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Free Damped Vibrations of Sandwich 

Shells of Revolution 

ALEKSANDR KORJAKIN, 1, * ROLANDS RIKARDS, 1 HOLM ALTENBACH 2 

AND ANDRIS CHATE 1 

1 Technical University Riga, Kalku iela 1, LV-1658, Riga, Latvia 

2 Martin-Luther-University Halle-Wittenberg, D-06099 Halle, Germany 

ABSTRACT: Using a zig-zag model the free damped vibrations of sandwich shells of 

revolution are investigated. As special cases the vibration analysis under consideration of 

damping of cylindrical, conical and spherical sandwich shells is performed. A specific 

sandwich shell finite element with 54 degrees of freedom is employed. Starting from 

the energy method the damping model is developed. Numerical examples for the free 

vibration analysis with damping based on the proposed finite element approach are 

discussed. Results for sandwich shells show a satisfactory agreement with various reference 

solutions. 

KEY WORDS: cylindrical shell, conical shell, spherical shell, sandwich structures, finite 

element analysis, free vibration, damping. 

INTRODUCTION 

DAMPING IS A significant dynamic parameter for vibration and sound control, 

dynamic stability, fatigue endurance and impact resistance. Many structural 

applications (large space structures, engine blades and high-speed machinery) require 

light weight and high dynamic performance. Therefore, a possible source of 

passive damping should add minimal parasitic weight and must be compatible 

with the structural configuration. Two potential damping sources satisfying the 

previous requirements are the introduction of constrained damping layers and the 

damping capacity of composites. 

Vibrational damping analysis of laminated composite beam and plates has been 

investigated by several authors [14,19,25,32,42]. The application to laminated 

*Author to whom correspondence should be addressed. E-mail: aleks@latnet.lv 

Journal of SANDWICH STRUCTURES AND MATERIALS, Vol. 3—July 2001 

171 

1530-7972/01/03 0171–26 $10.00/0 DOI: 10.1106/LB2E-22L4-7JA6-CAED 

© 2001 Technomic Publishing Co., Inc. 

Journal Online at http://techpub.metapress.com

172 ALEKSANDR KORJAKIN ET AL. 

shells is discussed, for example, in References [2,3,8,15,22,23,33,35,43,44,48]. 

Though the damping characteristics of laminated composite structures have been 

studied extensively, studies of sandwich composite structures are limited. The 

analysis of sandwich beams and constrained layer damping in beams has been performed 

in Reference [9]. A three-layered beam theory was employed where the 

continuity of the displacements and the transverse shear stresses is satisfied at the 

interfaces. The energy method is used to derive the governing equations of motion 

for transverse vibrations of curved sandwich beams [46]. Flexural vibrations of 

viscoelastic damped sandwich plates has been analyzed in Reference [17]. A finite 

element analysis associated with an asymptotic solution method for the harmonic 

flexural vibrations was proposed. The finite element method was employed for the 

analysis of the harmonic response of damped three-layer plates in Reference [20]. 

General non-linear equations of motion of viscoelastic damped sandwich plates 

and cylindrical panels were derived with the help of the principle of virtual work in 

Reference [49]. A special finite element approach was developed in Reference 

[36] to study the vibration and damping characteristics of three-layered conical 

shells with a viscoelastic core. A finite element based on the discrete layer theory 

and taking into account the shear deformations was used for analysis of spherical 

shells with a viscoelastic core in Reference [15]. Damping properties of three-layered 

shallow spherical shells have been studied in Reference [33]. Expressing the 

in-plane displacements in terms of auxiliary functions, the general solution of the 

equations of motion for non-axisymmetric modes was given in terms of Bessel’s 

functions. 

Different shell and plate theories can be used to analyze the sandwich structures. 

Good results can be obtained employing the so called zig-zag theories. In 

Reference [10] a multilayered plate element is proposed which meets computational 

requirements and includes both the zig-zag distribution along the thickness 

coordinate of the in-plane displacements and the interlaminar continuity for the 

transverse shear stresses. A simple layerwise higher-order zig-zag model is proposed 

for the bending of laminated composite plates. The model accounts for cubic 

variation of the transverse shear stresses across the laminate with zero values at 

the free surface [27]. A beam finite element based on a discrete layer laminated 

beam theory with the sublaminate first-order zig-zag kinematic assumptions 

is presented and assessed for thick and thin laminated beams [50]. Authors 

have used a modified form of DiSciuva’s linear zig-zag laminate kinematics, in 

which continuity interfacial transverse shear stresses are satisfied identically. A 

computationally efficient finite-element formulation for linear and non-linear 

analysis of flat sandwich panels is presented in Reference [11]. Mechanical accuracy 

is acquired by allowing a zig-zag in-plane displacement field in the thickness 

direction and by fulfilling interlaminate equilibrium at the interface between the 

core and skins for the transverse shear stress components. A high accuracy for prediction 

of stresses can be obtained employing a higher-order zig-zag theory [6].

Free Damped Vibrations of Sandwich Shells of Revolution 173 

The vibration and damping analysis of sandwich shells has not been attempted 

so far and it is a subject of the present investigation. The natural frequencies and 

the loss factors of cylindrical, conical and spherical shells are calculated. A sandwich 

shell finite element based on the zig-zag model and described in Reference 

[12] is used for the dynamic analysis with damping. The energy method (EM) is 

employed to model the damping. The background of the EM is in finding the energy 

dissipated in a natural mode by adding up the contributions from the individual 

components of deformation. Such approach has been used by several authors, 

for example, in References [1,8,29,51]. The EM has been used for damping analysis 

of laminated plates [40]. The present paper is an extension of a similar investigation 

in the case of laminated shells presented in Reference [26]. 

Governing Equations 

THEORETICAL BACKGROUND 

Considering a composite sandwich shell of uniform moderate thickness h with 

three anisotropic laminae, each of them may be arbitrarily oriented. The influence 

of the interphases between the laminae is neglected that means rigid connection 

between the layers. In addition, in the numerical analysis the typical sandwich assumptions 

(soft core, the thickness of the outer layers is much thinner in comparison 

with the core thickness) are taken into account. 

Displacements 

The kinematic equations for the first-order shear deformation theory (FOSDT) 

including the drilling rotation of the normal can be obtained from three-dimensional 

equations of the theory of elasticity using the Taylor series approximation of 

the vector of position function in the deformed state with respect to the coordinate 

x 3 (see, for example, References [7,39,41]). 

Let us introduce curvilinear coordinates x i ={x α ,x 3 } on the midsurface of the 

shell in the initial state which is defined by the triad {a α , a 3 }. The use of Greek indices 

means the values 1, 2. The unit vector a 3 denotes the normal to the middle 

surface of the shell. Curvilinear coordinates X i ={X α ,X 3 } are related to the deformed 

state which is defined by the triad {A α ,A 3 }. In the deformed state the vector 

A 3 may not be perpendicular to the midsurface of the shell. 

The representation of the displacement vector u of an arbitrary point of the shell 

with respect to the first-order approximation is 

α α α 

3 = + 3 

u( x , x ) v( x ) x ( 

x ) 

(1) 

Here v is the displacement vector of the midsurface and denotes the vector of


rotations at the midsurface 

= A −a 

3 3 

(2) 

In Equation (1) the vectors v and have three components, respectively 

α 

α 

3, 

α 3 

v= v a + wa = γ a + wa 

α 

(3) 

In this sense, the FOSDT is characterized by 6 independent degrees of freedom 

(FOSDT-6). The FOSDT version with five degrees of freedom (FOSDT-5) is usually 

used. The reasons for such limitation is the deformability of the thin-walled 

structure even in the case of moderate thickness: the magnitude of the assumed 

drilling of the normal is much smaller in comparison with the other two rotations. 

So in the case of the FOSDT-5 we assume γ = 0. For shells of revolution in this case 

the displacements u, v, and w in the x (meridian), ϕ (circumferential) and x 3 (normal, 

outward from the reference surface of the shell) directions at any point are expressed 

through the following approximations [31]: 

ux ( , ϕ , x, t) = u( x, ϕ , t) + xγ ( x, ϕ, t) 

3 0 3 

vx ( , ϕ , x, t) = v( x, ϕ , t) + xγ ( x, ϕ, t) 

3 0 3 

wx (, ϕ , x,) t = w(, xϕ,) 

t 

3 0 

x 

ϕ 

(4) 

where u 0 , v 0 , w 0 are the displacement components at the reference surface 

(midsurface) and γ x , γ ϕ are the rotations of the normal associated with the transverse 

shear deformations. Below the index “0” will be omitted. The approximations, 

Equation (4), are similar to Timoshenko’s proposal in the beam theory [45] 

and the analogue in the plate theory [18,30]. 

STRAIN-DISPLACEMENT RELATIONS 

The displacements of an arbitrary point of the shell continuum is approximated 

by two vectors. Therefore, the displacement field of the shell in the FOSDT-6 including 

extension of the normal is represented by six unknown functions—three 

displacements (v α ,w) and three rotations (γ α ,γ). These quantities are functions of 

the shell middle surface coordinates x α only. From these displacements and rotations 

the two-dimensional strains can be obtained. The three-dimensional Green’s 

strain tensor in the linear case is given by Reference [16,41]. 

2e ij = u, i ⋅ g j + u, 

j ⋅g 

i 

(5) 

Here (...), i denotes differentiation with respect to the coordinates x i , the dot means 

scalar product and g i is a triad of vectors for coordinates x i on the surfaces 

(x 3 = const) parallel to the middle surface of the shell. For these coordinates the


components of the metric tensor are given by g ij = g i ⋅ g j . With respect to Equation 

(1) from Equation (5) the kinematic equations for the strains in the shell can be deduced 

[41]: 

2 

αβ =Ω αβ + 3χ αβ + 3Φ αβ, 2 α3 = 2Ω α3 + 3κα3 

e x x e x 

(6) 

with 

and 

2 Ω = ε +ε , 2χ =κ +κ 

αβ αβ βα αβ αβ βα 

λ λ 

αβ bk α λβ bk β λα α3 

α α 

2 Φ = − − , 2Ω = γ +Ψ 

λ 

α3 

α bα λ 

κ =ϕ − γ 

λ 

αβ αβ αβ αβ αβ α λβ αβ αβ αβ 

ε = v −b w, κ = k −b ε , k = γ −b 

γ 

λ 

λ 

α α α λ α α α λ 

Ψ = w, + b v , ϕ = γ , + b γ 

(7) 

(8) 

Here (...) α denotes the covariant differentiation in the metric characterized by the 

fundamental tensor components of the midsurface a αβ = a α ⋅ a β , and b αβ are the curvature 

tensor components of the midsurface. All quantities in the Equations (7) 

and (8) depend on x α only. Neglecting the strains Φ αβ and κ α3 in Equations (6) the 

kinematics equations can be simplified: 

e =Ω + x χ , 2e 

= 2Ω =γ +Ψ 

αβ αβ 3 αβ α3 α3 

α α 

(9) 

It should be noted that the strains Φ αβ and κ α3 can be taken into account in the finite 

element analysis of the shell, since the number of unknown functions is the 

same—six. However, even for shallow shells, the influence of these strains is significant. 

Therefore, for simplicity these terms are hereafter omitted. 

For the analysis of shells of various shapes it is more convenient to use orthogonal 

coordinates, which conicide with the directions of principal curvatures. In this 

case the metric properties of the coordinate system associated with the midsurface 

of the shell are characterized by the Lamé’s coefficients A 1 and A 2 . These coefficients 

are connected with the components of the fundamental tensor of the 

midsurface: 

a 

αβ 

2 

⎡a11 0 ⎤ ⎡A 

1 0 ⎤ 

= ⎢ 

0 a 

⎥ = ⎢ ⎥ 

2 

⎣ 22 ⎦ ⎢⎣ 

0 A2 

⎥⎦ 

(10) 

The kinematics, Equations (7) and (8), in orthogonal coordinates of principal cur-


vatures are given by (indices of the physical components of tensors are given in parentheses) 

[41]: 

1 ∂v 1 ∂A Ω = + v − b w , (1↔2) 

(1) 1 

(11) 1 2 (2) (11) 

A1 ∂x 

A1A2 

∂x 

1 ∂v 

1 ∂A 

1 ∂v 

1 

2Ω = − + − 

∂A 

(1) 2 (2) 

1 

(12) v 

2 1 (2) v 

1 2 (1) 

A2 ∂x A1A2 ∂x A1 ∂x A1A2 

∂x 

1 ∂γ 1 ∂A 

χ = + γ 

(1) 1 

(11) 1 2 (2) 

A1 ∂x 

A1A2 

∂x 

⎛ 1 ∂v 

1 ∂A 

⎞ 

−b γ − b + v −b w⎟, (1↔2) 

⎠ 

(1) 1 

(11) (11) ⎜ 1 2 (2) (11) 

⎝ A1 ∂x 

A1A2 

∂x 

(11) 

1 ∂γ 1 ∂A 

1 ∂γ 1 ∂A 

2χ = + γ + − γ 

(1) 2 (2) 

1 

(12) 2 1 (2) 1 2 (1) 

A2 ∂x A1A2 ∂x A1 ∂x A1A2 

∂x 

⎛ 1 ∂v 

1 ∂A 

⎞ ⎛ 1 ∂v 

1 ∂A 

⎞ 

−b ⎜ − v ⎟ −b ⎜ − v 

⎝ ⎠ ⎝ 

⎟ 

⎠ 

(1) 2 (2) 

1 

(11) 2 1 (2) (22) 1 2 (1) 

A2 ∂x A1A2 ∂x A1 ∂x A1A2 

∂x 

1 ∂w 2 Ω = γ + + b v , (1↔2) 

(13) (1) 1 (11) (1) 

A1 

∂x 

Here, the notations v (1) = u, v (2) = v are used for the physical components of the displacements. 

For shells of revolution it is convenient to use the coordinates x 1 = α, x 2 = ϕ (see 

Figure 1). 

In this case the curvature tensor is given by: 

b 

αβ 

⎡ 1 ⎤ 

⎢− 

0 

R ⎥ 

1 

= ⎢ 

⎥ 

⎢ sin α ⎥ 

⎢ 0 − ⎥ 

⎣ r ⎦ 

(12) 

The Lamé’s coefficients of a shell of revolution are A 1 = R 1 and A 2 = r, where the radii 

can be functions of the meridian coordinate, r = r(α) and R 1 = R 1 (α). From these 

expressions the kinematic equations for various shaped shells of revolution can be 

obtained. Thus, the general strain-displacement relations for cylindrical shells 

(see Figure 2) [39] are


Figure 1. Geometry of a shell of revolution. 

Figure 2. Geometry of the sandwich cylindrical shell.


∂u 

∂γ x 

ε xx = + x3 =Ω xx + x3χxx 

∂x 

∂x 

1 ⎛ ∂v 

⎞ x ∂γ 

ε = + + =Ω + χ 

R⎝ 

⎜ 

∂ϕ ⎠ 

⎟ 

R ∂ϕ 

3 ϕ 

ϕϕ w 

ϕϕ x3 

ϕϕ 

1 ∂u 

∂v 

⎛ 1 ∂γ ∂γ ⎞ 

γ = + + + = Ω + χ 

R ∂ϕ ∂x ⎜ 

⎝R ∂ϕ ∂x 

⎟ 

⎠ 

x ϕ 

xϕ x3 2 xϕ 2x3 

xϕ 

∂w 

γ xx = γ 2 

3 x + = Ωxx 

3 

∂x 

1 ⎛∂w 

⎞ 

γ = γ + − v = 2Ω 

R 

⎜ 

⎝∂ϕ 

⎟ 

⎠ 

ϕx3 ϕ ϕx3 

(13) 

where R is the reference surface radius of the cylinder, ε xx , ε ϕϕ are the normal 

strains, and γ xϕ , γ xx , γ 

3 ϕx 

are the shear strains. In addition, Ω 

3 

xx , Ω ϕϕ , Ω xϕ denote 

the membrane strains, χ xx , χ ϕϕ , χ xϕ are the bending strains and Ω xx , Ω 

3 ϕx 

are the 

3 

transverse shear strains in the reference surface, respectively. The general 

strain-displacement relations for conical shells (see Figure 3) are given by 

∂u 

∂γ 

ε = + =Ω + χ 

∂x 

∂x 

x 

xx x3 xx x3 

xx 

1 ⎛ ∂v 

⎞ 

ε ϕϕ = usin 

wcos 

r 

⎜ β+ + β 

⎝ ∂ϕ 

⎟ 

⎠ 

x3 

⎡ ∂γ ϕ cosβ⎛ 

∂v 

⎞⎤ 

+ ⎢γx 

sinβ+ + + usinβ+ wcosβ ⎥ =Ω ϕϕ + x3χ 

r 

∂ϕ r 

⎜ 

⎝∂ϕ 

⎟ 

⎣ 

⎠⎦ 

ϕϕ 

1 ∂u ∂v v 

γ xϕ 

= + − sinβ 

r ∂ϕ ∂x r 

⎛1∂γ 

∂γ 

x ϕ γ ϕ cosβ∂v⎞ 

+ x3⎜ 

+ − sinβ+ = 2Ω xϕ 

+ 2x3χ 

⎝r ∂ϕ ∂x r r ∂x 

⎟ 

⎠ 

xϕ 

(14) 

∂w 

= 2Ω 

x 

γ xx = γ 

3 x + xx ∂ 

3 

1 ⎛∂w 

⎞ 

γ = γ + − v cosβ = 2Ω 

r 

⎜ 

⎝∂ϕ 

⎟ 

⎠ 

ϕx3 ϕ ϕx3 

where r = r(z) is the reference surface radius and β is the semi-vertex angle. The 

generalstrain-displacement relations for spherical shell (see Figure 4) are givenby


Figure 3. Geometry of the sandwich conical shell. 

Figure 4. Geometry of the sandwich spherical shell.


1 ⎛∂u 

⎞ x ⎡∂γ 

1 ⎛∂u 

⎞⎤ 

ε = + + + + =Ω + χ 

R⎝∂α ⎠ R 

⎢ 

∂α R⎝∂α 

⎠⎥ 

⎣ 

⎦ 

3 α 

αα ⎜ w⎟ ⎜ w⎟ 

αα x3 

αα 

1 ⎛ ∂v 

⎞ 

ε ϕϕ = ucosα+ + wsin 

α 

Rsinα 

⎜ 

⎝ ∂ϕ 

⎟ 

⎠ 

x ⎡ 

3 

⎛ ∂γ ϕ ⎞ 1 ⎛ ∂v 

⎞⎤ 

+ ⎢ γα 

cosα+ +γsin α + ucosα+ + wsin 

α ⎥ 

Rsinα ⎜ ⎟ 

R 

⎜ ⎟ 

⎣⎝ ∂ϕ ⎠ ⎝ ∂ϕ ⎠⎦ 

=Ω + x χ 

ϕϕ 

3 

ϕϕ 

1 ⎛∂u 

⎞ 1 ∂v 

γ αϕ = −v 

cosα + 

Rsinα ⎜ 

⎝∂ϕ ⎟ 

⎠ R ∂α 

x3 

⎡ 1 ⎛∂γ α ⎞ ∂γ ϕ 1 ⎛∂u⎞ 

1 ∂v⎤ 

+ ⎢ 

ϕ cos 

R sin 

⎜ −γ α ⎟ + + 

Rsin 

⎜ ⎟ + ⎥ 

⎣ α ⎝ ∂ϕ ⎠ ∂α α ⎝∂ϕ⎠ 

R ∂α ⎦ 

= 2Ω + 2x 

χ 

αϕ 

3 

αϕ 

(15) 

1 ⎛∂w 

⎞ 

γ = γ + ⎜ − u ⎟ = 2Ω 

R ⎝∂α 

⎠ 

αx3 α αx3 

1 ⎛∂w 

⎞ 

γ = γ + − v sinα = 2Ω 

Rsinα 

⎜ 

⎝∂ϕ 

⎟ 

⎠ 

ϕx3 ϕ ϕx3 

where R is the radius of the reference surface and α is the coordinate in the meridian 

direction of spherical shell. 

CONSTITUTIVE EQUATIONS 

Let us discuss the constitutive equations of sandwich shells. The constitutive 

equations related to the global shell coordinate system are given, for instance, in 

References [5,28,37]. The stress-strain relations for the anisotropic layers of the 

shell are 

= 

L 

Ae 

(16) 

where A L is the transformed reduced stiffness matrix of the Lth layer in the shell 

reference axes (L = 1,2,3). 

For mathematical modelling purposes the individual layer is considered to be 

homogeneous and orthotropic while the laminate is heterogeneous through the 

thickness and generally anisotropic. Therefore, the constitutive equations for the


sandwich shell can be split into two parts: 

 

= D e , = D e 

NM NM NM Q Q Q 

(17) 

with 

 

e 

T 

NM xx ϕϕ xϕ xx ϕϕ xϕ 

 

= { N , N , N , M , M , M } 

T 

Q 

= { Q , Q } 

x 

T 

NM xx ϕϕ xϕ xx ϕϕ xϕ 

= { Ω , Ω ,2 Ω , χ , χ ,2 χ } 

T 

Q 

e 

ϕ 

= {2 Ω , 2 Ω } 

x3 ϕ3 

(18) 

and 

D 

NM 

⎡Q11 Q12 Q16 B11 B12 B16 

⎤ 

⎢ 

Q12 Q22 Q26 B12 B22 B 

⎥ 

⎢ 

26 ⎥ 

⎢Q16 Q26 Q66 B16 B26 B66 

⎥ 

= ⎢ ⎥ 

⎢B11 B12 B16 D11 D12 D16 

⎥ 

⎢B 12 B22 B26 D12 D22 D ⎥ 

26 

⎢ 

⎥ 

⎢⎣B16 B26 B66 D16 D26 D66 

⎥⎦ 

(19) 

D 

Q 

⎡ 

2 

kQ 5 55 kkQ ⎤ 

4 5 45 

= ⎢ ⎥ 

2 

⎢⎣kkQ 

4 5 45 kQ 4 44⎥⎦ 

(20) 

The components of the in-plane (membrane), the coupling and the out-of-plane 

plate (bending and torsion) stiffness matrices Q ij , B ij , D ij , i, j = 1,2,6 are defined in 

Reference [26]. The shear correction coefficients k4 2 , k5 2 can be calculated by difference 

approaches [47]. Here the approach based on averaging of transverse shear 

strain complementary energy and outlined in Reference [40] is used. 

VARIATIONAL FORMULATION 

The governing differential equations of motion can be derived by using the 

Hamilton’s principle [39]: 

∫ 

t2 

t1 

( δU−δT −δ W ) dt = 0 

e 

(21) 

where δU, δT, δW e are the variations of the strain energy, the kinetic energy and the 

work of external forces, respectively, t denotes the time. The strain energy for the


sandwich shell is given by 

1 T 

T 

U = ∫ ( e D e + e D e ) dA 

2 Ashell 

NM NM NM Q Q Q shell 

(22) 

and the kinetic energy can be written in the form 

T 1 

p T 

= ∫ ∫ uu ̇ ̇ dx dA 

2 Ashell 

h 

3 

shell 

(23) 

here ρ is the mass density, (… • ) denotes differentiation with respect to time and 

u T ={v T , T }. A shell is the surface area of the shell and h is the thickness of the shell. 

Substituting Equation (1) into Equation (23) one can obtain [41] 

1 

T = ∫ [ ρ0vv ̇⋅ ̇+ρ1( v̇⋅ ̇ + ̇⋅ v̇) +ρ2γ⋅γ 

̇ ̇] 

dA 

2 Ashell 

(24) 

Here ρ i , i = 0,1,2 are the generalized densities of the shell, v T ={v (1) ,v (2) ,w} and 

T ={γ (1) ,γ (2) ,γ}. For the shallow shell Equation (24) can be expressed as [41] 

3 

∑ 

ρ = ρ [ x −x 

] 

0 k 3( k+ 

1) 3( k) 

k= 

1 

1 

3 

1 2 2 

∑ k x3( k+ 

1) x3( k) 

2 

k= 

1 

2 

3 

1 3 3 

∑ k x3( k+ 

1) x3( k) 

3 

k= 

1 

ρ = ρ [ − ] 

ρ = ρ [ − ] 

(25) 

ρ k is the density of the Lth layer. In the case of applied distributed load at the shell 

midsurface it is necessary to define the work of external forces W e [41]. 

Finite Element Formulation 

The finite element employed for the analysis of the present problem is the sandwich 

shell finite element CLW54 [12] based on the zig-zag model (see Figure 5). 

This model allows performance of more precise free vibration analysis for sandwich 

structures in comparison with the multilayered shell finite element presented 

in Reference [38]. This is important for sandwich structures with thick core layer 

and significant differences in elastic moduli between outer layers and core. In the 

shell finite element CLW54 each layer is considered as a simple laminated composite 

shell finite element. It is a triangular finite element (second order 

polynominal shape functions are used) with six nodes. In each nodal point there


Figure 5. Kinematical assumptions for sandwich finite element in the thickness direction. 

are five degrees of freedom: three displacements (u, v, w) and two rotations (γ x , γ ϕ ), 

where u, v and w are displacements in the directions of reference axes x 1 , x 2 and x 3 

of the shell. Stiffness and mass matrices of this finite element have been derived in 

detail in Reference [38]. Using the displacement continuity conditions between 

layers, the sandwich finite element was developed. A similar approach was used 

for developing the sandwich plate finite element PLW54 [12]. 

The sandwich shell finite element has 54 independent degrees of freedom. For 

the bottom layer (six nodal points) in every node there are three displacements 

( 1) ( 1) ( 1) 

( ) ( ) 

( u0 

, v0 

, w0 

) and two rotations ( γ 

1 , γ 

2 ), 

x1 x2 

while for the top layer there are 

two displacements ( ( 3 ) , ( 3 

u v 

) ) and two rotations ( γ 

( 3) , γ 

( 3) 

) only. 

0 

0 

x1 x2 

DISPLACEMENT REPRESENTATION 

In each layer of the sandwich structure the kinematics, Equation (4), are assumed. 

As the reference surface the midsurface of the bottom layer is used and displacements 

and rotations of all layers are defined in the coordinates of midsurface 

of the bottom surface. For the first (bottom) layer the displacements can be ex-


pressed as follows: 

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

1 3 x , 1 3 ϕ , 1 

u = u + x γ v = v + x γ w = w 

(26) 

For the third (top) layer the displacements are represented as 

(3) (3) (3) (3) 

3 3 x , 3 3 ϕ , 3 

u = u + x γ v = v + x γ w = w 

(27) 

The displacement continuity conditions between the layers of the sandwich shell 

are given as 

u = u , v = v , w = w ( i = 1,2) 

i i+ 1 i i+ 1 i i+ 

1 

(28) 

Taking into account these displacement continuity conditions, the displacements 

for the core can be presented as 

(2) (2) (2) (2) 

2 3 x , 2 3 ϕ , 2 

u = u + x γ v = v + x γ w = w 

(29) 

And the variables u (2) , v (2) , γ 

( 1) ( 1) ( 3) ( ) 

γ , γ , γ , γ 

3 as follows: 

x 

ϕ 

x 

ϕ 

( 2 ) ( ) 

x γ 2 

ϕ 

and depend from u (1) , v (1) , u (3) , v (3) , 

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

=+ 1 + 2 + 3γx 

− 3γx 

u Fu F u F F 

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

x Fu 4 Fu 4 F2 x F1 

x 

γ = − + + γ + γ 

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

=+ 1 + 2 + 3γϕ 

− 3γϕ 

v Fv F v F F 

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

ϕ Fv 4 Fv 4 F2 ϕ F1 

ϕ 

γ = + − − γ − γ 

(30) 

where 

(1) (1) 

h 

h 

1 

1 = 1 + 2, 2 = − , 

(2) 3 = 1 , 4 = 

(2) 

F F F F F F 

2h 

2 

The displacement field of the sandwich shell can be described by vector components 

h 

u 

T 

(1) (1) (1) (1) (1) (3) (3) (3) (3) 

0 0 0 x ϕ 0 0 x ϕ 

= { u , v , w , γ , γ , u , v , γ , γ } 

(31) 

For the case of a six nodal point finite element (with 9 degrees of freedom per 

node), the displacement variation over the finite element can be described by


u= N = [ N , N ,..., N ] =∑ N 

() e 1 2 6 () e i i 

i= 1 

6 

(32) 

where 

T 

() e = 1 2 6 

{ , ,..., } 

(33) 

are generalised displacements of the finite element and 

 

T (1) (1) (1) (1) (1) (3) (3) (3) (3) 

i = u0 v 

i 0 

w 

i 0 γ i x γ i ϕ 

u 

i 0 

v 

i 0 

γ i x 

γ 

i ϕi 

{ , , , , , , , , } 

(34) 

is the vector of the displacement parameters at node i. Here 

N 

i 

= Ni 

I 

9 

(35) 

where N i are the shape functions for node i expressed in terms of local triangular 

coordinate system (L 1 ,L 2 ,L 3 ) of the finite element and I 9 istheunit9×9matrix. 

UNDAMPED VIBRATION 

The Hamilton’s principle given by Equation (21) is discretised as 

∫ 

N 

t e 

2 

t1 

∑( δUl −δTl −δ Wel) dt = 0 

l= 

1 

(36) 

where N e is the total number of finite elements and the subscript l represents the 

contribution of the lth element. Equation (36) can be expressed in terms of the displacements 

and rotation angles (u 0 ,v 0 ,w 0 ,γ x ,γ ϕ ). Following the standard finite element 

technique, we obtain the discretised equations in terms of nodal degrees of 

freedom x: 

Mẋ̇ + Kx = F 

(37) 

where M, K, x, F are the global mass and stiffness matrices, the displacement and 

load vector, respectively. For free vibration problems F = 0 and Equation (37) becomes 

2 

( K− M) 

= 

0 

(38) 

where is the natural frequency and is the corresponding mode shape of the 

structure.


Damping Models 

ENERGY METHOD 

In this model a specific damping capacity (SDC) ψ is introduced as a parameter 

which characterises the damping properties of the material or structure [19]: 

∆U 

ψ = 2πη= 

(39) 

U 

where η is the loss factor, ∆U is the strain energy dissipated during one cycle of vibration, 

and U is the total strain energy of the entire of sandwich shell at maximum 

of the displacement during the same cycle. Following the definition of the SDC, 

the total energy dissipation can be expressed by [19] 

3 

∑ 

∆ U = ∆U 

L= 

1 

( L) 

(40) 

where ∆U (L) is the strain energy dissipated in the Lth layer 

( L) 

1 ZL 

T L 

∆ U = { 0} [ ] { 0} 

2 

∫∫ ε C dzdA 

A Z 

ψ ε 

L−1 

(41) 

Here 

⎡ 

L L L L 

ψ11C11 ψ12C12 

0 0 0 ⎤ 

⎢ 

⎥ 

L L L L 

⎢ψ21C21 ψ22C22 

0 0 0 ⎥ 

L ⎢ 

[ ] 

L L 

⎥ 

Cψ 

= ⎢ 0 0 ψ44C44 

0 0 ⎥ 

⎢ 

L L ⎥ 

⎢ 

0 0 0 ψ55C55 

0 

⎥ 

⎢ 

L L 

0 0 0 0 66C 

⎥ 

⎣ 

ψ 66 ⎦ 

(42) 

and ψ L ij are the SDC of the corresponding layer [19]. WhereC L ij are the plane-stress 

reduced components of the Lth layer in the material axis (1,2,3) and {ε 0 } T = 

{ε 1 ,ε 2 ,ε 23 ,ε 13 ,ε 12 } are strain components in the same axis. Using the energy 

method it is assumed that the natural frequencies and modes of the damped system 

are the same as those of the associated undamped system. This expression is obtained 

for general case when damping matrix is not symmetrical. It is assumed that 

L L 

ψ12 = ψ21 

in our algorithm. Solving the eigenproblem (38), the SDC to the nth 

mode can be calculated by 

ψ 

T 

∆U 

( n) H( n) 

( n) = 2πη ( n) 

= = 

U 

T 

( n) K( n) 

(43)


where η (n) is the modal loss factor and H is the global damping matrix for shell. A 

detailed description of global damping matrix H is given in Reference [12]. 

NUMERICAL EXAMPLES 

In the examples the symmetry of the cross section is assumed. So we have the 

same isotropic material properties for the inner and outer layer, the core is made 

from orthotropic material. For the inner and outer layer characteristics the index t 

is used; for the core, c. 

Free Undamped Vibration of a Sandwich Conical Shell 

The first problem considered is a conical sandwich shell with clamped-clamped 

boundary conditions. The geometry of the shell is characterised by the following 

quantities: the thickness of outer layers h t = 0.000535 m and core h c = 0.00762 m, 

the radius of the cone at its small end R = 0.5702 m, the semi-vertex angle 

β = 5.07°, the meridional length of the cone L = 1.8415 m, Figure 6. The following 

properties of the material are assumed: outer layers from glass epoxy E t = 25.08 

GPa, ν t = 0.20, p t = 2800 kg/m 3 and inner layer form aluminium honeycomb E1 c = 

0.529 GPa,G12= c 0.2204 GPa,G13= c c c 

0.126 GPa, ν = ν = 0.2, ρ c = 36.80 kg/m 3 . 

12 13 

Figure 6. The sandwich conical shell.


Table 1. Natural frequencies f nm for a sandwich 

clamped-clamped conical shell. 

n m CLW54 

Ramesh and 

Ganesan (1993) 

Wilkins 

(1970) 

1 1 — 299.9 — 

2 184.7 184.9 177.2 

3 127.4 128.3 126.0 

4 110.7 111.3 110.7 

5 127.0 127.3 126.7 

6 164.2 164.7 163.5 

7 212.7 213.8 212.3 

8 268.9 271.3 269.1 

2 1 — 521.0 — 

2 349.2 350.1 340.1 

3 250.0 252.3 254.3 

4 200.1 202.5 209.7 

5 188.0 190.2 197.7 

6 207.9 210.2 214.8 

7 250.4 252.9 254.5 

8 307.5 310.7 310.0 

The results for natural frequencies of sandwich conical shell obtained by using 

the presented finite element CLW54 and the results from the reference of [35,48] 

are shown in the Table 1. The values for modes m =1,n = 1 and m =1,n =2havenot 

been analyzed at this time due to a prohibitive amount of calculations. 

Vibration and Damping Analysis of 

Composite Sandwich Cylindrical Panel 

The free vibration frequencies and the corresponding loss factors are studied 

for a cylindrical panel under clamped conditions on the straight sides. The geometry 

of the panel is characterized by the following quantities: the thickness of 

the outer layers h t = 0.001 m and the thickness of core h c = 0.002; 0.004; 0.006; ...; 

0.018 m. The panel has the sizes 0.5 × 0.5 m in plane. The radius of the cylinder 

R = 0.5 m, Figure 7. The layers of the panel are made from materials with the 

following properties: for the aluminium outer layers, E t =70GPa;ν t = 0.20; 

p t = 2800 kg/m 3 ; and for the aluminium honeycomb core E1 

c = 0.529 GPa; 

G12 c = 0.2204 GPa;G13= c 0.126 GPa; ν c c 

12 = ν 23 = 0.2; ρ c = 280 kg/m 3 . The loss factor 

was assumed zero (η t = 0) for outer layers and η c = 0.4 for the core. The results obtained 

by the presented method for the first four natural frequencies and the corresponding 

loss factors of sandwich cylindrical panel are shown in Figure 8 and Figure 

9.


Figure 7. The sandwich cylindrical panel. 

Figure 8. Natural frequencies of the sandwich cylindrical panel for modes 1 – 1–1 – 4.


Figure 9. Modal loss factors of the sandwich cylindrical panel for modes 1 – 1–1 – 4. 

Vibration and Damping Analysis of a 

Composite Sandwich Conical Shell 

A three-layered composite conical shell has been considered. The shell has two 

variants of boundary conditions: clamped-clamped and simply supported-simply 

supported from both sides. The following geometric characteristics are presented: 

the thickness of outer layers h t = 0.0005 m and core h c = 0.004, 0.005, 0.006, ..., 

0.012 m, the radius of the cone at its small end R = 0.27 m, the semi-vertex angle 

β = 7.0°, the meridional length of the cone L = 0.84 m, Figure 6. The layers are with 

the following properties of material: outer from glass epoxy E t = 25.0 GPa, 

ν t = 0.20, p t = 2800 kg/m 3 and inner from aluminium honeycomb E1 c =0.529 GPa, 

E2 c c c 

= 0.226 GPa, ν 12 = ν 23 = 0.2, ρ c = 36.80 kg/m 3 . The loss factor was assumed 

zero (η t = 0) for outer layers and η c = 0.4 for the core. The results for the natural frequencies 

and modal loss factors of sandwich conical shell with different core 

thickness obtained by using finite element CLW54 are presented in Figures 10 and 

11 for clamped-clamped and Figures 12 and 13 for simply supported boundary 

conditions. 

Vibration and Damping Analysis of 

Composite Sandwich Spherical Shell 

A three-layered composite spherical shell has been considered. The shell is


Figure 10. Natural frequencies of the sandwich conical shell (clamped-clamped). 

Figure 11. Modal loss factors of the sandwich conical shell (clamped-clamped). 

Figure 12. Natural frequencies of the sandwich simply supported conical shell.


Figure 13. Modal loss factors of the sandwich simply supported conical shell. 

clamped from both sides and free from another both sides. The radius of shell is 0.5 

m, angles of expand are π/2 in the plane xz and yz of global coordinate system Figure 

14. The physical and geometrical parameters are the same as for example 3. 

The results for natural frequencies and modal loss factors of sandwich spherical 

shell with different core thickness obtained by using present finite element 

CLW54 are presented in Figures 15 and 16. 

Figure 14. The sandwich spherical shell.


Figure 15. Natural frequencies of the sandwich spherical shell. 

Figure 16. Modal loss factors of the sandwich spherical shell.


CONCLUSIONS 

The triangular sandwich shell finite element CLW54 with 54 degrees of freedom 

for damping analysis of sandwich shells is presented. The zig-zag model is 

employed for the finite element developed. A damping model on the base of the energy 

method has been used for the damping analysis of various sandwich shells of 

revolution. The vibration and damping characteristics of cylindrical, conical and 

spherical sandwich shells for different material and geometrical properties and 

boundary conditions has been studied. 

As seen in Table 1, the results obtained by the finite element CLW54 are in good 

agreement with those previously reported. Unfortunately no comparison is made 

with experimental values due to the difficulty in obtaining them. It has been seen 

from figures, that there is a proportional dependence between thickness of sandwich’s 

core and modal loss factor. It is very interesting to observe that despite 

slight decreasing of frequency1–2 for conical shell, Figure 10, the loss factor increases 

with increasing the mode number, Figure 11. 

It can be observed from Figure 8 that the natural frequencies of sandwich cylindrical 

panel increase with increasing of the core thickness. This influence appears 

obviously for higher frequencies. The same situation may be observed for the 

modal loss factor, Figure 9. There is almost a linear dependence between thickness 

of core and natural frequencies, Figures 10 and 12, and thickness of core and 

modal loss factors, Figures 11 and 13, for closed sandwich conical shells with 

clamped-clamped boundary conditions and with simply supported-simply supported 

boundary conditions. The same dependence is observed for spherical sandwich 

shells, Figures 15 and 16. As it is seen from Figure 10 and Figure 12, the significant 

dependence of natural frequencies from thickness of the core is observed 

for frequencies 1–4...1–8 for the sandwich conical shells. 

From the results obtained it can be concluded that the finite element CLW54 

can be used for solving design problems of sandwich shells. 

ACKNOWLEDGEMENTS 

The paper was prepared during short-time visits of the first two authors to 

the Martin-Luther-University Halle-Wittenberg. The visits were financially supported 

by the German Academic Exchange Service (DAAD). 

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