Vibrations and instability of double-nanowire-systems as electric current carriers

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1 Moern Physics Letters B Vol. 29, No pages c Worl Scientific Publishing Company DOI: /S Vibrations an instability of ouble-nanowire-systems as electric current carriers Keivan Kiani Department of Civil Engineering, K.N. Toosi University of Technology, Valiasr Ave., P. O. Box , Tehran, Iran k kiani@kntu.ac.ir keivankiani@yahoo.com Receive 6 February 215 Revise 3 March 215 Accepte 1 April 215 Publishe 2 September 215 Current-carrying nanowires are expecte to be builing blocks of the upcoming micro-/ nano-electromechanical evices, however, little is known on their ynamic interactions in a bunle. As a pivotal step towars realizing such a crucial mechanism, this work is evote to vibrations an instability of a ouble-nanowire-system as an electric current carrier. Using Biot Savart law, the Lorentz interactional forces between oubly parallel current-carrying nanowires are evaluate. Accounting for the surface elastic energy, equations of motion pertinent to the in-plane an out-of-plane vibrations are establishe. Using analytical techniques, the explicit expressions of both static an purely ynamic parts of the nanowires isplacements are obtaine. For each component of the transverse isplacement fiel, two major vibration moes are observe: in-phase an out-of-phase moes. The frequencies associate with these vibration moes are analytically calculate. Further, the conition correspons to the ynamic instability of the system is iscovere, an the roles of initial tensile force, electric current, an interwire istance on frequencies an stability of the system are aresse. Keywors: Vibration; instability; ouble nanowires; irect current; surface effect; Laplace transform. PACS Numbers: 46.4.f, 46.7.Hg, w 1. Introuction A nanowire is a nanostructure with the with in the range of a fraction of one nanometer to several nanometers. Owing to the superior physical an chemical properties of some special metallic nanowires, they are suggeste as ieal builing blocksforaiverserangeofapplicationsincluing chemicalanphysicalsensors, 1 4 optoelectronics, 5 7 energy harvesting, 8 1 an micro-/nano-electromechanical systems MEMS/NEMS In the latter application, an ensemble of current

2 K. Kiani carrying nanowires may be use. 15,16 Our knowlege regaring vibrations of magnetically affecte nanowires carry electric current is restricte to single nanowires. 17,18 However, no report on the ynamic behavior of multiple or even ouble nanowires with the purpose of carrying electric current is now available in the literature. As a key step towars better unerstaning of the vibration mechanisms of ensembles of current-carrying nanowires, we start the work by stuying a system of oubly parallel nanowires as an electrical current carrier. Generally the length to with ratio of nanowires is more than 1. It implies that for constraine nanowires, the bening rigiity is aequately negligible such that the flexural strain energy can be rationally exclue from the total strain energy of the nanostructure. This fact becomes more reasonable when the nanowire is acte upon by a consierable tensile force. As a result, a string moel woul be a suitable alternative for beam moels. On the other han, as the with-to-length ratio ecreases, the ratio of the number of surface s atoms to the number of bulk s atoms increases. Thereby, the share of the surface energy to total strain energy increases an the mechanical behavior of the nanowire is more influence by the surface s atoms. Such an issue is calle surface effect an the classical continuum theory CCT which cannot isplay this phenomenon. To overcome this eficiency of the CCT, a sophisticate nonclassical theory i.e. surface elasticity theory was propose by Gurtin Muroch in the previous century. In the newly evelope moel, the material is ivie into the bulk zone an surface layer. The volume of the bulk is fairlyequalto the volumeofthe materialan the thickness ofthe surface layer is so tiny that it can be neglecte in mechanical analysis of the problem. For the bulk material, the constitutive equations of the CCT or other avance continuum theories with the given engineering constants of the macro-scale material can be exploite; however, for the surface layer, the mechanical characteristics of the surface incluing surface ensity, resiual stress, Lame s constant of the surface layer which are introuce by the surface continuum theory SCT shoul be use. The governing equations of the surface layer in terms of the surface stresses are somehow ientical to those of the three-imensional elasticity moel of the bulk zone, nevertheless, the constitutive relations of the surface layer are completely ifferent from those of the bulk zone. So far, the SCT of Gurtin-Muroch has been extensively exploite in preicting various mechanical behaviors of nanowires In the present work, we utilize a string moel in the context of the SCT to investigate transverse vibrations of oubly parallel current-carrying nanowires in the vicinity of each other. When an electric current passes through a nanowire, a magnetic fiel generates aroun the nanowire which is isplaye by the Biot Savart law. This states that the generate magnetic fiel at a point is irectly proportional with the electric current an is inversely proportional with its istant from the nanowire. In the case of oubly parallel nanowires which carry electric currents, the nanowires exert mutually equal istribute force on each other that can be explaine by the Lorentz

3 Vibrations an instability of ouble-nanowire-systems formula. A close scrutiny of the present work shows that these forces inherently rely on the ifference of lateral isplacements of the nanowires. Therefore, the frequency such a couple system as well as its free vibration can be affecte by the electric current within the nanowires. One of the main objectives of the present work is to explain that uner what circumferences the ynamic instability occurs within the nanosystem. Subsequently, it is investigate that how such an extreme conition coul be appropriately controlle by changing the physical parameters of the nanosystem. So far, the influence of magnetic fiels on various aspects of vibrations of nanostructures has been aresse Concerningthe ynamic behaviorsofnanowires in a magnetic fiel, Kiani 38 stuie magneto-elasto ynamics of elastically reste nanowires acte upon by a longituinal magnetic shock. A two-imensional nonlocal elasticity moel is evelope an the ynamic elastic fiel within the nanowire is analytically etermine. In another work, Kiani 39 explore raial vibrations of nanowires subjecte to a magnetic fiel accounting for ey-current loss. Using a nonlocal elasticity moel, magneto-thermo-elastic fiels within the nanowire are explicitly etermine, an the roles of influential parameters on vibrations of the magnetically affecte nanowire were aresse. Recently, free an force vibrations of a current-carrying nanowire in the presence of a longituinal magnetic fiel has been examine by Kiani. 17,18 These explorations inicate that how variation of the parameters associate with the transversely applie loa, electric current, an magnetic fiel coul affect on the transverse vibrations of nanowires. A close survey of the literature reveals that the effect of generate magnetic fiel by an electric current in a nanowire on another current-carrying nanowire has not been aresse. In view of the potential applications of current-carrying nanowires in MEMS/NEMS inustry an the nee for more accurate esign of such nanoevices, the primary focus of this work is on the accurate evaluation of the generate magnetic forces in terms of transverse isplacement of nanowires. Herein, transverse vibrations an instability of two parallel nanowires use for carrying irect current are aresse. Using Biot Savart law an surface elasticity, the equations of motion escribe transverse vibration of long ouble nanowires are obtaine in the context of small eformation. Thereafter, an analytical solution is propose an the explicit expressions of transverse isplacements of the nanowires are extracte. Free vibration an ynamic instability of the nanoscale system are note. Aitionally, the roles of the initial tensile force, irect electric current, an interwire istance on the ynamic isplacements are examine. 2. Formulations of the Nanomechanical Problem 2.1. Evaluation of the Lorentz s forces between two eforme current-carrying nanowires Consier two lengthy parallel nanowires of length l b an interwire istance that carry electric currents I 1 an I 2 as shown in Fig. 1. The geometry an mechanical

4 K. Kiani T x T I 1 I 2 lb T z T y Fig. 1. Schematic representation of oubly parallel nanowires carry electric current. properties of the nanowires are ientical, an both ens of them are prohibite from any lateral movement. The origin of rectangular coorinate system has been attache to the left support of the first nanowire such that the x-axis is coincient with its revolutionary axis an the z-axis points ownwar. The unit base vectors associate with the x, y an z axes are represente by e x, e y, an e z, respectively. In all presente formulations, the subscripts 1 an 2 stan for the parameters associate with the first an the secon nanowire, respectively. The transverse isplacement fiels of the ith nanowire along the y an z axes in orer are represente by v i = v i x,t an w i = w i x,t where t is the time parameter. The eformations v i an w i are also calle in-plane an out-of-plane transverse isplacements, respectively, since the eformations v i occur in the plane passes through the revolutionary axes of the uneforme nanowires while w i represent the eformation occurring in the planes are perpenicular to the above-mentione plane. Herein, transverse vibrations of nanowires an their interactions are of interest. Because of passing the electric current through the nanowires, magnetic fiels are generate aroun the nanowires. Accoring to the Biot Savart law for lengthy nanowires, the magnetic fiel vector resulte from the first eforme nanowire at the location of the secon eforme nanowire can be evaluate as: B 1 = µ 2π I 1 e, 1 7 T.m where µ is the permeability of the free space which is given by µ = 4π 1 A, = is the istance between eforme nanowires, is a vector that specifies the location of the center of the first nanowire with respect to that of the secon nanowire, an e is its unit base vector. Using Lorentz s formula, the resulte magnetic fiel exerts a force on the ajacent current-carrying nanowire. Such a force per unit length of the secon nanowire can be calculate by: f 21 = I 2 B

5 Vibrations an instability of ouble-nanowire-systems Accoring to the geometry of the eforme nanowires, 17,18 v I j = I j e x +I j j x e y + w I j j x ; j = 1,2,an = + ve y + we z where v = v 2 v 1 an w = w 2 w 1. By substituting Eq. 1 into Eq. 2, f 21 = µ I 1 I 2 2π+ v 2 + w w2 x + v v 2 x + w w 2 e x x w v 1 x + v w 1 + v e y x v 2 w v 1 x x + v w 1 w e z. 3 x For small ynamic isplacements, by neglecting the proucts of isplacement an their first erivatives an excluing the longituinal component of the magnetic force, Eq. 3 is approximate by: f 21 = µ I 1 I 2 2π [ 1 v e y + w ] e z. 4 Similarly, it can be shown that the exerte force on the first current-carrying nanowire ue to the generate magnetic fiel by the secon nanowire is calculate by: f 12 = µ I 1 I 2 2π [ 1 v e y + w ] e z Equations of motion using surface elasticity Base on the surface elasticity, by neglecting the bening rigiity of the lengthy nanowires, the governing equations escribe transverse vibrations of the nanosystem which are as: 18,4 ρ b A b +ρ S 2 v i t 2 T +H 2 v i x 2 = q v i, 6a ρ b A b +ρ S 2 w i t 2 T +H 2 w i x 2 = q w i, 6b where ρ b, A b, ρ, S, T, H, q vi an q wi are the bulk ensity, cross-sectional area, surface ensity, pyrami of the nanowire s cross-section, initial tensile force within the nanowire, resiual surface stress uner unconstraine conitions, an the exerte Lorentz s forces along the y- an z-axes of the ith nanowire. By introucing Eqs. 4 an 5 to Eqs. 6a an 6b an taking into account the weight of the nanowires, the equations of motion of oubly parallel current-carrying nanowires in terms of transverse isplacements are expresse by: ρ b A b +ρ S 2 v 1 t 2 T +H 2 v 1 x 2 + µ I 1 I 2 2π 2 v 2 v 1 = µ I 1 I 2 2π, 7a

6 K. Kiani ρ b A b +ρ S 2 w 1 t 2 T +H 2 w 1 x 2 µ I 1 I 2 2π 2 w 2 w 1 = ρ b A b g, 7b ρ b A b +ρ S 2 v 2 t 2 T +H 2 v 2 x 2 µ I 1 I 2 2π 2 v 2 v 1 = µ I 1 I 2 2π, 7c ρ b A b +ρ S 2 w 2 t 2 T +H 2 w 2 x 2 + µ I 1 I 2 2π 2 w 2 w 1 = ρ b A b g. 7 Since the eformation fiels of the nanowires are resulte from both static an ynamic loas, therefore, we split up isplacements into static an ynamic parts as: v i x,t = vi s x+v i x,t, w ix,t = wi s x+w i x,t; i = 1,2, 8 where the superscripts s an are pertinent to the static an ynamic states, respectively. By introucing Eq. 8 to Eqs. 7a 7, the static eformation of the nanowires ue to the magneto-static Lorentz s force as well as nanowires weight force can be etermine from the following relations: T +H 2 v s 1 x 2 + µ I 1 I 2 2π 2 vs 2 vs 1 = µ I 1 I 2 2π, T +H 2 w s 1 x 2 µ I 1 I 2 2π 2 ws 2 ws 1 = ρ ba b g, T +H 2 v s 2 x 2 µ I 1 I 2 2π 2 vs 2 v s 1 = µ I 1 I 2 2π, 9a 9b 9c T +H 2 w2 s x 2 + µ I 1 I 2 2π 2 ws 2 ws 1 = ρ ba b g, with the following bounary conitions: 9 v s i = vs i l b = ; w s i = ws i l b = ; i = 1,2. 1 Further, the purely ynamic isplacements of the nanowires are governe by the following relations: ρ b A b +ρ S 2 v 1 t 2 T +H 2 v 1 x 2 + µ I 1 I 2 2π 2 v 2 v 1 =, ρ b A b +ρ S 2 w 1 t 2 T +H 2 w 1 x 2 µ I 1 I 2 2π 2 w 2 w 1 =, ρ b A b +ρ S 2 v 2 t 2 T +H 2 v 2 x 2 µ I 1 I 2 2π 2 v 2 v 1 =, ρ b A b +ρ S 2 w2 t 2 T +H 2 w 2 x 2 + µ I 1 I 2 2π 2 w 2 w1 =, with the following bounary conitions: v i,t = v il b,t =, w i,t = w il b,t =, 11a 11b 11c

7 Vibrations an instability of ouble-nanowire-systems an the following general initial conitions: v i x, = v ix, w i x, = w ix; i = 1,2, vi t x, = v w 13 i ix, t x, = ẇ ix, where v i x an w i x are the initial eformations in aition to those of static isplacements whereas v i x an ẇ i x enote their initial velocities. 3. An Analytical Solution In this part, it is aime to etermine the static an pure ynamic isplacements which are governe by Eqs. 9a 9 an 11a 11. To this en, the following imensionless quantities are consiere: ξ = x l b, τ = t l b E b ρ b, v i = v i l b, w i = w i l b, = l b, T = T E b A b, H = H E b A b, f = µ I 1 I 2 l 2 b 2π 2 E b A b, m = ρ S ρ b A b, fg = ρ ba b l b g E b A b Static analysis By introucing Eq. 14 to Eqs. 9a 9, the imensionless governing equations corresponing to the static eformation of the nanowires are given by: 14 T + H 2 v s 1 ξ 2 + f v s 2 v s 1 = f, T + H 2 w s 1 ξ 2 f w s 2 ws 1 = f g, T + H 2 v s 2 ξ 2 f v s 2 v s 1 = f, 15a 15b 15c with the following bounary conitions: T + H 2 w s 2 ξ 2 + f w s 2 ws 1 = f g, 15 v s i = v s i1 = ; w s i = w s i1 =. 16 By solving the couple Eqs. 15a an 15c as well as Eqs. 15b an 15 for v i s an w i; s i = 1,2, respectively, the explicit expressions of the imensionless static isplacements of two nanowires are erive as: fe v 1ξ s = v 2ξ s = 1 tan 2 sin f e ξ cos f e ξ, 17a w s 1 ξ = ws 2 ξ = 1 2 f w ξ1 ξ, 17b

8 K. Kiani where f e = f T + H an f w = f g T + H. Accoring to Eq. 17a, the magnitues of the in-plane transverse isplacements of ouble nanowires are the same, however, they have opposite irections. It is interprete by this fact that the nanowires ientically attract each other ue to the similar exerte magnetic forces see Eqs. 4 an Dynamic analysis By introucing Eq. 14 to Eqs. 11a 11, the imensionless governing equations of the nanoscale system in terms of purely ynamic isplacements take the following form: 2 v 1 τ 2 T s 2 v 1 ξ 2 + f s v 2 v 1 =, 2 w 1 τ 2 T s 2 w 1 ξ 2 f s w 2 w 1 =, 2 v 2 τ 2 T s 2 v 2 ξ 2 f s v 2 v 1 =, 2 w 2 τ 2 T s 2 w 2 ξ 2 + f s w 2 w 1 =, 18a 18b 18c 18 where T s = T + H 1+ m an f s = f 1+ m. Furthermore, the following bounary conitions see Eq. 12: v i,τ = v i 1,τ =, w i,τ = w i 1,τ =, 19 an the following initial conitions shoul be satisfie see Eq. 13: v iξ, = v i ξ, w iξ, = w i ξ; i = 1,2, v i τ ξ, = v i ξ, w i τ ξ, = w i ξ, 2 where v i ξ = l b ρb E b v i ξ an w i ξ = l b ρb E b ẇ i ξ. Now the imensionless isplacements of the nanowires, which are amissible with the bounary conitions, are consiere as follows: v i ξ,τ = a im τsinmπξ, w i ξ,τ = m=1 b im τsinmπξ; i = 1,2, 21 where a im an b im are the time-epenent parameters associate with the transverse isplacements of the mth vibration moe of the ith current-carrying nanowire. By substituting Eq. 21 into Eqs. 18a an 18c as well as Eqs. 18b an 18, an taking the Laplace transform of both sies of the resulting relations, one can m=1

9 Vibrations an instability of ouble-nanowire-systems arrive at: [ s 2 + T s mπ 2 f s fs f s s 2 + T s mπ 2 f s an [ s 2 + T s mπ 2 + f s f s ] L f s s 2 + T s mπ 2 + f s where L is the Laplace transform operator an 1 ] L { a1m a 2m { b1m b 2m a im = 2 v i sinmπξξ, a 1 im τ = 2 v i ξsinmπξξ, b im = 2 1 b 1 im τ = 2 w i sinmπξξ, w i ξsinmπξξ. By solving Eqs. 22 an 23 for L{a im } an L{b im }, where r 1m = } a 1m s+ a 1m τ = a 2m s+ a, 2m τ 22 } b 1m s+ b 1m τ = b 2m s+ b, 2m τ L{a im } = A i1ms 3 +A i2m s 2 +A i3m s+a i4m s 2 +r 2 1m s2 +r 2 2m, 25a L{b im } = B i1ms 3 +B i2m s 2 +B i3m s+b i4m s 2 +r 2 2m s2 +r 2 3m, 25b mπ 2 T s 2 f s, r 2m = mπ Ts, r 3m = mπ 2 T s +2 f s, 26 A 11m = a 1m, A 12m = a 1m τ, A 21m = a 2m, A 22m = a 2m τ, A 13m = a 1m Ts mπ 2 f s a2m f s, A 14m = a 1m τ T s mπ 2 f s a 2m τ f s, A 23m = a 2m T s mπ 2 f s a 1m f s, 27 A 24m = a 2m τ T s mπ 2 f s a 1m τ f s,

10 K. Kiani an B 11m = b 1m, B 12m = b 1m τ, B 21m = a 2m, B 22m = b 2m τ, B 13m = b 1m T s mπ 2 + f s +b 2m f s, B 14m = b 1m τ T s mπ 2 + f s + b 2m τ f s, B 23m = b 2m T s mπ 2 + f s +b 1m f s, 28 B 24m = b 2m τ T s mπ 2 + f s + b 1m τ f s. By splitting up the fractional expressions in Eqs.25a an25b to prouce simpler fractions, an taking the inverse Laplace transform of the resulting relations, the imensionless purely ynamic isplacements are evaluate as follows: v i ξ,τ Ai3m r1m 2 = A i1m r2m 2 r2 1m m=1 Ai3m r2m 2 + A i1m r1m 2 r2 2m w iξ,τ Bi3m r 2 = 1mB i1m r3m 2 r2 2m m=1 Bi3m r 2 + 3mB i1m r2m 2 r2 3m 4. Results an Discussion 4.1. Special cases cosr 1m τ+ cosr 2m τ+ cosr 2m τ+ cosr 3m τ r 1m = r 2m = r 3m : No ynamic interactions Ai4m r1m 2 A i2m r 1m r2m 2 r2 1m sinr 1m τ Ai4m r2m 2 A i2m r 2m r1m 2 r2 2m sinr 2m τ sinmπξ, 29a Bi4m r2mb 2 i2m r 2m r3m 2 r2 2m sinr 2m τ Bi4m r3mb 2 i2m r 3m r2m 2 r2 3m sinr 3m τ sinmπξ. 29b When f =, let us consier r 1m = r 2m = r 3m = r m ; hence, A ij+2m = rm 2 A ijm an B ij+2m = rm 2 B ijm; j = 1,2. By substituting these values into Eqs. 29a an 29b, the purely ynamic isplacements of the nanowires are expresse by: v iξ,τ = A i1m cosr m τ+ A i2m sinr m τ sinmπξ, 3a r m w iξ,τ = m=1 m=1 B i1m cosr m τ+ B i2m sinr m τ sinmπξ. r m b

11 Vibrations an instability of ouble-nanowire-systems In this case, as it is obvious from Eqs. 3a an 3b, vibrations of two nanowires become ecouple an each nanowire vibrates in accorance with its own initial conitions. In other wors, no interaction between nanowires exists since f = r 11 = : Dynamic instability This case correspons to f = π2 2 T. By substituting this expression into Eq. 29a an removing the ambiguity via L Hopital rule, the isplacements of the nanowires along the y-axis are erive as: v iξ,τ Ai31 = r A i41 Ai31 r21 2 r21 2 τ + A i11 r21 2 Ai3m r1m 2 A i1m r2m 2 + r2 1m m=2 Ai3m r 2 + 2mA i1m r1m 2 r2 2m cosr 1m τ+ cosr 21 τ+ cosr 2m τ+ Ai41 r 2 21 A i21 r21 3 Ai4m r1m 2 A i2m r 1m r2m 2 r2 1m sinr 1m τ Ai4m r2ma 2 i2m r 2m r1m 2 r2 2m sinr 21 τ sinπξ sinmπξ. sinr 2m τ As it is obvious from Eq. 31, the in-plane isplacements of the nanowires grow as time goes by an the ynamic instability occurs within the nanosystem. By increasing the isplacements of the nanowires, the resulting elastic fiels approach to their ultimate values until the nanoscale system collapses Free vibration an ynamic instability Let the purely ynamic isplacements of the nanowires can be expresse in the following harmonic form: v iξ,τ = a im e i mτ sinmπξ, 32a w iξ,τ = m=1 b im e i mτ sinmπξ; i = 1,2 m=1 32b where i = 1, a im an b im represent the amplitues of the ynamic response of the ith nanowire along the y- an z-axes, respectively In-plane vibration moes an frequencies By substituting Eq. 32 into Eqs. 18a an 18c an solving the resulting equations for m, the natural frequencies as well as amplitue ratios associate with the vibrations of the nanowires along the y-axis are obtaine as follows: a1m = 1, m I = mπ 2 T s 2 f s, 33a a 2m I

12 K. Kiani a1m a 2m II = 1, m II = mπ Ts, 33b where Eq. 33a enotes out-of-phase vibration moe of the current-carrying nanowires while Eq. 33b represents their in-phase vibration moe. As it is explaine earlier, the out-of-phase vibration enangers safety of the nanosystem when mi =. The most critical case is 1 I = in which leas to the following critical values of the initial tensile force, electric current, an interwire istance: T,cr = µ I 2 lb 2 π 3 2 H, I cr = π l b cr = Il b π 34a πt +H µ, 34b µ πt +H. 34c Equations 34a 34c state that when the existing initial tensile force in the nanowires is lower than its critical value or when the electric current is greater than its critical level or when the interwire istance is lower than its critical value, ynamic instability woul occur within the nanoscale system. In such a conition, any exerte lateral motion along the y-axis on each nanowire can lea to large isplacements in both nanowires. As it is seen in Eq. 33a, increasing of the surface stress effect, initial tensile force, or the interwire istance woul provie a more stable system, however, an increase of the electric current woul reuce the frequency until its value becomes zero an the nanosystem becomes ynamically unstable. To show the role of the influential factors on the funamental frequency, an inclusive parametric stuy is performe. Consier the silver nanowires with the following mechanical an geometry properties: E b = Pa, ρ b = 1,5 kg/m 3, ρ = 1 7 kg/m 2, τ =.89 N/m, l b = 15 nm, an r = 15 nm. In Figs. 2a 2c, the influences of the interwire istance an electric current on the funamental frequency of the nanosystem are emonstrate. To stuy the problem in a more general framework, the imensionless electric current, Ī, an the imensionless interwire istance, are efine by: Ī = I µ 2πE b A b an = /l b where I 1 = I 2 = I an f s = Ī/ 2. The plotte results have been provie for three levels of the initial tensile force within the nanowires, namely T =.5 1 3, an As it is obvious in Figs. 2a 2c, by increasing the initial tensile force within the nanowires, the funamental frequency of the nanosystem magnifies. The main reason of this fact is that the nanowires become stiffer as the initial tensile force increases; thereby, all the frequencies of the nanosystem increase. The plotte results reveal that the funamental frequency of the nanosystem increases as the interwire istance magnifies. However, for each level of the interwire istance, the funamental frequency rastically ecreases with the electric current until it approaches zero an the nanosystem becomes ynamically unstable. Such a fact is

13 Vibrations an instability of ouble-nanowire-systems a b c ϖ I I I Fig. 2. Variation of the imensionless funamental frequency in terms of the imensionless interwire istance an electric current for ifferent levels of the initial tensile force: a T =.5 1 3, b T = an c T = more obvious for lower levels of the interwire istance. Accoring to the emonstrate results in Figs. 2a 2c, the electric current correspons to the initiation of ynamic instability increases as the interwire istance increases Out-of-plane vibration moes an frequencies By substituting the equivalent values of w i from Eq. 32b into Eqs. 18b an 18 an solving the resulting set of equations for the unknowns b im, the if an only if conition to obtain a nontrivial solution results in: b1m = 1, m I = mπ Ts, 35a b1m b 2m b 2m II I = 1, m II = mπ 2 T s +2 f s. 35b Equations 35a an 35b isplay that there exists two moes of vibrations for oubly parallel nanowires along the z-axis. The first vibration moe i.e. with the lower frequency, Eq. 35a, shows the in-phase vibration pattern since the moes of two nanowires have ientical amplitues an the amplitues ratio is inepenent of the frequency as well as the interactional effects between the nanowires. As an example, consier two nanowires is which similarly eflecte only in the z irection

14 K. Kiani at time t =. If we let the nanosystem to vibrate freely with such an initial conition, two nanowires woul vibrate ientically i.e. with similar amplitues an zero phase elay with respect to each other. Equation 35a isplays that the vibration in this case is not affecte by the electric current within the nanowires such a fact can be reaily explaine via Eqs. 4 an 5 by letting w = ; therefore, the z-component of the Biot Savart force woul vanish. This is the main reason of this fact that the vibration of each current-carrying nanowire is not influence by the vibration of its neighboring one. The secon vibration moe, Eq. 35b, isplays the out-of-phase pattern of the vibration. In this case, both nanowires eflect in opposite irections an the interactional effects of two nanowires along the z-axis are incorporate into the corresponing frequency. When such interactions are negligible i.e. f, two nanowires vibrate inepenently an their ynamic isplacements coul be reaily evaluate from Eqs. 3a an 3b. Equation 35 also explains that no instability woul be generate in the nanoscale system ue to any cause of free or force vibration along the z-axis. 5. Conclusions By implementing a surface theory of elasticity, transverse vibrations of a oublenanowire-system for carrying electric current are investigate. For this purpose, the interactional Lorentz forces between the current-carrying nanowires accounting for their transverse isplacements are erive. Thereafter, the equations of motion of the nanosystem are obtaine in the context of small eformations. The explicit expressions of transverse isplacements of nanowires are erive. The instability of such a system plus to the influential factors on the frequencies of the system are isplaye in etail. The major obtaine results are summarize as follows: i For each component of transverse isplacement, two vibration moes are etecte: in-phase an out-of-phase moes. For those transverse isplacements in the plane of nanowires, ynamic instability woul be possible in the out-ofphase moes; however, for out-of-plane isplacements, no ynamic instability woul occur. The explicit expressions of the in-plane transverse isplacements correspon to the ynamic instability are also evaluate. ii Concerning the in-plane vibration, both in-phase an out-of-phase vibration moes an their corresponing frequencies are analytically obtaine. For a specifie value of a moe number, the frequencies of the out-of-phase moes are lower than those of the in-phase moes. iii Regaring the out-of-plane vibration, the analytical expressions of both inphase an out-of-phase vibration moes as well as their pertinent frequencies are erive. For a given moe number, the flexural stiffness of the system in the out-of-phase vibration moes is greater than that of the in-phase vibration moes

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