Characterization of free vibration of elastically supported double-walled carbon nanotubes subjected to a longitudinally varying magnetic field

Transcription

1 Characterization of free vibration of elastically supported double-walled carbon nanotubes subjected to a longitudinally varying magnetic field Keivan Kiani Acta Mechanica ISSN Volume 224 Number 12 Acta Mech : DOI /s

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3 Acta Mech 224, DOI /s Keivan Kiani Characterization of free vibration of elastically supported double-walled carbon nanotubes subjected to a longitudinally varying magnetic field Received: 8 March 2013 / Revised: 12 June 2013 / Published online: 18 July 2013 Springer-Verlag Wien 2013 Abstract Free transverse vibrations of elastically supported double-walled carbon nanotubes DWCNTs subjected to axially varying magnetic fields are examined. Using nonlocal Rayleigh beam theory, the explicit expressions of the governing equations are obtained and then numerically solved via an efficient numerical scheme. For magnetically affected DWCNTs with simply supported, fully clamped, simple-clamped, and clamped-free ends, the flexural frequencies as well as the corresponding vibration modes are evaluated for different varying magnetic fields. The influences of the small-scale parameter and the magnetic field strength on the dominant flexural frequencies of the DWCNTs are explained and discussed. The results indicate that the vibration characteristics of DWCNTs can be significantly affected by the axially varying magnetic field. The role of variation of the axial magnetic field on the vibrational mode patterns of both the innermost and outermost tubes is also revealed. For a special applied magnetic field, the alteration from coaxial to noncoaxial vibration pattern is also reported. The obtained results display that the flexural frequencies magnify with the magnetic field strength. Generally, the variation of the magnetic field strength has more influence on the variation of the frequencies of DWCNTs with higher small-scale parameters. This matter is mainly attributed to the incorporation of the size effect into the nonlocal Lorentz forces. 1 Introduction Thanks to the unprecedented physical and chemical properties of carbon nanotubes CNTs [1 3], they have been the focus of attention of scientists from various disciplines since the past two decades. Such excellent properties of CNTs have made researchers and companies to consider them in various fields such as nanotechnology, materials science, optics, electronics, medical and healthcare sciences, and civil engineering. In most of the above-mentioned engineering and technological applications, vibrations of CNTs should be realized and properly controlled. To date, vibration characteristics of CNTs have been theoretically investigated from various aspects including free vibration [4 12], flow-induced vibration [13 17], impact loadings [18 20], interactions with moving loads and nanoparticles [21 27], and nanomechanical sensor analysis [28 31], and an inclusive knowledge regarding such matters is beginning to come out. However, ways to control such vibrations have not been thoroughly examined. One efficient way for controlling the dynamic response of CNTs would be the application of an appropriate magnetic field. There are some research works dealing with the influence of a uniform magnetic field on the vibration characteristics of CNTs [32 34] and nanowires [35,36]. Most of the undertaken works validate this fact that exertion of a longitudinal magnetic field causes extra lateral stiffness within K. Kiani B Department of Civil Engineering, Islamic Azad University, Chalous Branch, Chalous, Mazandaran, Iran k_kiani@iauc.ac.ir; keivankiani@yahoo.com Tel.: Fax:

4 3140 K. Kiani K z1 x=0 K y1 x=0 K y2 x=0 K z2 x=0 C v /2 C /2 v H x x H x x K x=l z1 b K x=l y1 b K x=l y2 b K z2 x=l b Fig. 1 Continuum-based configuration of an elastically supported DWCNT subjected to an axially varying magnetic field the nanostructure; thereby, the flexural frequencies of the nanostructure would increase. However, the effect of the variation of the magnetic field across the CNT s length on its vibration characteristics has not been revealed. This matter encouraged the author to propose a numerical model based on the nonlocal continuum theory of Eringen [37 39] to study vibrations of double-walled carbon nanotubes DWCNTs immersed in a longitudinally varying magnetic field. Since the role of the DWCNT s boundary conditions on its vibrations is also of interest, the suggested model covers a wide range of boundary conditions for both the innermost and outermost tubes. In this paper, free dynamic response of lateral motion of an elastically supported DWCNT exposed to a longitudinally varying magnetic field is investigated in the context of the nonlocal continuum theory of Eringen. By exploiting Maxwell s equation, the Lorentz forces due to the applied steady magnetic field on the laterally deformed innermost and outermost tubes are determined. The strong form of the equations of motion is obtained. Since finding an analytical solution to the obtained equations is a very difficult task, an effective numerical scheme is proposed. For some special boundary conditions as well as axially varying magnetic fields, the frequencies and the corresponding vibration modes are assessed. In the continuing of our investigations, the influences of small-scale parameter, magnetic field strength, and the variation of the magnetic field along the nanostructure on the free dynamic response of the DWCNT are addressed and discussed in some detail. 2 Definition of the problem Consider an elastically supported DWCNT exposed to an axially varying magnetic field, H x x, asdemonstrated in Fig. 1. For continuum-based modeling of the problem, the innermost and outermost tubes of the DWCNT are substituted by equivalent continuum structures ECSs where most of their dominant frequencies are identical to their counterpart tubes. The investigations of Gupta and Batra [40] show that the ECS is a hollow isotropic tube whose mean radius and length are analogous to those of the parent tube, and its wall thickness is 0.34 nm. Throughout this study, the properties associated with the innermost and outermost tubes are specified via subscripts 1 and 2, respectively. The length, mean radius, cross-sectional area, moment inertia, density, and elasticity modulus of the ith tube are represented by l b, r mi, A bi, I bi, ρ bi,ande bi, respectively. Both the innermost and outermost tubes are interacted with each other during their deformations by the van der Waals vdw forces between their atoms. Until transverse vibration of the DWCNT has not occurred, no interactional forces would exist and both the innermost and outermost tubes are at their initial equilibrium states. When the DWCNT is subjected to a lateral deformation, the positions of the atoms of both tubes would change. As a result, interactional forces would result, which are calculated based on the Lennard-Jones potential function [41,42]. For lateral vibration of the DWCNT, such forces are commonly modeled via a laterally continuous spring system of constant C v. There are some theoretical works dealing with the determination of C v for both finite and infinite DWCNTs [43 46,22]. In order to cover a wide range of boundary conditions and to avoid unnecessary discussion on the imposition of essential boundary conditions in the used meshless methodology, both ends of each tube have been attached to lateral and rotary springs of constants K zi x k and K yi x k ; k = 1, 2, respectively, where x k = k 1l b. In the upcoming part, the exerted body forces on both innermost and outermost tubes with elastic supports due to the application of an axially varying steady magnetic field are methodically assessed. Subsequently, such

5 Characterization of free vibration 3141 forces are appropriately incorporated into the nonlocal equations of motion of the problem under study. The procedure for frequency analysis of the governing equation via reproducing kernel particle method RKPM is also explained. 3 Exerted body forces within the tubes of a DWCNT due to the presence of an axially varying magnetic field Maxwell s equations read [47 50] h = Ḋ + J, E = Ḃ,.D = ρ b,.b = 0, 1 where E, H, J, B, and D in order denote the electric field intensity, magnetic field, current density vector, magnetic field density, and the displacement current density. The overdot sign and represent the first derivative with respect to time and the nabla operator, respectively. The constitutive relations for the ECS are considered as D = ɛ E, B = η H, 2 where ɛ and η in order are the permittivity and the permeability of the CNT. It is assumed that the displacement current density as well as its derivative with respect to time would be negligible. Let E = E 0 +e and H = H 0 +h where e and h are the small disturbances associated with E 0 and H 0, respectively. For the problem under study, E 0 = 0. On the other hand, the generalized Ohm s law explains J-E-B relation by: J = σ E + v B where σ and v represent the electrical conductivity and the velocity filed vector of the DWCNT. By assuming DWCNT as a highly conducting medium i.e., a large value for σ, one can arrive at: e = u B where v = u, and u = ux, y, z, t is the displacement field of the DWCNT. Assuming h H 0, using the recent relation, and Eq. 2, one has the following: e = η u H 0. By using the latter relation as well as Eqs. 1and2, one can derive the following: h = u H 0. This expression states that the origin of the magnetic disturbance is the deformation of the DWCNT. Herein, such a deformation is caused by the propagation of transverse waves within the DWCNT. According to the Lorentz formulas, the exerted electromagnetic forces per unit volume of the medium are given by f m = J B. Using the recent relation, Eq. 1, and above-mentioned assumptions, the Lorentz force in terms of the displacement field of the DWCNT as well as the magnetic field is stated as: f m = η u H 0 H 0. 3 According to the Rayleigh beam theory, the components of the displacement field of the ECSs are expressed by: u xi x, z, t = u i x, t zw i,x x, t; u zi x, z, t = w i x, t; i = 1, 2, 4 where the x axis is coincident with the revolutionary axis of the DWCNT, and the z axis represents a lateral direction which is perpendicular to the x axis. Additionally, u xi and u zi in order are the dynamical longitudinal and transverse displacements of the ith tube, whereas u i and w i denote the dynamical longitudinal and transverse displacements of the neutral axis of the ith tube. Herein, H 0 = H x xi, wherei represents the unit vector associated with the x axis. Using Eqs. 3 and4, the only nonzero component of the magnetically applied force per unit length of the ECS of the ith tube, f zi, is evaluated as follows: f zi = A bi k.f m da = ηa bi H x H x w i,xx, where da is an infinitesimal area of the ECS s cross-section. 5

6 3142 K. Kiani 4 Free transverse vibration of elastically supported DWCNTs subjected to an axially varying magnetic field 4.1 Nonlocal equations of motion using NRBT The kinetic energy, T t, and elastic strain energy, Ut, of the elastically supported DWCNT as well as the work done by the axially applied magnetic field on the nanostructure, W t, based on the NRBT are stated as: T t = 1 2 Ut = 1 2 W t = 2 l b i=1 0 2 l b i=1 0 2 l b i=1 0 ρ bi A bi ẇ i x, t 2 + I bi ẇi,x x, t 2 dx, 6.1 w i,xx x, tmb nl i x, t + i 1C v w 2 x, t w 1 x, t K zi x k w i x k, t 2 + K yi x k w i,x x k, t 2 dx, 6.2 f zi w i dx, 6.3 where Mb nl i denote the nonlocal bending moment of the ith tube. According to the nonlocal continuum theory of Eringen [37 39], the nonlocal bending moments within the innermost and outermost tubes in this model based on the NRBT are expressed by [22,51 53] Mb nl i e 0 a 2 Mb nl i,xx = E bi I bi w i,xx ; i = 1, 2, 7 where a is an internal characteristic length. The value of e 0 is determined by adjusting the dispersion curves of the nonlocal model with those of an atomistic-based model when the experimentally observed data are not available. The parameter e 0 a is called small-scale effect parameter or small-scale parameter. By employing Hamilton s principle, after taking the required integration by parts, the equations of motion corresponding to the transverse vibration of an elastically supported DWCNT exposed to a longitudinally varying magnetic field are obtained as: ρ b1 A b1 ẅ 1 I b1 ẅ 1,xx Mb nl 1,xx + C v w 1 w 2 ηa b1 H x H x w 1,xx + 2 Kz1 x k w 1 K y1 x k w 1,xx δx xk = 0, 8.1 ρ b2 A b2 ẅ 2 I b2 ẅ 2,xx + M nl b 2,xx C v w 1 w 2 ηa b2 H x H x w 2,xx 2 Kz2 x k w 2 K y2 x k w 2,xx δx xk = 0, 8.2 where δ is the delta function. By combining Eq. 7 with Eqs. 8.1and8.2, the nonlocal equations of motion of the problem under study as a function of the deflections of the innermost and outermost tubes are derived as: ρ b1 A b1 ẅ1 e 0 a 2 ẅ 1,xx ρb1 I b1 ẅ1,xx e 0 a 2 ẅ 1,xxxx + Cv w1 w 2 e 0 a 2 w 1,xx w 2,xx + E b1 I b1 w 1,xxxx ηa b1 H x H x w 1,xx e 0 a 2 2 H x H x w 1,xx + Kz1 x,xx k w 1 δx x k e 0 a 2 w 1 δx x k,xx K y1 x k w 1,xx δx x k e 0 a 2 w 1,xx δx x k = 0, 9.1,xx

7 Characterization of free vibration 3143 ρ b2 A b2 ẅ2 e 0 a 2 ẅ 2,xx ρb2 I b2 ẅ2,xx e 0 a 2 ẅ 2,xxxx Cv w1 w 2 e 0 a 2 w 1,xx w 2,xx + E b2 I b2 w 2,xxxx ηa b2 H x H x w 2,xx e 0 a 2 2 H x H x w 2,xx + Kz2 x,xx k w 2 δx x k e 0 a 2 w 2 δx x k,xx K y2 x k w 2,xx δx x k e 0 a 2 w 2,xx δx x k = ,xx For the sake of generality in studying the problem, the following dimensionless parameters are considered: ξ = x l b,τ= 1 l 2 b E b1 I b1 ρ b1 A b1 t,μ= e 0a l b,λ 1 = l b r b1, w i = w i l b ; i = 1, 2, 10 where ξ is the dimensionless coordinate, w i is the dimensionless deflection field of the ith tube, τ is the dimensionless time, μ is the dimensionless small-scale parameter, r b1 denotes the gyration radius of the innermost tube, and λ 1 is the slenderness ratio of the innermost tube. Through introducing Eq. 10toEqs.9.1and9.2, the dimensionless equations of motion describing transverse vibration of elastically supported DWCNTs in the presence of a longitudinal magnetic field are derived as follows: w 1,ττ μ 2 w 1,ττξξ λ 2 1 ϱ 2 1 H w x1 1,ξξ μ2 H x1 H x1 w 1,ξξ K y1 ξ k w2,ττ μ 2 w 2,ττξξ ϱ 2 2 λ 2 1 where H w x2 2,ξξ μ2 H x2 H x2 w 2,ξξ w1,ττξξ μ 2 w 1,ττξξξξ + Cv w1 w 2 μ 2 w 1,ξξ w 2,ξξ + w 1,ξξξξ 2 K z1 ξ k w 1 δξ ξ k μ 2 w 1 δξ ξ k,ξξ,ξξ,ξξ + + w 1,ξξ δξ ξ k μ 2 w 1,ξξ δξ ξ k = 0, 11.1,ξξ w2,ττξξ μ 2 w 2,ττξξξξ Cv w1 w 2 μ 2 w 1,ξξ w 2,ξξ +ϱ3 2 w 2,ξξξξ 2 K z2 ξ k w 2 δξ ξ k μ 2 w 2 δξ ξ k,ξξ K y2 ξ k w 2,ξξ δξ ξ k μ 2 w 2,ξξ δξ ξ k,ξξ = 0, 11.2 ϱ1 2 = ρ b 2 A b2,ϱ2 2 ρ b1 A = ρ b 2 I b2,ϱ3 2 b1 ρ b1 I = E b 2 I b2, C v = C v lb 4, b1 E b1 I b1 E b1 I b1 ηa bi lb 2 H xi = E b1 I b1 H x, K zi ξ k = K z i x k l 3 b E b1 I b1, K yi ξ k = K y i x k l b E b1 I b A numerical solution to the governing equations via a meshless technique Seeking an analytical solution to the governing equations in Eqs. 11.1and11.2 is a very problematic task. This matter is mainly related to the variation of the axial magnetic field along the DWCNT s length as well as the existence of elastic supports for each tube at both ends. To overcome such a dilemma, RKPM is employed to discretize the unknown fields of the problem in the spatial domain. RKPM is an efficient numerical scheme from the meshless methods family. In contrast to finite element method FEM where the used shape function for each degree of freedom DOF is mesh dependent, the RKPM s shape functions are mesh independent. In other words, they depend on the positions of the particles. Further, the continuity of FEM s shape functions is generally limited by the chosen DOFs for the considered element. However, there is actually no limit for the order of continuity of the RKPM s shape functions. These two main privileges of the RKPM s shape functions with respect to those of the FEM have provided RKPM as a powerful and an efficient methodology for solving the problems that suffer from discontinuity, sharp variation, and progressive damage. To date, RKPM has been implemented for vibrational analysis of various structures, and a fairly good achievement has been reported [11,34,54 56]. For discretizing the governingequations,bothsides of Eqs and11.2 in order are premultiplied by δw 1 and δw 2. Subsequently, the sum of the resulting equations is integrated over the normalized length of the DWCNT. By taking successful integration by parts, one can arrive at

8 3144 K. Kiani 2 1 i=1 0 { ϱ 2i 2 1 δw i μ 2 δw i,ξξ w i,ττ + λ 2 1 ϱ2i 2 2 δw i,ξ w i,ξττ + μ 2 δw i,ξξ w i,ξξττ + 1 i+1 C v δw i μ 2 δw i,ξξ w 1 w 2 + ϱ3 2i 2 δw i,ξξ w i,ξξ + H xi δw i,ξ H xi w i,ξ + μ2 δw i,ξξ H xi H xi w i,ξξ } 2 + K zi ξ k δw i μ 2 δw i,ξξ w i + K yi ξ k δw i,ξ w i,ξ + μ 2 δw i,ξξ w i,ξξ δξ ξ k dξ = The unknown fields of the suggested model are discretized in the spatial domain as follows: w 1 ξ, τ = NP1 I =1 φw 1 I ξw 1I τ and w 2 ξ, τ = NP 2 I =1 φw 2 I ξw 2I τ where NP 1 /NP 2,φ w 1 I ξ/φ w 2 I ξ, andw 1I τ /w 2I τ in order are the number of RKPM particles, the RKPM shape function pertinent to the I th particle, and the nodal parameter value of the I th particle of the innermost/outermost tube. As a result, Eq. 13 could be rewritten in the following form: [ [Mb ] w 1w 1 [M b ] w 1w 2 ]{ } [ w1,ττ [Kb ] w 1w 1 [K [M b ] w 2w 1 [M b ] w 2w 2 + b ] w 1w 2 ]{ } { } w1 0 w 2,ττ [K b ] w 2w 1 [K b ] w 2w 2 =, 14 w 2 0 where the only nonzero submatrices in Eq. 14are w i = < w i1, w i2,...,w inpi > T ; i, j = 1, ϱ2i 2 1 φ w i I μ 2 φ w i I,ξξ φ w i J + λ 2 ϱ2 2i 2 φ w i I,ξ φw i J,ξ + μ2 φ w dξ, 15.2 i I,ξξ φw i J,ξξ [ Mb ] wi w i IJ = [ Kb ] wi w i IJ = [ Kb ] wi w j IJ = ϱ 2i 2 3 φ w i φ w i I,ξξ φw i J,ξξ + C v I μ 2 φ w i I,ξξ + H xi φ w i I,ξ H xi φ w i J,ξ + μ2 H xi φ w i 2 K z i ξ k φ w i I ξ k μ 2 φ w i I,ξξ ξ k + K yi ξ k C v φ w i I φ w i J I,ξξ H xi φ w i J φ w i J ξ k,ξξ dξ φ w i I,ξ ξ kφ w i J,ξ ξ k + μ 2 φ w i I,ξξ ξ kφ w i J,ξξ ξ k, 15.3 μ 2 φ w i I,ξξ φ w j J dξ; i = j For evaluating the flexural frequencies of the DWCNT, one can assume w i τ = w i0 e iϖτ ; i = 1, 2where w 10 and w 20 are the initial nodal parameter values of the innermost and outermost tubes, respectively, and ϖ denotes the dimensionless flexural frequency of the elastically restrained DWCNT under an axially varying magnetic field. By introducing these new forms of w i into Eq. 14, and by solving the resulting set of eigenvalue equations, the eigenvalues i.e., dimensionless flexural frequencies and the corresponding eigenvectors i.e., vibrational modes of the elastically supported DWCNT subjected to an axially varying magnetic field would be determined. 5 Results and discussion Consider a DWCNT subjected to an axially varying magnetic field modeled via ECSs with the following data: E bi = 1TPa,ν i = 0.2,ρ bi = 2500 kg/m 3, r m1 = 0.5 nm,andt b = 0.34 nm. According to Table 1, four different case studies are considered for the variation of the magnetic field along the DWCNT. For RKPM analysis of the problem, 31 particles with equal distances, linear base function, exponential window function, and dilation parameter equal to 3.2 are taken into account. The dimensionless values of the stiffness of the attached springs to both ends of the DWCNT are given in Table 2 for different boundary conditions. In this table, SS, CC, SC, and CF represent the fully simply supported, fully clamped supported, simple-clamped, and clamped-free i.e., cantilevered boundary conditions, respectively. For free vibration studies, the dimensionless frequency is defined by: n = ϖ n. In the following, the role of various axially varying magnetic

9 Characterization of free vibration 3145 Table 1 Under study cases for the exerted axially varying magnetic field Case H x1 x sin 1 H π x 0 l b H 2 0 x l b 3 H 0 1 sin π x l b 4 H 0 Table 2 Values of K yi and K zi for the considered boundary conditions SS CC SC CF K zi ξ K zi ξ K yi ξ K yi ξ fields, small-scale parameter, and strength of the applied magnetic field on the flexural frequencies as well as vibration mode shapes of the elastically supported DWCNT will be explained and scrutinized. In Tables 3, 4, 5, and6, the first five dimensionless frequencies of a DWCNT exposed to various axially varying magnetic fields are provided for SS, CC, SC, and CF boundary conditions. The results are provided for three levels of the strength of the applied magnetic field of the case 1 i.e., H 0 = 0, 10, and 20, four levels of the small-scale parameter i.e., e 0 a = 0, 1, 1.5, and 2 nm as well as λ 1 = 30. For such a level of the slenderness ratio of the innermost tube, the predicted results of the NRBT would be reliable with a good accuracy [23]. As it is seen in these Tables, for all cases of the applied magnetic fields, the predicted flexural frequencies of the DWCNT magnify with the strength of the applied magnetic field. For all considered boundary conditions and applied magnetic fields, the variation of the strength of the magnetic field is more influential on the variation of the frequencies of DWCNTs with higher small-scale parameters. It is mainly related to the incorporation of the small-scale parameter into the induced body force within the nanostructure due to the applied magnetic field. In the case of SS boundary condition see Table 3, for all considered cases of the applied magnetic field, the predicted frequencies would reduce as the magnitude of the small-scale parameter increases. This fact is more obvious for higher vibration modes. Concerning DWCNTs with SC boundary conditions see Table 5, for small-scale parameter larger than 1 nm, the predicted frequencies would generally lessen as the size effect becomes highlighted. Both local and nonlocal continuum theories predict that both simply supported and simply clamped DWCNTs subjected to the axially applied magnetic field of case 4 have the highest frequencies. Moreover, variation of the magnetic field strength of the case 4 has the most significant influence on the variation of the flexural frequencies of the DWCNT. Such a fact is more apparent for those of higher vibration modes. Based on Table 3, both the classical and nonlocal continuum theories state that the applied magnetic field in case 3 leads to the lowest frequencies. Further, such a case has the lowest effect on the variation of the frequencies of DWCNT with SS and SC conditions. As a result, among various forms of the axially applied magnetic field, the uniform one would be the most effective way for altering the frequencies of the DWCNT with both SS and SC ends. For fully clamped DWCNTs subjected to a longitudinally varying magnetic field see Table 4, both local and nonlocal continuum theories predict that the frequencies of the DWCNT would increase as the magnitude of the strength of the applied magnetic field magnifies. According to Table 4, the classical continuum theory predicts that the DWCNT subjected to the magnetic field of case 1 has the highest fundamental flexural frequencies among all considered cases. Additionally, the predicted second and higher frequencies by the classical elasticity theory for case 4 are larger than for other cases. However, the proposed nonlocal model explains that the case 4 of the applied magnetic field would lead to the highest frequencies for all vibration modes. Further, it is predicted that the variation of the strength of the applied magnetic field in case 4 has the highest impact on the variation of the frequencies of the DWCNT. Such contradictions between the predicted results by the above-mentioned continuum models reveal that the classical continuum theory would not be trustable for predicting the vibration behavior of nanostructures subjected to an axially varying magnetic field.

10 3146 K. Kiani Table 3 First five dimensionless frequencies of a DWCNT with SS boundary condition subjected to various axially varying magnetic fields for different levels of magnetic field strength as well as small-scale parameter Case e 0 a = 0 nm e 0 a = 1 nm e 0 a = 1.5 nm e 0 a = 2 nm H 0 = 0 H 0 = 10 H 0 = 20 H 0 = 0 H 0 = 10 H 0 = 20 H 0 = 0 H 0 = 10 H 0 = 20 H 0 = 0 H 0 = 10 H 0 = Table 4 First five dimensionless frequencies of a DWCNT with CC boundary condition subjected to various axially varying magnetic fields for different levels of magnetic field strength as well as small-scale parameter Case e 0 a = 0 nm e 0 a = 1 nm e 0 a = 1.5 nm e 0 a = 2 nm H x1 = 0 H x1 = 10 H x1 = 20 H x1 = 0 H x1 = 10 H x1 = 20 H x1 = 0 H x1 = 10 H x1 = 20 H x1 = 0 H x1 = 10 H x1 = For practical applications, when controlling the frequencies of DWCNTs with clamped ends is of concern, exploiting a uniformly applied magnetic field would be the most efficient solution. Regarding DWCNTs with CF conditions see Table 6, the variation of the strength of the magnetic field has the most significant influence on the variation of the fundamental frequencies of the DWCNT in case 1. This fact holds true for the predicted results by both local and nonlocal continuum theories. Furthermore, the fundamental frequencies of the DWCNT of the cases 1 and 2 in order are the highest and the lowest ones among the considered cases. As it is seen in Table 6, the variation of the magnetic field strength of case 4 has the most pronounced effect on the variations of the second and higher-level frequencies of the DWCNT. Additionally, such frequencies in case 4 are the highest ones among all studied cases. In brief, when adjusting the second and higher flexural frequencies of cantilevered DWCNTs is of concern, a uniformly axial magnetic field could be efficiently employed. However, when altering the fundamental frequency of such nanostructures

11 Characterization of free vibration 3147 Table 5 First five dimensionless frequencies of a DWCNT with SC boundary condition subjected to various axially varying magnetic fields for different levels of magnetic field strength as well as small-scale parameter Case e 0 a = 0 nm e 0 a = 1 nm e 0 a = 1.5 nm e 0 a = 2 nm H x1 = 0 H x1 = 10 H x1 = 20 H x1 = 0 H x1 = 10 H x1 = 20 H x1 = 0 H x1 = 10 H x1 = 20 H x1 = 0 H x1 = 10 H x1 = Table 6 First five dimensionless frequencies of a DWCNT with CF boundary condition subjected to various axially varying magnetic fields for different levels of magnetic field strength as well as small-scale parameter Case e 0 a = 0 nm e 0 a = 1 nm e 0 a = 1.5 nm e 0 a = 2 nm H x1 = 0 H x1 = 10 H x1 = 20 H x1 = 0 H x1 = 10 H x1 = 20 H x1 = 0 H x1 = 10 H x1 = 20 H x1 = 0 H x1 = 10 H x1 = is of particular interest, application of the axially magnetic field of case 1 would be the most influential way among the cases studied herein. In Figs. 2, 3, 4,and5, the influence of various axially magnetic fields on the first four vibration modes of the DWCNTs with SS, CC, SC, and CF conditions is demonstrated. The solid and dashed lines are associated with the innermost and the outermost tube, respectively. In the case of a DWCNT with SS boundary condition see Fig. 2, those magnetic fields that are symmetric with respect to the midspan point of the DWCNT would result in symmetric first and third modes as well as anti-symmetric second and fourth vibration modes. In the case of the DWCNT subjected to a uniformly applied magnetic field case 4, the profiles of the first three vibration modes of the DWCNT generally remain unchanged. However, in other cases, the vibration mode shapes are totally different from the DWCNT in the absence of the magnetic field. Interestingly, the application of the uniformly distributed magnetic field to the DWCNT would result in noncoaxial deflections of the innermost

12 3148 K. Kiani Fig. 2 First four flexural mode shapes of the DWCNT with SS boundary condition subjected to axially varying magnetic fields: a case 1, b case 2, c case 3, d case 4; λ 1 = 30, H 0 = 20, e 0 a = 2nm Fig. 3 First four flexural mode shapes of the DWCNT with CC boundary condition subjected to axially varying magnetic fields: a case 1, b case 2, c case 3, d case 4; λ 1 = 30, H 0 = 20, e 0 a = 2nm and outermost tubes in the fourth mode of vibration. As it is seen in Fig. 2, in the cases of both first and second vibrational modes, the amplitudes of the innermost tube are slightly larger than those of the outermost tube, and the DWCNT exhibits coaxial vibration modes. For higher vibration modes, the discrepancies between the deflections of the innermost and outermost tubes are more obvious. In such cases, the amplitude of the innermost is apparently larger than that of the outermost tube. In Fig. 3, the first four modes of lateral vibrations of the innermost and outermost tubes are plotted for the considered axially varying magnetic fields. For the first three modes, the amplitude of the innermost tube is generally larger than that of the outermost tube. Further, for higher modes, the discrepancies between the amplitudes of the innermost and outermost tubes are more apparent. Except the third vibration mode of the case 4, the DWCNT generally exhibits a coaxial vibration pattern. The predicted first and second vibrational modes of the DWCNT exposed to the magnetic field of case 4 are commonly identical to those of the DWCNT in the absence of the magnetic field. Concerning the fourth mode of vibration, the amplitudes of the innermost tube are generally larger than those of the innermost tube for both cases 2 and 3. However, the amplitudes of the outermost tube are generally larger than those of the outermost tube for both cases 1 and 4. In Fig. 4, the plots of the first four vibration modes of the DWCNT with SC boundary conditions are provided for different axially varying magnetic fields. Except the fourth mode of the case study 4, the amplitude of the innermost tube is larger than that of the outermost tube. As the mode number increases, the discrepancies between the amplitudes of the innermost tube and those of the outermost tube generally magnify for all studied cases. Except the third vibration mode of the case study 4, the DWCNT subjected to the axial magnetic field demonstrates a coaxial vibration pattern.

13 Characterization of free vibration 3149 Fig. 4 First four flexural mode shapes of the DWCNT with SC boundary condition subjected to axially varying magnetic fields: a case 1, b case 2, c case 3, d case 4; λ 1 = 30, H 0 = 20, e 0 a = 2nm Fig. 5 First four flexural mode shapes of the DWCNT with CF boundary condition subjected to axially varying magnetic fields: a case 1, b case 2, c case 3, d case 4; λ 1 = 30, H 0 = 20, e 0 a = 2nm For the cantilevered DWCNT exposed to an axially varying magnetic field, the first four vibrational modes are given in Fig. 5. As it is seen, the vibration patterns of the first mode of the DWCNT for the cases 1-3 are almost different from those of the DWCNT in the absence of the magnetic field. Moreover, the first three mode patterns of the DWCNT subjected to the uniform magnetic field are generally analogous to those of the non-exposed DWCNT. Except the fourth mode of the case study 4, the cantilevered DWCNT commonly exhibits a coaxial vibration pattern. Furthermore, except the third vibration mode of the case 4, the amplitudes of the innermost tube are larger than those of the outermost tube. 6 Concluding remarks Free transverse vibrations of elastically supported DWCNTs exposed to axially varying magnetic fields are studied using nonlocal Rayleigh beam theory. The dimensionless-nonlocal equations of motion are derived and numerically solved via RKPM. However, the formulations of the problem have been provided for a wide range of boundary conditions, we restrict our studies to magnetically induced DWCNTs with fully simple, fully clamped, simple-clamped, and clamped-free ends. For different axially varying magnetic fields, the frequencies and the corresponding vibration modes of the DWCNT with different boundary conditions are obtained. The influences of the small-scale parameter as well as the magnetic field strength on the characteristics of the free vibration of the DWCNT are carefully addressed and explained. The results show that the predicted frequencies magnify with the magnetic field strength. For most of the considered boundary conditions, the

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