Mechanics of Materials 142 (2020) 103279 Contents lists available at ScienceDirect Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat Research paper Study on size-dependent vibration and stability of DWCNTs subjected to moving nanoparticles and embedded on two-parameter foundations Mostafa Pirmoradiana, Ehsan Torkanb, Davood Toghraiea, a b T ⁎ Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr Branch, Khomeinishahr 84175-119, Iran Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran A R T I C LE I N FO A B S T R A C T Keywords: Double-walled carbon nanotubes Dynamic stability Incremental harmonic balance method Nanoparticle delivery Van der Waals effect Eringen's nonlocal elasticity theory Parametric resonance is an important phenomenon that may be evinced in applying carbon nanotubes for the delivery of nanoparticles. This paper aims to investigate dynamics instability of double-walled carbon nanotubes (DWCNTs) surrounded by elastic medium and excited by a sequence of moving nanoparticles. The DWCNT is modeled as two Euler-Bernoulli beams interacting between them through van der Waals (vdW) forces. Based on Eringen's nonlocal elastic theory to consider the small-scale effects, the governing equations are derived by using Hamilton's principle. All inertial terms of the moving nanoparticles are taken into account. In addition, the van der Waals force between the constitutive atoms of the moving nanoparticle and those of the nanotube is considered. By utilization of the Galerkin method, the partial differential equations (PDEs) of motion are reduced to couple ordinary differential equations with time-varying coefficients describing a parametrically excited nanosystem. Then, an incremental harmonic balance (IHB) method is implemented to calculate the instability regions of the DWCNT. The results show that considering the vdW effects, increasing the amplitude of the static axial tensile force, reducing the amplitude of axial oscillating force, and increasing the stiffness of the elastic medium improve stability of the system. A comparison between the results with those reported in the literature is performed to verify the precision of the presented analyses. 1. Introduction Nearly after three decades of carbon nanotube (CNT) discovery by Iijima (1991), researchers have not stopped investigating for identification of their new properties to use them in nanostructural applications. This is due to the unique properties of CNTs, such as especial physical, mechanical, chemical, thermal and electronic characteristics. Among too many proposed industrial applications for CNTs, applying them as Hydrogen storages, fluid conveyance, and nanoparticle delivery systems are attended thoroughly notable because of their great mechanical properties besides their unique molecular structures, that is, cylindrical shape with inner hollow space (Ajayan and Zhou, 2001). The issue of drug delivery can be understood by considering the interior hollow space of CNTs as the container, and the fullerenes, e.g. C60, C70, C80, and C84 as the nanoparticles. Accordingly, numerous researchers have studied different aspects of applying CNTs as drugs delivery systems. For example, Rezapour and Araghi (2019b) analyzed dynamics of a viscoelastic single-walled carbon nanotube (SWCNT) in nanoparticle delivery and showed that the interaction and friction force effects between nanoparticle and SWCNTs reveals significant influence ⁎ on dynamic response of the system. In another study, Rezapour and Araghi (2019a) concerned dynamic behavior of CNTs delivering a nanoparticle with constant velocity. In his study, Kiani (2014) employed the nonlocal Rayleigh beam theory to study nonlinear vibrations of SWCNTs as nanoparticle delivery systems. The results showed that a nonlinear analysis is necessary especially for large amounts of the mass and velocity of the moving nanoparticle. Another study by Lee and Chang (2010) studied dynamic behavior of SWCNTs for nanoparticle delivery. Their numerical results show that increasing the non-local parameter decreases the dynamic displacement of SWCNT while increasing the velocity of nanoparticle increases the maximum displacement. When applications like drug delivery systems or systems conveying fluids are considered, investigating vibrational response of CNTs as the containers is unavoidable. It is because of the nanostructure-nanoparticle interaction and the small-scale effect. Therefore, dynamic analysis and the study of induced dynamic instability of a CNT due to passage of nanoparticles or fluids through it become important and are of great academic as well as practical concern. Molecular dynamics (MD) simulation, hybrid atomistic-continuum mechanics, experimental Corresponding author. E-mail address: Toghraee@iaukhsh.ac.ir (D. Toghraie). https://doi.org/10.1016/j.mechmat.2019.103279 Received 11 June 2019; Received in revised form 8 December 2019; Accepted 10 December 2019 Available online 11 December 2019 0167-6636/ © 2019 Elsevier Ltd. All rights reserved. Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. These forces influence vibrational characteristics of CNTs according to the nonlocal interaction of carbon atoms of different layers on each other. The effect of vdW interaction on the vibration characteristics of MWCNTs was studied by modeling it as a radius-dependent function (He et al., 2006). The natural frequencies were calculated for MWCNTs with various number of tubes and radii. It was reported that the vdW interaction plays a significant role on the vibration of MWCNTs with small radii. Wang et al. (2012) investigates rigorous vdW interaction effect on vibration characteristics of MWCNTs under a transverse magnetic field. Their results showed that the rigorous vdW force considerably influences the frequency of MWCNTs. Also, Budarapu et al. (2014) estimated the natural frequencies of MWCNTs embedded in an elastic medium by modeling the interaction between adjacent nanotubes through the vdW forces. Nonlinear vibration of DWCNTs embedded in an elastic medium was investigated by Ansari and Hemmatnezhad (2012). They showed that the governing equations of layers were coupled due to the vdW interlayer forces. Tylikowski (2008) investigated the effect of vdW interaction force on dynamic stability of CNTs under a time-dependent axial load. Fu et al. (2009) investigated the nonlinear dynamic instability of DWNT taking into account the effects of vdW forces. Their results show that when the vdW forces are sufficiently strong, the DWNT can be assumed as a single column. Ke and Wang (2011) studied vibration and stability of fluid-conveying DWNTs and obtained the resonance frequencies of the system. Their results show that the critical flow velocity of the fluidconveying DWNTs increases with an increase in the length scale parameter. Lei et al. (2012) studied the vibrational frequencies of DWCNTs, as latent materials for drug carriers, using the nonlocal Timoshenko beam model. Their results show that the vibrational frequency is significantly influenced by the aspect ratio, vibration mode and the nonlocal parameter. Investigating dynamics of CNTs under periodic excitations caused by time-dependent axial loads (Tylikowski, 2008), magnetic and electrical fields (Wang et al., 2012), fluid flows (Ke and Wang, 2011) and repetitive delivery of drugs (Lee and Chang, 2010) is of great interest. In these cases, it is expected that inter-layer radial displacements of MWNTs would come to play an important role. A review on the open literature shows that no comprehensive investigation has been done on the instability of MWCNTs excited by a series of moving nanoparticles until now. Addressing the necessity to bridge to this technical gap, the present study is dedicated to examine dynamic instability of DWCNTs loaded by successive nanoparticles through drug delivery process. In addition, the van der Waals force between the constitutive atoms of the moving nanoparticle and those of the nanotube, which has been ignored in most of previous studies, is considered. To this end, two consisting layers of a DWCNT are simulated using Euler–Bernoulli beams based on the nonlocal continuum theory. An elastic layer introducing the vdW interaction force between two adjacent tubes connects the beams. In addition, using a confined spring connecting the nanoparticle to the innermost CNT the vdW effect is taken into account. After applying the Hamilton's principle to find the nonlocal partial differential equations of the motion, the Galerkin procedure is used to discretize the unknown fields in the spatial domain. Then, the incremental harmonic balance method is applied to explore stability characteristics of the system and the effects of nonlocal parameter, the elastic medium stiffness and the vdW interaction forces on the system stability are investigated, comprehensively. research and continuum mechanics (Farokhi and Ghayesh, 2017, 2018; Ghayesh and Farajpour, 2018; Kazemirad et al., 2013; Farokhi et al., 2013; Ghayesh et al., 2013c; Ghayesh, 2018b,c ; Ghayesh and Farokhi, 2015) are four common methods which are used to study mechanical behavior of CNTs. The first two methods are especially complex and time-consuming and just may be used for systems with small number of atoms. In addition, performing exact and reliable experiments at nanoscale is difficult and expensive. However, because of dependency of nanostructures mechanical behavior on the length scale, application of classical continuum mechanics may lead to erroneous results. Consequently, in recent years significant investigation has been done to explore dynamic behavior of CNTs by employing several nonclassical continuum theories including the nonlocal elasticity (Eringen, 1983), the coupled stress (Yang et al., 2002), the surface stress (Gurtin and Murdoch, 1978) and the strain gradient (Lam et al., 2003) theories, where the scale effect is considered in analyses. Some investigations done to find vibrational characteristics of CNTs using the non-classical continuum theories are as the following.the study by Lü et al. (2015) investigated the transverse vibration of simply supported DWCNTs both conveying moving nanoparticles. Effect of some system parameters was analyzed on the tubes dynamics. Their results show that the maximum transverse deflections of both coupled tubes can be reduced because of the time lag. In another study, Hashemi and Khaniki (2018) examined nonlocal continuum model of simply supported Euler–Bernoulli nanobeams under a moving nanoparticle using Eringen's nonlocal theory. Beam layers were coupled by Winkler elastic medium. Their results show that small-scale parameter has an important role on dynamic response of nanobeams under moving nanoparticles. The work of Kiani and Roshan (2019) investigated transverse vibrations of doubly parallel nanotubes acted by doubly lagged-moving nanoparticles applying the nonlocal Rayleigh and higher-order beam models. They considered the nonlocal inertial force as well as the lag of moving nanoparticles. The effect of nonlocality, shear deformation, lag effect, and kinematic properties of the moving nanoparticles on the dynamic deflections of the tubes was studied. Furthermore, Karličić et al. (2017) using the nonlocal continuum theory studied the nonlinear vibrations of SWCNTs influenced by a time-varying axial load and a longitudinal magnetic field. Using the method of multiple scales the amplitude-frequency relationship was derived. They approximated an analytical expression for nonlinear frequency. They presented instability regions for the linear vibration of the system. It was shown that the magnetic field, the nonlocal parameter, and stiffness coefficient of the viscoelastic medium have important effects on the vibration and instability behavior of the nanobeam. In their study, Pourseifi et al. (2015) evaluated active vibration control of nanotubes under action of a moving nanoscale particle. The effects of the moving nanoparticle velocity, small scale effect parameter and slenderness ratio of nanotube on the dynamic deflection were investigated. They showed the efficiency of the control algorithm in suppressing the vibrations of the nanostructure. Hołubowski et al. (2019) based on non-local elasticity theory studied dynamics of SWCNTs under distributed random loads. They examined the influence of load standard deviation and nonlocal parameters on dynamic response and showed that random load perturbations should not be neglected in dynamic analyses. Based on different beam theories and using the nonlocal continuum theory of Kiani and Wang (2012) studied the interaction of a moving nanoparticle with a single-walled carbon nanotube. Examining forced vibration of a simply supported SWCNT excited by a moving harmonic load, Şimşek (2010) investigated the effects of aspect ratio, nonlocal parameter, and velocity and the excitation frequency of the moving load on the dynamic response of SWCNTs. Based on the number of consisting rolled graphene sheets, CNTs are divided into single-, double- and multi-walled carbon nanotubes. While the outer layer of MWCNTs may keep the inner tubes away from chemical interactions with outside environs, each of the nested tubes interacts with the adjacent nanotubes through the vdW interlayer forces. 2. Model development A schematic of a DWCNT which has been modeled as a double-tube pipe with length of l, inner tube of radius r1, outer tube of radius r2, Young's modulus of E, the density of ρ, and Poisson's ratio of ν is shown in Fig. 1. The surrounding medium is described by a Pasternak foundation modeled with shear constant ks and spring constant kw. The DWCNT is simply-supported at both ends. A nanoparticle of mass m 2 Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. where t is time and ui(x, t) and wi(x, t) represent axial and transverse displacement components of the ith tube, respectively. Assuming smalldeflection and linear vibration of the DWCNT, the only nonzero strain component based on Euler-Bernoulli beam theory can be written as: εxx = ∂u¯ i ∂ui ∂ 2w = − z 2i ∂x ∂x ∂x (5) Considering that the CNT deforms within its linear elastic range, Hooke's law is predominant as follows σxx = Eεxx . (6) The strain energy (PE)s stored in the DWCNT is given by: PEs = 1 2 Fig. 1. Schematic of a DWCNT carrying moving nanoparticles. ∫0 l i = 1, 2. σxi εxi dx , (7) Substituting Eqs. (5) and (6) into Eq. (7) yields to: moves through the inner tube of the DWCNT at a constant speed of V from left to right. It is assumed that when the nanoparticle leaves the DWCNT, another one enters and moves immediately. The consecutive transition of moving nano-particles provides a varying mass system which leads to have a so-called “open” system. PEs = ∫t δ (L)o dt + 1 ∫t δH dt =0 1 where δH = δW + → u ·δ→ r )( VB − → u ). → n ds ∫∫(ρm → Bo (t ) PEf = ∫t (δ KE − δ PE + δW )dt =0 , 1 w¯ i (x , z , t ) = wi (x , t ) 1 2 ∫0 PE vdW = − 2 2 ⎝ ⎢ ⎣ A1 ⎠ ⎜ ⎟ ⎝ ⎜ A2 −z 2 ⎤ ∂ 2w 2 ⎞ dA ⎥ dx , ∂x 2 ⎠ ⎥ ⎦ ⎟ l 2 ⎛k w w2 − ks ∂ w2 ⎞ w2 dx ∂x 2 ⎠ ⎝ ⎜ ⎟ (9) 1 2 ∫0 p12 w1 dx − l 1 2 ∫0 l p21 w2 dx (10) where p12 is the force applied to the outer tube by the inner one and p21 is the force applied to the inner tube by the outer one. It should also be noted that p12 = −p21 = pvdW . For small-deflection and considering just linear vibrations of the system, the vdW force at any point should be a linear function of the distance between two adjacent nanotubes at that point. So, the vdW pressure per unit length of the DWCNT is given by: (1) (2) p VdW = c (w2 − w1) (11) where c is the intertube interaction coefficient and can be estimated by: c = 2r1 ∂ 2U ∂Δ2 Δ = Δe (12) The interlayer cohesive energy U can be stated in terms of the interlayer spacingΔ as: Δ 4 Δ 10 U (Δ) = kL ⎡ ⎛ 0 ⎞ − 0.4 ⎛ 0 ⎞ ⎤ ⎢ ⎝Δ⎠ ⎥ ⎣⎝ Δ ⎠ ⎦ (13) where kL is calculated to be 0.4089101874 J/m and Δ0 = 0.34 nm (Ansari et al., 2012). Also, the equilibrium interfacial spacing is Δe = Δ0 . The work done by the moving nanoparticle in contact with the inner tube of the DWCNT is given by: 2 (3) where KE is the kinetic energy of the nanostructure and PE introduces its potential energy including strain energy of the nanotube PEs, the potential energy resulted by the Pasternak foundation PEf, and the potential energy due to vdW interaction forces PEvdW. Based on the well-known Euler–Bernoulli beam theory (Ghayesh, 2011, 2018a, 2019; Gholipour et al., 2015; Ghayesh et al., 2013a,b,d, 2016; Ghayesh and Farajpour, 2019), the displacement fields are expressed as: u¯ i (x , z , t ) = ui (x , t ) − ⎡ Moreover, the potential energy resulted by vdW forces between the tubes may be expressed as: The Lagrangian of the system, which has been confined by the open control volume, is denoted by(L)o = (KE − PE)o and δWrepresents the u = D→ r /Dt is virtual work done by external forces on the system. Also → → the material time derivative of the nano-particle position vector, r , and → → VB is the velocity of control surface. Moreover, n introduces the normal vector on the differential surface element with area ds. In other term, the last term in Eq. (2) may be supposed as a virtual momentum → → → u ·δ→ r ) which arises with a rate ( VB − u )· n across the transport (ρm → open surface Bo(t). For the case of simply supported or cantilevered → boundary conditions, there is no virtual displacement at ends (δ r = 0) . Therefore, the second term in right side of Eq. (2) will vanish. Finally, applying all above-mentioned consideration on Eq. (1) leads to: t2 l (8) For an open system, the entrances and exits of nano-particles across the DWCNT can be perceived as intermittent momentum transfer through the CNT. To derive the governing equations of such problems, the conventional Hamilton's principle cannot be applied, since the set of particles involved is flowing across the boundaries and does not permit a system formulation approach (Torkan et al., 2019a,b; Torkan et al., 2017; Torkan and Pirmoradian, 2019) So, the extended Hamilton's principle (Pirmoradian and Karimpour, 2017; Pirmoradian et al., 2015, 2018) for a system of changing mass is written as t2 ∫0 ⎢∫ ⎛ ∂∂ux1 − z ∂∂xw21 ⎞ dA + ∫ ⎛ ∂∂ux2 where A1 and A2 are the cross-sectional areas of inner and outer tubes, respectively. The surrounding elastic medium is simulated by Pasternak foundation, which is generally referred as a two-parametric mechanical model which assumes the foundation is formed from some separate springs and a shear layer over the springs. The potential imposed to the system due to this type of foundation can be written as: 2.1. Derivation of motion equations t2 E 2 W=− ∫0 l F (x , t ) w1 dx (14) where F(x, t) is the loading function. Considering all inertial effects of the moving nanoparticle, this function is defined as: F (x , t ) = mg − m ∂w (x , t ) z i∂x = ⎡m g − V 2 ⎣ ( (4) 3 ( ∂2w 1 ∂x 2 d2w1 dt2 − 2V ∂x ∂t1 − ∂2w − k v w1 δ¯ (x − x m) ∂2w ) 1 ∂t 2 ) − k w ⎤⎦ δ¯ (x − Vt ) v 1 (15) Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. where Dirac-delta function δ̄ is used to determine the position of the nanoparticle throughout the DWCNT. Also, kv is a linear spring constant, which is utilized to simulate the vdW force between the moving nanoparticle and the DWCNT. In addition, the kinetic energy KE of the DWCNT is given by: KE = ρ 2 ∫0 ⎨∫ ⎡⎢ ⎛⎝ ∂∂ut1 ⎞⎠ l ⎧ ⎩ 2 A1 ⎣ ∂w + ⎛ 1 ⎞ ⎤ dA1 + ⎝ ∂t ⎠ ⎥ ⎦ 2 ∫ ⎡⎢ ⎛⎝ ∂∂ut2 ⎞⎠ 2 A2 ⎣ ρA1 ( ( +µ2 c (16) ρA2 ∂2u1 ∂N1 − = 0, ∂t 2 ∂x (17) ∂2u2 ∂N2 − = 0, ∂t 2 ∂x (18) ρA1 ∂ 2w 1 ∂2M1 ∂ 2w ∂ 2w 1 ∂ 2w 1 ⎞ − − ⎡m ⎛g − V 2 21 − 2V − − k v w1⎤ δ¯ ⎢ ⎝ ⎥ ∂t 2 ∂x 2 x x t ∂ ∂ ∂ ∂t 2 ⎠ ⎣ ⎦ (x − Vt ) − c (w2 − w1) = 0, (19) ρA2 ∂ 2w 2 ⎞ ∂2M2 ∂ 2w 2 + c (w2 − w1) = 0. + ⎛k w w2 − ks − ∂x 2 ⎠ ∂x 2 ∂t 2 ⎝ ⎜ ⎜ ρA2 ∫ σxi dAi , Mi = V (20) (22) ρA2 ∂2w 2 ∂t 2 ∂2w 2 ∂x 2 (24) ∂2u2 ∂ 2u ∂ 4u − EA2 22 − µ2 ρA2 2 2 2 = 0, 2 ∂t ∂x ∂x ∂t (25) − ∂ 4w ∂2w1 ∂x 2 − µ2 ρA2 ∂ 4w ∂2w1 ∂t 2 ) − k w ⎞⎠ δ¯ (x − Vt ) ) − k ⎞⎠ δ¯ (x − Vt ) ∂ 4w v 1 ∂x 2∂t 2 1 ∂2w1 v ∂x 2 )=0 ∂ 4w 2 ∂x 2∂t 2 + EI2 ( ∂2w 2 ∂x 2 (26) ∂ 4w 2 ∂x 4 − + k w w 2 − ks ∂2w1 ∂x 2 )=0 ∂2w 2 ∂x 2 − µ2 k w ∂2w 2 ∂x 2 + µ2 ks ∂ 4w 2 ∂x 4 ∞ w1 (x , t ) = ∑ φ1i (x ) q1i (t ) w2 (x , t ) = ∑ φ2i (x ) q2i (t ) i=1 (28) ∞ i=1 (29) M (t ) q¨ 1 (t ) + C (t ) q˙ 1 (t ) + K (t ) q1 (t ) + T q 2 (t ) = F (t ) (30) ˜ q (t ) + T ˜ q (t ) = 0 ˜ q¨ (t ) + K M 2 1 2 (31) where q1 (t ) = {q11 (t ), q21 (t ), ...}T , q2 (t ) = {q12 (t ), q22 (t ), ...}T are the modal coordinates vectors and components of the matrices can be achieved as: (23) ∂2u1 ∂ 2u ∂ 4u − EA1 21 − µ2 ρA1 2 1 2 = 0, 2 ∂t ∂x ∂x ∂t ∂2w where φ(x) is the spatial mode shape, q(t) introduces the time dependent modal amplitude and i represents the number of modes. In order to reduce the coupled PDEs of motion to corresponding coupled ordinary differential equations, Eqs. (28) and (29) are substituted into Eqs. (26) and (27), then the results are multiplied by φ1j(x) and φ2j(x) respectively, and at last integrating is performed over the domain [0, L] leading to the following equations: where σij and σijnl denote local and nonlocal stresses, respectively, ∇2 is the Laplacian operator, and µ = e0 a is the nonlocal parameter in which a is an internal characteristic length and e0 is a constant. Using Eqs. (17)–(20) and (22) and performing some arithmetic operations, the final nonlocal form of PDEs governing the motion can be obtained as: ρA1 ( (21) τ inwhere σij(X) is the nonlocal stress tensor in point X, K(|X − X ′|,˜) troduces the nonlocal modulus, Cijkl is the elasticity tensors of forthorder, ɛkl presents the linear strain tensor and τ̃ is the material constant. To avoid the difficulties of spatial integrating in Eq. (22), Eringen (1983) presented an equivalent differential constitutive relation using Green's function as (Oveissi et al., 2019; Liu and Lv, 2018): (1 − µ2 ∇2 ) σijnl = σij − 2V ∂x ∂t1 − The motion equations describing flexural and longitudinal vibrations of the DWCNT were derived in last section. Generally, the flexural modes occur at low frequencies and longitudinal modes are excited at high operating frequencies. In addition, the transverse modes are more prone to get excited by common types of external excitations like what happens for the motion of a moving nanoparticle inside a CNT. Accordingly, in the following the lateral vibration of the DWCNT excited by constitutive motion of moving nanoparticles is investigated. In order to discretize the equations of motion by applying the Galerkin method, the transverse displacements of the DWCNT tubes are considered as (Pirmoradian et al., 2014, 2018, 2019; Torkan et al., 2018): ⎟ ∫ K( X − X ′ , τ˜)Cijkl εkl (X ′) d V(X ′) ∂ 4 w1 ∂x 4 2.2. Discretization of motion equations In classical local elasticity theory, stress at an arbitrary point depends only on the strain at same point, whereas, in nonlocal elasticity theory, the stress at a point is a function of the strains at that point and also on strains at all other points of the continuum as following (Eringen, 1983, 2002): σij (X ) = + EI1 (27) ⎟ ∫ σxi z dAi , ∂2w1 ∂x 2 +c (w2 − w1) − µ2 c where resultants stress are defined as: Ni = ∂ 4 w1 ∂x 2∂t 2 +µ2 ⎛m −V 2 41 − 2V 3 1 − ∂x ∂x ∂t ⎝ − c (w2 − w1) where A1 and A2 are the cross-sections of the inner and outer tubes. Substituting Eqs. (8)–(10), (14) and (16) into Eq. (3) and taking variation with respect to u1, u2, w1 and w2, the governing coupled PDEs of motion are derived as: ρA1 − µ2 ρA1 −⎛m g − V 2 ⎝ ⎫ ∂w 2 + ⎛ 2 ⎞ ⎤ dA2 ⎬ ⎝ ∂t ⎠ ⎥ ⎦ ⎭ dx ∂2w1 ∂t 2 4 Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. Mij = ρA1 ∫0 φ1i φ1j dx − µ2 ρA1 ∫0 l ∂2φ1i φ (x )dx ∂x 2 1j l ∂φ1i Cij = 2mV ∂x Kij = ∫0 EI1 φ1j − ∂ 4φ1i l ∂3φ 2µ2 mVµ2 31i φ1j ∂x ∂x 4 φ1j dx + mV 2 dx − µ2 c ∫ ∂2φ1i l ∂2φ1i φ dx 0 ∂x 2 1j ∂2φ1i +k v φ1i φ1j − µ2 k v ∂x 2 φ1j − mµ2 V 2 ∂x 2 + m φ1i φ1j − µ2 m ∂ 4φ1i ∂x 4 Tij = −c ∫ φ2i φ1j dx + µ2 c ∫0 l ∂2φ2i Fj = mg φ1j −ks ∫0 ∂ 4φ2i ∂x 4 ∂x 2 +(1 − 2ζ Ω2 sin2 (τ ) − 2ζµ*Ω2π 2 sin2 (τ ) + 2k v*sin2 (τ ) + 2k v*µ*π 2 sin2 (τ ) + c1* + µ*π 2c1*) Q1 −(c1* + µ*π 2c1*) Q2 = 2 ζg *sin(τ ) (37) φ1j dx Ω2 (1 + µ*π 2) Q″2 + (1 + k w* + µ*π 2k w* + ks* + µ*π 2ks* + c2* + µ*π 2c2*) Q2 −(c2* + µ*π 2c2*) Q1 = 0 φ2j (x )dx + k w ∫0 φ2i φ2j dx − µ2 k w ∫0 l ∂2φ2i l − µ2 c ∫0 Ω2 (1 + π 2µ* + 2ζ sin2 (τ ) + 2ζπ 2µ*sin2 (τ )) Q″1 +(4ζ Ω2 cos(τ ) sin (τ ) + 4ζµ*Ω2π 2 cos(τ ) sin (τ )) Q′1 l l ∂2φ2i φ dx 0 ∂x 2 2j l ∂2φ2i φ dx ∂x 2 2j as: φ1j φ1j + c ∫0 φ1i φ1j ˜ ij = ρA2 ∫ φ φ dx − µ2 ρA2 ∫ M 2i 2j l T˜ij = EI2 ∫0 ∂x 2 φ1j , l 0 l 0 ∂2φ1i + µ2 ks ∫0 l ∂ 4φ2i φ dx ∂x 4 2j l ∂2φ2i φ dx ∂x 2 2j l l K˜ ij = −c ∫0 φ1i φ2j dx + µ2 c ∫0 ∂2φ1i ∂x 2 ∂x 2 + c ∫0 φ2i φ2j dx (38) φ2j dx where the prime superscript indicates the non-dimensional time derivative. l 2.3. Consecutive loading φ2j dx (32) The coefficients of Eq. (37) change with time as long as the nanoparticle moves inside the CNT. Once the particle exits the CNT, the time-varying coefficients will disappear resulting in removal of any growth of vibrational amplitude. Therefore, a sequential transition of nanoparticles with period of T = l/ V is considered to investigate the conditions of system dynamic instability. To reflect this statement in the motion equations, the Fourier expansion of the time-varying coefficients of Eq. (37) is written as: By selecting the following mode shape functions φ1i (x ) = 2 l φ2i (x ) = 2 l ( ) sin ( ) iπx l sin iπx l (33) which satisfy the boundary conditions of simply supported boundary conditions (Oveissi et al., 2016, 2018) and substituting them into expressions for aforementioned matrices components, the governing equations of the modal coordinates for the fundamental mode of the system (i = 1) are obtained as follows: ⎡ρA1 + µ2 ρA1 ⎣ +⎡4mV ⎣ π l2 ( ) + 2µ m cos ( ) sin ( ) + 4mµ V π2 l2 + 2 l sin2 m πVt l πVt l +⎡EI1 4 − 2mV 2 3 sin2 l l ⎢ ⎣ π4 k π2 +2µ2 v 3 l = mg 2 l πVt l π2 ( ) πVt l ( )+c+µc sin ( ) , πVt l sin2 2 2 π2 l3 2 π3 l4 ( ) ⎤⎦ q¨ cos ( ) sin ( ) ⎤ q˙ ⎦ sin2 πVt l π4 l2 1 πVt l − 2mµ2 V 2 l5 sin2 π2 Ω2 (1 + π 2µ* + ζ (1 − cos(2τ )) + ζπ 2µ* (1 − cos(2τ ))) Q″1 + (2ζ Ω2sin (2τ ) + 2ζβ 2µ*π 2sin (2τ )) Q′1 + (1 − ζ Ω2 (1 − cos(2τ )) − ζµ*Ω2π 2 (1 − cos(2τ )) + k v* (1 − cos(2τ )) + k v*µ*π 2 (1 − cos(2τ )) + c1* + µ*π 2c1*) Q1 πVt l ( ) πVt l − (c1* + µ*π 2c1*) Q2 = + 2 lv sin2 k ( ) + ⎡EI2 ⎣ π4 l4 + k w + µ2 k w −⎡c + µ2 c 2 ⎤ q1 = 0 l ⎦ ⎣ π2 l2 + ks ⎤ q − ⎡c + µ2 c ⎤ q l2 ⎦ 2 ⎥ 1 ⎣ ⎦ π2 l2 + µ2 ks π4 l2 + c + µ2 c π2 ⎤q l2 ⎦ 2 π2 (35) It should be noted that in most of mechanical loading problems, the most amount of energy is stored in modes with lower orders (frequencies) like what happens in Fourier components which are larger in magnitude for lower frequencies and smaller for higher frequencies. In other words, the energy which is required to excite higher modes of vibration may not be available in many applications. By defining the following non-dimensional parameters Δ m ζ = ρA l , 1 Δ kw l 4 , π 4EI2 k w* = Δ Vl π Ω= ρA EI , Δ ks l 2 , π 2EI2 ks* = Δ πVt , l τ= Δ kv l 3 , π 4EI1 k v* = Δ q Q= l3/2 , + ∑k = 1 ∞ 4 π (1 − 4k2) cos(2kτ ) ) Ω2 (1 + π 2µ* + ζ (1 − cos(2τ )) + ζπ 2µ* (1 − cos(2τ ))) Q″1 +(2ζ Ω2sin (2τ ) + 2ζ Ω2µ*π 2sin (2τ )) Q′1 +(1 − ζ Ω2 (1 − cos(2τ )) − ζµ*Ω2π 2 (1 − cos(2τ )) + k v* (1 − cos(2τ )) +k v*µ*π 2 (1 − cos(2τ )) + c1* + µ*π 2c1* − (γ + λ cos(2τ )) − µ*π 2 Δ c l4 , π 4EI1 c1* = Δ c l4 , π 4EI2 c2* = 2 π According to some uncertainty due to reaction force of non-ideal supports or undesired external loadings, an axial force involving static and periodic parts(P0 + P cos(2τ )) is applied to the DWCNT. It is assumed that the frequency of the oscillating part is twice the frequency of the nanoparticles passage (Karimpour et al., 2016). This term has to be entered alongside the CNT stiffness term due to the recognized effect of axial forces on flexibility of systems. In addition, it is noteworthy that although investigating external resonance of the system may be of interest because of appearance of the direct forcing term in right side of Eq. (39), the moving nanoparticles inertia terms individually render the modal Eq. (39) parametrically excited where dynamic instability can occur due to parametric resonance. Accordingly, the right side of Eq. (39) is omitted by refraining the gravity effect of passing nanoparticles. Finally, considering the small-scale effects, the non-dimensional coupled homogeneous motion equations of the DWCNT surrounded by an elastic medium and excited by moving nanoparticles are obtained as follows: π2 (34) π2 ⎤ q¨ l2 ⎦ 2 ( (39) πVt l πVt l ⎡ρA2 + µ2 ρA2 ⎣ 2 ζg * 1 (γ + λ cos(2τ ))) Q1 −(c1* + µ*π 2c1*) Q2 = 0 Δ ρAg l3 g* = 4 , π EI1 (36) (40) the governing equations of motion in dimensionless form are resulted 5 Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. Fig. 2. Stable and unstable regions in the parameters plane. Fig. 3. Phase plane of the DWCNT carrying moving nanoparticles for parameters belonging to the unstable region. Ω2 (1 + µ*π 2) Q″2 + (1 + k w* + + ks* + + c2* + −µ*π 2 (γ + λ cos(2τ ))) Q2 − (c2* + µ*π 2c2*) Q1 = 0 µ*π 2k w* where Δ γ = P0/ Pcr , Δ λ = P / Pcr , µ*π 2ks* and Pcr = π 2EI / l 2 . µ*π 2c2* The incremental procedure is the first step of the IHB method. If (ζ*, Ω*) is an initial point located on one instability boundary corresponding to a periodic response of Q*(τ ) = [Q1, Q2]T , the next point can be measured by adding the related increments as the following: − (γ + λ cos(2τ )) Q (τ ) = Q*(τ ) + ΔQ (τ ), (41) ζ = ζ * + Δζ , Ω = Ω* + ΔΩ, (42) where Δζ, ΔΩ and ΔQ(τ) are small increments. By substituting Eq. (42) into Eqs. (40) and (41), and omitting the nonlinear terms of small increments, the linear incremental equation is extracted as follows: 3. Solution procedure Ω* (M1 + ζ *M2 (τ ))ΔQ″ + 2ζ *Ω* C1 (τ )ΔQ′ 2 As stated in the last section, the consecutive passage of nanoparticles through the DWCNT results in presence of periodic coefficients in motion equations. Based on Floquet theory for periodic systems with period T, there exist periodic solutions of period T and 2T on the transition curves in the parameters plane which separate stable and unstable regions. Therefore, any method capable to find solutions with period of T or 2T for the differential equations governing the problem may be applied to find these boundary curves. In this paper, the IHB method is utilized to determine the transition curves in the mass-velocity plane of the passing nanoparticles. 2 + ⎜⎛K1 + ζ *Ω* K2 (τ ) + K3 (τ ) ⎟⎞ ΔQ ⎠ ⎝ 2 =R − ⎧Ω* (M2 (τ ) Q*″ + 2C1 (τ ) Q*′ + K2 (τ ) Q*) ⎫ Δζ ⎬ ⎨ ⎭ ⎩ −{2Ω*((M1 + ζ *M2 (τ )) Q*″ + 2ζ *C1 (τ ) Q*′ + ζ *K2 (τ ) Q*)}ΔΩ 2 (43) where R is the residual vector and its value will approach to zero when Q*(τ) tends to its exact value. The values of R and the matrices of M1, 6 Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. Fig. 4. Phase plane of the DWCNT carrying moving nanoparticles for parameters belonging to the stable region. Fig. 5. Effect of vdW interacting force between layers on the unstable region. 2 −c2* − π 2µ*c2* ⎤ ⎡1 + c1* + π µ*c1* 2 * * * K1 = ⎢ 1 + k w* + µ*π 2k w* + ks* + µ*π 2 ⎥ ⎥, ⎢ −c1 − π µ c1 2 ⎥ ⎢ ks* + c2* + µ*π c2* ⎦ ⎣ M2, C1, K1, and K2 are R = −Ω* (M1 + ζ *M2 (τ )) Q*″ − 2ζ *Ω* C1 (τ ) Q*′ 2 2 − ⎜⎛K1 + ζ *Ω* K2 (τ ) + K3 (τ ) ⎟⎞ Q*, ⎠ ⎝ 1 + π 2µ* 0 ⎤ ⎡ M1 = , ⎢ 0 1 + π 2µ* ⎥ ⎦ ⎣ −1 + cos(2τ ) − π 2µ* (1 − cos(2τ )) 0 ⎤ K2 = ⎡ , ⎢ 0 0⎥ ⎣ ⎦ 2 M2 1 − cos(2τ ) + π 2µ* (1 − cos(2τ )) =⎡ ⎢ 0 ⎣ sin(2τ ) + π 2µ*sin(2τ ) 0 ⎤ , C1 = ⎡ ⎢ 0 0⎥ ⎦ ⎣ 0 0 ⎡ −(γ + λ cos(2τ )) − π 2µ* ⎤ ⎢ ⎥ + ( cos(2 )) γ λ τ ⎢ ⎥ ⎢ * ⎥ +k v (1 − cos(2τ )) + k v*µ* 0 ⎢ ⎥. K3 = ⎢ π 2 (1 − cos(2τ )) ⎥ ⎢ ⎥ ⎢0 −(γ + λ cos(2τ )) − π 2µ* ⎥ ⎢ ⎥ (γ + λ cos(2τ )) ⎢ ⎥ ⎣ ⎦ ⎤, ⎥ ⎦ (44) The harmonic balance procedure is the second step of the IHB method which obtains periodic solutions of Eqs. (40) and (41) by expanding Q* and ΔQ* to finite Fourier series and applying the Galerkin method. For periodic solutions with period of 2T, Qi* (τ ) and ΔQi(τ) are 7 Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. Fig. 6. Effect of vdW force between nanoparticle and inner tube on the unstable region. Fig. 7. Effect of nonlocal parameter on the unstable region. expressed in the following form: Qi* (τ ) = ∑p = 1,3,5, … [aip cos(pτ ) + bip sin(pτ )] = Todd ai, odd, Q*′ = Y′A, i = 1, 2 ΔQ*′ = Y′ΔA, ⎡ π 0 (45) in which: Todd = [cos(τ ), cos(3τ ), …, sin(τ ), sin(3τ ), …] ai, odd = [ai1, ai3, ai5, …, bi1, bi3, …]T 2 where Y = diag (T) and coefficients A and ΔA are expressed as: A = [a1T , a T2 ] , T ΔA = [Δa1T , Δa T2 ] T 2 ⎣ 2 ⎤ ΔQ′ + ⎜⎛K1 + ζ *Ω* K2 (τ ) + K3 (τ ) ⎟⎞ ΔQ⎥ dτ ⎝ ⎠ ⎦ 2 ⎡ = ∫ δ (ΔQ)T ⎢R − ⎧Ω* (M2 (τ ) Q*″ + 2C1 (τ ) Q*′ + K2 (τ ) Q*) ⎫ Δζ ⎬ ⎨ 0 ⎭ ⎩ ⎣ −{2Ω*((M1 + ζ *M2 (τ )) Q*″ + 2ζ *C1 (τ ) Q*′ + ζ *K2 (τ ) Q*)}ΔΩ] dτ π (46) Using the following terse form: ΔQ = YΔA (49) ∫ δ (ΔQ)T ⎢Ω* (M1 + ζ *M2 (τ ))ΔQ″ + 2ζ *Ω* C1 (τ ) i = 1, 2 Q* = YA, ΔQ*″ = Y″ΔA Applying the Galerkin method to Eq. (43) for one period leads to ΔQi (τ ) = ∑p = 1,3,5, … [Δaip cos(pτ ) + Δbip sin(pτ )] = Todd Δai, odd, Δai, odd = [Δai1, Δai3, Δai5, …, Δbi1, Δbi3, …]T Q*″ = Y″A, (50) (47) Substituting Eqs. (47) and (49) into Eq. (50), the following linear equations are achieved in terms of parameters ΔA, Δζ and ΔΩ ˜ SΔA ΔA + SΔζ Δζ + SΔβ ΔΩ = R (48) where one may have: 8 (51) Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. Fig. 8. Effect of oscillatory axial force amplitude on the unstable region. Fig. 9. Effect of static axial force amplitude on the unstable region. Eq. (51) is used to find the instability boundaries for terms A, ζ*, and Ω*. Since the number of unknown variables exceeds the number of equations by two, without reducing the generality of problem solving, an element of A is equated to one and its corresponding increment in ΔA is set to zero. Also, by choosing ζ as the active parameter in solving procedure, Δζ is considered to be zero. Then, the number of equations and unknowns is equal in Eq. (51) and it can be solved using a recursive algorithm. The details of the algorithm can be found in references (Torkan et al., 2017; Pirmoradian et al., 2018). SΔA 2 ⎡ 2 = ∫ Y T ⎢Ω* (M1 + ζ *M2 (τ )) Y″ + 2ζ *Ω* C1 (τ ) 0 ⎣ π 2 ⎤ Y′ + ⎜⎛K1 + ζ *Ω* K2 (τ ) + K3 (τ ) ⎟⎞ Y⎥ dτ ⎠ ⎦ ⎝ SΔζ = ∫ Y T ⎡Ω* (M2 (τ ) Y″ + 2C1 (τ ) Y′ + K2 (τ ) Y) ⎤ A dτ ⎥ ⎢ 0 ⎦ ⎣ π 2 SΔβ = ∫ Y T [2Ω*((M1 + ζ *M2 (τ )) Y″ + 2ζ *C1 (τ ) Y′ + ζ *K2 (τ ) Y)] A dτ π 0 ˜ R 4. Results and discussion 2 ⎡ 2 = − ∫ Y T ⎢Ω* (M1 + ζ *M2 (τ )) Y″ + 2ζ *Ω* C1 (τ ) 0 ⎣ π The present parametric-type excitation problem, which was produced by successive passage of nanoparticles, may lead to instability in transverse vibrations of the DWCNT. Accordingly, the amplitude of DWCNT vibrations can be bounded for some system parameters or it may grow without any bound for some other parameters. Actually, dynamic instability arises for some regions in the mass-velocity plane of the transiting nanoparticles. This study emphasizes on finding these 2 ⎤ Y′ + ⎜⎛K1 + ζ *Ω* K2 (τ ) + K3 (τ ) ⎟⎞ Y⎥ A dτ ⎠ ⎦ ⎝ (52) 9 Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. Fig. 10. The unstable region for tensile (λ = 1, γ = 0.1) and compressive (λ = −1, γ = −0.1) axial loadings. Fig. 11. Effect of springs stiffness of elastic foundation on the unstable region. parameters e0 a = 0.5 nm , k w = 0 , ks = 0 , λ = 1 and γ = 0 . Moreover, in this step of analyses the vdW forces are neglected. Stable and unstable regions are shown in Fig. 2. From a dynamic point of view, selecting the nanoparticle parameters (mass and speed) from the unstable region will lead to unbounded increase of the DWCNT vibrational amplitude with time. In addition, as shown in Fig. 2, the convergence analysis of the IHB method has been performed to find the suitable number of harmonic terms of Eq. (46). As can be seen, the results of three and four harmonic terms converge. So, for the rest of the study, the first three harmonic terms of Eq. (46) are considered in the analyses. To approve the regions obtained in Fig. 2, phase plane of the system response is plotted for parameters picked from stable and unstable regions by solving the governing Eqs. (40) and (41) by using Runge-Kutta method. The first point (Ω = 0.91, ζ = 0.13) is selected in the vicinity of the boundary from the unstable region, and the second point (Ω = 0.94, ζ = 0.13) is selected near the boundary but from the stable region. The phase plane plots are shown in Figs. 3 and 4. As shown in Fig. 3, the amplitude of transverse vibrations of the DWCNT for the selected parameters from the unstable region increases and instability occurs. However, Fig. 4 illustrates the bounded amplitude of the DWCNT oscillations representing its stability for the selected regions. The stable regions are separated from unstable ones by transition curves from which picking system parameters results in presence of a periodic solution of period T or 2T for the system. The regions associated with solutions of period 2T are wider than those of period T. This makes them more important in practical applications (Torkan et al., 2019a; Bolotin, 1962). In fact, these regions are correlated to happening of principle parametric resonance in the mentioned system. This kind of resonance occurs in self-excited systems when the exciting frequency is near twice the fundamental natural frequency of the system (Pirmoradian and Karimpour, 2017; Pirmoradian et al., 2014; Torkan et al., 2018; Nayfeh and Mook, 1979). The geometric properties considered for the DWCNT in the present analyses are like those used in Ansari et al. (2012), i.e., the nanotube density is ρ = 1300 Kg/m3, the Poisson's ratio is ν = 0.3, and its Young's modulus is considered to be E = 1.1 TPa . Also, the inner radius is assumed to be r1 = 0.82 nm , the outer radius is r2 = 1.5 nm , the thickness is t = 0.34 nm and the length of nanotube is considered as l = 45 nm . By using the IHB method, the curves corresponding to 2T-period solutions were obtained and plotted in Fig. 2. In this figure, the horizontal and vertical axes represent the dimensionless passing frequency of nanoparticles and the mass ratio, respectively. The curves are obtained for 10 Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. Fig. 12. Effect of shear layer stiffness of elastic foundation on the unstable region. Fig. 13. Stability diagram of the SWCNT under the influence of axial harmonic loading. moving nanoparticles. The effect of nonlocal parameter on the DWCNT dynamic instability is studied in Fig. 7. The parameters are considered to be k w = 0 , ks = 0 , λ = 1 and γ = 0 . The figure shows that when the nonlocal parameter increases, the unstable region shifts to lower frequencies of the passing nanoparticles and consequently the DWCNT will become more unstable. This is because increasing the nonlocal parameter reduces the interaction forces between the CNT atoms which results in having a softer nanostructure. This emphasizes the significance of considering size-dependent theories such as nonlocal elasticity theory in the analyses of nanostructures. The effect of the oscillatory axial force amplitude, λ, on the parametric region of the Ω − ζ plane is shown in Fig. 8. In this analysis, the system parameters are set as e0 a = 0.5 nm , k w = 0 , ks = 0 , and γ = 0 . As can be seen, decreasing the amplitude of the axial oscillatory force does not change the origin of the unstable region but causes two boundary curves to come closer. Therefore, the area of unstable region is reduced leading to a more stable system. In the analyzes performed so far, the static axial force amplitude parameters. The results of numerical simulations indicate high precision of the IHB method in predicting dynamic behavior of the system. The effect of vdW forces on the DWCNT instability is investigated in Fig. 5. The value of parameters are selected as e0 a = 0.5 nm , k w = 0 , ks = 0 , λ = 1, and γ = 0 . The figure shows that considering the vdW forces in the analysis leads to a more stable system which can be seen as a movement of the unstable region to higher frequencies of nanoparticles passage. It may be inferred that inclusion of the vdW forces makes the DWCNT stiffer. Also, Figs. 2 and 5 show that by increasing the relative mass parameter ζ, the DWCNT will experience parametric resonance for a wider range of nanoparticle transition frequencies which means that increasing the mass of the nanoparticle will increase the probability of the DWCNT instability. Fig. 6 shows the effect of vdW force between the moving nanoparticle and the inner tube of the DWCNT. It should be noted that this term appears in motion equations as a time-varying coefficient in stiffness matrix. As can be seen, when the vdW force is considered in the analysis, the unstable zone will become smaller and stability of the DWCNT is guaranteed for a wider range of masses and speeds of the 11 Mechanics of Materials 142 (2020) 103279 M. Pirmoradian, et al. parameter, γ, was considered to be zero. The effect of this parameter on the dynamic stability of the DWCNT under consecutive transition of moving nanoparticles is studied in Fig. 9. The figure shows that when the amplitude of the static compressive axial force grows, the unstable region shifts to lower frequencies of the passing nanoparticles. In fact, increasing the amplitude of the compressive force reduces the natural frequency of the DWCNT and makes it softer. In addition, a comparison between the results of compressive and tensile axial loadings is performed and shown in Fig. 10. It can be concluded that the DWCNT is more stable in the case of tensile axial loading. The effect of foundation stiffness on the dynamic instability of the DWCNT is examined in Fig. 11. The system parameters are set to be e0 a = 0.5 nm , ks = 0 , λ = 1, and γ = 0.2 in the analysis. The figure shows that the elastic foundation stiffness has a progressive effect on the system stability. So that, an increase in the value of foundation stiffness moves the unstable region to higher frequencies of the nanoparticles passage and so the DWCNT will become unstable for higher values of the nanoparticle velocities. Fig. 12 illustrates the effect of the shear layer stiffness on the stability of the system. Again, when the shear layer stiffness increases, the critical velocities of the moving nanoparticles will also increase. • • • • • • analyses, the DWCNT carrying moving nanoparticles becomes more stable. By increasing the nonlocal parameter, the unstable region shifts to lower frequencies of the passing nanoparticles which means the critical velocities of the nanoparticles decrease. As the amplitude of the static tensile axial force increases, the instability of the DWCNT occurs for higher velocities of the passing nanoparticles. By reducing the amplitude of the oscillatory axial force, the width of the unstable region decreases and subsequently the DWCNT will become more stable. The DWCNT is more stable in the case of static tensile axial force in comparison with the case of compressive force. Increasing the stiffness of the springs and the shear layer of the foundation improves stability of the system, so that the DWCNT will experience instability for higher values of the nanoparticles velocities. The results of numerical simulations show that the IHB method is accurate in analyzing stability of nonlocal transverse vibrations of DWCNTs excited by moving nanoparticles. Declaration of Competing Interest 5. Validation study The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. In order to certify the results of the implemented method, a simplification on the problem is performed which makes us able to compare the results of the present study with those of existing literature. For this purpose, by neglecting the passing nanoparticles and eliminating one tube from the model, a single-walled carbon nanotube under axial oscillatory force is considered. The geometric and mechanical properties are chosen as E = 1 TPa , ρ = 1300 Kg/m3, l = 45 nm , outer diameter d1 = 1.64 nm , and inner diameter d 0 = 0.96 nm like those were considered in reference Karličić et al. (2017). The SWCNT is embedded in a Winkler foundation with stiffness of k w = 106 N/m2 , and the nonlocal parameter value is set to zero. The nanotube is subjected to axial oscillatory force Pcos (2ωt) with the oscillation frequency of 2ω. Fig. 13 compares the boundaries obtained by the IHB method used in this paper by those of reference (Karličić et al., 2017). The horizontal axis represents the ratio of the excitation frequency to the natural frequency of the SWCNT and the vertical axis is the axial oscillatory force amplitude. The nice agreement between the results confirms the accuracy of the IHB method in the study of DWCNTs stability under loading. References Ajayan, P.M., Zhou, O.Z., 2001. Applications of Carbon Nanotubes. Springer, pp. 391–425. Ansari, R., Gholami, R., Darabi, M.A., 2012. Nonlinear free vibration of embedded double-walled carbon nanotubes with layerwise boundary conditions. Acta Mech. 223, 2523–2536. Ansari, R., Hemmatnezhad, M., 2012. Nonlinear finite element analysis for vibrations of double-walled carbon nanotubes. Nonlinear Dyn. 67, 373–383. Bolotin, V., Dynamic Stability of Elastic Systems. (1962). Budarapu, P.R., Yb, S.S., Javvaji, B., Mahapatra, D.R., 2014. Vibration analysis of multiwalled carbon nanotubes embedded in elastic medium. Front. Struct. Civ. Eng. 8, 151–159. Eringen, A.C., 1983. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710. Eringen, A.C., 2002. Nonlocal Continuum Field Theories. Springer Science & Business Media. Farokhi, H., Ghayesh, M.H., 2017. Nonlinear resonant response of imperfect extensible Timoshenko microbeams. Int. J. Mech. Mater. Des. 13, 43–55. Farokhi, H., Ghayesh, M.H., 2018. Supercritical nonlinear parametric dynamics of Timoshenko microbeams. Commun. Nonlinear Sci. Numer. Simul. 59, 592–605. Farokhi, H., Ghayesh, M.H., Amabili, M., 2013. Nonlinear resonant behavior of microbeams over the buckled state. Appl. Phys. A 113, 297–307. Fu, Y., Bi, R., Zhang, P., 2009. Nonlinear dynamic instability of double-walled carbon nanotubes under periodic excitation. Acta Mech. Solida Sin. 22, 206–212. Ghayesh, M.H., 2011. Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance. Int. J. Mech. Sci. 53, 1022–1037. Ghayesh, M.H., 2018a. Dynamics of functionally graded viscoelastic microbeams. Int. J. Eng. Sci. 124, 115–131. Ghayesh, M.H., 2018b. Nonlinear vibration analysis of axially functionally graded sheardeformable tapered beams. Appl. Math. Model. 59, 583–596. Ghayesh, M.H., 2018c. Functionally graded microbeams: simultaneous presence of imperfection and viscoelasticity. Int. J. Mech. Sci. 140, 339–350. Ghayesh, M.H., 2019. Viscoelastic dynamics of axially FG microbeams. Int. J. Eng. Sci. 135, 75–85. Ghayesh, M.H., Amabili, M., Farokhi, H., 2013a. Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams. Int. J. Eng. Sci. 71, 1–14. Ghayesh, M.H., Amabili, M., Farokhi, H., 2013b. Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. 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Based on Eringen's nonlocal elasticity theory and applying Euler-Bernoulli beam theory, the dynamic formulation of the system was extracted. Then, the coupled ODEs with periodic coefficients describing a parametrically excited system were obtained. In order to present a real model, all inertial effects of the nanoparticles were considered in the governing equations. In addition, the van der Waals force between the constitutive atoms of the moving nanoparticle and those of the inner tube was considered. The IHB method was utilized to calculate the boundaries of instability. The effects of different parameters like nonlocal parameter, vdW forces, amplitude of axial oscillatory and static forces, and the stiffness of Pasternak foundation on the system stability were investigated. The results of this study are in agreement with those of the available literature. 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