Dynamic response of multiple nanobeam system under a moving nanoparticle Alexandria Engineering Journal (2017) xxx, xxx–xxx HO ST E D BY Alexandria University Alexandria Engineering Journal www elsevi[.]
Alexandria Engineering Journal (2017) xxx, xxx–xxx H O S T E D BY Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com Dynamic response of multiple nanobeam system under a moving nanoparticle Shahrokh Hosseini Hashemi a,b, Hossein Bakhshi Khaniki a,* a b School of Mechanical Engineering, Iran University of Science and Technology, Narmak 16842-13114, Tehran, Iran Center of Excellence in Railway Transportation, Iran University of Science and Technology, 16842-13114 Narmak, Tehran, Iran Received 22 September 2016; revised November 2016; accepted 15 December 2016 KEYWORDS Dynamic response; Analytical solution; Moving particle; Nanobeam; Multi-layered nanobeam Abstract In this article, nonlocal continuum based model of multiple nanobeam system (MNBS) under a moving nanoparticle is investigated using Eringen’s nonlocal theory Beam layers are assumed to be coupled by winkler elastic medium and the nonlocal Euler-Bernoulli beam theory is used to model each layer of beam The Hamilton’s principle, Eigen function technique and the Laplace transform method are employed to solve the governing equations Analytical solutions of the transverse displacements for MNBs with simply supported boundary condition are presented for double layered and three layered MNBSs For higher number of layers, the governing set of equations is solved numerically and the results are presented This study shows that small-scale parameter has a significant effect on dynamic response of MNBS under a moving nanoparticle Sensitivity of dynamical deflection to variation of nonlocal parameter, stiffness of Winkler elastic medium and number of nanobeams are presented in nondimensional form for each layer Ó 2016 Faculty of Engineering, Alexandria University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction One of the most important nano-structures used in nanodevices such as oscillators, clocks and sensor devices is Nanobeams Lots of researches have been done in order to achieve the behavior of engineering structures such as beams, tubes, plates and shells in various scales and static conditions such as bending [1–11] and buckling [12–16] Also in dynamic manners, with highlighting free and forced vibration analysis there have been plenty of researches [17–28] By investigation of nanocars in 2005 at Rice University, lots of researches had * Corresponding author E-mail address: h_bakhshi@mecheng.iust.ac.ir (H.B Khaniki) Peer review under responsibility of Faculty of Engineering, Alexandria University started in different subjects to understand the behavior of nanocars Shirai et al [29] studied a controlled molecular motion on surfaces through the rational design of surfacecapable molecular structures called nanocars They showed that the movement of the nanocars is a new type of fullerene-based wheel-like rolling motion, not stick-slip or sliding translation, due to evidence including directional preference in both direct and indirect manipulation and studies of related molecular structures Sasaki et al [30] reported the synthesis of a new nanovehicle, a porphyrin-based nanotruck The porphyrin inner core was designed for possible transportation of metals and small molecules across a surface Akimov et al [31] developed molecular models describing the thermally initiated motion of nanocars, nanosized vehicles composed of two to four spherical fullerene wheels chemically coupled to aplanar chassis, on a metal surface The simulations were aimed http://dx.doi.org/10.1016/j.aej.2016.12.015 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 at reproducing qualitative features of the experimentally observed migration of nanocars over gold crystals as determined by scanning tunneling microscopy Sasaki et al [32] reported the synthesis of two nanocars by a process resembling an assembly line where front and rear portions are attached using hydrogen bonding and metal complexation Vives and Tour [33] presented a new class of nanovehicles incorporating trans-alkynyl(dppe) ruthenium-based wheels A four-wheeled nanocar and a three-wheeled trimer were synthesized for future studies at the single molecule level Khatua et al [34] monitored the mobility of individual fluorescent nanocars on three surfaces: plasma cleaned, reactive ion etched, and aminefunctionalized glass Using single-molecule fluorescence imaging, the percentage of moving nanocars and their diffusion constants were determined for each substrate It was shown that the nanocar mobility decreased with increasing surface roughness and increasing surface interaction strength Ganji et al [35] studied nanocars motion on one-dimensional substrate surfaces which provided an important contribution to the practical goal of designing nanoscale transporters First principles of VdW-DF calculations were performed to study the interaction between the nanocar and the graphene/graphyne surface The accuracy of this method was validated by experimental results and the MP2 level of theory Moreover, by having a moving nano-car, nano-truck, three-wheeled trimer or nanoparticle on a surface, it is necessary to know the behavior of the surface while the moving mass is crossing through Understanding the dynamical behavior of the surface in which the mass is moving through was an important issue for scientist which led to a new branch of studies S ß imsßek [36] studied the forced vibration of a simply supported single-walled carbon nanotube (SWCNT) subjected to a moving harmonic load by using nonlocal Euler–Bernoulli beam theory The time-domain response was obtained by using both the modal analysis method and the direct integration method Kiani [37] examined the dynamic response of a SWCNT subjected to a moving nanoparticle in the framework of the nonlocal continuum theory of Eringen He also studied [38] the vibration of elastic thin nanoplates traversed by a moving nanoparticle involving Coulomb friction using the nonlocal continuum theory of Eringen Yas and Heshmati [39] studied the vibrational properties of functionally graded nanocomposite beams reinforced by randomly oriented straight single-walled carbon nanotubes (SWCNTs) under the actions of moving load S ß imsßek [40] presents an analytical method for the forced vibration of an elastically connected double-carbon nanotube system (DCNTS) carrying a moving nanoparticle based on the nonlocal elasticity theory Yas and Heshmati [41] studied the free and forced vibrations of nonuniform functionally graded multi-walled carbon nanotubes (MWCNTs) polystyrene nano-composite beams Different MWCNTs distribution in the thickness direction was introduced to improve fundamental natural frequency and dynamic behavior of non-uniform polymer composite beam under action of moving load Ghorbanpour et al [42] investigated an analytical method of the small-scale parameter on the vibration of single-walled Boron Nitride nanotube (SWBNNT) under a moving nanoparticle SWBNNT was embedded in bundle of carbon nanotubes (CNTs) which was simulated as Pasternak foundation Governing equation was derived using Euler–Bernoulli beam model, Hamilton’s S.H Hashemi, H.B Khaniki principle and nonlocal piezo-elasticity theory Heshmati and Yas [43] studied the dynamic response of functionally graded multi-walled carbon nanotube (MWCNT) polystyrene nanocomposite beams subjected to multi-moving loads The effect of uniform, linear symmetric and unsymmetric MWCNT distributions through the thickness direction on dynamic behavior was studied Hong et al [44] analyzed the vibration of single-walled carbon nanotube embedded in an elastic medium under excitation of a moving nanoparticle based on the Winkler spring model and the Euler–Bernoulli beam model Ghorbanpour and Roudbari [45] investigated the nonlocal longitudinal and transverse vibrations of coupled boron nitride nanotube (BNNT) system under a moving nanoparticle using piezoelastic theory and surface stress based on EulerBernoulli Luă et al [46] focused on the investigation of the transverse vibration of double carbon-nano-tubes (DCNTs) which were coupled through elastic medium Both tubes were conveying moving nano-particles and their ends were simply supported The system equations were discretized by applying Galerkin expansion method, and numerical solutions were obtained Li and Wang [47] investigate the nonlinear dynamic response characteristics of GP (graphene/ piezoelectric) laminated films in sensing moving transversal load induced by externally moving adhesive particles or molecules, based on the nonlocal elasticity theory and Von Ka´rma´n nonlinear geometric relations With respect to all the researches done in this manner, predicting the behavior of MNBS systems under a moving load, is an important issue in designing nanosensors which are under different type of loadings Also, with the presentation of nanorace [48] between nanocars in 2016 and the studies around the optimization of the behavior of nanovehicles of different surfaces, understanding the reaction of MNBS systems under a moving nanocar/particle and preventing the undesirable behaviors could be key steps in having a more efficient system By the knowledge of the authors, there is no study based on dynamic analyzing multi-layered elastic nanobeams carrying a moving load or nanoparticle In this paper as shown in Fig 1, dynamical behavior of multi-layered nanobeams under a moving particle is presented while the beam layers are assumed to be coupled by winkler elastic medium and the small-scale effect is modeled by nonlocal Euler-Bernoulli beam theory for each layer Different parameter sensitivities are discussed and results for multiple nanobeam systems (MNBS) are presented Problem formulation Displacement relations and longitudinal strain for multi-layered Euler–Bernoulli beams are given by the following: ui ¼ z @wi x; tị @x vi ẳ wi ẳ wi x; tị eixx ẳ 1ị dui @ wi ẳ z dx @x Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 Dynamic response of multiple nanobeam system Figure Schematic representation of multi-layered nanobeam with a moving nanoparticle on top layer where i presents the number of the layer u, v and w are the displacement components, x is the longitudinal coordinate measured from the left end of the beam, z is the coordinate measured from the midplane of the beam and exx is the normal strain The strain energy U could be written as Z Z L Ui ¼ rixx eixx dAdx ð2Þ A in which A and L are the cross-sectional area and length of the beam and rxx is the axial stress while the strain energy due to the shearing strain is zero By substituting Eq (1) into Eq (2), the strain energy may be expressed as Z Z Z L @ wi L @ wi Ui ¼ rixx z dAdx ẳ Mi dx 3ị @x @x2 2 A where Mi is the bending moment defined as Z Mi ẳ zrixx dA 4ị A Also the kinetic energy Ti is given by Z (Z " 2 2 # ) L @ui @wi Ti ẳ q ỵ dA dx @t @t A ð5Þ where q is the mass density of the beam By excluding the rotary inertia effect, Eq (5) may be expressed as Z Z 2 L @wi Ti ẳ q dAdx 6ị @t A Transverse load on each layer with respect to Fig is defined as follows: > < mp gdðx xp Þ Kn1 ðwn wn1 Þ; i ¼ n Fi ẳ Ki wiỵ1 wi ị Ki1 wi wi1 Þ; i n > : K1 w2 w1 ị; i ẳ ð7Þ in which Ki is the stiffness of Winkler elastic medium between ith and (i + 1)th layer, and mp and xp are the mass and position of nanoparticle moving on the upper layer of MNBS as shown in Fig By this definition external energy could be written as follows: Z L Qi ẳ Fi wi dx 8ị Governing equation of motion is achieved by Hamilton’s principle as Z t dðTi Ui Qi Þdt ¼ 0 Z t Z L @wi @wi d2 dwi ẳ qA ỵ Fdwi dxdt d ỵM 9ị @t @t dx2 0 With integrating by parts and since dW is arbitrary in < x < L, the governing equation of motion is obtained as @ Mi @ wi Fi ¼ qA 2 @x @t ð10Þ To add the small-scale effects in nanoscale beams, Eringen’s nonlocal elasticity [49] is employed to model the nonlocal continuum based of MNBS The classical elasticity does not conflict the atomic theory of lattice dynamics and experimental observation of phonon dispersion by defining the stress at a reference point x in an elastic continuum depends only on the strain at that point while The basic assumption in the nonlocal elasticity theory is that the stress at a point is observed to be a function of not only the strain at that point but also on strains at all other points of a body Eringen’s nonlocal elasticity involves spatial integrals which represent weighted averages of the contributions of strain tensors of all points in the body to the stress tensor at the given point Basic equations for a linear homogenous nonlocal elastic body are given as Z rnl ij sðjx x0 j; aÞrlij ðx0 ÞdVðx0 Þ; ¼ 8x V V rlij ðxÞ Figure Free diagram of the ith layer of MNBS subjected to external forces Fi and Fi1 ¼ Cijkl ekl eij ẳ ui;j ỵ uj;i ị 11ị Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 S.H Hashemi, H.B Khaniki where rlij and elij are the local stress and strain tensors, Cijkl is the fourth-order elasticity tensor, |x x0 | is the distance in Euclidean form and a(|x x|, s) is the nonlocal modulus or attenuation function incorporating into constitutive equations the nonlocal effects at the reference point x produced by local strain at the source x0 a is the material constant which is defined as (e0a/l) depends on the internal (e.g lattice parameter, granular distance, distance between C–C bonds) and external (e.g crack length, wavelength) lengths Due to the difficulty of solving the integral constitutive Eq (13) can be simplified to equation of differential form to fully gain the integral form results for simply supported boundary conditions [50,51] and it is written as ð1 a2 l2 r2 ịr ẳ t 12ị For a one dimensional elastic material, Eq (12) can be simplified as @2 e0 aị2 rxx xị ẳ Eexx xị ð13Þ @x where (e0a) is the scale coefficient which leads to small-scale effect and E is the Young’s modulus of the nanobeam Multiplying Eq (15) by zdA and integrating the result over the area A lead to M ðe0 aị2 d2 M d2 w ẳ EI dx2 dx2 And by substituting Eq (12) into Eq (14), we have @ wi @ wi ỵ e aị qA ỵ F Mnl i i ẳ EI @x2 @t2 ð14Þ ð15Þ Thus, the governing equation of motions for multi-layered nanobeam with moving nanoparticle can be expressed in terms of transverse displacement for nonlocal constitutive relations as @ wn ỵ mgdx xm ị Kn1 wn wn1 Þ @t2 @ wn @2 @ wn ỵ EI ẳ a2 qA ỵ mgdx xm ị Kn1 wn wn1 ị @x @x @t qA @ wi ỵ Ki wiỵ1 wi ị Ki1 wi wi1 ị @t2 @ wi @2 @ wi þ EI ¼ a2 qA þ Ki wiỵ1 wi ị Ki1 wi wi1 ị i n @x @x @t qA @ w1 ỵ K1 w2 w1 Þ @t2 @ w1 @2 @ w1 ỵ EI ẳ a2 qA ỵ K1 ðw2 w1 Þ @x @x @t qA ð16Þ Solution procedure ipx gi xị ẳ sin L Moreover, for the external force made by the moving particle could be represented as F0 x; tị ẳ mp gdðx xp Þ X gi ðxÞWi ðtÞ ð19Þ where d(x xp) is the Dirac delta function which could be expressed in terms of sinusoidal as follows 2X ip ip sin dðx xp ị ẳ xp sin x 20ị L iẳ1 L L where xp denotes the coordinate of moving particle from the left end of the upper layer Substituting Eqs (17)–(20) into Eq (16) leads to set of Equations as follows: @ Wn ỵ2mgL3 sinipXp ịKn1 L4 Wn Wn1 ịỵipị4 EIWn @t2 @ Wn ẳ a2 qAL4 ipị2 2ipị2 mgL3 sinipXp ị @t2 i @ wi ỵKn1 L4 ipị2 Wn Wn1 ị qAL4 ỵKi L4 wiỵ1 wi Þ @t @ w i Ki1 L4 ðwi wi1 ịỵipị4 EI @x a2 @ wi ẳ ipị2 qAL4 ỵKi L4 wiỵ1 wi ị @t L @ W1 ỵK1 L2 W2 W1 Þ Ki1 L4 ðwi wi1 Þ qAL2 @t2 2 2a @ W1 ỵK1 L W2 W1 ị ỵipị EIW1 ẳ ipị qAL @t2 L qAL4 ð21Þ By defining new set of parameters as follows pffiffiffiffi x w e0 a L A qAL2 X¼ ; W¼ ; a¼ ; f ¼ pffiffi ; c2 ¼ ; L L L E I mgL2 KL2 ; jẳ P2 ẳ EI EI 22ị The set of equations of motions could be represented in a matrix form as W1 > > > > > > v1 ỵ j1 j1 0 > > > W2 > > > > > > j1 j1 ỵ v ỵ j2 > > j2 0 > > W3 > > > > > = < : > > j2 j2 ỵ v3 ỵ j3 j3 @ þ6 c2 > @t > : > > > > > > > > > > : > > 0 j j ỵ v ỵ j j > > n2 n2 n1 n1 n1 > > > >W > > > n1 > 0 0 j j þ v > > n1 n1 n ; : Wn 9 w1 > > > > > > > > > > > > > > > > > w2 > >0> > > > > > > > > > > > > > > > > > > > > w3 > >0> > > > > > > > > = < : = < : > ¼ > > > > : : > > > > > > > > > > > > > > > > > > > > > > > > > > > > : > > : > > > > > > > > > > > > > > > : > > wn1 ; >0; : Ci wn In modal form, the total transverse dynamic deflection wi(x, t) is written as wi x; tị ẳ 18ị 23ị where vi and C are defined as 17ị jẳ1 where Wi(t) are the unknown time-dependent generalized coordinates and gi(x) are the eigenmodes of an undamped simply-supported beam which are expressed as vi ¼ ipị4 ỵ ipị2 La 2 mgL3 sinipXp ị Ci ẳ 2 EI 24ị Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 Dynamic response of multiple nanobeam system With assuming that the moving particle starts moving from the left end of the first beam at t = by having a constant velocity through the path, the dimensionless location of the moving nanoparticle would be Xp ¼ V t L ð25Þ where V is the velocity of the moving nanoparticle At t = L/V moving mass reaches the end of the beam By having the same material of elastic modulus, mass density, uniform cross section A and same continuous linear Winkler elastic medium of stiffness per length K the set of equations of motions could be represented as w1 > > > > > > > w2 > vỵj j 0 > > > > > > > j vỵ2j j 0 > > > > w3 > > > > > > > = < j vỵ2j j : @ ỵ6 c > @t > : > > > > > > > > > > > j vỵ2j j > > : > > > > > > wn1 > > > 0 0 j vỵj > > ; : wn 9 w1 > > > > > > > > > > > > > >0> > > > w2 > > > > > > > > > > > > > > > > > > > > w3 > 0> > > > > > > > > > > > < : = = ¼ > > : > > > >0> > > > > > > > > > > > > > > > > : 0> > > > > > > > > > > > > > > > > > > > > w > > > n1 > > : > > > : ; ; wn C ð26Þ For solving Eq (26) in time domain, Laplace transform method is employed so the following set of equations is obtained: 6 6 6 6 c2 S ỵ v ỵ j j j 0 c2 S2 ỵ v ỵ 2j j 0 j c2 S2 þ v þ 2j j j c2 S2 ỵ v ỵ 2j 0 9 LfW1 g > > > > > > > > > > > > > > > > > > > > LfW2 g > > > > > > > > > > > > > > > > > > > > > LfW3 g > > > > > > > > > > > > > < < = = : ¼ > > > > : > > > > > > > > > > > > > > > > > > > > : > > > > > > > > > > > > > > > > > > > > LfW g > > > > > n1 > > > : ; > : 2P2 ipV2p > ; LfWn g ipVp ị ỵS2 0 j 3.1 Double layered MNBS with moving nanoparticle For two layered MNBS, Eq (27) could be rewritten as ) " # ( LfW1 g c2 S2 ỵ v ỵ j j ẳ ipV 2P2 ipV ị2pỵS2 LfW2 g j c2 S2 ỵ v ỵ j p ð28Þ By inversing the coefficient matrix Eq (28) may be written as ( LfW1 g 7 7 7 7 c2 S þ v þ j ¼ By inversing the coefficient matrix and evaluating the inverse Laplace transform of Eq (27), the results of transverse displacement with respect to time for each layer could be achieved Further solution depends on the number of layers used for MNBS where the process is presented for double layered and three layered MNBS with moving nanoparticle and for MNBS with more layers the procedure is the same 38 < c2 S2 ỵ v þ j : 2P j ipVp ðipVp Þ2 þS2 = ð29Þ ; By evaluating the inverse Laplace transforms and doing some calculations, the values of W1(t) and W2(t) are obtained as ( W1 tị ẳ P2 2j sinnpVtị V2 c2 n2 p2 vịV2 c2 n2 p2 ỵ 2j þ vÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 2jv ) v c sinh t npV c sinh c t npV c ỵ p 2 2 ỵ p 2 2 2j vðV c n p 2j vÞ vðV c n p 2j vÞ ð30Þ pffiffiffiffiffiffiffiffiffiffi 2jv csinh t npV c W1 tị ẳ P2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2j vðV c n p ỵ 2j ỵ vị p v csinh c t npV sinnpVtịV2 c2 n2 p2 ỵ j ỵ vị = ỵ 31ị ỵ p vV2 c2 n2 p2 þ 2j þ vÞ ðV2 c2 n2 p2 vÞðV2 c2 n2 p2 ỵ 2j ỵ vị; ( Substituting Eqs (30) and (31) into Eq (17) the result for transverse displacement of double layered nanobeams with moving nanoparticle with respect to time is achieved as w1 X;tị ẳ X P2 iẳ1 ( 2jsinnpVtị V2 c2 n2 p2 vịV2 c2 n2 p2 ỵ 2j ỵ vị p p 2jv ) v csinh t npV csinh c t npV c ỵ p 2 2 ỵ p 2 2 sinðipXÞ ð32Þ vðV c n p 2j vÞ 2j vðV c n p 2j vÞ pffiffiffiffiffiffiffiffiffiffi 2jv c sinh t npV c w2 X; tị ẳ P2 p 2j vV2 c2 n2 p2 ỵ 2j ỵ vị iẳ1 p v c sinh c t npV ỵ p vV2 c2 n2 p2 ỵ 2j ỵ vị 2sinnpVtịV2 c2 n2 p2 ỵ j ỵ vị ỵ 2 2 sinipXị 33ị V c n p vịV2 c2 n2 p2 ỵ 2j ỵ vị X 27ị c4 S4 ỵ 2jc2 S2 ỵ 2c2 vS2 ỵ 2jv ỵ v2 LfW2 g c2 S2 ỵ v ỵ j 4 j 0 j ) ( 3.2 Three layered MNBS with moving nanoparticle Following the same procedure, for three layered MNBS, Eq (27) is expressed as Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 S.H Hashemi, H.B Khaniki 38 c2 S2 ỵ v1 ỵ j j > < LfW1 g > = LfW2 g j j c2 S2 ỵ v2 ỵ 2j > > : ; LfW3 g j c2 S2 ỵ v3 ỵ j > > < = ẳ 34ị > : 2P2 ipVp > ; 2 ipV ị ỵS p ni ẳ Roots of equation c4 n6 ỵ jc4 ỵ vc4 ỵ 3jc2 ỵ 2vc2 ịn4 ỵ 2j2 c2 ỵ 5jvc2 ỵ 2v2 c2 þ j2 þ 3jv þ v2 Þn2 þ 3j2 v ỵ 4jv2 ỵ v3 ị 40ị D2i ẳ V6 c4 i6 p6 jV4 c4 i4 p4 vV4 c4 i4 p4 3jV4 c2 i4 p4 2vV4 c2 i4 p4 ỵ 2j2 V2 c2 i2 p2 ỵ 5jvV2 c2 i2 p2 ỵ 2v2 V2 c2 i2 p2 ỵ j2 V2 i2 p2 ỵ 3jvV2 i2 p2 3j2 v Inversing the coefficient matrix in Eq (34) leads to where D1 is 4jv2 v3 ð41Þ 9 38 2 2 2 2 jS2 ỵ j ỵ vị j2 > > > < = < LfW1 g > = c S ỵ jc S ỵ vc S ỵ 2jS ỵ vS ỵ j ỵ 3vj ỵ v LfW2 g ẳ jS2 ỵ j þ vÞ ðc2 S2 þ v þ jÞðS2 þ j þ vÞ jðc2 S2 þ v þ jÞ > > > : 2P2 ipVp > ; : ; D1 LfW3 g jc2 S2 ỵ v ỵ jị c4 S4 ỵ 3jc2 S2 ỵ 2vc2 S2 ỵ j2 ỵ 3vj ỵ v2 j2 ipVp ị2 ỵS2 35ị D1 ẳ c4 S6 þ jc4 S4 þ vc4 S4 þ 3jc2 S4 þ 2vc2 S4 ỵ 2j2 c2 S2 ỵ 5vjc S ỵ 2v c S ỵ j S ỵ 3vjS ỵ v S 2 2 2 2 2 ỵ 3vj2 ỵ 4v2 j ỵ v3 36ị By evaluating the inverse Laplace transforms, the values of W1(t), W2(t) and W3(t) are obtained as " !# X P2 j2 ni t W1 tị ẳ sinipVtị ipV e g1i =g2i ð37Þ D2i n i In the same way by Substituting Eqs (36) and (37) into Eq (17) the result for transverse displacement of three layered nanobeams with moving nanoparticle with respect to time is calculated as w1 ðX; tị ẳ ( " !#) X X P2 j2 2sinipVtị ipV eni t g1i =g2i sinipXị D2i iẳ1 n 42ị i w2 X; tị ẳ ( " !#) X X P2 j sinðipXÞ 2ðV2 c2 i2 p2 þ j þ vÞsinðipVtÞ ipV eni t g3i =g4i D2i iẳ1 n i " X Pj W2 tị ẳ 2V2 c2 i2 p2 ỵ j ỵ vị sinipVtị ipV eni t g3i =g4i D2i n !# ð43Þ ð38Þ i 91 2ðV4 c4 i4 p4 3jV2 c2 i2 p2 2vV2 c2 i2 p2 ỵ j2 ỵ 3jv ỵ v2 ị sinipVtị > > < =C " # BP C sinðipXÞ X nt w3 ðX; tị ẳ i @D2i > A e > g5i ỵ g6i ỵ g7i ị=g8i ipV : ; iẳ1 ni B X ð44Þ ni P2 2ðV4 c4 i4 p4 3jV2 c2 i2 p2 2vV2 c2 i2 p2 ỵj2 D2i " #) X eni t g ỵg6i ỵg7i ị=g8i þ3jvþv ÞsinðipVtÞipV ni 5i n 3.3 Higher layered MNBS with moving nanoparticle W3 tị ẳ 39ị i where g1i to g8i are defined in Appendix A and ni and D2i are defined as For more than three layered MNBS with moving nanoparticle, the same calculation procedure is done which causes long complex equations due to the inverse of matrix coefficient For having a more accurate results and forbidding the errors, numerical solution is used to obtain the deflection of each layer in higher number of layers by solving Eq (45) numerically 991 2 3> w1 > > > > > > > > > j : : : c S þvþj > > > > > > B C > > > > > > > > > > > > B C > > > > > > w2 > 2 > > > > > S ỵ v ỵ 2j j : : j c B C > > > > > > > > > > > > B > > > > > > C > > > > > w3 > 2 > > > > > > B C S þ v þ 2j j : 0 j c > > > > > > > > > > 7> C = < = X = < : > < B : B 1 C BL C sinðipXÞ ¼ inv B 7> C : > > > i¼1 B > > : > > > > > > 7> C > > > > > > > > > > > B C > > > > > > : : > > > > > > B C : > > > > > > > > > > > > B C > > > > > > > > > > > > B C : : j c S ỵ v ỵ 2j j > > > > > > wn1 > > > > > > @ A > > > > > > > ; : > 2 ; : ;> : 2P2 ipV2p > S ỵ v ỵ j : : : j c wn ðipVp ị ỵS2 45ị Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 Dynamic response of multiple nanobeam system Table Nondimensional maximum deflection of double layered carbon nanotubes Second layer j 10 100 1000 First layer S ß imsßek [40] e0a = (nm) S ß imsßek [40] e0a = (nm) 1.66832 1.5648 1.01093 0.84742 1.68662 1.57122 1.01187 0.84823 0.01348 0.13173 0.77004 0.85255 0.01420 0.13410 0.77099 0.85259 0.8 0.6 0.6 0.5 0.4 0.4 0.2 0.3 0.2 0.5 X 0.35 W 2(X,T) / Wst W 1(X,T) / Wst 0.7 0.4 0.3 0.3 0.25 0.2 0.2 0.1 0.15 0.1 T 0.5 0.5 0 0.1 X (a) 0.05 0.5 0 T (b) Figure Dynamic respond of double layered nanobeam with nondimensional nonlocal parameter = 0.1: (a) second layer nondimensional transverse displacement and (b) first layer nondimensional transverse displacement 0.7 1.2 1 0.8 0.5 0.6 0.4 1 0.5 X 0.2 T 0.8 0.6 0.6 0.5 0.4 0.4 0.2 0.3 0.2 0.5 0.5 0 W 2(X,T) / Wst W 1(X,T) / Wst 1.5 (a) X 0.1 0.5 0 T (b) Figure Dynamic respond of double layered nanobeam with nondimensional nonlocal parameter = 0.3: (a) second layer nondimensional transverse displacement and (b) first layer nondimensional transverse displacement Results and discussions For different number of layers, stiffness and nonlocal parameter the dynamical behavior of MNBS under a moving nanoparticle is illustrated Wang and Wang [52] has shown that the value of e0a should be smaller than 2.0 nm for carbon nanotubes and also the exact value of nonlocal parameter is not exactly known The external characteristic length varies so the nonlocal or scale coefficient parameter is assumed to be a ¼ e0l a ¼ to The geometrical and mechanical properties of the multi-layered nanoribbons are considered as [47]: E = 1.0 TPa, q = 2.25 g/cm3, t = 0.34 nm, L = 10t In order to achieve a nondimensional dynamical deflection parameter, m gL3 p static deflection [46] is assumed as wst ¼ 48EI The presented analysis, describes the dynamical behavior of simply supported Euler–Bernoulli Multi-layered nanobeam Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 S.H Hashemi, H.B Khaniki 1.5 2.2 2 1.8 1.6 1.5 1.4 1.2 0.5 0.8 W 2(X,T) / Wst W 1(X,T) / Wst 2.5 1.5 0.5 0.5 0.6 0.5 0 0.4 0.2 0.5 X 1 0.5 0.5 X T 0 (a) T (b) Figure Dynamic respond of double layered nanobeam with nondimensional nonlocal parameter = 0.5: (a) second layer nondimensional transverse displacement and (b) first layer nondimensional transverse displacement 0.15 0.1 0.1 0.08 0.05 0.06 0.04 0.5 X 0.5 0 0.25 0.2 0.15 0.1 0.1 X (a) 0.4 0.4 0.3 0.2 0.2 0.5 0.5 0 0.5 0.6 0.05 0.5 T 0.2 0.3 0.02 0.6 0.8 0.4 W 1(X,T)/Wst 0.12 W 2(X,T)/Wst W 3(X,T)/Wst 0.14 0.2 T X (b) 0.5 0 0.1 T (c) Figure Dynamic respond of three layered nanobeam with nondimensional nonlocal parameter = 0.1: (a) first layer nondimensional transverse displacement, (b) second layer nondimensional transverse displacement, and (c) third layer nondimensional transverse displacement 0.35 0.5 0.8 0.6 0.5 0.4 1 0.5 X 0.5 0 (a) T 0.2 0.8 0.45 0.4 0.6 0.35 0.4 0.3 0.25 0.2 0.2 1 0.5 X 0.5 0 (b) T 0.15 0.1 0.3 0.4 W 3(X,T)/Wst W 1(X,T)/Wst 1.5 W 2(X,T)/Wst 0.25 0.3 0.2 0.2 0.15 0.1 0.1 1 0.05 0.5 X 0.5 0 T 0.05 (c) Figure Dynamic respond of three layered nanobeam with nondimensional nonlocal parameter = 0.3: (a) third layer nondimensional transverse displacement, (b) second layer nondimensional transverse displacement, and (c) first layer nondimensional transverse displacement carrying a moving nanoparticle The Eigen function technique and the Laplace transform method are employed to solve the governing equations of the nanobeams In order to verify the validation of present solution procedure the number of layers is assumed to be two (double layered) and the analysis is done for carbon nanotubes to compare the present solution with the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle presented by S ß imsßek [40] In Table the maximum non-dimensional deflection of the first and second layer of double carbon nanotube for various values of stiffness and nonlocal parameter is presented and compared to those achieved by S ß imsßek [40] which shows a great equality in the results Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 Dynamic response of multiple nanobeam system 1.2 1 0.8 0.5 0.6 X 0 0.6 0.4 0 (a) 0.8 0.6 0.6 0.5 0.4 0.4 0.2 0.3 0.2 0.2 0.1 0.5 X 0.7 0.3 0.5 T 0.5 0.5 0.2 0.5 0.7 1 0.4 0.5 0.8 st 1.5 0.9 1.5 1.4 W (X,T)/W 1.6 0.8 W 2(X,T)/Wst W 1(X,T)/Wst 1.8 2.5 0.5 T X (b) 0.5 0 0.1 T (c) Figure Dynamic respond of three layered nanobeam with nondimensional nonlocal parameter = 0.5: (a) third layer nondimensional transverse displacement, (b) second layer nondimensional transverse displacement, and (c) first layer nondimensional transverse displacement 2.5 k=1 k=5 k = 10 k = 50 k = 100 k = 500 k = 1000 1.5 0.8 W max /W st W max /W st k=1 k=5 k = 10 k = 50 k = 100 k = 500 k = 1000 0.9 0.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 T T (a) (b) Figure Nondimensional maximum deflection of double layered MNBS with respect to nondimensional time parameter from the time which nanoparticle enters the surface of the first layer till it leaves for different stiffness parameters: (a) second layer and (b) first layer 1.4 1.2 0.8 0.6 0.4 0.5 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.2 0 k=1 k=5 k = 10 k = 50 k = 100 k = 500 k = 1000 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0 0.7 k=1 k=5 k = 10 k = 50 k = 100 k = 500 k = 1000 0.6 W max /W st 1.6 W max /W st 0.7 k=1 k=5 k = 10 k = 50 k = 100 k = 500 k = 1000 W max /W st 1.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 T T T (a) (b) (c) Figure 10 Nondimensional maximum deflection of three layered MNBS with respect to nondimensional time parameter from the time which nanoparticle enters the surface of the first layer till it leaves for different stiffness parameters: (a) third layer, (b) second layer, and (c) first layer Dynamic response of double layered MNBS with nondimensional nonlocal parameter = 0.1, 0.3, 0.5 is shown in Figs 3–5 by having a nanoparticle moving on the upper layer Results are shown with respect to the nondimensional time parameter which in time T = nanoparticle enters the system and at T = leaves it Dynamical deflection is presented for each layer separately The same analysis has been done for three layered MNBS and the results of each layer for different nonlocal parameter are presented in Figs 6–8 It can be seen that for both cases, increasing the nonlocal term leaded to a Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 10 S.H Hashemi, H.B Khaniki Double layered First layer, α = 0.5 Second layer, α = 0.1 2 Second layer, α = 0.3 Second layer, α = 0.5 1.5 W max /W st W max /W st Three layered 2.5 First layer, α = 0.1 First layer, α = 0.3 Four layered First layer, α = 0.1 First layer, α = 0.3 1.8 First layer, α = 0.1 First layer, α = 0.3 First layer, α = 0.5 Second layer, α = 0.1 1.6 First layer, α = 0.5 Second layer, α = 0.1 Second layer, α = 0.3 Second layer, α = 0.5 Third layer α = 0.1 1.5 W max /W st 2.5 Third layer, α = 0.3 Third layer, α = 0.5 Second layer, α = 0.3 Second layer, α = 0.5 Third layer α = 0.1 1.4 1.2 Third layer, α = 0.3 Third layer, α = 0.5 Fourth layer α = 0.1 0.8 Fourth layer, α = 0.3 Fourth layer, α = 0.5 0.6 0.5 0.5 0 0.4 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 X X X (a) (b) (c) Figure 11 Nondimensional maximum deflection of MNBS for different amounts of nondimensional nonlocal parameter: (a) double layered, (b) three layered, and (c) four layered 7 W max /W st W max /W st k = 50 k = 100 k = 500 k = 1000 k = 5000 0.1 k = 50 k = 100 k = 500 k = 1000 k = 5000 0.2 0.3 0.4 0.5 α 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 α 0.6 0.7 0.8 0.9 (b) (a) Figure 12 Nondimensional maximum deflection of double layered MNBS for different stiffness parameters with respect to nonlocal parameter: (a) first layer and (b) second layer 4 3.5 W max /W st W max /W st 3.5 4.5 k = 50 k = 100 k = 500 k = 1000 k = 5000 2.5 1.5 1.5 0.5 0.5 α 2.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 k = 50 k = 100 k = 500 k = 1000 k = 5000 W max /W st 4.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (a) k = 50 k = 100 k = 500 k = 1000 k = 5000 α (b) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 α (c) Figure 13 Nondimensional maximum deflection of three layered MNBS for different stiffness parameters with respect to nonlocal parameter: (a) first layer, (b) second layer, and (c) third layer higher deformation in MNBS system In order to show the effects of the Winkler elastic medium between each layer on the dynamical behavior and the maximum deflection of each layer, stiffness parameter varies from to 1000 As shown in Figs and 10, by having j = each layer almost acts independently from others By increasing the stiffness parameter to higher orders, deflection is more shared between layers also by having more layers and deflection will decrease in the layer carrying the nanoparticle which is caused by the incorporation of other layers in the MNBS system Also in Fig 11 by changing the nonlocal parameter, maximum deflection parameter is presented through the beam Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 Dynamic response of multiple nanobeam system 11 3 2.5 W max /W st W max /W st 2.5 k = 50 k = 100 k = 500 k = 1000 k = 5000 1.5 1.5 0.5 0.5 0.1 k = 50 k = 100 k = 500 k = 1000 k = 5000 0.2 0.3 0.4 0.5 α 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 (a) α 0.6 0.7 0.8 0.9 0.7 0.8 0.9 (b) 3.5 k = 50 k = 100 k = 500 k = 1000 k = 5000 4.5 3.5 W max /W st 2.5 W max /W st 0.5 1.5 k = 50 k = 100 k = 500 k = 1000 k = 5000 2.5 1.5 1 0.5 0.1 0.5 0.2 0.3 0.4 0.5 α 0.6 0.7 0.8 0.9 0.1 0.2 (c) 0.3 0.4 0.5 α 0.6 (d) Figure 14 Nondimensional maximum deflection of four layered MNBS for different stiffness parameters with respect to nonlocal parameter: (a) first layer, (b) second layer, (c) third layer, and (d) fourth layer It is shown that increasing the nonlocal parameter in all multi-layered nanobeams will increase the deflection beside the number of beams Variation of nonlocal parameter has the most effect on the first layer which decreases by going downward to other layers Such kind of behavior is seen due to the reduction in natural frequencies of the nanobeams which makes them more flexible and the top layer that shares the most roles in deformation due to the moving load/nanoparticle, shows the most sensibility to changes in nonlocal parameter Varying stiffness parameter for multi-layered nanobeams with different nonlocal parameters and different number of layers leads to unique deflections In Fig 12 double layered nanobeams deflection is presented for different stiffness and nonlocal parameter which show that the maximum deflection will increase by increasing the nonlocal parameter and this behavior stays the same for every stiffness parameter For higher number of layers, the same behavior is obtained which is presented in Figs 13–15 for three, four and five layered nanobeams It is shown that the sensibility of the maximum deflection on the stiffness parameter will decrease by increasing the stiffness parameter For all nonlocal parameters and all multi layered nanobeams increasing the stiffness parameter, will cause a lower deflection in the first layer and higher deflection in the last one Increasing the number of layers and the stiffness parameter will cause the top layers to act like the first layer against changing the nonlocal parameter For example in four layered nanobeams, the second layers acts almost independent of the stiffness parameter while by increasing the number of layers to five, the second layer, in higher number of nonlocal parameter acts like the first layer which means increasing the stiffness parameter will decrease the maximum deflection Also having higher number of layers helps to share the force caused by the moving particle which leads to having a smaller magnitude of deflection in each layer for all nonlocal and stiffness parameters Conclusion In this work, dynamic response of multi-layered nanobeam carrying a moving nanoparticle is studied Eringen’s nonlocal elasticity theory is used to model nonlocal the continuum based multiple nanobeam The moving nanoparticle is modeled by a moving point at the top surface of upper layer of MNBS In order to model the coupling between layers of MNBS, Winkler elastic medium model is used Using Hamilton’s principle, general equations of motions are achieved for all multi layered nanobeam systems To solve this equations, Eigen function technique and the Laplace transform method are employed which led to the solution of dynamical transverse deflection of each layer of nanobeams during the movement of Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 12 S.H Hashemi, H.B Khaniki 2.5 W max /W st W max /W st k = 50 k = 100 k = 500 k = 1000 k = 5000 1.5 2.5 k = 50 k = 100 k = 500 k = 1000 k = 5000 W max /W st 2.5 1.5 0.5 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 α α (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 α (b) 2.5 (c) k = 50 k = 100 k = 500 k = 1000 k = 5000 4.5 W max /W st W max /W st 1.5 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 k = 50 k = 100 k = 500 k = 1000 k = 5000 1.5 3.5 k = 50 k = 100 k = 500 k = 1000 k = 5000 2.5 1.5 0.5 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 α 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 α (d) (e) Figure 15 Nondimensional maximum deflection of four layered MNBS for different stiffness parameters with respect to nonlocal parameter: (a) first layer, (b) second layer, (c) third layer, (d) fourth layer, and (e) fifth layer nanoparticle on the top side of the upper layer Results are achieved by varying parameters such as nonlocal parameter, Winkler elastic medium stiffness and number of layers It was founded that: Small-scale effect plays a significant role in the magnitude of the dynamical deflection In all the MNBSs with a moving nanoparticle, increasing the nonlocal parameter causes a bigger deflection in each layer Variation in the nonlocal parameter has the most efficacy on the upper layer of MNBS in which the nanoparticle is moving through This effect decreases by going through the other layers and reaches to the least effect in the last layer In all the MNBSs with a moving nanoparticle, increasing the Winkler elastic medium stiffness leads to a smaller deflection in the upper layer of MNBs In higher number of MNBS, dynamical deflection in (n 1) th layer of MNBS acts like the nth layer to the changes in Winkler elastic medium stiffness Having a great number of layers will make the upper layers to behave in the same way Dynamical deflection of layers shows more sensitivity to the changes in Winkler elastic medium stiffness when the stiffness has a lower magnitude Increasing the stiffness parameter to a great number will cause a merging in the answer of dynamical deflection of each layer and the deflection will lose its sensitivity to the changes in stiffness For all values of stiffness and nonlocal parameters increasing the number of layers will lead to a smaller dynamical deflection in all layers while the nanoparticle is passing through Appendix A Parameters g1i to g8i are defined as follows: g1i ¼ c4 V4 i4 p4 ỵ n4i c4 ỵ j2 2c2 ỵ 1ị ỵ v2 2c2 ỵ 1ị c2 V2 i2 p2 n2i c2 ỵ jc2 ỵ vc2 ỵ 3j ỵ 2vị ỵ n2i c2 jc2 ỵ vc2 ỵ 3j ỵ 2vị ỵ jv5c2 ỵ 3ị A1ị g2i ẳ 3n4i c4 ỵ 2jc4 n2i ỵ 2vc4 n2i ỵ 6jc2 n2i ỵ 4vc2 n2i þ 2j2 c2 þ 5jvc2 þ 2v2 c2 þ j2 ỵ 3jv ỵ v2 ịni A2ị g3i ẳ c2 V2 i2 p2 n2i c2 ỵ j ỵ vịc2 V2 i2 p2 n2i c2 jc2 vc2 3j 2vị ỵ vc2 ỵ 1ị4j2 ỵ v2 ị ỵ c2 n2i j ỵ vịc2 n2i ỵ jc2 ỵ vc2 þ 3j þ 2vÞ þ v2 jð3c2 þ 4Þ þ j2 2c2 ỵ 1ị A3ị g4i ẳ 3c4 n4i ỵ 2jc4 n2i ỵ 2vc4 n2i ỵ 6jc2 n2i ỵ 4vc2 n2i ỵ 2j2 c2 ỵ 5jvc2 ỵ 2v2 c2 ỵ j2 ỵ 3jv ỵ v2 ịni g5i ẳ A4ị 44 j c V i p ỵ 3jvc4 V4 i4 p4 ỵ v2 c4 V4 i4 p4 ni j3 c4 V2 i2 p2 j2 vc4 V2 i2 p2 3j3 c2 V2 i2 p2 11j2 vc2 V2 i2 p2 9jv2 c2 V2 i2 p2 2v3 c2 V2 i2 p2 ỵ 2j4 c2 þ 2j3 vc2 þ j2 v2 c2 þ j4 þ 6j3 v ỵ 11j2 v2 ỵ 62 jv3 ỵ v4 Þ ðA5Þ Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 Dynamic response of multiple nanobeam system 13 g6i ¼ c4 n3i ðV4 c4 i4 p4 3jV2 c2 i2 p2 2vV2 c2 i2 p2 ỵ j2 ỵ 3jv ỵ v2 ị A6ị g7i ẳ c2 ni ẵV4 c4 i4 p4 3j þ 2vÞ V2 c2 i2 p2 ðj2 c2 þ 9j2 ỵ 12jv ỵ 4v2 ị ỵ j3 c2 ỵ 3ị ỵ j2 vc2 ỵ 11ị ỵ 9jv2 ỵ 2v3 A7ị g8i ẳ 3c4 n4i ỵ 2jc4 n2i ỵ 2vc4 n2i ỵ 6jc2 n2i ỵ 4vc2 n2i ỵ 2j2 c2 ỵ 5jvc2 ỵ 2v2 c2 ỵ j2 ỵ 3jv þ v2 ðA8Þ References [1] A Bouchafa, M.B Bouiadjra, M.S.A Houari, A Tounsi, Thermal stresses and deflections of functionally graded sandwich plates using a new refined hyperbolic shear deformation theory, Steel Compos Struct 18 (6) (2015) 1493– 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http://dx.doi.org/10.1016/j.aej.2016.12.015 ... Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2016.12.015 Dynamic response of multiple nanobeam system. .. Euler–Bernoulli Multi-layered nanobeam Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria Eng J (2017),... cases, increasing the nonlocal term leaded to a Please cite this article in press as: S.H Hashemi, H.B Khaniki, Dynamic response of multiple nanobeam system under a moving nanoparticle, Alexandria