

drds

r r

c C r r r r s s

� �

� � � � � � � �

�

� ��

� � (A.25)

1 1 1

1 1

1 1

1 1

11 00

66 11

222 00

'55 00

'12 00

( , ) ( ) sin cos ( )

/ ( , ) ( ) cos cos ( )

cos / ( , ) ( ) cos cos ( )

( , ) ( ) cos sin ( )

cos ( ( ) cos

m lr m q l s

lr m q l s

fguu lr m q l s

q lr m q l s

lr m

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C r

� � � � �

� � � �

� � � � �

� � � � �

� � �

� �

�

� �

�

�

K

1 1 1 1

0 0

( )sin ) cos ( )

L H

m lr m q l s

drds

r r r s s� � � � �

� ��

� � (A.26)

1 1

1 1

1 1

1 1 1

11 00

66 11

222 00

55 00

00

( , ) ( ) ( ) cos cos

/ ( , ) ( ) ( ) cos cos

cos / ( , ) ( ) ( ) cos cos

( , ) ( ) ( )sin sin

[ ( ,0) ( , ) ]

lr l r q q

lr l r q q

lr l r q qfguu

q q lr l r q q

uo uL l

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

c C R r s r r s s

C R r k R r L k

� � � �

� � � �

� � � � �

� � � � � �

�

�

�

��

�

�

K

1

1

1 1 1

0 0

'12 00

( ) ( )( 1)

cos ( ( ) ( ) ( ) ( )) cos cos

L H

q qr l r

lr l r lr l r q q

drds

r r

c C r r r r s s

�

� � � � � � �

�

� ��

� � (A.27)

1 1 1

1 1

1 1

1 1 1

11 00

66 11

222 00

55 00

00

( , )sin sin cos cos

/ ( , ) cos cos cos cos

cos / ( , ) cos cos cos cos

( , ) cos cos sin sin

[ ( ,0) ( ,

m m m m q q

m m q q

m m q qFGuu

q q m m q q

uo

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

S R r k R r L

� � � � � �

� � � �

� � � � �

� � � � � �

�

�

��

�

�

K ��

1

1

1 1 1 1

0 0

12 00

) ]cos cos ( 1)

cos ( sin cos cos sin )cos cos

L H

q quL m m

m m m m m m q q

drds

k r r

c S r r r r s s

� �

� � � � � � � � �

�

� ��

��

� � (A.28)

1 1 1

1 1

1 1

1 1

11 00

66 11

222 00

'55 00

12 00

( , )sin sin cos ( )

/ ( , ) cos cos cos ( )

cos / ( , ) cos cos cos ( )

( , ) cos cos sin ( )

cos ( sin co

m m m m q l s

m m q l s

Fguu m m q l s

q m m q l s

m m

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

c S r

� � � � � �

� � � �

� � � � �

� � � � �

� � �

�

�

� �

�

K ��

1 1 1 1

0 0

s cos sin )cos ( )

L H

m m m m q l s

drds

r r r s s� � � � � �

� ��

� � (A.29)

1 1

1 1

1 1

1 1 1

11 00

66 11

222 00

55 00

00

( , ) sin ( )cos cos

/ ( , ) cos ( ) cos cos

cos / ( , ) cos ( )cos cos

( , ) cos ( )sin sin

[ ( ,0) ( , )

m m l r q q

m l r q q

m l r q qFguu

q q m l r q q

uo

c S R r s r r s s

c s R r s r r s s

c S R r s r r s s

c S R r s r r s s

S R r k R r L k

� � � � �

� � � �

� � � � �

� � � � � �

� �

�

��

�

�

K ��

1

1

1 1 1

0 0

12 00

]cos ( )( 1)

cos ( sin ( ) cos ( )) cos cos

L H

q quL m l r

m m l r m l r q q

drds

r r

c S r r r r s s

� �

� � � � � � � �

�

� ��

��

� � (A.30)

1 1 1

1 1

1 1

1 1 1

11 00

66 11

222 00

55 00

12 00

( , )sin sin ( )cos

/ ( , ) cos cos ( )cos

cos / ( , ) cos cos ( ) cos

( , ) cos cos ( )sin

cos ( sin c

m m m m ls q

m m ls q

fGuu m m ls q

q m m ls q

m m

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s s

c S R r s r r s s

c S r

� � � � � �

� � � �

� � � � �

� � � � �

� � �

�

�

� �

�

K ��

1 1 1 1

0 0

os cos sin ) ( ) cos

L H

m m m m ls q

drds

r r r s s� � � � � �

� ��

� � (A.31)

1 1 1

1 1

1 1

1 1

11 00

66 11

222 00

' '55 00

12 00

( , ) sin sin ( ) ( )

/ ( , ) cos cos ( ) ( )

cos / ( , ) cos cos ( ) ( )

( , ) cos cos ( ) ( )

cos ( sin cos

m m m m ls l s

m m ls l s

fguu m m ls l s

m m ls l s

m m

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

c S r

� � � � � �

� � � �

� � � � �

� � � �

� � �

�

�

� �

�

K ��

1 1 1 1

0 0

cos sin ) ( ) ( )

L H

m m m m ls l s

drds

r r r s s� � � � � �

� ��

� � (A.32)

1 1

1 1

1 1

1 1 1

11 00

66 11

222 00

'55 00

00

( , )sin ( ) ( ) cos

/ ( , ) cos ( ) ( ) cos

cos / ( , ) cos ( ) ( ) cos

( , ) cos ( ) ( )sin

[ ( ,0) ( , )

m m l r ls q

m l r ls q

m l r ls qfguu

q m l r ls q

uo u

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

S R r k R r L k

� � � � �

� � � �

� � � � �

� � � � �

� �

�

��

�

�

K ��

1

1

1 1 1

0 0

12 00

]cos ( )( 1)

cos ( sin ( ) cos ( )) ( ) cos

L H

q qL m l r

m m l r m l r ls q

drds

r r

c S r r r r s s

� �

� � � � � � � �

�

� ��

� � (A.33)

1 1 1

1 1

1 1

1 1 1

'11 00

66 11

222 00

55 00

00

( , ) ( )sin cos cos

/ ( , ) ( ) cos cos cos

cos / ( , ) ( ) cos cos cos

( , ) ( ) cos sin sin

[ ( ,0) ( ,

m lr m q q

lr m q q

lr m q qfGuu

q q lr m q q

uo

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

S R r k R r L

� � � � �

� � � �

� � � � �

� � � � � �

� �

�

��

�

�

K��

1

1

1 1 1 1

0 0

'12 00

) ] ( ) cos ( 1)

cos ( ( ) cos ( )sin )cos cos

L H

q quL lr m

lr m m lr m q q

drds

k r r

c S r r r r s s

� �

� � � � � � � �

�

� ��

� � (A.34)

1 1 1

1 1

1 1

1 1

11 00

66 11

222 00

'55 00

'12 00

( , ) ( ) sin cos ( )

/ ( , ) ( ) cos cos ( )

cos / ( , ) ( ) cos cos ( )

( , ) ( ) cos sin ( )

cos ( ( ) cos

m lr m q l s

lr m q l s

fguu lr m q l s

q lr m q l s

lr

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

c S r

� � � � �

� � � �

� � � � �

� � � � �

� �

� �

�

� �

�

�

K��

1 1 1 1

0 0

( )sin )cos ( )

L H

m m lr m q l s

drds

r r r s s� � � � � �

� ��

� � (A.35)

1 1

1 1

1 1

1 1 1

11 00

66 11

222 00

55 00

00

( , ) ( ) ( ) cos cos

/ ( , ) ( ) ( ) cos cos

cos / ( , ) ( ) ( ) cos cos

( , ) ( ) ( )sin sin

[ ( ,0) ( , ) ]

lr l r q q

lr l r q q

lr l r q qfguu

q q lr l r q q

uo uL

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

c S R r s r r s s

S R r k R r L k

� � � �

� � � �

� � � � �

� � � � � �

�

�

��

�

�

K��

1

1

1 1 1

0 0

'12 00

( ) ( )( 1)

cos ( ( ) ( ) ( ) ( )) cos cos

L H

q qlr l r

lr l r lr l r q q

drds

r r

c S r r r r s s

� �

� � � � � � �

�

� ��

� � (A.36)

where 2 2

0 0

cos( ) cos( ) sin( ) sin( ),i j i j

ij iji j i jd n d n d n d nC d S d

d d d d

� ��

� ��

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List of Collected Table and Figure Captions

Table 1 The non-dimensional frequency parameters /L G� �) � of isotropic conical

shells with different thickness-to-radius ratios H/R1 (Lcos�=2 m, R1=1m, �=30°, E=168 GPa,

�=5700 kg/m3, and �=0.3; boundary condition: C-F)

Table 2 The non-dimensional frequency parameters /L G� �) � of isotropic conical

shells with different thickness-to-height ratios H/L (R1/L=0.25, �=30°, E=168 GPa, �=5700

kg/m3, and �=0.3; boundary condition: F-F and C-C)

Table 3 The fundamental frequencies (Hz) of the isotropic conical shells with different

semi-vertex � ( R1=1 m, Lcos�=2 m, E=168 GPa, �=5700 kg/m3, �=0.3)

Table 4 The fundamental frequencies (Hz) of isotropic conical shells with various boundary

conditions (R1=1 m, Lcos�=2 m, E=70 GPa, �=2707 kg/m3, �=0.3)

Table 5 The fundamental frequencies (Hz) of orthotropic conical shells (type I) with various

boundary conditions (R1=1 m, Lcos�=2 m)

Table 6 The fundamental frequencies (Hz) of orthotropic conical shells (type II) with various

boundary conditions (R1=1 m, Lcos�=2 m)

Table 7 The fundamental frequencies (Hz) of orthotropic conical shells (type III) with various

boundary conditions (R1=1 m, Lcos�=2 m).

Fig. 1 The coordinate system and geometry of a conical shell.

Fig. 2 Mode shapes of Al conical shells with C-C boundary condition (H/R1=1, Lcos�=2m,

R1=1m, �=60°)

Fig. 3 Mode shapes of orthotropic conical shells (Type I) with F-C boundary condition (H/R1=1,

Lcos�= 2m, R1=1m, �=60°)

Fig. 4 Mode shapes of orthotropic conical shells (Type II) with S3-S3 boundary condition

(H/R1=1, Lcos�= 2m, R1=1m, �=60°)

Fig. 5 Mode shapes of orthotropic conical shells (Type III) with E2-E2 boundary condition

(H/R1=1, Lcos�= 2m, R1=1m, �=60°)

Fig. 6 Effect of the semi-vertex angle � on the natural frequencies (Hz) of isotropic conical

shells with different boundary condition (H/R1=0.5, Lcos�= 2m, R1=1m).

Fig. 7 Effect of the semi-vertex angle � on the natural frequencies (Hz) of orthotropic conical

shells with different boundary condition (H/R1=0.5, Lcos�= 2m, R1=1m): (a) C-C; (b) S1-S1;(c)

S3-S3; (d) E2-E2.

Fig. 8 Effect of the thickness-to-radius ratio H/R1 on the natural frequencies (Hz) of isotropic

conical shells with different boundary condition (Lcos�= 2m, R1=1m, �=30°).

Fig. 9 Effect of the thickness-to-radius ratio H/R1 on the natural frequencies (Hz) of orthotropic

conical shells with different boundary condition (Lcos�= 2m, R1=1m, �=30°): (a) C-C; (b)

S1-S1;(c) S3-S3; (d) E2-E2.

R

Fig. 1 The coordinate system and geometry of a conical shell

Fig.

Lcos�=2m

2 Mode sh

m, R1=1m, �

Mode shap

Mode shap

hapes of isotr

�=60°)

pes 1-2

pes 4-5

ropic conica

al shells with

Mode

Mode sha

h C-C boun

shape 3

apes 6-7

ndary conditi

ions (H/R1=1,

Fig.

(H/R1=1,

3 Mode sha

, Lcos�=2m,

Mode shap

Mode shap

apes of ortho

R1=1m, �=6

pes 1

pes 4-5

otropic (Type

60°)

e I) conical s

mode sh

Mode

shells with F

hapes 2- 3

shape 6

F-C boundaryy conditions

s

Fig.

(H/R1=1,

4 Mode sha

, Lcos�=2m,

Mode sha

Mode sh

apes of ortho

R1=1m, �=6

apes 1

apes 4-5

otropic (Type

60°)

e II) conical

mode shap

Mode shap

shells with S

pes 2- 3

pes 6-7

S3-S3 bounda

ary conditionns

Fig.

condition

5 Mode sha

ns (H/R1=1, L

Mode shap

Mode shap

apes of ortho

Lcos�=2m, R

pes 1

pes 4-5

otropic (Type

R1=1m, �=6

e III) conica

60°)

Mode sha

Mode

al shells with

apes 2- 3

shape 6

h E2-E2 boun

ndary

Fig. 6 Effect of the semi-vertex angle � on the natural frequencies (Hz) of isotropic

conical shells with different boundary condition (H/R1=0.5, Lcos�= 2m, R1=1m).

10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Semi-vertex angle �

Nat

ural

Fre

quen

cy

C-CS1-S1

S3-S3

E2-E2

(a) (b)

10 20 30 40 50 60 70 800

100

200

300

400

500

Semi-vertex angle �N

atur

al fr

eque

ncy

Type IType IIType III

10 20 30 40 50 60 70 800

50

100

150

200

250

300

350


Nat

ural

freq

uenc

y


10 20 30 40 50 60 70 800

50

100

150


Nat

ural

freq

uenc

y


(c) (d)

Fig. 7 Effect of the semi-vertex angle � on the natural frequencies (Hz) of orthotropic

conical shells with different boundary condition (H/R1=0.5, Lcos�= 2m, R1=1m):(a) C-C; (b)

S1-S1; (c) S3-S3; (d) E2-E2

10 20 30 40 50 60 70 800

100

200

300

400

500


Nat

ural

freq

uenc

y


Fig. 8 Effect of the thickness-to-radius ratio H/R1 on the natural frequencies (Hz) of

isotropic conical shells with different boundary condition (Lcos�= 2m, R1=1m, �=30°).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

H/R1

Nat

ural

freq

uenc

y

C-CS1-S1

S3-S3

E2-E2

(a) (b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100

150

200

250

300

350

400

450

H/R1

Nat

ural

freq

uenc

y


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100

150

200

250

300

H/R1

Nat

ural

freq

uenc

y


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 140

60

80

100

120

140

160

H/R1

Nat

ural

freq

uenc

y Type IType IIType III

(c) (d)

Fig. 9 Effect of the thickness-to-radius ratio H/R1 on the natural frequencies (Hz) of

isotropic conical shells with different boundary condition (Lcos�= 2m, R1=1m, �=30°): (a) C-C;

(b) S1-S1; (c) S3-S3; (d) E2-E2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100

150

200

250

300

350

400

H/R1

Nat

ural

freq

uenc

y


Highlights

> A three-dimensional solution method is presented for the free vibration of isotropic

and orthotropic conical shells.

> The method is universally applicable to general boundary conditions.

> New results for shells with elastically restrained edges are presented.

> The effects of geometrical parameters on the frequency of the shells are investigated.

Table 1

The non-dimensional frequency parameters /L G� �� of isotropic conical shells with

different thickness-to-radius ratios H/R1 (Lcosα=2 m, R1=1m, α=30°, E=168 GPa, ρ=5700 kg/m3,

and μ=0.3; boundary condition: C-F).

H/R1 M× Mode 1 2 3 4 5 6 7 8 9 10

0.1 10×

0.3111 0.3111 0.3196 0.3196 0.4645 0.4645 0.5453 0.5453 0.6979 0.6979

11×

0.3109 0.3109 0.3194 0.3194 0.4644 0.4644 0.5452 0.5452 0.6978 0.6978

12×

0.3109 0.3109 0.3193 0.3193 0.4643 0.4643 0.5451 0.5451 0.6977 0.6977

13×

0.3108 0.3108 0.3191 0.3191 0.4643 0.4643 0.5449 0.5449 0.6976 0.6976

14×

0.3108 0.3108 0.3191 0.3191 0.4642 0.4642 0.5449 0.5449 0.6975 0.6975

0.2 10×

0.3876 0.3876 0.5133 0.5133 0.5734 0.5734 0.8441 0.8441 0.9656 1.2821

11×

0.3874 0.3874 0.5132 0.5132 0.5732 0.5732 0.8440 0.8440 0.9656 1.2819

12×

0.3873 0.3873 0.5131 0.5131 0.5731 0.5731 0.8439 0.8439 0.9656 1.2818

13×

0.3871 0.3871 0.5131 0.5131 0.5730 0.5730 0.8438 0.8438 0.9656 1.2817

14×

0.3871 0.3871 0.5130 0.5130 0.5729 0.5729 0.8438 0.8438 0.9656 1.2817

0.5 10×

0.6024 0.6024 0.6601 0.6601 1.0105 1.0123 1.0123 1.3380 1.6725 1.6725

11×

0.6022 0.6022 0.6599 0.6599 1.0105 1.0122 1.0122 1.3379 1.6724 1.6724

12×

0.6020 0.6020 0.6598 0.6598 1.0105 1.0121 1.0121 1.3378 1.6724 1.6724

13×

0.6019 0.6019 0.6596 0.6596 1.0105 1.0121 1.0121 1.3377 1.6724 1.6724

14×

0.6018 0.6018 0.6595 0.6595 1.0105 1.0120 1.0120 1.3377 1.6724 1.6724

Table(s)

Table 2

The non-dimensional frequency parameters /L G� �� of isotropic conical shells with

different thickness-to-height ratios H/L (R1/L=0.25, α=30°, E=168 GPa, ρ=5700 kg/m3, and μ=0.3;

boundary condition: F-F and C-C)

n Mode F-F C-C H/L=0.25 H/L=1 H/L=0.25 H/L=1 Ref.[44] present Ref.[44] present Ref.[44] present Ref.[44] present 0 1 1.928 1.928 1.274 1.274 3.050 3.049 3.172 3.171 2 2.956 2.956 1.880 1.880 3.226 3.226 3.311 3.310 3 3.523 3.523 3.218 3.218 4.734 4.733 4.865 4.865 4 3.650 3.650 3.650 3.650 5.742 5.740 5.356 5.351 5 5.521 5.518 4.238 4.236 6.329 6.328 5.573 5.570 6 5.661 5.661 4.488 4.487 7.455 7.447 6.302 6.300 1 1 2.158 2.158 1.769 1.768 2.483 2.483 3.001 3.000 2 2.965 2.965 1.961 1.961 4.296 4.295 3.651 3.650 3 3.477 3.476 3.622 3.622 4.840 4.839 5.205 5.203 4 5.169 5.168 3.964 3.963 5.463 5.462 5.499 5.498 5 5.348 5.348 4.181 4.180 7.293 7.289 5.662 5.656 6 5.770 5.766 4.585 4.584 7.605 7.599 6.124 6.121 2 1 0.618 0.618 0.786 0.786 2.555 2.555 3.205 3.203 2 1.468 1.468 1.077 1.076 4.900 4.900 4.248 4.248 3 3.005 3.004 2.775 2.774 5.534 5.532 5.332 5.330 4 3.697 3.696 2.799 2.799 6.394 6.393 5.834 5.828 5 4.614 4.613 4.099 4.097 7.729 7.721 6.049 6.045 6 5.480 5.479 4.316 4.315 8.994 8.991 6.387 6.381 3 1 1.497 1.496 1.687 1.687 3.323 3.323 3.770 3.766 2 3.096 3.095 2.200 2.198 5.530 5.530 4.986 4.984 3 4.225 4.224 3.668 3.667 6.575 6.573 5.583 5.581 4 4.792 4.791 3.733 3.732 7.934 7.932 6.252 6.244 4 1 2.509 2.508 2.538 2.537 4.440 4.439 4.414 4.411 2 4.448 4.448 3.159 3.157 6.568 6.566 5.690 5.687 3 5.439 5.438 4.405 4.404 7.756 7.755 6.103 6.100 4 6.499 6.497 4.516 4.515 9.008 9.001 6.703 6.693

Table 3

The fundamental frequencies (Hz) of the isotropic conical shells with different semi-vertex α (R1=1

m, Lcosα=2 m, E=168 GPa, ρ=5700 kg/m3, μ=0.3).

α H/R1 F-F F-C C-F C-C present ANASYS present ANASYS present ANASYS present ANASYS 30° 0.10 22.17 22.19 174.75 174.98 72.12 72.19 249.35 249.57 0.20 41.47 41.51 214.81 214.90 89.83 89.82 346.86 346.95 0.50 86.76 86.85 305.18 305.37 139.66 139.66 520.60 520.71 1.00 131.68 131.85 343.83 343.87 186.42 186.36 635.06 635.59 45° 0.10 13.33 13.33 137.70 137.90 40.00 40.04 169.56 169.72 0.20 25.15 25.17 175.03 175.13 51.29 51.28 238.80 238.87 0.50 55.06 55.12 261.66 261.84 83.96 83.94 378.38 378.44 1.00 91.18 91.28 293.86 293.95 116.56 116.49 484.83 485.11 60° 0.10 6.98 6.98 82.22 82.34 16.60 16.57 89.28 89.35 0.20 13.19 13.20 107.09 107.17 22.16 22.13 126.38 126.41 0.50 29.74 29.76 167.10 167.16 39.64 39.63 210.14 210.13 1.00 53.17 53.22 199.09 199.12 55.00 54.94 293.21 293.24

Table 4

The fundamental frequencies (Hz) of isotropic conical shells with various boundary conditions

(R1=1 m, Lcosα=2 m, E=70 GPa, ρ=2707 kg/m3, μ=0.3).

α H/R1 S1-S1 S2-S2 S3-S3 C-S2 C-S3 E1-E1 E2-E2 E3-E3 15° 0.10 272.90 228.22 58.07 281.14 257.69 284.90 279.96 241.33 0.20 342.91 275.07 66.42 294.82 352.34 373.93 352.63 329.99 0.50 387.67 362.48 81.82 303.71 495.47 461.22 344.88 351.43 1.00 403.82 378.18 94.66 315.89 563.83 493.88 334.42 352.97 30° 0.10 215.10 180.61 86.17 206.46 203.01 226.65 218.27 193.34 0.20 270.12 223.57 99.80 209.87 283.37 304.16 289.99 268.78 0.50 305.01 308.84 129.45 219.62 397.19 384.54 316.13 297.24 1.00 319.36 332.78 158.70 233.93 468.06 416.43 307.97 327.07 45° 0.10 146.49 128.30 82.26 131.04 137.77 155.34 147.76 133.14 0.20 181.77 162.79 96.48 133.41 192.34 212.23 201.58 185.81 0.50 210.60 240.89 131.55 140.35 267.65 285.76 268.43 216.24 1.00 221.41 276.18 178.16 151.13 342.36 314.72 264.00 269.20 60° 0.10 78.23 72.66 53.34 66.88 72.13 82.59 78.06 70.97 0.20 96.03 94.13 63.16 67.95 102.19 114.63 108.40 97.86 0.50 113.20 152.72 88.48 71.14 139.34 172.35 178.21 120.20 1.00 121.64 199.89 132.26 76.33 195.69 199.61 199.90 164.51 75° 0.10 23.92 22.21 17.87 19.70 21.09 24.59 23.21 21.24 0.20 29.96 29.81 21.33 19.90 28.29 34.22 32.19 28.82 0.50 34.01 51.17 30.16 20.50 39.17 56.10 55.50 36.91 1.00 37.67 83.95 47.16 21.51 60.32 78.39 88.71 53.76

Table 5

The fundamental frequencies (Hz) of orthotropic conical shells (type I) with various boundary

conditions (R1=1 m, Lcosα=2 m).

α H/R1 C-C S1-S1 S2-S2 S3-S3 C-S2 E1-E1 E2-E2 E3-E3 15° 0.10 170.79 157.42 113.18 27.53 154.37 168.75 163.95 147.11 0.20 231.90 198.18 139.52 31.60 183.44 227.13 215.29 203.45 0.50 340.72 225.03 193.85 39.92 189.33 312.33 307.72 286.18 1.00 414.43 220.78 202.38 46.72 196.93 355.86 337.84 346.24 30° 0.10 135.02 124.34 89.00 40.82 116.54 133.78 128.50 117.81 0.20 185.28 154.25 112.71 47.50 130.83 182.21 172.55 162.94 0.50 279.64 175.98 163.75 63.50 136.92 260.58 260.54 227.41 1.00 355.93 174.45 176.73 78.91 145.84 305.14 309.75 280.42 45° 0.10 91.32 84.48 63.02 38.88 76.33 90.75 86.98 80.48 0.20 126.38 102.96 81.65 45.82 83.17 124.88 118.91 111.11 0.50 196.43 117.04 126.74 64.65 87.50 187.65 188.01 153.41 1.00 265.93 121.54 146.17 90.10 94.22 232.67 252.01 199.17 60° 0.10 47.74 44.99 35.72 25.13 39.34 47.58 45.77 42.40 0.20 66.53 54.56 47.09 29.88 42.36 66.07 63.39 58.48 0.50 106.23 60.63 79.70 43.17 44.35 103.48 103.39 81.07 1.00 153.30 65.86 105.99 67.25 47.59 141.20 151.87 111.13 75° 0.10 13.96 13.63 11.28 8.39 11.53 13.94 13.48 12.51 0.20 19.40 15.62 14.99 10.05 12.40 19.35 18.72 17.09 0.50 31.43 17.46 26.55 14.56 12.78 31.17 30.86 24.02 1.00 47.90 19.50 44.83 23.47 13.41 46.61 47.90 34.53

Table 6

The fundamental frequencies (Hz) of orthotropic conical shells (type II) with various boundary



Table 7

The fundamental frequencies (Hz) of orthotropic conical shells (type III) with various boundary



three-dimensional vibration analysis of isotropic and orthotropic conical shells with elastic...

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