VIBRATION OF FUNCTIONALLY GRADED SANDWICH BEAMS EXCITED BY a MOVING h ARMONIC POINT LOAD DAO ĐỘNG của dầm SANDWICH có cơ TÍNH BIẾN THIÊN CHỊU KÍCH ĐỘNG của lực điều hòa DI ĐỘNG

Table 1 lists the fundamental frequency parameter μ of the FG sandwich beam for various values of the core thickness to the beam height ratio hC/h and the material index n.. The effect

VIBRATION OF FUNCTIONALLY GRADED SANDWICH BEAMS EXCITED

BY A MOVING H ARMONIC POINT LOAD

DAO ĐỘNG CỦA DẦM SANDWICH CÓ CƠ TÍNH BIẾN THIÊN CHỊU KÍCH

ĐỘNG CỦA LỰC ĐIỀU HÒA DI ĐỘNG

Van Tuyen BUI1a, Quang Huan NGUYEN2b, Thi Thom TRAN2b, Dinh Kien NGUYEN2b

1ThuyLoi University, Hanoi, Vietnam

2Institute of Mechanics,VAST, Hanoi, Vietnam

a tuyenbv@tlu.edu.vn; b ndkien@imech.ac.vn

ABSTRACT

The vibration of functionally graded (FG) sandwich beams excited by a moving harmonic point load is studied by the finite element method (FEM) The beams are assumed

to be formed from a homogeneous metallic soft core and two symmetrical FG layers Based

on the first-order shear deformation beam theory, a finite beam element is formulated by using the exact shape functions The implicit Newmark method is employed in computing the dynamic response of the beams The numerical results show that the formulated element is capable to access accurately the dynamic characteristics of the beam by using just several elements A parametric study is carried out to highlight the material distribution, the core thickness to the beam height ratio and the loading parameters on the vibration characteristics

Keywords: FG sandwich beam, moving load, vibration, dynamic response, FEM

TÓM TẮT

Dao động của dầm sandwich có cơ tính biến thiên (FG) chịu kích động của lực điều hòa

di động được nghiên cứu bằng phương pháp phần tử hữu hạn (FEM) Dầm được giả định có

một lõi kim lọại và hai lớp ngoài FG, đối xứng qua mặt giữa dầm Phần tử dầm dựa trên lý thuyết biến dạng trượt bậc nhất được xây dựng trên cơ sở các hàm dạng chính xác Đáp ứng động lực học của dầm được tính bằng phương pháp tích phân trực tiếp Newmark Kết quả số

chỉ ra rằng phần tử xây dựng trong bài báo có khả năng đánh giá chính xác các đặc trưng động

lực học của dầm chỉ bằng một vài phần tử Ảnh hưởng của sự phân bố vật liệu, tỷ số giữa độ dày của lõi và chiều cao dầm cũng như các tham số của lực di động tới các đặc trưng dao động của dầm được khảo sát chi tiết

Từ khóa: dầm sandwich FG, lực di động, dao động, đáp ứng động lực học, FEM

1 INTRODUCTION

Functionally graded (FG) sandwich material is a new type of composite which is widely used as structural material in recent years This new composite has many advantages, including the high strength-to-weight ratio, good thermal resistance and no delaminating problem which often met in the conventional composites Investigations on the vibration analysis of FG sandwich beams have been extensively carried out recently Mohanty et al [1] proposed a finite element procedure for static and dynamic stability analysis of FG sandwich Timoshenko beams Bui et al [2] used the meshfree radial point interpolation method to study the vibration response of a cantilever FG sandwich beam subjected to a time-dependent tip load Adopting the refined shear deformation theory, Vo et al [3] investigated the free vibration and buckling of FG sandwich beams In [4], Vo et al presented a finite element model for the free vibration and buckling analyses of FG sandwich beams

Analysis of beams subjected to moving loads is a classical problem in structural mechanics, and it has been a subject of investigation for a long time This problem becomes a hot topic in the field of structural mechanics since the date of invention of FG materials by Japanese scientists in 1984 A combination of strong and light weight ceramics with traditional ductile metals remarkably enhances the vibration characteristics of the structures The investigations on the dynamic response of FG beams [5-8] in recent years have shown that the dynamic deflections of an FG metal-ceramic beam considerably reduces comparing to that of the pure beam In addition, an FG beam induced by a soft core may improve the dynamic behavior of the structure when it subjected to moving loads

The present work aims to study the vibration of an FG sandwich beam excited by a moving harmonic load, which to the authors’ best knowledge has not been investigated so far The beam in this work is assumed to be formed from a homogeneous metallic soft core and two symmetrical FG skin layers Based on the first-order shear deformation beam theory, a finite element beam formulation is derived and employed in computing the dynamic response

of the beam A parametric study is carried out to highlight the effect of the material distribution, the ratio of core thickness to beam height as well as the loading parameters on the vibration characteristics of the beam

2 MATHEMATICAL FORMULATION

Figure 1 shows a simply supported FG sandwich beam with length L, height h, width b, core thickness hC in a Cartesian co-ordinate system (x, z) The beam is assumed to be subjected to a harmonic load P=P0cos( Ωt), moving from left to right at a constant speed v

Figure 1: FG sandwich beam under a moving harmonic load

The beam is assumed to be formed from a metallic soft core and two FG layers with the volume fraction of the constituent materials follows a power-law function as follows

C

C C

C C

2

2

n

c

n C

z h

z h

h h

−







and V m =1-V c Here and afterwards, the subscripts ‘c’ and ‘m’ are used to indicate the

‘ceramic’ and ‘metal’, respectively In Eq.(1), n is the material power-law index, defining the

variation of the constituent materials through the beam thickness From Eq.(1) one can see that the top and bottom surfaces of the beam are pure ceramic, and the core is full metal The

effective property P(z) (such as Young’s modulus, shear modulus and mass density) can be

evaluated by Voigt model and having the form

C C

C

C C

2

2

n

m

n

z h

z h

h h

−





 − 



(2)

where P c and P m are the material properties of ceramic and metal, respectively Based

on the first-order shear deformation beam theory, the displacements u1 and u3 in x and z

directions at any point of the beam are given by

1 3

( , , ) ( , ) ( , ), ( , , ) ( , ),

u x z t u x t z x t

u x z t w x t

θ

= (3)

in which u(x,t) and w(x,t) respectively are the axial and transverse displacements of the

corresponding points on the mid-plane; θ(x,t) is the rotation of the cross section The strains

and stresses based on Hook’s law resulted from Eq (3) are as follows

, , , ,

x x x xz x

= = (4) where εx, σx, γxz, τxz are respectively the axial strain, axial stress, and the shear strain

and shear stress; G(z)=E(z)/[2(1+ν)] is the shear modulus The Poisson’s ratio is assumed to

be constant in the present work

From Eq (4), the strain energy of the beam can be can written in the form

11 , 22 , 33 , 0

1

2

L

U= ∫ A u +A θ +ψA w −θ dx (5)

In Eq (5), ψ is the shear correction factor, equals to 5/6 for the rectangular section

herein; A11, A22, A33 respectively are the axial, bending and shear rigidities, defined as

A A =∫E z z dA A =∫G z dA (6) Using Eq (2), one can write the rigidities in Eq (6) in explicit forms as follows

22

1

1

c m m

m

c m m

n

n

+

+

+

+

(7)

Eq (3) gives the kinetic energy of the beam in the form

0

1

2

L

T = ∫ I u +w +I θ dx (8)

11 22

( , ) ( )(1, )

A

I I =∫ρ z z dA (9) are the mass moments, and as the rigidities these mass moments can be computed

explicitly

The potential of the moving load P is simply given by

0 cos( ) ( , ) ( )

V = −P Ωt w x t δ x−vt (10) where δ is the delta Diract function

Applying Hamilton’s principle to Eqs (5), (8) and (10), one can obtain the equations of

motion for the beam in the form

11 11 ,

11 33 , , 0

22 22 , 33 ,

0

0

xx

x xx

I u A u







(11)

Eq (11) has the same form as of homogeneous beams subjected to a moving harmonic

load, but the rigidities and mass moments are now defined by Eqs (6) and (9), respectively It

should be noted that due to the material property is symmetrically with respect to the

mid-plane, the coupling axial stretch and bending terms are not appeared in the governing

equations as in case of FG beams

3 FINITE ELEMENT FORMULATION

The finite element method is used herein in solving Eq (11) To this end, the beam is

assumed being divided into a numbers of two-node beam elements with length of l The

vector of nodal displacements for a generic element (i,j) has the following components

i i i j j j

where and hereafter a superscript ‘T’ denotes the transpose of a vector or a matrix By

introducing the shape functions for the displacement field, we can write the displacements and

rotation inside the element as follows

,

where Nu, Nw, Nθ are the matrices of the shape functions for u, w and θ, respectively

The exact linear, quadratic and cubic polynomials previously deried by Nguyen et al [7] by

solving the static equilibrium of a beam segment are employed herein to interpolate u, θ and

w, respectively Using Eqs (12) and (13), one can write the strain energy, kinetic energy and

the potential of the external load in term of the nodal displacement vector as follows

w

0

)

w

nELE nELE

uu

nELE nELE

uu w

U T

θθ γγ

θθ

δ

(14)

where nELE is the total number of the elements; k, m, f respectively are the element

stiffness, mass matrices and load vector of the element The stiffness matrices kuu, kθθ, kγγ

and the mass matrices muu, mww, mθθ in Eq (14) have the folloing forms

uu u u ww w x w x

θθ θ θ

ψ

(15)

Using Eqs (14) and (15) one can rewrite equations of motion for the beam in terms of

finite element analysis as follows

in which M, K and F respectively are the global mass, stiffness matrices and load

vector These matrices and vector are obtained by assembling the element mass, stiffness and

load vector m, k and f derived above in the standard way of the finite element analysis Eq

(16) can be solved by the direct integration Newmark method Here, the average acceleration method which ensures the unconditional stability [9] is employed

4 NUMERICAL RESULTS

The derived element formulation has been implemented into a computer code and employed in analysis of the FG sandwich beam subjected to a moving harmonic point load A simply supported beam composed of Aluminum (Al – metal phase) core and Aluminum-Alumina (Al-Al2O3) FG layers is considered in this section The material data for Aluminum

and Alumina are as follows: E m=70 GPa, ρm= 2702 kg/m3 for Aluminum, and E c=390 GPa,

ρ c= 3960 kg/m3 for Alumina The amplitude of the moving load is taken by P0=100kN A

total of 500 steps are used for Newmark method in all the computations reported below Table 1 lists the fundamental frequency parameter μ of the FG sandwich beam for

various values of the core thickness to the beam height ratio hC/h and the material index n

The frequency parameter in the Table is defined as follows

2

m

L

ρ ω

µ =

in which ω1 is the fundamental frequency of the beam, and ρ m and E m are the mass density and Young’s modulus of the core material, respectively

The effect of the material index n and the core thickness to the beam height ratio hC/h on the

fundamental frequency of the beam is clearly seen from the Table At a given value of the

hC/h, the fundamental frequency of the beam is smaller for the beam associated with a higher

index n The effect of the hC/h ratio on the frequency is similar to that of the index n, and at a given value of the index n, the frequency also decreases by raising the hC/h ratio The decrease in the fundamental frequency by raising the index n can be explain by the fact that,

as seen from Eq (1), the beam associated with a higher index n contains more percentage of

metal As a result, the rigidities of the beam, defined by Eq (6), will be decreased, and this leads to the lower frequency of the beam The reduction in the fundamental frequency of the

beam by raising the hC/h ratio can be explained by the same reason as by raising the index n, that is the rigidities of the beam reduced by raising the hC/h ratio

It should be noted that the volume fraction of the constituent materials defined in this paper is different from some works published before, e.g the work by Vo et al [3] In Ref [3], the volume fraction of metal is defined first, and in this case, the volume fraction of metal

decreases by raising the index n As a results, the frequency parameter in Ref [3] increases by increasing the index n The authors have computed the frequency of the beam by using the

definition of the volume fraction in Ref [3] (not shown herein), and a good agreement was obtained

Table 1: Fundamental frequency parameter μ of FG sandwich beam

In Table 2, the maximum frequency parameter, max(fD), of the beam is given for

various values of the the core thickness to the beam height hC/h ratio, and the material index

n The efrequency factor fD is defined as follows

D

0

max( ( / 2, ))

w

where w0 is the deflection of simply supported homogeneous beam made of the core material under a static load P0 at the mid-span, that is

3 0 0

48 m

P L w

E I

The maximum deflection factor, as seen from Table 2 steadily increases when raising

the index n and the hC/h ratio The increase in the maximum deflection factor can also be explained by the reduction in the rigidities of the beam by increasing the index n and the core

thickness to beam height ratio

Table 2: Maximum dynamic deflection factor max(f D ) of FG sandwich beam

Index n

hC/h

0.2 0.9731 0.9748 0.9879 1.0526 1.2277

0.5 1.0169 1.0200 1.0403 1.1180 1.2953

Figure 2: Time-histories for mid-span deflection of FG sandwich beam at various values

of the index n, moving speed v, hc/h ratio and excitation frequency Ω

In Figure 2, the time-histories for the mid-span deflection of the beam are depicted for

various values of the material index n, moving speed v, core thickness to beam height ratio

hC/h and excitation frequency Ω In the figure, DT is the total time necessary for the load to

pass the beam The following comments can be made from the figure

• The dynamic response of the beam is governed by many parameters, including the

material distribution, core thickness to beam height ratio, moving speed and excitation

frequency These factors not only change the maximum amplitude of the dynamic deflection, but the time at which the maximum deflection attains also

• The dynamic deflection considerably increases when raising the moving speed The beam executes less vibration cycles when it subjects to a higher speed moving load than when

it subjects to a lower moving speed load At a given value of the material index, the core thickness to the beam height ratio and the moving speed, the dynamic deflection rapidly increases when raising the excitation frequency towards to fundamental frequency of the beam Different from the moving speed, the number of vibration cycles which the beam executes when it subjects to a higher frequency moving load is larger

Figure 3: Relation between the deflection parameter and the moving speed of FG

sandwich beam with various values of index n and hc/h ratio

In order to examine the effect of the material distribution and the core thickness on the dynamic response of the beam, the relation between the deflection parameter fD and the

moving speed v of the FG sandwich beam under a moving point load (Ω=0) is depicted in

Figure 3 for different values of the index n and the core thickness to beam height ratio hC/h

The following comments can be drawn from Figure 3

• At a given value of the index n and the core thickness to beam height ratio, the curve

represented the relation between the deflection factor and the moving speed of the FG sandwich beam is similar to that of homogeneous beams [10] When the moving speed is larger than a certain value, the parameter fD steadily increases and it reaches a peak value before descending For the lower values of the moving speed, the parameter fD in Figure 3

both increases and decreases with increasing v This phenomenon is associated with the

oscillations as seen from the time-histories depicted in Figure 2 as explained by Olsson in Ref [10]

• Regardless of the moving speed, the deflection factor fD increases when raising the

material index n or the core thickness to the beam height This phenomenon, as explained above, due to the decrease of the beam rigidities when raising the index n and the hC/h ratio

5 CONCLUSIONS

The paper investigated the vibration of FG sandwich beam excited by a moving harmonic point load by using the finite element method The beam is assumed to be formed from a homogeneous metallic soft core and two symmetrical FG layer A beam element based on the first-order shear deformation beam theory was formulated and employed in the investigation The direct integration Newmark method has been used in computing the dynamic response of

the beam The numerical results have shown that the vibration characteristics of the beam, including the fundamental frequency and dynamic deflection factor, are strongly affected by the material distribution, the core thickness to the beam height ratio, the speed and frequency

of the moving force The dynamic deflection factor increases by raising the index n and the

core thickness to beam height ratio The moving speed and the excitation frequency not only alters the amplitude of the dynamic deflection but it also changes the vibration cycles which the beam executes

REFERENCES

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AUTHORS’ INFORMATION

1 Bui Van Tuyen (e-mail: tuyenbv@tlu.edu.vn) is a Lecture at the ThuyLoi University

His current research is finite element modeling of FG structures subjected to moving loads

2 Nguyen Quang Huan (e-mail: nqhuan@.mail.ac.vn), Tran Thi Thom (ttthom@.mail.ac.vn) and Nguyen Dinh Kien (ndkien@.mail.ac.vn) are research

members at the Institute of Mechanics, VAST Their interested topic is development of finite element formulations for analysis of solids and structures