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

Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV 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 ThuyLoi University, Hanoi, Vietnam Institute of Mechanics,VAST, Hanoi, Vietnam a tuyenbv@tlu.edu.vn; bndkien@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 dầm sandwich có tính biến thiên (FG) chịu kích động lực điều hòa di động nghiên cứu phương pháp phần tử hữu hạn (FEM) Dầm giả định có lõi kim lọại hai lớp FG, đối xứng qua mặt dầm Phần tử dầm dựa lý thuyết biến dạng trượt bậc xây dựng sở hàm dạng xác Đáp ứng động lực học dầm tính phương pháp tích phân trực tiếp Newmark Kết số phần tử xây dựng báo có khả đánh giá xác đặc trưng động lực học dầm vài phần tử Ảnh hưởng phân bố vật liệu, tỷ số độ dày lõi chiều cao dầm tham số lực di động tới đặc trưng dao động dầm khảo sát chi tiết Từ khóa: dầm sandwich FG, lực di động, dao động, đáp ứng động lực học, FEM 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 750 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV 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 MATHEMATICAL FORMULATION Figure shows a simply supported FG sandwich beam with length L, height h, width b, core thickness h C in a Cartesian co-ordinate system (x, z) The beam is assumed to be subjected to a harmonic load P=P cos(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   2z + h C  − h h  C  = Vc     z − hC  h−h C     n    n for − h / ≤ z ≤ − hC / for − hC / ≤ z ≤ hC / for (1) hC / ≤ z ≤ 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 751 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV P( z ) n   z + hC  ( Pc − Pm )   + Pm   hC − h    Pm  n  z − hC   ( Pc − Pm )  h − h  + Pm C    for − h / ≤ z ≤ − hC / for − hC / ≤ z ≤ hC /      (2) hC / ≤ z ≤ h / for 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 u and u in x and z directions at any point of the beam are given by u1 ( x= , z , t ) u ( x, t ) − zθ ( x, t ), (3) u3 ( x, z , t ) = w( x, t ), 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 εx = u, x − zθ, x , γ xz = w, x − θ (4) = σ x E= ( z ) ε x , τ xz G ( z ) γ xz where ε x , σ x , γ xz , τ xz 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 L  A11u, x + A22θ, x + ψ A33 ( w, x − θ ) dx ∫ = U (5) In Eq (5), ψ is the shear correction factor, equals to 5/6 for the rectangular section herein; A 11 , A 22 , A 33 respectively are the axial, bending and shear rigidities, defined as = ( A11 , A22 ) E ( z )(1, z )dA , A ∫= ∫ G ( z )dA (6) 33 A A Using Eq (2), one can write the rigidities in Eq (6) in explicit forms as follows Ec + nEm n +1 (h − hC )( Ec − Em )  (h − hC ) 2h(h − hC ) h   bhC3 h + hhC + hC2 = − + + A22  + n+2 n +   12  n+3 A= bhC Em + (h − hC ) 11 A= bhC Gm + (h − hC ) 33   Em  (7) Gc + nGm n +1 Eq (3) gives the kinetic energy of the beam in the form = T with L  I11 (u + w ) + I 22θ  dx ∫ ( I11 , I 22 ) = ∫ ρ ( z )(1, z )dA (8) (9) A 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 V= − P0 cos(Ωt ) w( x, t ) δ ( x − vt ) where δ is the delta Diract function 752 (10) Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Applying Hamilton’s principle to Eqs (5), (8) and (10), one can obtain the equations of motion for the beam in the form I11u − A11u, xx =  − ψ A33 ( −θ, x + w, xx= I11w ) P0 cos(Ωt )δ ( x − vt ) (11) I 22θ − A22θ, xx + ψ A33 ( −θ + w, x ) = 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 midplane, the coupling axial stretch and bending terms are not appeared in the governing equations as in case of FG beams 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 d = {ui wi θi wj θ j } T uj (12) 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 = u N= N= N wd ud , θ θd , w (13) where N u , N w , N θ 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 nELE T nELE T d kd = ∑ ∑ d ( k uu + kθθ + k γγ ) d i 1= i1 = nELE  T  nELE  T d ( m uu + m ww + mθθ )d =  T ∑ d md = ∑ i 1= = i = U (14) −dT P0 cos(Ωt )NTw δ ( x − vt ) = −dT f V= 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 k uu , k θθ , k γγ and the mass matrices m uu , m ww , m θθ in Eq (14) have the folloing forms l l l T T T k uu = ∫ Nu , x A11Nu , x dx, kθθ = ∫ Nθ , x A22 Nθ , x dx, k γγ =− ∫ (N w , x Nθ ) ψ A33 (N w, x − Nθ )dx 0 l = m uu l N I N dx, m I N dx, mθθ ∫ Nθ I ∫= ∫ N= T u 11 u T w, x 11 ww (15) l T w, x 22 Nθ dx Using Eqs (14) and (15) one can rewrite equations of motion for the beam in terms of finite element analysis as follows  + KD = MD F (16) 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 753 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV 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 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 AluminumAlumina (Al-Al O ) FG layers is considered in this section The material data for Aluminum and Alumina are as follows: E m =70 GPa, ρ m = 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 P =100kN A total of 500 steps are used for Newmark method in all the computations reported below Table lists the fundamental frequency parameter μ of the FG sandwich beam for various values of the core thickness to the beam height ratio h C /h and the material index n The frequency parameter in the Table is defined as follows µ= ω1 L2 ρm h Em (17) in which ω 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 h C /h on the fundamental frequency of the beam is clearly seen from the Table At a given value of the h C /h, the fundamental frequency of the beam is smaller for the beam associated with a higher index n The effect of the h C /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 h C /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 h C /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 h C /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 h C /h n 1/5 1/3 1/2 4/5 0.2 5.3926 5.4790 5.4842 5.3757 4.5456 0.5 5.3540 5.3631 5.3138 5.1468 4.3060 5.2395 5.1682 5.0675 4.8512 4.0342 4.9806 4.8351 4.6940 4.4499 3.7178 4.4054 4.2236 4.0774 3.8597 3.3343 754 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV In Table 2, the maximum frequency parameter, max(f D ), of the beam is given for various values of the the core thickness to the beam height h C /h ratio, and the material index n The efrequency factor f D is defined as follows fD = max( w( L / 2, t )) w0 (18) where w is the deflection of simply supported homogeneous beam made of the core material under a static load P at the mid-span, that is w0 = P0 L3 48 Em I (19) The maximum deflection factor, as seen from Table steadily increases when raising the index n and the h C /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 h C /h Index n 1/10 1/8 1/4 1/2 3/4 0.2 0.9731 0.9748 0.9879 1.0526 1.2277 0.5 1.0169 1.0200 1.0403 1.1180 1.2953 1.1752 1.1812 1.2152 1.3115 1.4690 1.2467 1.2531 1.2888 1.3836 1.5245 1.3462 1.3526 1.3874 1.4730 1.5875 Figure 2: Time-histories for mid-span deflection of FG sandwich beam at various values of the index n, moving speed v, h c /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 h C /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 755 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV 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 h c /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 f D and the moving speed v of the FG sandwich beam under a moving point load (Ω=0) is depicted in Figure for different values of the index n and the core thickness to beam height ratio h C /h The following comments can be drawn from Figure • 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 f D steadily increases and it reaches a peak value before descending For the lower values of the moving speed, the parameter f D in Figure both increases and decreases with increasing v This phenomenon is associated with the oscillations as seen from the time-histories depicted in Figure as explained by Olsson in Ref [10] • Regardless of the moving speed, the deflection factor f D 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 h C /h ratio 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 756 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV 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 [1] Mohanty, S.C., Dash R.R., & Rout, T., Static and dynamic stability analysis of a functionally graded Timoshenko beam International Journal of Structural Stability and Dynamics, 2012, Vol 12(4), DOI: 10.1142/S0219455412500253 [2] Bui, T.Q., Khosravifard, A., Zhang, Ch., Hematiyan, M.R., & Golub, M.V., Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method, Engineering Structures, 2013, Vol 47, p 90-104 [3] Vo, T.P., Thai, H.T., Nguyen, T.K., Maheri, A., & Lee, J., Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory, Engineering Structures, 2014, Vol 64, p 12-22 [4] Vo, T.P., Thai, H.T., Nguyen, T.K., Maheri, A., Inam, F., & Lee, J., A quasi-3D theory for vibration and buckling of functionally graded sandwich beams, Composite Structures, 2015, Vol 119, p 1-12 [5] Şimşek, M., & Kocatürk, T., Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load, Composite Structures, 2009, Vol 90 (4), p 465–473 [6] Şimşek, M., Vibration analysis of a functionally graded beam under a moving mass by using different beam theories, Composite Structures, 2010, Vol 92 (4), p 904-917 [7] Nguyen, D.K., Gan, B.S., & Le, T.H., Dynamic response of non-uniform functionally graded beams subjected to a variable speed moving load, Journal of Computational Science and Technology, JSME, 2013, Vol 7(1), p 12-27 [8] Le, T.H., Gan, B.S., Trinh, T.H., & Nguyen, D.K., Finite element analysis of multi-span functionally graded beams under a moving harmonic load Mechanical Engineering Journal, Bulletin of the JSME, 2014, Vol 1(3), p 1-13 [9] M Géradin, D Rixen, Mechanical Vibrations Theory and Application to Structural Dynamics, Second edition, John Willey & Sons, Chichester, 1997 [11]M Olsson, On the fundamental moving load problems, Journal of Sounds and Vibration, 1991, Vol 145(2), p 299-307 AUTHORS’ INFORMATION 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 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 757