12165


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 21, No 4, 2023, pp. 591 - 614  

https://doi.org/10.22190/FUME231004045E 

© 2023 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

FINITE ELEMENT ANALYSIS FOR DYNAMIC RESPONSE OF 

VISCOELASTIC SANDWICHED STRUCTURES INTEGRATED 

WITH ALUMINUM SHEETS 

Yasser Hamed Elmoghazy1,2, Babak Safaei1,2,3, Saeid Sahmani4 

1Nanotechnology and Multifunctional Structures Research Center (NMSRC),  

Eastern Mediterranean University, Famagusta, Turkey 
2Department of Mechanical Engineering, Eastern Mediterranean University, Famagusta, Turkey 
3Department of Mechanical Engineering Science, University of Johannesburg, South Africa 

4School of Science and Technology, The University of Georgia, Tbilisi, Georgia 

Abstract. Passive vibration attenuation of modern mechanical structures is one of the 

most essential technologies applied to the arsenal of modern mechanical structures. In 

this work, dynamics analysis is performed on viscoelastic (VE) sandwich beam and plate 

by using finite element method. The proposed structure is composed of a VE core and 

aluminum face sheets as substrate layers on both sides of the structure. Small-strain VE 

material is modeled based on complex constant moduli model and numerical method is 

used to develop the finite element (FE) shear model based on first-order shear 

deformation theory (FSDT) and Hamilton principle. In modal analysis, model-effective 

mass analysis is performed to investigate its dominant mode shape and sweet spot at 

resonance using C3D20 and CPS8 elements. The former is indicated as a 3-D element 

with 20 nodes while the latter is indicated as a 2-D plane strain element with eight nodes. 

In harmonic analysis, resonance frequency is obtained based on mode superposition 

method to evaluate the steady-state response of VE sandwich beam via maximum 

deformation. Thereafter, the results are compared against analytical solutions from the 

literature. Moreover, a parametric study shows that the natural frequency of the beam 

did not change with the increase in core thickness. However, normalized loss factor of 

VE sandwich beam 𝜂𝑛 𝜂𝑣⁄  is directly proportional to VE damping factor 𝜂𝑣 and the 

thickness of VE core layer. Based on frequency response function, the results of 

resonance frequencies for VE beam are in the range of modal natural frequency. 

Key words: Viscoelastic, Sandwich Structure, Dynamics Analysis, Complex Constant 

Moduli, Normalized Loss Factor 

 
Received: October 04, 2023 / Accepted November 26, 2023   
Corresponding author: Babak Safaei 

Department of Mechanical Engineering, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 

10, Turkey 
Department of Mechanical Engineering Science, University of Johannesburg, Gauteng 2006, South Africa 

E-mail: babak.safaei@emu.edu.tr 



592  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

   1. INTRODUCTION 

The key parameters to obtain an optimum design of modern mechanical structures with 

ideal performance are lightweight and noise control which directly lead to the wide 

applications of thin-walled structures. These structures can be represented as thin beams 

and thin plates that are extensively used in many engineering applications such as aerospace 

[1], automobile industry [2], and civil engineering [3]. Since thin-walled structures are "light" 

and "thin”, it is more likely to augment the induced vibrations under external excitations, 

which compromises the stability of such structures. As a result, the vibration of thin-walled 

structures must be controlled. One of the approaches to the problem is to use active 

piezoelectric elements which may offer an optimal damping performance of thin-walled 

structures since it includes the ability of sensing and actuation control [4]. Applying a 

damping layer to thin-walled structures to control their vibrations is a widely utilized 

strategy in engineering practice [5]. Among them, viscoelastic (VE) materials, which have 

good damping qualities, are most widely applied in structural vibration as a passive damping 

treatment [6, 7]. A typical passive damping treatment structure consists of two elastic layers 

and a VE core sandwiched between constraining layer and base beam. The VE core allows 

vibration energy to dissipate through a cyclic deformation, while the main advantage of 

constraining layer is to increase periodic shear deformation and stress resistance of the 

core, hence directly leading to an increase in damping energy within VE core [8].  

In engineering practice, sandwich structures have complex geometric shapes and boundary 

conditions thus, such problems cannot be effectively solved by the conventional analytical 

process, accordingly finite element (FE) approach is commonly used which may facilitate 

the computational complexity of such problem [9–12]. Arvin et al. [13] developed four 

nodes VE sandwich beam element based on higher-order shear deformation theory to 

explore the steady-state and transient responses of a VE composite sandwich beam. Wang 

and Inman [14] developed a 2D FE model to study the dynamic performance of a hybrid 

composite sandwich beam by using VE Golla–Hughes model (GHM). Lin and Rao [15] 

proposed FE model for vibration performance of VE multiple-layered sandwich beam by 

incorporating VE Biot model into FE model. Reddy and Panda [16] developed FE model to 

explore the nonlinear frequency response for harmonically excited VE sandwich beam based 

on harmonic balance method considering different VE constitutive models. Jahan et al. [17] 

performed FE modal study to examine the effectiveness and influential parameters of a passive 

rail damper system using lab-scaled model. Araújo et al. [18] proposed FE model to explore the 

dynamic response of active-passive hybrid sandwich plate using mixed layerwise approach. 
Although FE models have been widely used in the dynamic modeling of complex 

structures, some FE models are difficult to implement, thus some researchers have tried to 
ensure accurate modeling based on searching for ways to improve computational 
efficiency, such that, some models between the analytical method and the FE method, 
called semi-analytical semi-numerical method. Lewandowski and Baum [19] developed 
mixed model to anticipate the free vibration response of VE sandwich beam using virtual 
work principle and VE Zener model. Daya and Potier-Ferry [20] established a shell FE 
model using homotopy algorithm and asymptotic method to solve the nonlinear eigenvalue 
problem of VE sandwich structures. Koutsawa et al. [21] investigated the non-linear behavior 
of a laminated glass beam by applying asymptotic numerical technique to establish a diamond 
toolbox. The performance of the diamond toolbox was demonstrated numerically and 
experimentally. Also, Lampoh et al. [22] performed sensitivity studies on the non-linear 
mechanical behavior of VE laminated glass beam. 



593  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

Recently, small-size composite structures with VE material have been explored by 

many researchers. For instance, Loghman et al. [23] used finite difference method coupled 

with Galerkin bases to examine the size-dependent vibration behavior of a VE micro-beam 

employing modified couple stress theory (MCST). Li et al. [24] analytically and experimentally 

investigated the nonlinear vibration response of multiple layered VE composite sandwich 

plate and the equation of motion derived based on von Karman approach. Their analytical 

model was developed based on strain energy principle and validated against experimental 

nonlinear vibration characteristics. Rahmani et al. [25] utilized a comprehensive mathematical-

mechanical model based on higher-order, nonlocal Eringen’s theory, and MCST theory to 

explore the wave propagation of rotating VE nanobeams including the influence of many 

parameters on wave propagation frequencies, such as nonlocal parameter and thermal gradient. 

Yuan et al. [26] studied the dynamic stability of FGM composite microcells integrated with 

a nonlinear VE foundation under size-dependent gradient. Jalaei et al. [27] proposed the 

transient response of small-size visco-porous FG structure exposed to transverse loads and 

magnetic field based on Eringen’s nonlocal strain gradient theory. Youzera et al. [28] 

proposed an applicable analytical model for the forced vibration analysis and nonlinear 

associated loss factor of VE-FGM composite beam by using zig-zag theory coupled with 

HSDT. The governing sixth-order differential equilibrium equation of motion is derived 

using virtual work principle. Karmi et al. [29] established a numerical approach based on 

modal stability procedure (MSP) to explore the vibroacoustic behavior of sandwich beam 

integrated with frequency and temperature-dependent VE core.  

Hereby, research methodology for dynamic analysis of VE sandwich structures is 

proposed by using finite element method. In the pre-processing step, the integral FE shear 

beam element is developed based on first-order shear deformation theory (FSDT) since the 

VE core layer demonstrates considerable shear deformation behavior under different 

loading conditions. The beam’s equation of motion is derived based on Hamilton principle. 

The numerical results are compared against analytical models from previous literature. 

Further, the steady-state frequency response is obtained for optimizing the design and 

ensuring the structural integrity of such system. This work aims to achieve an appropriate 

design for both VE beam and plate with integrated Aluminum sheets by investigating their 

damping characteristics, including the effects of VE loss factor 𝜂𝑣 and thickness of the VE 

core layer. 

2. SHEAR MODEL KINEMATICS RELATIONSHIPS 

The geometric configurations and deformation of typical VE sandwich beams are 

demonstrated in Fig. 1. Here. ℎ1, ℎ2 and H are the thickness of upper layer, lower layer and 

the total thickness of the VE sandwich beam, respectively, and L is the span of VE sandwich 

beam. The shear strain of VE layer 𝛾𝑥𝑧 and rotation of VE sandwich beam 𝜃 could be expressed 

as [30]: 

 𝛾𝑥𝑧 = 𝜃 + 𝜑𝑥 (1) 

where 𝜑𝑥 is the shear angle of VE core since the two elastic layers are treated as Euler-

Bernoulli beams neglecting shear deformation. 



594  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

To simplify FE modeling and formulation the following basic assumptions are proposed: 

1. Shear deformation is negligible for elastic layers. 

2. The vibrations of the VE sandwich beam is dissipated through periodic shear 

deformation of the VE layer. 

3. All layers of VE sandwich beam have the same transverse displacement 𝑤. 

4. All layer interaction is constrained. 

5. VE material is considered as small strain linear VE. 

 

Fig. 1 Passive damping treatment illustration for typical VE sandwich beam with a deformed 

shear configuration 

Axial and transverse displacements 𝑢(𝑖)(𝑥, 𝑧, 𝑡), 𝑤(𝑖)(𝑥, 𝑧, 𝑡) respectively, at any point 

within the elastic layers could be expressed as [31]: 

 
𝑢(𝑖)(𝑥, 𝑧, 𝑡) = 𝑢𝑖(𝑥, 𝑡) − 𝑧𝑖

𝜕𝑤𝑖(𝑥,𝑡)

𝜕𝑥

𝑤(𝑖)(𝑥, 𝑧, 𝑡) = 𝑤𝑖(𝑥, 𝑡)
}  𝑖 = 𝑐, 𝑏 (2) 

where 𝑢𝑖(𝑥, 𝑡) and 𝑤𝑖  (𝑥, 𝑡) are the axial and transverse displacements, at the centroid point 

of elastic layers, respectively. The relative kinematics relations of VE core shear strain 𝛾𝑥𝑧 

and axial displacement 𝑢𝑣  can be derived using local coordinate system and FSDT [32] as: 

 𝑢𝑣 =
1

2
(𝑢𝑐 + 𝑢𝑏) +

1

4
(ℎ1−ℎ2)

𝜕𝑤

𝜕𝑥
 (3) 

 𝛾𝑥𝑧 =
1

𝐻−(ℎ1+ℎ2)
[(𝑢𝑐 − 𝑢𝑏) + (

ℎ1+ℎ2

2
+ (𝐻 − (ℎ1+ℎ2))

𝜕𝑤 

𝜕𝑥
] (4) 



595  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

3. FINITE ELEMENT MODELLING 

3.1. Degrees of Freedom and Shape Functions of Proposed Beam Model 

The Integral shear beam element with two nodes i and j is shown in Fig. 2. Each node 

had four DOF with lateral displacement 𝑤 along z-direction, rotation 𝜃 about x-axis, and axial 

displacements of elastic layers 𝑢𝑐 and 𝑢𝑏. The total set of element nodal displacements {𝑑𝑒} are 

given as: 

 {𝑑𝑒} = {
𝑑𝑖
𝑑𝑗
} = {𝑤𝑖 𝜃𝑖 𝑢𝑏𝑖 𝑢𝑐𝑖 𝑤𝑗 𝜃𝑗 𝑢𝑏𝑗 𝑢𝑐𝑗}𝑇 (5) 

θi θj wi wj 

uci 

ubi 

ucj 

ubj 

i j

zc 

zb 

u1 

u2 

 

Fig. 2 Integral shear beam element 

The transverse displacement field 𝑤(𝑥), rotation field 𝜃(𝑥) =
𝜕𝑤

𝜕𝑥
, and axial displacement 

field of elastic layers 𝑢𝑏(𝑥) and 𝑢𝑐(𝑥) are expressed as a cubic polynomial function: 

 𝑤(𝑥) = 𝑐0 + 𝑐1𝑠 + 𝑐2𝑠
2 + 𝑐3𝑠

3 (6) 

 𝜃(𝑥) = 𝑐1 + 2𝑐2𝑠 + 3𝑐3𝑠
2  (7) 

 𝑢𝑏(𝑥) = 𝑐4 + 𝑐5𝑠, 𝑢𝑐(𝑥) = 𝑐6 + 𝑐7𝑠 (8) 

where 𝑠 = 𝑥/𝑙𝑒 represents the normalized coordinate with 𝑙𝑒 being equivalent element 

length. From Eqs. (6), (7), and (8) The element displacement field within  the element could 

be expressed in matrix form as: 

 {

𝑤
𝜃
𝑢𝑏
𝑢𝑐

} = [𝑇]{𝑐} =

[
 
 
 
 
1 𝑠 𝑠2 𝑠3 0 0 0 0

0
1

𝑙𝑒

2𝑠

𝑙𝑒

3𝑠2

𝑙𝑒
0 0 0 0

0 0 0 0 1 𝑠 0 0
0 0 0 0 0 0 1 𝑠]

 
 
 
 

{
 
 

 
 
 𝑐0
𝑐1
𝑐3
⋮
𝑐7}
 
 

 
 

 (9) 

where 𝑠 = 𝑥/𝑙𝑒 represents the normalized coordinate with 𝑙𝑒 being equivalent element 

length. {𝑐} and [𝑇] are the coefficient matrix and polynomial matrix, respectively which 

can be represented by element nodal displacements {𝑑𝑒} explicitly in matrix form as [32]:  

 {𝑑} = [𝑇]{𝑐} = [𝑇][𝑄−1]{𝑑𝑒} = [𝑁]{𝑑𝑒} (10) 

where N is the shape function matrix corresponding to the four nodal DOF of each node in 

the form of 4×8 matrix and 𝑄 is 8×8 matrix that depends on the applied boundary 

condition. Using Eq. (10) and applying the C-F boundary condition, the 𝑄  matrix is 

obtained in a matrix form as: 



596  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

 𝑄 =  

[
 
 
 
 
 
 
 
 
1 1 1 1 0 0 0 0

0
1

𝑙𝑒

2

𝑙𝑒

3

𝑙𝑒
0 0 0 0

0 0 0 0 1 1 0 0
0 0 0 0 0 0 1 1
1 0 0 0 0 0 0 0

0
1

𝑙𝑒
0 0 0 0 0 0

0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0]

 
 
 
 
 
 
 
 

 (11) 

The shape function is obtained after the decoupling-inversion operation using 𝑄−1 as: 

 [𝑁𝑥 𝑁𝑥
′ 𝑁1 𝑁2]

𝑇 = 𝑇𝑄−1  (12) 

The generalized shape function matrix is obtained as a column matrix [13]: 

𝑁𝑥 =

[
 
 
 
 
 
 
 
 
 
1 − 3𝑠2 + 2𝑠3

𝑥 −
2𝑥2

 𝑙𝑒
+

𝑥3

 𝑙𝑒
2

0
0

3𝑠2 − 2𝑠3

𝑥3

 𝑙𝑒
2 −

𝑥2

 𝑙𝑒

0
0 ]

 
 
 
 
 
 
 
 
 
𝑇

, 𝑁𝑥
′ =  

[
 
 
 
 
 
 
 
 
 6(

𝑥2

 𝑙𝑒
3 −

𝑥

 𝑙𝑒
2)

1 − 4𝑠 + 3𝑠2

0
0

−6(
𝑥2

 𝑙𝑒
3 −

𝑥

 𝑙𝑒
2)

−2𝑠 + 3𝑠2

0
0 ]

 
 
 
 
 
 
 
 
 
𝑇

, 𝑁1 =

[
 
 
 
 
 
 
 
0
0

1 − 𝑠
0
0
0
𝑠
0 ]

 
 
 
 
 
 
 
𝑇

, 𝑁2 =

[
 
 
 
 
 
 
 
0
0
0

1 − 𝑠
0
0
0
𝑠 ]
 
 
 
 
 
 
 
𝑇

 

The displacements filed within the element could be then determined by the nodal 

displacement of element through shape functions interpolation: 

𝑤 = 𝑁𝑥𝑑
𝑒, 𝜃 = 𝑁𝑥

′𝑑𝑒 , 𝑢𝑏 = 𝑁1𝑑
𝑒, 𝑢𝑐 = 𝑁2𝑑

𝑒, 𝑢𝑣 = 𝑁3𝑑
𝑒, 𝛾 = 𝑁4𝑑

𝑒 

where 𝑁3 and 𝑁4 were derived based on the relative kinematics relations 𝛾𝑥𝑧 and 𝑢𝑣 in Eqs. 

(3) and (4) as follows: 

 𝑁3 =
1

2
[(𝑁1 +𝑁2) + (

ℎ1+ℎ3

2
)𝑁𝑥

′] (13) 

 𝑁4 =
1

𝐻−(ℎ1+ℎ2)
[(𝑁2 − 𝑁1) + (

ℎ1+ℎ3

2
+ (𝐻 − (ℎ1+ℎ2))𝑁𝑥

′]  (14) 

3.2. Element Stiffness Matrix 

The principle of strain energy can be applied to obtain element stiffness matrix of VE 

sandwich beam. In general, the elastic strain energy is expressed as: 

 𝑈 = ∫
𝜎2

2𝐸

 

𝑉
𝑑𝑉 (15) 

3.2.1. Strain Energy due to Axial Loading 

Based on Eq. (15), the strain energy due to axial loading of elastic layer 𝑈𝑎 is: 

 𝑈𝑎 =
𝐸𝐴

2
∫ 휀2𝑑𝑥  (16) 



597  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

The definition of axial strain was defined as the rate change of the displacement: 

 휀 =
𝜕𝑢

𝜕𝑥
 (17) 

Substituting Eq. (17) into Eq. (16) yields [32]: 

 𝑈𝑎 =
1

2
𝐸𝐴∫ (

𝜕𝑢

𝜕𝑥
)

𝑙𝑒
0

2
𝑑𝑥 =

1

2
𝑑𝑒𝑇𝐾𝑒𝑖

𝑒 𝑑𝑒   (18) 

 𝑈𝑎𝑐 =
1

2
𝐸𝑐𝐴𝑐 ∫ (

𝜕𝑢𝑐

𝜕𝑥
)

𝑙𝑒
0

2
𝑑𝑥 =

1

2
𝑑𝑒

𝑇
𝐾𝑒𝑐
𝑒 𝑑𝑒   (19) 

 𝑈𝑎𝑏 =
1

2
𝐸𝑏𝐴𝑏 ∫ (

𝜕𝑢𝑏

𝜕𝑥
)

𝑙𝑒
0

2
𝑑𝑥 =

1

2
𝑑𝑒

𝑇
𝐾𝑒𝑏
𝑒 𝑑𝑒   (20) 

The element stiffness matrix of base and constraining layers, corresponding to axial 

strain energy 𝐾𝑎𝑏
𝑒  and 𝐾𝑎𝑐

𝑒 , respectively can be expressed as: 

 𝐾𝑎𝑏
𝑒 = 𝐸𝑏𝐴𝑏 ∫ [

𝜕𝑁1

𝜕𝑥
]

𝑙𝑒
0

𝑇

[
𝜕𝑁1

𝜕𝑥
] 𝑑𝑥 (21) 

 𝐾𝑎𝑐
𝑒 = 𝐸𝑐𝐴𝑐 ∫ [

𝜕𝑁2

𝜕𝑥
]

𝑙𝑒
0

𝑇

[
𝜕𝑁2

𝜕𝑥
] 𝑑𝑥 (22) 

3.2.2. Strain Energy due to Bending Loading 

The elastic strain energy due to flexural bending in the beam is given by:  

 𝑈𝑏 = ∫
𝑀2(𝑥) 𝑦2

2𝐸𝐼2

 

𝑉
𝑑𝑉 (23) 

The flexural bending of the element is given by: 

 
𝑀(𝑥)

𝐸𝐼
=

𝜕2𝑤

𝜕𝑥2
 (24) 

Substituting Eq. 24 into Eq. 23, the bending strain energy of elastic layers is given as: 

 𝑈𝑏 =
𝐸

2
∬(

𝜕2𝑤

𝜕𝑥2
)
2

𝑦2𝑑𝐴 𝑑𝑥 (25) 

 𝑈𝑏 =
1

2
𝐸𝐼 ∫ (

𝜕2𝑤

𝜕𝑥2
)
2

𝑑𝑥 =
1

2
𝑑𝑒𝑇𝐾𝑏

𝑒𝑑𝑒 (26) 

 𝑈𝑏𝑐 =
1

2
𝐸𝑐𝐼𝑐 ∫ ((

𝜕2𝑤

𝜕𝑥2
)
2

)
𝑙𝑒
0

2

𝑑𝑥 =
1

2
𝑑𝑒𝑇𝐾𝑏𝑐

𝑒 𝑑𝑒 (27) 

 𝑈𝑏𝑏 =
1

2
𝐸𝑏𝐼𝑏 ∫ ((

𝜕2𝑤

𝜕𝑥2
)
2

)
𝑙𝑒
0

2

𝑑𝑥 =
1

2
𝑑𝑒𝑇𝐾𝑏𝑏

𝑒 𝑑𝑒 (28) 

The element stiffness matrix of constraining and base layers, corresponding to bending 

strain energy 𝐾𝑏𝑐 
𝑒  and 𝐾𝑏𝑏

𝑒 , respectively can then be expressed as: 

 𝐾𝑏𝑏
𝑒 = 𝐸𝑏𝐼𝑏 ∫ [

𝜕𝑁𝑥

𝜕𝑥
]

𝑙𝑒
0

𝑇

[
𝜕𝑁𝑥

𝜕𝑥
] 𝑑𝑥 (29) 

 𝐾𝑏𝑐
𝑒 = 𝐸𝑐𝐼𝑐 ∫ [

𝜕𝑁2

𝜕𝑥
]

𝑙𝑒
0

𝑇

[
𝜕𝑁2

𝜕𝑥
] 𝑑𝑥 (30) 



598  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

3.2.3. Strain energy due to shear loading 

Strain energy due to shear loading can be derived as  

 𝑈𝑠ℎ𝑒𝑎𝑟 = ∫
𝜏2

2𝐺
𝑑𝑣 =

𝐺𝛾2

2
∫𝑑𝑣  (31) 

 𝑈𝑠ℎ𝑒𝑎𝑟 =
𝐺∗𝛾2

2
∬𝑑𝐴 𝑑𝑥  (32) 

 𝑈𝑠ℎ𝑒𝑎𝑟 =
1

2
𝐺∗𝐴𝑣 ∫ 𝛾2

𝑙𝑒
0

𝑑𝑥 =
1

2
𝑑𝑒𝑇𝐾𝑠

𝑒𝑑𝑒   (33) 

𝐾𝑠
𝑒 is the shear element stiffness matrix associated with shear strain energy such that: 

 𝐾𝑠
𝑒 = 𝐺∗𝐴𝑣 ∫ 𝑁4

𝑇𝑙𝑒
0

𝑁4𝑑𝑥 (34) 

The total elastic element stiffness matrix 𝐾𝑒
𝑒 can be expressed as:  

 𝐾𝑒
𝑒 = 𝐾𝑒𝑏

𝑒 + 𝐾𝑒𝑐
𝑒 + 𝐾𝑏 𝑏

𝑒 + 𝐾𝑏 𝑐 
𝑒   (35) 

The element stiffness matrix then could be expressed as: 

 𝐾𝑒⏟
⌊8×8⌋

= [
[𝐾𝑒𝑏

𝑒 ] [𝐾𝑏𝑏
𝑒 ]

[𝐾𝑏𝑐
𝑒 ] [𝐾𝑒𝑐

𝑒 ]
] + 𝐾𝑠

𝑒⏟
⌊8×8⌋

  (36) 

where 𝐾𝑒 is the total element stiffness matrix obtained as the summation of elastic element 

stiffness matrix 𝐾𝑒
𝑒 and shear stiffness element matrix 𝐾𝑠

𝑒 of VE sandwich beam. 

3.3. Element Mass Matrix 

To obtain the element mass matrix, the principle of kinetic energy is applied. By 

considering the axial and rotational loading associated with kinetic energy, the total kinetic 

energy 𝑇 of VE sandwich beam is evaluated as: 

 𝑇 =
1

2
(𝜌𝑐𝐴𝑐 + 𝜌𝑣𝐴𝑣 + 𝜌𝑏𝐴𝑏) ∫ �̇�

2 𝑑𝑥 +
1

2
(𝜌𝑐𝐴𝑐 + 𝜌𝑣𝐴𝑣 + 𝜌𝑏𝐴𝑏) ∫ �̇�

2 𝑑𝑥 (37) 

The kinetic energy corresponding to the axial and bending loading of each layer is: 

 𝑇𝑎𝑐 =
1

2
𝜌𝑐𝐴𝑐 ∫ (

𝜕𝑢𝑐

𝜕𝑡
)

𝑙𝑒
0

2

𝑑𝑥 =
1

2
�̇�𝑒

𝑇
𝑀𝑒𝑐
𝑒 �̇�𝑒 (38) 

 𝑇𝑏𝑐 =
1

2
𝜌𝑐𝐴𝑐 ∫ (

𝜕𝑤

𝜕𝑡
)

𝑙𝑒
0

2

𝑑𝑥 =
1

2
�̇�𝑒

𝑇
𝑀𝑏𝑐
𝑒 �̇�𝑒   (39) 

 𝑇𝑎𝑏 =
1

2
𝜌𝑏𝐴𝑏 ∫ (

𝜕𝑢𝑏

𝜕𝑡
)

𝑙𝑒
0

2

𝑑𝑥 =
1

2
�̇�𝑒

𝑇
𝑀𝑒𝑏
𝑒 �̇�𝑒   (40) 

 𝑇𝑏𝑏 =
1

2
𝜌𝑏𝐴𝑏 ∫ (

𝜕𝑤

𝜕𝑡
)

𝑙𝑒
0

2

𝑑𝑥 =
1

2
�̇�𝑒

𝑇
𝑀𝑏𝑏
𝑒 �̇�𝑒   (41) 

  𝑇𝑎𝑣 =
1

2
𝜌𝑣𝐴𝑣 ∫ (

𝜕𝑢𝑣

𝜕𝑡
)

𝑙𝑒
0

2

𝑑𝑥 =
1

2
�̇�𝑒

𝑇
𝑀𝑒𝑐
𝑒 �̇�𝑒 (42) 

  𝑇𝑏𝑣 =
1

2
𝜌𝑣𝐴𝑣 ∫ (

𝜕𝑤

𝜕𝑡
)

𝑙𝑒
0

2

𝑑𝑥 =
1

2
�̇�𝑒

𝑇
𝑀𝑏𝑣
𝑒 �̇�𝑒  (43) 

where 𝑇𝑎𝑐, 𝑇𝑎𝑏and 𝑇𝑎𝑣  are kinetic energies of the constraining, base, and core layers that 

correspond to axial loading, respectively. 𝑇𝑏𝑐, 𝑇𝑏𝑏  and 𝑇𝑏𝑣 are kinetic energies of constraining, 

base, and core layers corresponding to bending loading, respectively. 



599  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

The element mass matrices associated with elastic layers and VE core layer corresponding 

to axial and bending loading 𝑀𝑎𝑐
𝑒 , 𝑀𝑏𝑐

𝑒 , 𝑀𝑎𝑏
𝑒 , 𝑀𝑏𝑏

𝑒 , 𝑀𝑎𝑣
𝑒  and 𝑀𝑏𝑣

𝑒  are expressed  using the 

element shape functions as: 

 𝑀𝑎𝑐
𝑒 = 𝜌𝑐𝐴𝑐 ∫ 𝑁2

𝑇𝑙𝑒
0

𝑁2𝑑𝑥,𝑀𝑏𝑐
𝑒 = 𝜌𝑐𝐴𝑐 ∫ 𝑁𝑥

𝑇𝑁𝑥
𝑙𝑒
0

𝑑𝑥 (44) 

 𝑀𝑎𝑏
𝑒 = 𝜌𝑏𝐴𝑏 ∫ 𝑁1

𝑇𝑁1
𝑙𝑒
0

𝑑𝑥,𝑀𝑏𝑏
𝑒 = 𝜌𝑏𝐴𝑏 ∫ 𝑁𝑥

𝑇𝑁𝑥
𝑙𝑒
0

𝑑𝑥  (45) 

 𝑀𝑎𝑣
𝑒 = 𝜌𝑣𝐴𝑣 ∫ 𝑁3

𝑇𝑁3
𝑙𝑒
0

𝑑𝑥,𝑀𝑏𝑣
𝑒 = 𝜌𝑣𝐴𝑣 ∫ 𝑁𝑥

𝑇𝑙𝑒
0

𝑁𝑥𝑑𝑥 (46) 

The element mass matrix 𝑀𝑒 is given as: 

 𝑀𝑒⏟
⌊8×8⌋

= 𝑀𝑎𝑐
𝑒 + 𝑀𝑏𝑐

𝑒 + 𝑀𝑎𝑏
𝑒 + 𝑀𝑏𝑏

𝑒 +𝑀a𝑣
𝑒 +𝑀𝑏𝑣

𝑒   (47) 

3.4. Finite Element Equation of Motion  

After assembling the global stiffness and mass matrix of the structure, the Hamilton 

variation principle is used to derive the dynamic equation of the VE sandwich beam: 

 ∫ 𝛿(𝑇 − 𝑈)
𝑡2
𝑡1

𝑑𝑡 + ∫ 𝛿𝑤
𝑡2
𝑡1

𝑑𝑡 = 0  (48) 

After applying Eq. (48), The VE sandwich beam equation of motion is expressed as: 

 𝑀𝑒�̈�𝑒 + 𝐾𝑒
𝑒𝑑𝑒 + 𝐾𝑣

𝑒𝑑𝑒 = 𝐹𝑒 (49) 

3.5. Linear Vibration Eigenvalue Problem (Constant Complex Modulus) 

The complex constant elastic modulus of VE material E is expressed as: 

 𝐸∗ = 𝐸′ + 𝑗𝐸′′  (50) 

where 𝐸′ is the real part, also called storage modulus, and 𝐸′′ is the imaginary part which is 

called energy dissipation modulus. 𝜂𝑣 = 
𝐸′′

𝐸′
 is VE loss factor, which is the ratio of energy 

dissipation modulus to storage energy modulus.  

The loss factor of VE material physically represents the ratio of the energy consumed 

from VE material to total energy.  

Many researchers have used constant shear complex modulus to obtain a simple dynamic 

equation by applying a simple harmonic excitation force [33] and Eq. (49) could be expressed as: 

 𝑀𝑒�̈�𝑒 + 𝑑𝑒𝐾𝑒 = 0 (51) 

where 𝐾𝑒, 𝑀𝑒 are the total stiffness matrix and total mass matrix of the VE sandwich beam, 

respectively. Then, the eigenvalue problem is expressed as  

 (𝐾𝑒 − 𝜔∗2𝑀𝑒)𝑋 = 0  (52) 

where 𝜔∗ is the complex frequency, and X is the complex eigenmodes of VE sandwich 

beam. Natural frequency 𝜔𝑛 and model loss factor 𝜂𝑛 are calculated as follows: 

 𝜔𝑛 = √𝑅𝑒𝜔
∗ (53) 



600  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

4. NUMERICAL RESULTS AND VALIDATION 

4.1. Normal Mode Analysis (Free Vibration) 

Normal modal analysis is a prerequisite step in modeling the dynamic behavior of VE 

sandwich structures. It is a semi-final step that gives theoretical values for eigenvalues and 

eigenvectors that have no physical meaning unless normal model analysis is paired with 

harmonic or time response analysis. The mechanical and geometrical properties of elastic 

and core layers of VE sandwich beam are summarized in Table 1.  

Table 1 Materials mechanical properties and geometrical specifications of VE sandwich 

beam [33] 

 Aluminum layers VE core layer 

Young modulus (MPa) 6.9× 104 1.794×(1+0.3i) 

Poisson’s ration 0.3 0.3 

Loss factor - 0.1,0.6,1,1.5 

Density (kg/m3) 2766 968.1 

Length (mm) 177.8 177.8 

Width (mm) 12.7 12.7 

Thickness (mm) 1.52 0.127 

4.1.1. Case Study #1: C-F VE Sandwich Beam  

To study the linear vibration of VE sandwich structures, a cantilever beam was discretized 

into 30 elements. In this case, VE material is independent from Young’s modulus E. The 

mechanical and geometrical properties of elastic and core layers of VE sandwich beam are 

summarized in Table 1.  

A MATLAB code was developed to solve the eigenvalues problem. In addition, 2D 

and 3D finite element models were built using FE solver software Abaqus to show the 

efficiency of the numerical approach.  

For mesh modeling, 2D and 3D FE models were built using Abaqus software. The 

eight-node continuum quadratic brick element CPS8 is used to discretize the 2D model, 

while the 20-node C3D20 was used to discretize the 3D model.  

The sizing control technique is used with a maximum mesh global size of 0.5 mm and 

maximum deviation factor of 0.1. Table 1 gives the specifications of CPS8 and C3D20. 

Based on Euler-Bernoulli beam theory, the natural frequencies of a base beam without 

VE core and  constraining layer are calculated as: 

 𝜔𝑛 =
𝐴𝑛

2𝜋𝐿2
√
𝐸𝐼

𝜌𝐴
 (54) 

where 𝐸 is the modulus of elasticity, 𝐼 is the cross-sectional moment of inertia of the base 

beam, 𝜌 is material density, and 𝐴 is the cross-sectional area of the beam. According to Eq. 

(54), the first fundamental natural frequencies of the base beam without VE core and 

constraining layer are calculated as 38.83Hz and 247.1Hz.  



601  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

It is worth pointing out that, with the addition of a VE layer and a constrained layer, the 

stiffness and mass are increased accordingly. From Eq. (54), an increase in structure 

stiffness leads to an increase in the natural frequency of the structure, while an increase in 

the additional mass leads to a decrease in the natural frequency.  

According to Table 1, the density of VE core is about half of elastic layer's density, 

however, the stiffness of the VE core is almost negligible compared to stiffness of the 

elastic layers. Accordingly, the effect of the additional mass on the natural frequency is 

much greater than the effect the additional stiffness.  

For the sake of comparison, the first six natural frequencies and normalized loss factor 

of VE sandwich beam were calculated using  three different elements based on shear energy 

consumption assumption when the loss factor 𝜂𝑣= [0.1, 0.6, 1.5]. 

Table 2 Mesh specifications of CPS8 and C3D20 elements implemented in Abaqus 

Element Description Schematic NDOF/node 

CPS8 

▪ Continuum 

biquadratic. 

▪ 8-nodes. 

▪ Plane stress 

 

2 Translations 

C3D20 

▪ Continuum 

quadratic 

brick. 

▪ 20-nodes. 

 

3 Translations 

Table 3 shows that FE shear models, real eigenmodes (RM) and asymptotic numerical 

method (ANM) all had good accuracy in calculating the first six natural frequency of VE 

sandwich beam under C-F boundary condition against the analytical solution [34]. Table 4 

shows normalized loss factor of C-F VE sandwich beam with different 𝜂𝑣. 

In comparison, the accuracy of FE shear model was the highest compared with the 

analytical method [34], the error range was 0-0.15% followed by C3D20 element with an 

error range of 0-5.918% and an average error of 0.2%; CPS8 element was the third with an 

error range of 0-6.565% and an average error of 0.858% and RM method had the lowest 

accuracy, with the lowest error value of 0%, the highest error value of 8.892% and an 

average error of 0.97%.  



602  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

Although RM approach had a slightly higher average error than ANM method, RM 

method could not be deemed accurate. Because RM approach only considers the real part 

of natural frequency, which contradicts reality. Although the prediction results of natural 

frequency by RM method were close to analytical solutions when the loss factor 𝜂𝑣 was 

small but when the loss factor 𝜂𝑣 was increased to 1.5, the prediction error of natural 

frequency reached 8.892%. On the other hand, the prediction error of ANM for natural 

frequency was very consistent and did not fluctuate much, thus the calculated loss factor 

of ANM for natural frequency was generally accurate. As a result, the prediction accuracies 

of the five numerical approaches for the natural frequency of C-F VE sandwich beam in 

this work were ranked as follows: Integral shear element> C3D20 > CPS8 > ANM> RM. 

Table 3 Linear frequencies of the C-F VE sandwich beam with different 𝜼𝒗 

𝜼𝒗 Analytical [34] CPS8 FE Model RM [35] C3D20 ANM [36] 

 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 

0.1 

64.1 64.1 64.1 64.1 64.5 64.5 

296.4 296.5 296.3 296.6 298.6 298.9 

743.7 743.6 743.4 744.3 750.1 746.5 

1393.9 1392.6 1393.4 1395.2 1408.3 1407.7 

2261.1 2256.7 2260.2 2263.4 2288.4 2286.2 

3343.6 3332.8 3342.3 3347.3 3388.7 3385.7 

0.6 

65.5 64.3 65.4 64.1 64.7 65.9 

298.9 297.4 298.8 296.6 299.5 303.1 

745.5 744.5 745.2 744.3 751 752.3 

1394.9 1393.1 1394.3 1395.2 1409 1412.7 

2261.7 2257 2260.8 2263.4 2289 2290.6 

3344 3333 3342.6 3374.3 3389 3389.5 

1 

67.4 64.7 67.4 64.1 65.1 67.8 

302.8 299.1 302.7 296.6 301 309.1 

748.6 746.1 748.4 744.3 753 761.1 

1396.6 1394 1396 1395.2 1410 1420.6 

2262.8 2257.6 2261.9 2263.4 2289 2297.9 

3345 3333.3 3343.4 3374.3 3389 3395.9 

1.5 

69.8 65.5 69.9 64.1 65.9 67.8 

308.8 302.3 308.7 296.6 304 309.1 

754 749 754.2 744.3 756 761.1 

1399.7 1395.8 1399.1 1395.2 1412 1420.6 

2265 2258.7 2264.1 2263.4 2290 2297.9 

3346 3333.9 3344.7 3374.3 3390 3395.9 

 Average Error (%) ≈ 0.858 0.03 0.94 0.2 0.97 



603  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

Table 4 shows that the accuracy of the FE shear element models implemented in this 

study is very high in predicting model loss factor for the first six modes of VE sandwich 

beam structure. With an error range of 0% to 11.45% and an average error of 2.38%, ANM 

[36] was the most accurate one. In addition, C3D20 element had an error range of 0-14.7%, 

with an average of 4.19%, which placed it second. RM [35], which had a minimum error 

of 0% and a maximum error of 8.892%, had the lowest accuracy. 

The results showed a significant improvement in the overall damping performance of the 

C-F VE sandwich beam as the loss factor 𝜂𝑣was increased. When low frequencies were close 

to loading frequencies, they could generate huge amplitudes, causing what is known as 

resonance. 

Table 4 Normalized loss factor of C-F VE sandwich beam with different 𝜼𝒗 

𝜼𝒗 Analytical [34] CPS8 RM [35] C3D20 ANM [36] 

 𝜂𝑛 𝜂𝑣⁄  𝜂𝑛 𝜂𝑣⁄  𝜂𝑛 𝜂𝑣⁄  𝜂𝑛 𝜂𝑣⁄  𝜂𝑛 𝜂𝑣⁄  

0.1 

0.281 0.283 0.283 0.282 0.281 

0.242 0.242 0.243 0.241 0.242 

0.154 0.154 0.154 0.153 0.154 

0.088 0.089 0.089 0.088 0.089 

0.057 0.057 0.057 0.056 0.057 

0.039 0.039 0.039 0.038 0.039 

0.6 

0.246 0.281 0.283 0.280 0.247 

0.232 0.241 0.243 0.239 0.224 

0.152 0.154 0.154 0.152 0.150 

0.088 0.089 0.089 0.088 0.088 

0.057 0.057 0.057 0.056 0.057 

0.039 0.039 0.039 0.038 0.039 

1 

0.202 0.277 0.283 0.276 0.204 

0.217 0.238 0.243 0.236 0.201 

0.150 0.153 0.154 0.152 0.142 

0.088 0.089 0.089 0.088 0.086 

0.057 0.057 0.057 0.056 0.057 

0.038 0.039 0.039 0.038 0.037 

1.5 

0.153 0.271 0.283 0.270 0.155 

0.197 0.232 0.243 0.231 0.176 

0.146 0.152 0.154 0.150 0.131 

0.087 0.088 0.089 0.087 0.083 

0.056 0.057 0.057 0.056 0.056 

0.038 0.039 0.039 0.038 0.039 

 Average Error (%) ≈ 5.42 6.02 4.19 2.38 

Fig. 3 shows relative changes in normalized deflection amplitudes for the first three 

eigenmodes of C-F VE sandwich beam with a different loss factor 𝜂𝑣= [0.1, 0.6, 1.5]. It 

could be noticed that both real and imaginary parts had different amplitudes but the same 

mode shapes.  

For high loss factor 𝜂𝑣, the imaginary part of the complex eigenmodes was significant 

which affected the damping performance of the whole structure.  



604  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

 

 

Fig. 3  The complex normalized eigenmodes of C-F VE sandwich beam with 𝜼𝒗= [0.1, 0.6, 1.5] 

(a) mode 1; (b) mode 2; (c) mode 3 

4.1.2. Case Study #2: S-S VE Sandwich Beam 

For the same mechanical and geometrical properties shown in Table 1, the linear 

vibration characterizations of S-S supported boundary VE sandwich beam are considered. 

In this study, based on Rao analytical model [37], the prediction accuracy of the natural 

frequency and loss factor of the VE sandwich beam structure are compared. It is seen from 

Tables 5 and 6 that all models, had high accuracy in calculating the first six natural 

frequencies of VE sandwich beam in comparison with analytical solution [37]. For S-S 

boundary condition, CPS8 had 0.59% average relative error and C3D20 had the lowest 

accuracy of 3.6%. On the other hand, from Table 6 it was noticed that both models had 

relatively high accuracy in predicting the first six loss factors of S-S VE sandwich beam 

compared with analytical solution [37]. As a result, it means that the prediction accuracy 

of the numerical approaches for the natural frequency of S-S VE sandwich beam in this 

work is ranked as follows: CPS8> C3D20. The dynamic characteristics of VE sandwich 

beam were studied, and the natural frequency of VE sandwich beam system was larger 

under S-S boundary conditions. The fundamental loss factor of VE sandwich beams with 

C-F boundary conditions was higher than that for VE sandwich beams with S-S boundary 

(c) 

(a) (b) 



605  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

conditions. For high values of loss factor of VE material, the imaginary part of the stiffness 

matrix was considered which implied the existence of not only storage modulus but, also 

loss modulus. As can be noticed from Fig. 4, the first normalized fundamental complex 

vibration modes of S-S VE sandwich beam with different damping factor of VE core 𝜂𝑣= 

[0.1,0.6,1.5] using the 2D plan stress element CPS8 implemented in Abaqus. The results 

show that the relative amplitude difference between real and imaginary modes was 

increased as the loss factor increased so that the imaginary part of the complex fundamental 

vibration modes was significant and could not be neglected.  

The flowchart of the employed methodology for dynamics analysis of sandwich beam 

is shown in Fig. 5. In the pre-processing stage, material model and tie constraints between 

the layers are defined. Also, the process involves assigning appropriate element models 

and defining C-C and S-S boundary conditions. In post-processing, the free vibration 

response results are interpreted based on tracking model effective mass. In addition, the 

parametric study stage involves investigation of VE parameters' effects on free vibration 

response of sandwich beam. In the last stage, steady state forced vibration response with 

different values of 𝜂𝑣 is obtained by utilizing mode superposition method that implies the 

representation of each eigenmode as an independent pattern of motion. 

Table 5 Linear frequencies of S-S VE sandwich beam with different 𝜼𝒗 

𝜼𝒗 Analytical [37] CPS8 element C3D20 element 

 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 

0.1 

148.51 148.44 148.55 

488.47 488.19 489.797 

1034.69 1033.5 1040.57 

1795.13 1791.56 1408.3 

2771.49 2763 2288.4 

3964.28 3946.95 3388.7 

0.6 

150.71 149.23 149.34 

489.75 489.01 490.61 

1035.38 1034.01 1041.08 

1795.54 1791.89 1811.05 

2771.76 2763.23 2802.92 

3964.47 3947.11 4016.7 

1 

154.42 150.62 150.73 

492.06 490.48 492.08 

1036.63 1034.95 1042.02 

1796.3 1792.5 1811.65 

2772.27 2763.64 2803.33 

3964.83 3947.41 4016.99 

1.5 

160.72 153.16 153.27 

496.49 493.29 494.89 

1039.07 1036.78 1043.84 

1797.78 1793.68 1812.82 

2773.25 2764.45 2804.12 

3965.52 3947.99 4017.55 

 Average Error (%) ≈ 0.59 3.60 



606  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

Table 6 Normalized loss factor of S-S VE sandwich beam with different 𝜼𝒗 

𝜼𝒗 Analytical [37] CPS8 element C3D20 element 

 𝜂𝑛 𝜂𝑣⁄  𝜂𝑛 𝜂𝑣⁄  𝜂𝑛 𝜂𝑣⁄  

0.1 

0.107 0.1069 0.1063 

0.065 0.0651 0.0644 

0.043 0.0433 0.0426 

0.031 0.0307 0.03 

0.3328 0.3468 0.3465 

0.1943 0.195 0.1945 

0.6 

0.1068 0.1068 0.1062 

0.0652 0.0651 0.0644 

0.0433 0.0433 0.0426 

0.0308 0.0307 0.03 

0.305 0.3405 0.3402 

0.192 0.1938 0.1933 

1 

0.106 0.1066 0.106 

0.065 0.0651 0.0644 

0.043 0.0433 0.0426 

0.031 0.0307 0.03 

0.2626 0.3292 0.27 

0.1871 0.1916 0.231 

1.5 

0.1059 0.1063 0.15 

0.065 0.065 0.087 

0.0433 0.0432 0.056 

0.0308 0.0307 0.038 

0.0410 0.0432 0.0510 

3346 3333.9 3390 

 Average Error (%) ≈ 1.8 6.28 

4.1.3. Case Study #3: Four-Sided S-S Sandwich Plate 

The material and geometrical specifications of SSSS VE plate structure are shown in 

Table 7. The natural frequencies and loss factors of the structure are calculated using the 

developed FE model implemented in Abaqus and compared with the results obtained by 

Johnson and Kienholz [38], who used modal strain energy method (MSE) implemented in 

NASTRAN commercial software to calculate the natural frequency and loss factor of VE 

sandwich structures.  

The VE sandwich plate in this study is discretized into 2200×120 and 1050×255 

meshed areas using C3D20 element along with the length and width directions during the 

calculation. In adition, the mesh specification of C3D20 element elements implemented in 

Abaqus is shown in Table 2.  



607  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

 
 

 

Fig. 4 The complex normalized eigenmodes of S-S VE sandwich beam with 𝜂𝑣= [0.1, 0.6, 1.5] 

(a) mode 1; (b) mode 2; (c) mode 3 

Table 7 Mechanical and geometrical specifications of VE SSSS VE sandwich plate [38] 

Properties Elastic layers VE core layer 

Young modulus (Pa) 6.89× 1010 1.794×106 

Poisson’s ration 0.3 0.49 

Density (kg / m3) 

Loss factor 

2740 

- 

999 

0.5 

Length (mm) 348 348 

Width (mm) 304.8 304.8 

Thickness (mm) 0.762 0.254 

The mode shapes of the VE sandwich plate associated with the first fundamentals 

natural frequency obtained using C3D20 element are shown in Fig. 6. The sweet spot could 

be visualized which showed the maximum deformation location points. 

(c) 

(a) (b) 



608  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

 

Fig. 5 Flow chart of the proposed dynamics analysis methodology 

 Pre-processing 

Meshing 

Test 

No 

Yes 

No 

Normal mode analysis 

Problem definition 
-Mechanical and geometrical 

specifications 
  

Kinematics formulation 
-FSDT 

Degree of freedom 
-Interpolation shape function  

FEM Mechanical assembly 

-𝐾𝑒  

-𝑀𝑒  
 

  
  Discretization 

(Ke −ω∗2Me)X = 0 

Integral beam element 

FE Model  
-Abaqus/Explicit 

Boundary conditions 

Model meshing 

Contact 
-Tie constrains 

CPS8/C3D20 element 

Complex frequency 𝝎∗ 

 

Material Loss factor 𝜼𝒗 

VE Core Thickness 

Model frequency 𝝎𝒏 

Model loss factor 𝜼𝒏 𝝎𝒏 = √𝑹𝒆𝝎∗ 

Harmonic load excitations 

FRF 

Steady-state response 

Parametric study 



609  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

          

     

Fig. 6 Fundamental eigenmodes shape of viscoelastic sandwich SSSS plate: (a) First Mode, 

(b) Second Mode, (c) Third Mode, (d) Forth Mode 

Tables 8 and 9 show the calculated results of natural frequency and normalized loss 

factor, respectively. It is clear from Tables 8 and 9 that C3D20 element had good accuracy 

in predicting the damping characteristics of VE sandwich SSSS plate against analytical 

solution with an average error in predicting the fundamental natural frequency of less than 

0.4% and an average error of less than 6.5% in predicting the damping loss factor  𝜂𝑛 𝜂𝑣⁄  
of VE sandwich plate. In addition, it can be noticed from Tables 8 and 9 that, while 

increasing the mesh size the error slightly decreases but this increases the time computation 

of the whole FE process. Thus, there exists a trade-off between mesh refinement and 

computational efficiency in the FE analysis. 

Table 8 Linear complex natural frequencies of VE sandwich plate obtained for different 

mesh sizes of C3D20 element 

 

Mode 

MSE method [38] C3D20 2200×120 C3D20 1050×255 

Natural frequency 

𝜔𝑛 (Hz)  

Natural frequency 

𝜔𝑛 (Hz)  

Error 

(%) 

Natural frequency 

𝜔𝑛 (Hz)  

Error 

(%) 

1 57.4 57.3 0.174 57.2 0.348 

2 113.2 113.5 0.265 113.3 0.088 

3 129.3 129.5 0.155 129.1 0.155 

4 179.3 177.7 0.892 177.1 1.227 

Average   0.372  0.454 

Table 9 Normalized loss factor of VE sandwich plate obtained for different mesh sizes of 

C3D20 element 

 

Mode 

MSE method [38] C3D20 2200×120 C3D20 1050×255 

Loss factor 

𝜂𝑛 𝜂𝑣⁄  

Loss factor 

𝜂𝑛 𝜂𝑣⁄  

Error 

(%) 

Loss factor 

𝜂𝑛 𝜂𝑣⁄  

Error 

(%) 

1 0.352 0.366 3.98 0.366 3.963 

2 0.376 0.394 4.79 0.396 5.342 

3 0.376 0.390 3.72 0.391 4.038 

4 0.306 0.346 13.07 0.348 13.835 

Average   6.39  6.794 

a) First Mode b) Second Mode c) Third Mode d) Forth Mode  



610  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

4.2. Parametric Studies 

It is required to examine the effects of various parameters on the vibration characteristics of 

VE sandwich beam. The thickness of the core and VE core loss factor are varied which affects 

natural frequency and mode shape. 

4.2.1 Effect of VE Material Loss Factor on the Dynamic Characteristics of VE 

Sandwich Beam 

To study the influence of VE loss factor on the dynamic characteristics of natural 

frequency, VE loss factor is varied in the range of 0.1 to 1.5 and the change trends of natural 

frequency and normalized loss factor corresponding to the fundamental modes of C-F VE 

sandwich beam are illustrated in Fig. 7. The fundamental natural frequencies of VE 

sandwich beams only increased minimally when the loss factor was increased from 0.1 to 

1.5, indicating that 𝜂𝑣 have little effect on the natural frequencies of VE sandwich beams. 

Constant complex module 𝐸′(1 + 𝑗𝜂𝑣), showed that the complex elastic modulus 𝐸∗ of VE 

material is directly proportional to the loss factor. In addition, the stiffness of VE layer and 

the stiffness of the entire VE sandwich beam increased. Since the mass of VE sandwich 

beam did not vary, the fundamental frequency of the beam was increased. However, as 

shown in Table 1, the elastic modulus of VE layer was four times lower than the elastic 

modulus of elastic layers, thus the increase in stiffness induced by the loss factor of VE 

layer could only make VE sandwich C-F beam stiffer. Under the condition of constant 

mass, the natural frequencies of VE sandwich C-F beam are only increased slightly with a 

small increase in stiffness. In addition, when the loss factor of VE material 𝜂𝑣 is increased, 

the normalized loss factor 𝜂𝑛 𝜂𝑣⁄  of VE sandwich beam increased linearly, implying that 

the loss factor of VE material 𝜂𝑣 had a significant impact on the loss factor of VE sandwich 

beam and energy dissipation capacity of VE material is characterized by its loss factor.  

 

Fig. 7 The effect of 𝜂𝑣 on the dynamic characteristics of C-F VE sandwich beam 



611  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

4.2.2. Effect of VE Core Layer Thickness on the Dynamic Characteristics of VE 

Sandwich Beam 

Core thickness is varied from 0.5 to 6.5 mm to study its influence on the dynamic 

characteristics 𝜔𝒏 and normalized loss factor 𝜂𝑛 𝜂𝑣⁄ . Indeed, the natural frequency of the 

system slightly decreased as the thickness of VE layer increased, as seen in Fig. 8. Increase 

in the stiffness of VE sandwich beam system generated by increasing the thickness of the 

VE layer was much smaller than the increase in the whole C-F VE sandwich beam stiffness 

since the elastic modulus of VE layer was four times lower than elastic modulus of elastic 

layers and the mass of the core was about 50% of elastic layers mass. The changing of the 

normalized loss factor 𝜂𝑛 𝜂𝑣⁄  for C-F beam was more significant than the changing trend 

of natural frequency, which was related to the damping nature of VE material. 

 

Fig. 8 The effect of core thickness on the dynamic characteristics of C-F VE sandwich beam 

4.3. Frequency Response Analysis  

Harmonic analysis is required to ensure that the system will successfully overcome 

resonance and harmful effects and mode superposition method is used based on linear 

iterations of mode shapes to obtain the frequency response function (FRF). The frequency 

response curves were then generated to capture the steady-state vibration response of VE 

sandwich beam across specific frequency range. 

4.3.1. Steady-State Dynamics of VE Sandwich Beam/Plate 

To excite the first three fundamental mode shapes, a harmonic concentrated force was 

applied at point p = (59.27, 6.3) mm located along beam length, with magnetite of 𝐹0 = 100 
N. The system linear frequency response (FRF) is shown in Fig. 9 with different loss factor 

𝜂𝑣= [0.3, 0.6, 1]. It could be noticed from Fig. 9 that the effect of the loss factor was 

significant, as the system had minimum amplitude response when loss factor 𝜂𝑣= 1. The 
maximum response of the system was within the range of the first fundamental natural 
frequency that had the highest value of effective mass along the direction of applied load.  



612  Y. ELMOGHAZY, B. SAFAEI, S. SAHMANI 

In addition, Fig. 9 shows that the system tends to vibrate with maximum amplitude at 
natural frequency values. This occurs since the natural frequency of the structure matches 
the excitation frequency, leading to a resonance condition that can potentially cause failure 
effects in the system. 

 

Fig. 9 Linear frequency response curve of fundamental complex modes for C-F VE 

sandwich beam with 𝜂𝑣= [0.3, 0.6, 0.1] 

5. CONCLUSIONS 

In the present work, dynamic analysis was examined to evaluate the damping 
characteristics of VE sandwich structures, including natural frequency and normalized loss 
factor 𝜂𝑛 𝜂𝑣⁄  based on longitudinal shear deformation assumption. The obtained numerical 
results were compared against analytical model to validate the proposed FE models. The 
numerical results showed that all numerical FE models are applicable for approximating 
the damping characteristics of VE sandwich beam. The integral shear element model was 
developed using FE formulation had the lowest error range among other models. In 
addition, the linear complex natural frequency of VE sandwich plate were obtained for 
different mesh sizes of C3D20 element. The element shows high accuracy in predicting 
damping characteristics of VE sandwiched plate structure. Finally, a parametric study was 
performed to study the effect of various parameters on the dynamic characteristics of VE 
sandwich beam. The obtained keys conclusion is demonstrated as follows: 

▪ Increasing VE layer thickness decreased the natural frequency and loss factor of the 
VE sandwich beam. 

▪ The loss factor of VE layer 𝜂𝑣 had a significant effect on the overall loss factor of 
the structure, whereas it had no significant effect on natural frequency. 

▪ The calculated resonance frequencies for VE beam are in the range of modal natural 
frequency. 



613  Dynamics Analysis of Viscoelastic Sandwiched Structures Integrated with Aluminum Sheets 

 

In summary, the proposed study provides valuable insights into the dynamics of VE 

sandwiched structures integrated with aluminum sheets, offering a comprehensive 

understanding of the factors affecting their damping characteristics.  

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