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European Journal of Mechanics A/Solids 54 (2015) 132e145

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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/ejmsol

Vibration insight of a nonlocal viscoelastic coupled multi-nanorodsystem

Danilo Karli�ci�c a, Predrag Kozi�c a, Tony Murmu b, *, Sondipon Adhikari c

a Faculty of Mechanical Engineering, University of Ni�s, A. Medvedeva 14, 18000 Ni�s, Serbiab School of Engineering, University of the West of Scotland, Paisley PA12BE, UKc College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK

a r t i c l e i n f o

Article history:Received 14 October 2014Accepted 30 June 2015Available online 9 July 2015

Keywords:Nonlocal effectsComplex eigenvalueMultiple-nanorod system

* Corresponding author.E-mail address: [email protected] (T. Murm

http://dx.doi.org/10.1016/j.euromechsol.2015.06.0140997-7538/© 2015 Elsevier Masson SAS. All rights re

a b s t r a c t

Longitudinal vibration of viscoelastic multi-nanorod system (VMNS) is studied. Based on the D' Alem-bert's principles, nonlocal and viscoelastic constitutive relations, the system of m partial differentialequations are derived which described the motion of the presented nano-system. ClampedeClampedand ClampedeFree boundary conditions and two different chain systems, namely “Clamped-Chain” and“Free-Chain” are illustrated. The method of separations of variables and trigonometric method are uti-lized for solutions. The analytical expressions for critical viscoelastic parameters and asymptotic fre-quencies are presented. The predicted results are validated with results obtained by direct numericalsimulations and results from literature. The effects of nonlocal parameter, number of nanorods, visco-elastic material constant and parameter of viscoelastic layer on the complex eigenvalue are discussed indetails.

© 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction

Recently, growing interest in the dynamic response of nano-structures elements like a nanorods, nanobeams or nanoplates,plays an important role in the development of nanodevices.Therefore, the issue of vibration behavior of nanostructures ele-ments has become very important from the practical point of viewand it has wide application in nanotechnology. The nanodevicesinclude biosensors (Ziegler, 2004; Sotiropoulou and Chaniotakis,2003; Wang, 2005; Wang et al., 2003; Shen et al., 2012; Ali et al.,2009; Chowdhury et al., 2011), mass sensors (Lee et al., 2010;Mehdipour et al., 2011; Murmu and Adhikari, 2011), nano-resonators (He et al., 2005; Liu et al., 2011), gas sensors (Basu andBhattacharyya, 2012; Llobet, 2013), nanoopto-mechanical system(Hierold et al., 2007; Lu et al., 2007) etc. Nanomaterial's such ascarbon nanotubes (CNTs) (Iijima, 07 November 1991), boron nitridenanotubes (BNNTs) (Chopra et al., 18 August 1995), zinc oxidenanotubes (ZnO) (Liu and Zeng, 2009) and graphene sheet (Geimand Novoselov, 2007) are the basis material of many

u).

served.

nanostructures and nanodevices. These nanomaterial's haveextraordinarily properties resulting from their nanoscale di-mensions (Guz et al., 2007; Gouadec and Colomban, 2007; Kuoet al., 2005; Dresselhaus et al., 2004; Ruoff et al., 2003). Perform-ing controlled experiments at the nano-level is very difficult andexpensive. Therefore, development appropriate mathematicalmodels based on Eringen's continuum theory, which takes intoaccount size effect and atomic forces is very important. By ignoringthese effects in the development of mathematical models ofnanoscale structures can cause completely incorrect solutions andhence erroneous designs. According to a paper (Eringen and Edelen,1972), Eringen derived a constitutive relation in integral form,based on the assumption that the stress at the point is function ofthe strain at all points of the elastic body. Since then, many re-searchers have contributed to the development of nonlocal con-tinuum theory and application in mathematical modeling ofnanostructures.

Studying the static and dynamical behavior of elastic nanorod,nanobeam and nanoplates subject of many papers (Ansari et al.,2010; Akg€oz and Civalek, 2013; Aydogdu and Filiz, 2011; Wanget al., 2006). One of the first applications of the nonlocal contin-uum theory in nanotechnology is the work presented by Peddiesonet al. (2003). They used the nonlocal elasticity theory to develop

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D. Karli�ci�c et al. / European Journal of Mechanics A/Solids 54 (2015) 132e145 133

nonlocal EulereBernoulli beam for different boundary conditions.Also, they are considering application of cantilever beam as micro-electromechanical actuator. Lately, nonlocal theories for theEulereBernoulli, Timoshenko, Reddy and Levinson beams arederived by Reddy (Reddy, 2007) in a unique way using Hamilton'sprinciple and nonlocal constitutive relation of Eringen. The authoris obtain analytical solution of banding, vibration and buckling andshowed the effect of nonlocal parameter on deflections, bucklingload and natural frequencies. In the paper, presented by Reddy andPang (Reddy and Pang, 2008), the equations of motion ofEulereBernoulli and Timoshenko beam theories are reformulatedby Eringen nonlocal theory, and then used to evaluate staticbending, vibrations, and buckling response of carbon nanotubeswith several boundary conditions. The influences of nonlocalparameter and aspect ratio on the natural frequency, staticdeflection and buckling load are considered. The small scale effecton the axial vibration of a tapered nanorod based on the nonlocalelasticity theory is studied by Danesh et al. (2012). The governingequations are solved by using the differential quadrature methodfor three type of boundary conditions, clampedeclamped (CeC),clampedefree (CeF) and fixed-attached spring boundary condi-tions. Also, it is show that the nonlocal effect plays an importantrole in the axial vibration of nanorods. The free vibration of double-nanorod system is investigated by Murmu and Adhikari (2010).Based on Eringen's nonlocal elasticity theory and methods of sep-arations of variables, they obtained analytical solutions for naturalfrequencies for two types of boundary conditions, Clampede-Clamped and ClampedeFree. A carbon nanotube embedded in anelastic medium was modeled by Aydogdu (2012) as a nanorodsurrounded with elastic layers by using the Eringen's nonlocalelasticity theory. The author compared the longitudinal frequenciesfor the nonlocal and classical continuum models. Narendar andGopalakrishnan (2011) considered the nonlocal effects in theaxial wave propagationwithin the system of two nanorods coupledwith an elastic layer. The authors studied the influence of small-scale (nonlocal) parameter and stiffness of the layer on axialwave propagation. Hsu et al. (2011) investigated the longitudinalfrequencies of cracked nanobeams for different boundary condi-tions and using the theory of nonlocal elasticity. Awide study of thelongitudinal, transversal and torsional vibration and instability wasconducted by Kiani (2013) for a system of SWCNTs. Şimşek (2012)used a Galerkin approach to obtain the natural frequencies forthe longitudinal vibration of axially functionally graded taperednanorods. The author performed the analysis for nanorods with avariable cross-section, differently tapered ratios, material proper-ties and boundary conditions. Longitudinal vibration of nanorods,which takes the nonlocal long-range interactions into account, wasexamined by Huang (2012). Chang (2012) considered the small-scale effects to investigate the axial vibration of elastic nanorods.The author used the differential quadrature method to solve themodel equations. Filiz and Aydogdu (2010) analyzed the longitu-dinal vibration of carbon nanotubes with heterojunctions using thenonlocal elasticity for different lengths, diameters and chirality ofheterojunctions. Karli�ci�c et al. (2015) performed a detailed analysisof the free longitudinal vibrational response of the systemwith twocoupled viscoelastic nanorods and investigated the influence ofdifferent physical parameters on complex natural frequencies.Recently, Adhikari et al. (2013) examined the free and forced lon-gitudinal vibration of the nonlocal nanorod by using two types ofnonlocal damping models. The authors obtained the partial dif-ferential equation of motion in terms of axial displacements andthen solved by analytical and finite element method. Exactanalytical solutions for cut-off frequency are also obtained whenthe number of mode in the complex natural frequency tends to theinfinity.

Damping properties appear in all nanostructures systems andhelp to better define suppression vibration behavior. Understand-ing their source is an important issue, not only for design and ap-plications in nanoengineering practice but also to understand theinner workings of the nanomaterial's and nanostructures elements.Therefore, different technologies have been developed to investi-gated the damping effects on the vibration characteristics ofdamped or viscoelastic nanostructures (Imboden and Mohanty,2014). Viscoelastic materials displaying both solid-like and fluidlike characteristics, are common in polymeric structures. Energydissipation or portion of energy storage from fluid-like part isirrecoverable and can be separated from energy of deformationusing a complex modulus, which is represented by real and imag-inary parts named storage and loss modulus, respectively. Thus,should be paid a more attention to the study of the dynamicbehavior of the nanostructures with viscoelastic properties. Theapplication of the nonlocal continuum theory to describe the in-ternal and external damping effects in the structure elements at thenanoscale level have started recently. Lei et al. (2013a) proposedtwo type nonlocal dumped viscoelastic model of nanobeam basedon nonlocal viscoelastic constitutive relations for vibration analysis.A transfer functionmethods is applied to obtain analytical solutionsof free vibration for EulereBernoulli nanobeam with differentboundary conditions. Also, the influences of material and geometricparameters on the complex eigenvalue are investigated. In thepaper by Lei et al. (2013b) the dynamical behavior of nonlocalviscoelastic damped nanobeam has been investigated by using theKelvineVoigt viscoelastic model, velocity-dependent externaldamping and Timoshenko beam theory. The authors showed thatnonlocal damped beams have maximum frequencies, calledasymptotic frequencies, and also possess an asymptotic criticaldamping factor. The numerical results are presented on carbonnanotube example. In the paper by Paola et al. (2013) the dynamicsof a nonlocal Timoshenko beam is presented. Nonlocal effects aremodeled as long-range volume forces and moments mutuallyexerted by non-adjacent beam segments, that contribute to theequilibrium of any beam segment along with the classical localstress resultants. Also, model is provided with elastic and viscouslong-range volume forces and moments which are linearlydependent on the product of the volumes of the interacting beamsegments and on generalized measures of their relative motion,based on the pure deformation modes of the beam. The numericalresults are presented for different values of nonlocal parameters.Vibration behavior of boron nitride nanotubes coupled by visco-Pasternak layer under a moving nanoparticle was proposed byGhorbanpour Arani and Roudbari (2013) who investigated thenonlocal piezoelastic surface effect. Pouresmaeeli et al. (2013) re-ported on vibration characteristics of simply supported viscoelasticorthotropic nanoplates resting on viscoelastic foundation. The au-thors are obtained closed form solutions of complex frequencieswhich includes influence of nonlocal parameter and structuraldamping of the nanoplate and foundation. They showed that thefrequency significantly decreases with increasing the structuraldamping.

By browsing the literature, the authors have found that someinteresting papers about physics of multiple system of nanorods(Lao et al., 2002; Wen et al., 2003; Schulz et al., 2005). Nanorodsgrowing from nanowire core can be viewed as multi-nanorodsystem Fig. 1. Mechanical modeling of those systems can be ofgreat progress for their application and comprehension since ex-periments on nano-scale level cannot be well controlled. Therefore,this paper represents an extension of work Karli�ci�c et al. (2015), forsystems of multiple coupled nanorods with viscoelastic properties.In the following of this work, it is presented an analytical solution ofaxial vibrations of a viscoelastic multi-nanorod system embedded

Fig. 1. ZnO side nanorods growing on central nanowire cores (Wen et al., 2003) ePhysical model.

D. Karli�ci�c et al. / European Journal of Mechanics A/Solids 54 (2015) 132e145134

in viscoelastic medium. We assume that the system underconsideration is composed of a set of m nonlocal, parallel andidentical viscoelastic nanorods coupled by viscoelastic layers, withgiven stiffness and dumping parameters. By applying the D'Alembert's principle, nonlocal and viscoelastic constitutive re-lations the set ofm coupled partial differential equations of motionare derived. The closed form solutions for complex eigenvalues areobtain by using method of separations of variables and trigono-metric method for different number of nanorods and differentboundary conditions. Also it is obtain analytical expressions for theasymptotic frequencies and critical damping factor for undampedand damped VMNS. In order to validate of present analyticalresearch, the obtained results are compared with results obtainedby numerical methods and results reported in literature. The in-fluence of the nonlocal parameter, number of nanorods andviscoelastic material constant on the real and imaginary part ofcomplex eigenvalues of the system is also determined throughnumerical experiment for boundary conditions and “Chain”systems.

The present study is very useful in designing of nano-electromechanical devices, and it is also useable in analyzingdamping effect on the dynamic excitation systems at the nano-level.

2. Mathematical model of VMNS

2.1. Brief introduction in nonlocal constitutive relations

The basic assumption in the nonlocal elasticity theory, thatthe stress at a point x is observed to be a function not only on astrain at that point x but also on strains at all other points of abody. Based on this, Eringen has introduced material parameterin the constitutive relations which takes into account the sizeeffect. The integral form (Eringen and Edelen, 1972) of nonlocallinear constitutive relation for a three-dimensional body aregiven as

sijðxÞ ¼Z

aðjx� x0j; tÞCijklεklðx0ÞdVðx0Þ; cx2V ; (1a)

sij;j ¼ 0; (1b)

εij ¼12�ui;j þ uj;i

�; (1c)

where Cijkl is the elastic modulus tensor for classical isotropicelasticity; sij and εij are stress and strain tensors, respectively and uiis displacement vector. With a(jx�x0j,t) we denote the nonlocalmodulus or attenuation function which incorporates nonlocal ef-fects into the constitutive equation at the referencepoint x produced by local strain at the source x0. The above absolutevalue of difference jx�x0j denotes the Euclidean metric. Theparameter t ¼ (e0a)/l where l is the external characteristic length(crack length, wave length), a describes internal characteristiclength (lattice parameter, granular size and distance between CeCbounds) and e0 is a constant appropriate to each material that canbe identified from atomistic simulations or by using dispersivecurve of the BorneKarman model of lattice dynamics.

The main disadvantage of the integral nonlocal constitutiverelations is their complexity form for using to solve particularproblems in nanomechanics. According to the paper (Eringen,1983), Eringen is presented a differential form of constitutive re-lations, for the one-dimensional case follows

sxx � md2sxx

dx2¼ Eεxx; (2a)

sxz � md2sxz

dx2¼ Ggxz; (2b)

where E and G are elastic modulus and shear modulus of the beam,respectively; m ¼ ðe0aÞ2 is the nonlocal parameter (length scales),sxx, sxz are normal and shear nonlocal stresses, respectively, andεxx ¼ vu/vx is axial deformation. As the exact value of nonlocalparameter is scattered and depends on various parameters, in thepresent study, the free vibration analysis of CMNRS is carried outassuming for e0a to be from 0 to 2 [nm]. If e0a ¼ 0, i.e. there is noinfluence of non-localness (same as in macro-scale modeling) weget back to normal stressestrain relation. The constitutive relationfor nonlocal viscoelastic body can be obtained by combiningnonlocal elasticity and viscoelasticity theory (Lei et al., 2013b).Therefore, for one-dimensional nonlocal viscoelastic solids,constitutive relations for KelvineVoigt viscoelastic model are givenby

sxx � md2sxx

dx2¼ Eðεxx þ td _εxxÞ; (3a)

sxz � md2sxz

dx2¼ Gðgxz þ td _gxzÞ; (3b)

where td is the viscous damping coefficient of nanorod. If td ¼ 0, i.e.there is no influence of internal viscosity we get back to nonlocalelastic constitutive relation. In the next section, we derived partialdifferential equations based on nonlocal viscoelastic constitutiverelations (3a) for coupled system of nanorods.

2.2. The dynamic equations of VMNRS

Let us consider a system of m viscoelastic nanorods coupledby linear viscoelastic layer for two types of boundary conditions,ClampedeClamped and ClampedeFree, as shown in Fig. 2. Also, onthese figures shows two different cases of connections VMNRSwitha fixed base, so-called chain systems. In the first case of chainsystem i.e. “Clamped-Chain”, it is assumed that the first and last

Fig. 2. The VMNRS for different boundary conditions: a) Clamped-Chain system with CeC boundary conditions, b) Clamped-Chain system with CeF boundary conditions, c) Free-Chain system with CeC boundary conditions, d) Free-Chain system with CeF boundary conditions e Mechanical model.



nanorod are connected with fixed base by viscoelastic layers ofstiffness k0 and km and damping b0 and bm (Fig. 2a) and b)).

For the second case of chain system or “Free-Chain”, it isassumed that the first and last nanorods without connection withfixed base i.e. parameters of the first and last viscoelastic layer areequal to zero (k0 ¼ km ¼ 0 and b0 ¼ bm ¼ 0) (Fig. 2c) and d)). Theother nanorods are also joined by an axially distributed visco-elastic layers with stiffness and damping per length denote ask1 ¼ k2 ¼ … ¼ ki ¼ … ¼ km�1 ¼ k andb1 ¼ b2 ¼ … ¼ bi ¼ … ¼ bm�1 ¼ b, respectively. The system of m

m€ui � e d2uidx2

þ tdd2 _uidx2

!þ kiðui � uiþ1Þ þ bið _ui � _uiþ1Þ þ ki�1ðui � ui�1Þ þ bi�1ð _ui � _ui�1Þ

¼ m d2

dx2½m€ui þ kiðui � uiþ1Þ þ bið _ui � _uiþ1Þ þ ki�1ðui � ui�1Þ þ bi�1ð _ui � _ui�1Þ�; i ¼ 1;2;…;m (6)

nanorods is referred to as nanorod 1, nanorod 2 and so on to m-thnanorod. Also, it should be noted that the all nanorods are made ofsame viscoelastic materials with coefficient of internal dampingtd, elastic modulus E, mass density r, uniform cross-section of areaA and length L. The axial displacement of the i-th nanorods isui(x,t).

Consider now the axial motion of an infinitesimal element of i-th nanorod of VMNRS, as shown in Fig. 3. Based on D' Alembert'sprinciple, summing all forces in the x-direction gives equation ofmotion in following form

dNidx

dxþ Fidx� Fi�1dx ¼ €uidm; (4)

where Fi and Fi�1 are external forces which results fromviscoelasticlayers; dm is mass of the infinitesimal element; Ni(x,t) is the stressresultant, defined as

Fi ¼ kiðuiþ1 � ui Þ þ bið _uiþ1 � _ui Þ; Fi�1¼ ki�1ðui � ui�1 Þ þ bi�1ð _ui � _ui�1 Þ; (5a)


þ tdd2 _uidx2

!þ kðui � uiþ1Þ þ bð _ui � _uiþ1Þ þ kðui � ui�1Þ

¼ m d2

dx2½m€ui þ kðui � uiþ1Þ þ bð _ui � _uiþ1Þ þ kðui � ui�1Þ þ bð _ui � _u

m€um � e d2umdx2

þ tdd2 _umdx2

!þ kum þ b _um þ kðum � um�1Þ þ bð _um � u

¼ m d2

dx2½m€ui þ kum þ b _um þ kðum � um�1Þ þ bð _um � _um�1Þ�;

dm ¼ rAdx; (5b)

Niðx; tÞ ¼ZA

sxxðx; tÞdA: (5c)

By substituting Eq. (5c) into the Eq. (3a), using that result andEqs. (5a), (5b) and Eq. (4), we get the following equation of motionexpressed in terms of the displacement ui(x,t) for axial vibration ofthe i-th nanorod

and

EA ¼ e ¼ constant; (7a)

rA ¼ m ¼ constant; (7b)

where e and m denotes axial rigidity and mass per unit length,respectively.

Setting the axial displacement u0(x,t) and umþ1(x,t) to zero in Eq.(6), we can obtain equations of motion for the “Clamped-Chain”system, where k0 ¼ km ¼ k andb0 ¼ bm ¼ b, (Fig. 2a) and b)) asfollows

m€u1 � e d2u1dx2

þ tdd2 _u1dx2

!þ kðu1 � u2 Þ þ bð _u1 � _u2Þ þ ku1 þ b _u1

¼ m d2

dx2½m€u1 þ kðu1 � u2Þ þ bð _u1 � _u2Þ þ ku1 þ b _u1�;

(8a)

þ bð _ui � _ui�1Þ

i�1Þ�; i ¼ 2;…;m� 1 (8b)

_m�1Þ

(8c)

Fig. 3. The i-th nanorod (left) of VMNRS and differential element of the i-th nanorod (right).


For the case of “Free-Chain” system (Fig. 2c) and d)) we canobtain equations of motion, by setting k0 ¼ km ¼ 0 and b0 ¼ bm ¼ 0,in Eq. (6), yields

m€u1 � e d2u1dx2

þ tdd2 _u1dx2

!þ kðu1 � u2Þ þ bð _u1 � _u2Þ

¼ m d2

dx2½m€u1 þ kðu1 � u2Þ þ bð _u1 � _u2Þ�; (9a)


þ tdd2 _uidx2

!þ kðui � uiþ1Þ þ bð _ui � _uiþ1Þ þ kðui � ui�1Þ þ bð _ui � _ui�1Þ

¼ m d2

dx2½m€ui þ kðui � uiþ1Þ þ bð _ui � _uiþ1Þ þ kðui � ui�1Þ þ bð _ui � _ui�1Þ�; i ¼ 2;…;m� 1; (9b)

m€um � e d2um þ td

d2 _um!

þ kðum � um�1Þ þ bð _um � _um�1Þ
dx2 dx2
¼ m d2

dx2½m€ui þ kðum � um�1Þ þ bð _um � _um�1Þ�

(9c)

The boundary conditions for ClampedeClamped (see Fig. 2a)and c)) and ClampedeFree (see Fig. 2b) and d)), may be written asfollows

ClampedeClamped:

uið0; tÞ ¼ uiðL; tÞ ¼ 0; (10)ClampedeFree:

uið0; tÞ ¼ NiðL; tÞ ¼ 0; (11)

where Ni(L,t), (i ¼ 1,2,..,m) are stress resultants on the right side of aset of m nanorods.

By substituting Eqs. (4) and (5) into Eq. (3a) we get the stressresultant force Ni as follows

Niðx; tÞ ¼ md2Nidx2

þ e�duidx

þ tdd _uidx

�

¼ m ddx

½m€ui þ kðui � uiþ1Þ þ bð _ui � _uiþ1Þ þ kðui � ui�1Þ

þ bð _ui � _ui�1Þ� þ e�duidx

þ tdd _uidx

�:

(12)

Introducing Eq. (13) into the boundary conditions (12), yields

NiðL; tÞ ¼ mddx

½m€uiðL; tÞ þ kðuiðL; tÞ � uiþ1ðL; tÞÞ þ bð _uiðL; tÞ� _uiþ1ðL; tÞÞ þ kðuiðL; tÞ � ui�1ðL; tÞÞ þ bð _uiðL; tÞ

� _ui�1ðL; tÞÞ� þ e�duiðL; tÞ

dxþ td

d _uiðL; tÞdx

�¼ 0:

(13)

3. Solution of the problem

3.1. Complex natural frequencies

Using the method of separation of variables the general solu-tions of dynamics equation of motion Eq. (6) and boundary con-ditions Eqs. (10) and (11) are taken in the form

uiðx; tÞ ¼X∞n¼1

Uin sin anxeiunt ; (14)

where for ClampedeClamped boundary conditions, (Murmu andAdhikari, 2010) and (Graff, 1975, pp. 91), we have


an ¼ npL ; n ¼ 1;2;…;∞; (15)

for ClampedeFree boundary conditions (Murmu and Adhikari,2010; Graff, 1975), we have

an ¼ ð2n� 1Þp2L ; n ¼ 1;2;…;∞; (16)

where i ¼ �1, Uin is the amplitude and un is natural frequency in n-th mode of vibration.

By substituting general solutions (14) into the Eq. (6), we obtainsystem of m algebraic equations as

�vi�1nUi�1n þ SinUin � vinUiþ1n ¼ 0; i ¼ 1;2;3;…;m; (17)

where

Sin ¼ a2neð1þ iuntdÞ �mu2n�1þ ma2n

�þ vin þ vi�1n; (18a)

vin ¼ ki�1þ ma2n

�þ iunbi

�1þ ma2n

�; (18b)

vi�1n ¼ ki�1�1þ ma2n

�þ iunbi�1

�1þ ma2n

�: (18c)

For a homogenous system of algebraic equations (17), analyticalexpressions for complex eigenvalues of VMNRS can be determinedusing the trigonometric method (Ra�skovi�c, 1953, 1963, 1957;

unð1=2Þ ¼ i�a2netd þ 2blnð1� cos 4Þ

2mln

±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4mln

�a2neþ 2klnð1� cos 4Þ

�� a2netd þ 2blnð1� cos 4Þ�2q

2mln(25)

Stojanovi�c et al., 2013). According to the paper by Ra�skovi�c(1963), solution of the i-th algebraic equation is assumed in thefollowing form

Uin ¼ N cosði 4Þ þM sinði4Þ; i ¼ 1;2;3;…;m: (19)

where unknown parameter 4 which will be determined in thefollowing. Introducing Eq. (19) into the i-th algebraic equation ofsystem (17), and assuming that the all nanorods have the samematerial and geometrical properties and connected with the sameviscoelastic layers, we get two trigonometric equations

Nf � vncos½ði� 1Þ4� þ Sncosði4Þ � vncos½ðiþ 1Þ4�g ¼ 0; i ¼ 2;3;…;m� 1; (20a)

Mf � vnsin½ði� 1Þ4� þ Snsinði4Þ � vnsin½ðiþ 1Þ4�g ¼ 0; i ¼ 2;3;…;m� 1; (20b)

where

Sn ¼ a2neð1þ iuntdÞ �mu2n�1þ ma2n

�þ 2vn; (21a)

vn ¼ k�1þ ma2n

�þ iunb

�1þ ma2n

�; (21b)

under the condition that the constants M and N are not simulta-neously equal to zero.

After some algebra, Eq. (20) can be written as

ðSn � 2vn cos 4ÞN cosði4Þ ¼ 0; (22a)

ðSn � 2vn cos 4ÞM sinði4Þ ¼ 0: (22b)From Eqs. (20) and (22), we can conclude that, N s 0 and

cos(i4) s 0 or M s 0 and sin(i4) s 0 in order the system had anoscillatory behavior, for i ¼ 2,3,…,m�1. Now it gets frequencyequation

Sn ¼ 2vn cos 4: (23)Introducing Eq. (21) into Eq. (23), we get the frequency equation

in the following form

�mlnu2n þ iha2netd þ 2blnð1� cos 4Þ

iun

þha2neþ 2klnð1� cos 4Þ

i¼ 0;

(24)

where ln ¼ ð1þ ma2nÞ.Now we can express the complex natural frequency from

quadratic equation (24), in the following form

Equation (25) is a general solution of the complex natural fre-quencies of VMNRS for m coupled nanorods.

3.2. Chain systems of VMNRS

In this subsection we will discusses about the upper and lowerboundary condition for both chain system, “Clamped-Chain” and“Free-Chain”, and method for determination of the unknown 4. It isalso assumed that all four cases of VMNRS shown Fig. 2, composed

of m identical nanorods with same material and geometricalproperties and coupled by viscoelastic layers with same charac-teristics (stiffness and damping coefficients). Introducing assumedsolutions Eq. (14) into a set of m partial differential equations for


“Clamped-Chain” Eq. (8) and “Free-Chain” Eq. (9), we obtain twohomogenous systems of algebraic equations in matrix form as

for Clamped-Chain system:

26666666666664

Sn �vn 0 0 0 0 0 0 0�vn Sn �vn 0 0 0 0 0 0… … … … … … … … …

0 0 0 Sn �vn 0 0 0 00 0 0 �vn Sn �vn 0 0 00 0 0 0 �vn Sn 0 0 0… … … … … … … … …

0 0 0 0 0 0 �vn Sn �vn0 0 0 0 0 0 0 �vn Sn

37777777777775

266666666666666664

U1nU2nU3n…

Ui�1nUinUiþ1n…

Um�2nUm�1nUmn

377777777777777775

¼

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

000…

000…

000

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

(26)

for Free-Chain system:

26666666666664

Sn � vn �vn 0 0 0 0 0 0 0�vn Sn �vn 0 0 0 0 0 0… … … … … … … … …

0 0 0 Sn �vn 0 0 0 00 0 0 �vn Sn �vn 0 0 00 0 0 0 �vn Sn 0 0 0… … … … … … … … …

0 0 0 0 0 0 �vn Sn �vn0 0 0 0 0 0 0 �vn Sn � vn

37777777777775

266666666666666664

U1nU2nU3n…

Ui�1nUinUiþ1n…

Um�2nUm�1nUmn

377777777777777775

¼

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

000…

000…

000

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

(27)

where expressions for Sn and vn are defined in Eq. (21).Now we determine the unknown parameter 4, from the upper

and lower boundary conditions, i.e. equation (19) must satisfy thefirst and the last equation of system of algebraic equations Eq. (26)for “Clamped-Chain” and Eq. (27) for “Free-Chain”.

Let us first consider the case of “Clamped-Chain” system. Ifintroducing expressions U1n ¼ Ncos 4 þ M sin 4 andU2n ¼ Ncos(24) þ M sin(24) into first equation and Um�1n ¼ Ncos[(m�1)4]þM sin[(m�1)4] and Um�1n¼Ncos(m4)þM sin(m4) intolast equation of the system (26), we obtain system of algebraicequations

1�cos 4 �sin 4cos½ðmþ1Þ4��cosðm4Þ sin½ðmþ1Þ4�� sinðm4Þ

¼00sinðm4Þ¼0; (32)

N½Sn cos 4� vn cosð24Þ� þM½Sn sin 4� vn sinð24Þ� ¼ 0;(28a)

N½Sn cosðm4Þ � vn cos½ðm� 1Þ4�� þM½Sn sinðm4Þ� vn sin½ðm� 1Þ4�� ¼ 0:

(28b)

Non-trivial solutions for the constants N andM can be obtained,which yields the following trigonometric equation:

1 0cos½ðmþ 1Þ4� sin½ðmþ 1Þ4�

¼ 0 0 sin½ðmþ 1Þ4� ¼ 0;

(29)

from which we obtain solutions for unknown 4 as

4cc;s ¼sp

mþ 1; s ¼ 1;2;…;m: (30)

where 4cc,s is unknown parameter 4 for “Clamped-Chain” system.Using the procedures described for previously case, we again

introducing the assumed solution U1n ¼ Ncos 4 þ M sin 4 andU2n ¼ Ncos (24) þ M sin (24) into first equation and Um�1n ¼ Ncos[(m�1)4]þM sin[(m�1)4] andUm�1n¼Ncos(m4)þM sin(m4) intolast equation of the system (27), after some algebrawe obtain againsystem of algebraic equations in following form

N½ðSn� vnÞcos 4� vncosð24Þ�þM½ðSn� vnÞsin 4� vnsinð24Þ� ¼0;(31a)

N½ðSn � vnÞcosðm4Þ � vncos½ðm� 1Þ4�� þM½ðSn � vnÞsinðm4Þ� vnsin½ðm� 1Þ4�� ¼ 0:

(31b)

Non-trivial solutions for the constants N andM can be obtained,which yields trigonometric equation in the following form

from which we obtain solutions for unknown 4 as

4fc;s ¼spm; s ¼ 0;1;…;m� 1: (33)

where 4fc,s is unknown parameter 4 for “Free-Chain” system.Now substituting expression for 4cc,s into Eq. (25) and 4fc,s into

Eq. (25), we obtain expressions for complex natural frequencies forboth cases, respectively,

“Clamped-Chain” system

ucc;nð1=2Þ ¼ i"a2netd þ 2bln

�1� cos 4cc;s

�2mln

#±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4mln

�a2neþ 2kln

�1� cos 4cc;s

�� a2netd þ 2bln�1� cos 4cc;s��2q

2mln; (34)


and “Free-Chain” system

ufc;nð1=2Þ ¼ i24a2netd þ 2bln

�1� cos 4fc;s

�2mln

35±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4mln

ha2neþ 2kln

�1� cos 4fc;s

�i�ha2netd þ 2bln

�1� cos 4fc;s

�i2r2mln

: (35)

It should be noted that real part of the complex eigenvaluesrepresents the natural frequency of the systemwhile the imaginarypart represents dumping of the system.

3.3. Asymptotic analysis

Suppose that the number of mode tends to infinity i.e. n / ∞introducing in frequency equation (24), an asymptotic equation forthe complex natural frequencies can be obtain as

un/∞ð1=2Þ ¼ i�etd þ 2bmð1� cos 4Þ

2mm

±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4mm½eþ 2kmð1� cos 4Þ� � ½etd þ 2bmð1� cos 4Þ�2

q2mm

; (36)

where 4 can take the values for both chain systems Eq. (30) or Eq.(33). Expression (36) is representing complex natural frequencieswhich are independent of the boundary conditions of the system.

In the next case we consider asymptotic equation for the com-plex natural frequencies when a number of nanorods tend to in-finity, i.e.m/∞ introducing in frequency equations Eq. (30) or Eq.(33), which implies

um/∞nð1=2Þ ¼ ia2netd2mln

±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4mlna2ne�

�a2netd

�2q2mln

: (37)

From Eq. (37), we can concluded that the asymptotic complexnatural frequencies are independent of influence of viscoelasticlayers, and it is also represent a lowest complex natural frequencyof the system for both chain system, “Clamped-Chain” and “Free-Chain”.

Now, consider the case when the number of modes and thenumber of nanorods tends to the infinity, i.e. introducing m / ∞into the Eq. (36), we get the asymptotic complex natural frequencyas

un/∞;m/∞ð1=2Þ ¼ ietd2mm

±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4mme� ðetdÞ2

q2mm

: (38)

Expression (38) is representing the fundamental complexnatural frequency of the system when the number of nanorods

and the number of modes tends to the infinite, and also is thesame for both boundary conditions (ClampedeClamped andClampedeFree) and both chain systems (“Clamped-Chain” and“Free-Chain”).

The critical damping rations are obtained by setting the naturalfrequencies to zero, from Eqs. 36e38 may be written as

tcr;n/∞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4mm½eþ 2kmð1� cos 4Þ�

p� 2bmð1� cos 4Þ

e;

(39a)

tcr;m/∞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4mlna2ne

qa2ne

; (39b)

t cr;m/∞n/∞

¼ffiffiffiffiffiffiffiffiffiffiffiffi4mme

pe

: (39c)

The obtained analytical expression of the critical damping ra-tions is function of the material parameters of nanorods and in-dependent of boundary condition and chain systems, for the casewhen the number of nanorods and the number of modes tends tothe infinite.

4. Numerical results and discussion

The analytical model for VMNRS presented here is thegeneralized theory, which includes the damping effect of thesystem, and therefore represents a more realistic case. Thismodel can be applied for the axial vibration analysis of m coupledcarbon nanotubes system, ZnO nanorods system and it is alsouseful for the study of vibration behavior of multiple-nanobeamand nanoplates system for nanoresonator application. The firstpart of this section is relates to the comparison of the resultsobtained by applying trigonometric method with available datain the literature (Murmu and Adhikari, 2010) for special case ofVMNRS when m ¼ 2, and also with the results obtained by using

Table 1Validation of first four complex eigenvalues of the “Free-Chain” VMNRS for CeF boundary conditions and different values of viscoelastic constant td and nonlocal parameter e0a.

CeF e0a ¼ 0 nm e0a ¼ 0.5 nm e0a ¼ 1 nm e0a ¼ 1.5 nm e0a ¼ 2 nmMurmu and Adhikari (2010)td ¼ 0 nsb ¼ 0 Nns/nmk ¼ 8 N/nm

1 1.5708 1.2353 0.8436 0.6137 0.47642 4.2974 4.1864 4.0880 4.0468 4.02833 4.7124 1.8411 0.9782 0.6601 0.49724 6.1812 4.4033 4.1179 4.0541 4.0308

“Free-Chain” VMNRS e Trigonometric methodtd ¼ 0.001 nsb ¼ 0.01 Nns/nmk ¼ 8 N/nmm ¼ 2

1 1.5708 þ 0.0012i 1.2353 þ 0.0007i 0.8435 þ 0.0003i 0.6136 þ 0.0001i 0.4764 þ 0.0001i2 4.2973 þ 0.0112i 4.1864 þ 0.0107i 4.0879 þ 0.0103i 4.0467 þ 0.0101i 4.0282 þ 0.0101i3 4.7123 þ 0.0111i 1.8410 þ 0.0016i 0.9782 þ 0.0004i 0.6600 þ 0.0002i 0.4972 þ 0.0001i4 6.1811 þ 0.0211i 4.4033 þ 0.0116i 4.1178 þ 0.0104i 4.0540 þ 0.0102i 4.0307 þ 0.0101i

td ¼ 0.004 nsb ¼ 0.01 Nns/nmk ¼ 8 N/nmm ¼ 2

1 1.5707 þ 0.0049i 1.2353 þ 0.0030i 0.8435 þ 0.0014i 0.6136 þ 0.0007i 0.4764 þ 0.0004i2 4.2973 þ 0.0149i 4.1863 þ 0.0130i 4.0879 þ 0.0114i 4.0467 þ 0.0107i 4.0282 þ 0.0104i3 4.7121 þ 0.0444i 1.8410 þ 0.0067i 0.9782 þ 0.0019i 0.6600 þ 0.0008i 0.4972 þ 0.0004i4 6.1809 þ 0.0544i 4.4033 þ 0.0167i 4.1178 þ 0.0119i 4.0540 þ 0.0108i 4.0307 þ 0.0104i


the numerical methods for the general case of VMNRS whenm > 2. The following parameter has been considered in thecomparative analysis proposed in ref. Murmu and Adhikari(2010): L ¼ 1 [nm], m ¼ 10�9 ½kg=m�, e0a ¼ 0�2 [nm], stiffnesscoefficient K ¼ 8 N/nm, and two different values of viscoelasticparameters td ¼ 0.001 [ns] and 0.004 [ns]. The complex eigen-value of two coupled viscoelastic nanorods system obtained bytrigonometric method for different values of nonlocal andviscoelastic parameters are presented in Table 1. This tableillustrate that the both parts of complex eigenvalues i.e. naturalfrequency and damping of the system decrease with increase ofnonlocal parameter. Also, it can be concluded that the increasingof viscoelastic parameters have very small effect on the naturalfrequency, but on the system damping has a significant influence.The present results for natural frequencies are in excellentagreement with work proposed by Murmu and Adhikari (2010).In order to confirm the accuracy of the trigonometric method forgeneral case of VMNRS, the obtained results for the complexeigenvalue will be compared with the results obtained by usingthe numerical methods. These results are presented in Table 2.For this case, we consider the system which consists of three, fiveand ten (m ¼ 3, 5, 10) identical nanorods coupled by viscoelasticlayers. The analytical solutions obtained from Eqs. (34) and (35)are validated with results obtained by numerical solutions ofhomogenous system of algebraic equations given in Eqs. (26) and(27), for both boundary conditions. Based on the presented re-sults we can concluded that the values of complex eigenvalueobtained by analytical and numerical methods in excellentagreement.

However, for a more general and detailed vibration analysis ofVMNRS, a system of m coupled viscoelastic single walled carbonnanotube is used as an example in the second part of this section.Also, the influence of nonlocal parameter, the number of nano-tubes, the number of mode and viscoelastic parameter on thecomplex eigenvalues for both boundary conditions and both chainsystem are analyzed numerically. The following values are used forthe numerical study: diameter of nanorod d ¼ 1.1 [nm], length

Table 2Validation of complex eigenvalues of the VMNRS when number of nanorods is greater th

s ¼ 1, n ¼ 1 Trigonometric methodm ¼ 3 m ¼ 5 m ¼ 10

CeC Clamped Chain 2.22038 þ 0.00305i 1.54513 þ 0.00146i 0.94442 þFree Chain 2.8712 þ 0.00512i 1.81647 þ 0.00203i 1.01337 þ

CeF Clamped Chain 2.21659 þ 0.00304i 1.53967 þ 0.00145i 0.93547 þFree Chain 2.86827 þ 0.00511i 1.81183 þ 0.00202i 1.00504 þ

L ¼ 10 d, Young's modulus E ¼ 1.1 [TPa], the mass densityr ¼ 2300 [kg/m3], parameters of viscoelastic layers, stiffnessk ¼ 10 [N/nm] and damping b ¼ 0.01 [Nns/nm]. The parametervalues of KelvineVoigt damping coefficient td and nonlocalparameter m are given in follow.

In order to demonstrate the effects of the number of nanorodsand number of mode on the real and imaginary parts of complexeigenvalues of the VMNRS for both boundary conditions, surfacehave been plotted as function of these two parameters n2[1, 50] and m2[2, 50] shown on Figs. 4 and 5. From these figureswe can notice that the effects of the above mentioned parame-ters on the real and imaginary part of the complex values arevery similar for all cases. In general, the influence of increasingthe number of mode n causing an increase in the real andimaginary parts, but increases of the number of nanorods m inthe VMNRS causes a reduction of their values. However, it wasfound that, when the number of modes and the number ofnanorods tends to the larger values (n / ∞, m / ∞), this leadsto the asymptotic values of the real and imaginary part of thecomplex eigenvalues, whence we can conclude that it is inde-pendent of the number of nanorods, boundary conditions and“Chain” systems. It should be noted that asymptotic values of thereal part of complex eigenvalue represents a finite value thenatural frequency, beyond which vibration of the system isimpossible.

The real and imaginary parts of complex eigenvalues or nat-ural frequencies and damping ration of a VMNRS are shown as afunction of the nonlocal parameter in Figs. 6e9, for differentvalues of viscoelastic parameter td ¼ 0.001 and 0.003 [ns] andnumber of nanorods m¼ 2, 5, 15. The presented results suggestthat the both parts of the complex eigenvalue significantlyinfluenced by nonlocal parameter m for ClampedeClampedboundary conditions, but influence are reduced for Clampede-Free boundary conditions. Comparison of the results obtained fornatural frequencies and damping rations by variation of theviscoelastic parameter td, it is obvious that the influence on thedumping ration is very a significant, but on the natural

an two (m > 2), for both boundary conditions and chain systems.

Numerical method

m ¼ 3 m ¼ 5 m ¼ 100.00052i 2.22038 þ 0.00305i 1.54513 þ 0.00146i 0.94442 þ 0.00052i0.00061i 2.8712 þ 0.00512i 1.81647 þ 0.00203i 1.01337 þ 0.00061i0.00051i 2.21659 þ 0.00304i 1.53967 þ 0.00145i 0.93547 þ 0.00051i0.00060i 2.86827 þ 0.00511i 1.81183 þ 0.00202i 1.00504 þ 0.00060i

Fig. 4. The real and imaginary parts of the complex eigenvalues for ClampedeClamped boundary conditions: a) “Clamped-Chain” system, b) “Free-Chain” system, for s ¼ 1,m ¼ 2 [nm2], td ¼ 0.001 [ns].


frequencies can be see much smaller influence. From this figureswe can note a quite linear influence on the imaginary parts ofcomplex eigenvalue. As expected, the both parts of complex ei-genvalues are very sensitive to variation of the number ofnanorods. The results show that increases of the number ofnanorods in the system causes a reduction in both, natural fre-quencies and damping ratio. This is valid for all presented cases.To see the effect of the boundary conditions on natural fre-quencies and damping ratio for the same “Chain” systems, wecan consider results presented on two figures Figs. 6 and 8 orFigs. 7 and 9. It can be observed that the both parts of complexeigenvalues have a much higher values for CeC boundary con-ditions than for the CeF one. Moreover, the natural frequenciesfor CeC boundary conditions more sensitive to the influence ofthe viscoelastic parameter td.

5. Conclusion

This work presents an analytical and numerical investigation ofthe vibration behavior of the viscoelastic coupled multi-nanorodssystem. The set of m partial differential equations of motion are

obtained based on D' Alembert's principle and nonlocal Kel-vineVoigt viscoelastic constitutive relations for both boundaryconditions and both “Chain” systems. The closed form solutions ofcomplex eigenvalues are derived in a unique manner by combiningtwo analytical methods, namely, separations of variables and trig-onometric method. We have derived analytical expressions of theasymptotic values of natural frequency and damping ratio ofVMNRS, when the number of nanorods and the number of modestends to the infinite. It is found that the both asymptotic valuesdepend only on the material characteristics of VMNRS. In order todemonstrate the accuracy of proposed trigonometric method, wecompared the analytical results with results obtained by numericalmethods for the general case of VMNRS. It is shown that theapplication of these analytical methods provides excellent agree-ment when compared with the numerical results. We have alsovalidated the analytical results obtained for the special case ofVMNRS whenm ¼ 2, with the results found in the literature. In thepresented numerical analyses, the influence of number of nano-rods, number of mode, nonlocal parameter and viscoelastic con-stant on the both parts of complex eigenvalues are investigated.From these observations we can conclude that the real parts or a

Fig. 5. The real and imaginary parts of the complex eigenvalues for ClampedeFree boundary conditions: a) “Clamped-Chain” system, b) “Free-Chain” system, for s ¼ 1, m ¼ 2 [nm2],td ¼ 0.001 [ns].


natural frequency was significantly influenced by nonlocalparameter while on the damping ration have smaller influence forboth boundary conditions. Also, it can be noted that the increase inviscoelastic parameter cause increase in the damping ratio for all

Fig. 6. The a) real and b) imaginary part of the complex eigenvalues for C

considered cases. Regarding the influence of the number of nano-rods and the number of modes on the complex eigenvalues, it wasfound that when the values of the both parameters tend to theinfinite their influence are vanishes.

lampedeClamped boundary conditions and “Clamped-Chain” system.

Fig. 7. The a) real and b) imaginary part of the complex eigenvalues for ClampedeClamped boundary conditions and “Free-Chain” system.

Fig. 8. The a) real and b) imaginary part of the complex eigenvalues for ClampedeFree boundary conditions and “Clamped-Chain” system.

Fig. 9. The a) real and b) imaginary part of the complex eigenvalues for ClampedeFree boundary conditions and “Free-Chain” system.


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Vibration insight of a nonlocal viscoelastic coupled multi-nanorod system1. Introduction2. Mathematical model of VMNS2.1. Brief introduction in nonlocal constitutive relations2.2. The dynamic equations of VMNRS

3. Solution of the problem3.1. Complex natural frequencies3.2. Chain systems of VMNRS3.3. Asymptotic analysis

4. Numerical results and discussion5. ConclusionReferences

european journal of mechanics a/solidsengweb.swan.ac.uk/~adhikaris/fulltext/journal/ft219.pdf ·...

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