dynamic stability of a nonlinear multiple-nanobeam system

Nonlinear Dyn (2018) 93:1495–1517https://doi.org/10.1007/s11071-018-4273-3

ORIGINAL PAPER

Dynamic stability of a nonlinear multiple-nanobeam system

Danilo Karlicic · Milan Cajic ·Sondipon Adhikari

Received: 17 July 2017 / Accepted: 6 April 2018 / Published online: 26 April 2018© The Author(s) 2018

Abstract We use the incremental harmonic balance(IHB) method to analyse the dynamic stability prob-lem of a nonlinearmultiple-nanobeam system (MNBS)within the framework of Eringen’s nonlocal elastic-ity theory. The nonlinear dynamic system under con-sideration includes MNBS embedded in a viscoelas-tic medium as clamped chain system, where everynanobeam in the system is subjected to time-dependentaxial loads. By assuming the von Karman type of geo-metric nonlinearity, a system of m nonlinear partialdifferential equations of motion is derived based on theEuler–Bernoulli beam theory andD’Alembert’s princi-ple. All nanobeams in MNBS are considered with sim-ply supported boundary conditions. Semi-analyticalsolutions for time response functions of the nonlinearMNBS are obtained by using the single-mode Galerkindiscretization and IHB method, which are then vali-dated byusing the numerical integrationmethod.More-over, Floquet theory is employed to determine the sta-bility of obtained periodic solutions for different con-figurations of the nonlinear MNBS. Using the IHB

D. Karlicic · M. CajicMathematical Institute of the Serbian Academy of Scienceand Art, Kneza Mihaila 36, Belgrade 11001, Serbiae-mail: [email protected]

M. Cajice-mail: [email protected]

S. Adhikari (B)College of Engineering, Swansea University, SingletonPark, Swansea SA2 8PP, UKe-mail: [email protected]

method, we obtain an incremental relationship with thefrequency and amplitude of time-varying axial load,which defines stability boundaries. Numerical exam-ples show the effects of different physical and materialparameters such as the nonlocal parameter, stiffnessof viscoelastic medium and number of nanobeams onFloquet multipliers, instability regions and nonlinearamplitude–frequency response curves of MNBS. Thepresented results can be useful as a first step in the studyand design of complex micro/nanoelectromechanicalsystems.

Keywords Multiple-nanobeam system · Geometricnonlinearity · Nonlocal elasticity · Instability regions ·IHB method · Floquet theory

1 Introduction

Most dynamic stability studies in the literature are con-sidering single or double-micro/nanostructure-basedsystems [1–4]. Investigation of the nonlinear dynamicbehaviour of such systems has been an exciting per-spective in the last decade due to possible applicationsin designprocedures of differentmicro/nanoengineeringsystems [5], especially when these systems are aimedto be exploited as vibrating nanodevices such as res-onators, nanosensors, or other nanoelectromechanicalsystems. However, it has been shown that organizednanostructure architectures made of vertically alignedforests, yarns, and sheets of carbon nanotubes (CNTs)

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1496 D. Karlicic et al.

give an exciting perspective to scale up the proper-ties of individual CNTs and realize new functionali-ties [6]. The lack of reliable nonlinear dynamic mod-els of multiple-nanostructure-based systems makes afuture investigation in this field as an attractive task forresearchers. A seminal idea of this work is to fill thisgap in the literature and propose newmodels and proce-dures of a solution to investigate the dynamic stabilityof the geometrically nonlinear multiple-nanobeam sys-tem (MNBS). The presented model of MNBS consistsof multiple individual simply supported nanobeamsthat are parallel to each other and placed into a vis-coelasticmedium. In general, suchmodel can representsome nanocomposite material composed of verticallyalignedCNTs array placed in somepolymer. Therefore,the presented model of the nonlinear MNBS might beimportant to investigate the aligned arrays of CNTs [7]and CNT/polymer composites [8], especially stabilityof CNTs embedded in the polymer matrix. In addition,due to their remarkable physical and chemical charac-teristics, various applications and mathematical mod-els of CNTs are suggested in the literature [9]. Basedon experimental results, it is shown that small-scaleeffects play a significant role in the static and dynamicbehaviour ofmicro/nanostructures. Sincemodels basedon classical continuum theory are scale-free, they needto be modified introducing some new assumptions inorder to take into account size effects that are presentat small scales. Eringen [10,11] extended the classicalcontinuum theory by introducing integral and differen-tial forms of constitutive equations with a single mate-rial parameter, such that it takes into account forcesbetween atoms and internal length scale. Moreover,it has been shown that the results obtained by usingthe nonlocal continuum models are in good agreementwith the results obtained via molecular dynamics sim-ulations. Many authors have analysed the mechanicalbehaviour of different types of nanostructures withinthe framework of nonlocal continuum mechanics.

A pioneering study where nonlocal elasticity the-ory is proposed to model micro/nanostructures is thework by Pedinson et al. [12]. They analysed the staticand dynamic behaviour of a simple micro/nanobeamand proposed a possible application to real MEMSand NEMS. Reddy [13–15] has derived new equationsof motion for Euler–Bernoulli, Timoshenko, Reddyand Levinson beam theories based on the Eringen’snonlocal elasticity theory and then obtained analyt-ical solutions for bending, vibrations, and buckling

response of beams for simply supported boundary con-ditions. The new shear deformation beam theory pro-posed by Huu-Tai [16] is used to analyse the free vibra-tion and stability of short nanobeams using Eringen’snonlocal elasticity theory. Aydogdu [17] employedthe Euler–Bernoulli, Timoshenko, Levinson, Reddy,and Aydogdu beam theories to analyse the bending,buckling, and vibration of nanobeams in an analyt-ical manner. Ansari et al. [18,19] compared resultsobtained from nonlocal continuum mechanics withthe results obtained by molecular dynamic simula-tions for simple models of nanostructures and con-cluded that the results obtained from both theories arein good agreement. More complex nanoscale systemsconsist of two and more organized nanostructures suchas rods, beams, and plates, which are usually cou-pled through some medium. The dynamic behaviourof such systems is interesting to observe and it isstill not fully explored in the literature. A double-nanostructure-based system is the simplest model of acoupled multiple-nanostructure systems, which can becomposed of two nanobeams, nanorods or nanoplatescoupled through a medium with elastic or viscoelas-tic properties. Murmu and Adhikari conducted detailedvibration studies of a double-nanorod, nanobeam, andnanoplate systems [20–23], where partial differentialequations of motion are obtained based on the D’Alembert’s principle and nonlocal elasticity theory andsolved by using analytical methods. They investigatedthe influence of small-scale effects and other physicalparameters on natural frequencies and critical buck-ling loads and compared analytical results with resultsobtained by molecular dynamics simulations. One ofthe first dynamic behaviour studies of multi-layeredgraphene sheet system is the paper by He et al. [24]and Liew et al. [25], where the authors derived anexplicit formula for natural frequencies by utilizingthe classical elasticity model. Using the methodologythat was proposed for chain-like mechanical systemsby Raškovic [26] and multiple coupled structural ele-ments by Hedrih [27], Karlicic et al. [28] presented astraightforward method to obtain analytical solutionsfor natural frequencies and critical buckling loads ofmultiple nanorods, nanobeams, and nanoplates sys-tems based on the Eringen’s nonlocal elasticity theoryand trigonometricmethod. To this time, themechanicalbehaviour of different nanostructures such as the longi-tudinal vibration of nanorods [29], transverse vibration

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Dynamic stability of a nonlinear multiple-nanobeam system 1497

of nanobeams [30–32], nanoplates [33] and nanoshells[34] with elastic properties are given in the literature.

Forced and parametric vibration problems of com-plex coupled nanostructure-based systems such as cou-pled nanobeams, nanorods, nanoplates have attractedmuch attention of the scientific community. Simsek[35] investigated the influence of a moving load ondifferent types of single-walled carbon nanotubes.Karaoglu and Aydogdu [36] investigated the forcedvibration of carbon nanotubes via nonlocal Euler–Bernoulli beam model. Ansari et al. [37,38] analysedthe forced vibration of nanobeams as mechanical mod-els of carbon nanotubes by using different numericaltechnics and compared the results with those obtainedby the molecular dynamic simulations. In addition,Kiani [39–42] has examined the influence of differ-ent physical fields and shear effects on the free andforced vibration response of nanoscale systems com-posed of multiple nanobeams coupled in membraneand forest configurations. Also, Kiani [43–45] con-ducted several studies on the influence of moving loadson double-carbon nanotube system based on the non-local elasticity theory. Arani et al. [46] investigatedthe nonlinear dynamic stability of a double-graphenesheet with integrated actuators and sensors based on theGurtin–Murdoch elasticity theory. Further, in series ofpapers Wang et al. [47,48] investigated the nonlinearbehaviours of double-nanoplate systems for forced andparametric excitations. Recently, Pavlovic et al. [49]observed the stochastic stability problem by determin-ing the stability boundaries and regions of almost surestability of a linear multi-nanobeam system embeddedin a viscoelastic medium by using the Monte Carlosimulation method.

In general, dynamic stability analysis of nanobeam-like structures can play a significant role in designprocedures of future nanodevices. For axially loadednanobeams, where loads are time-dependent harmonicfunctions, a failure due to dynamic instability mightoccur formuch smaller amplitudes of load than the fail-ure induced by static buckling. These instability condi-tions usually lead to the failure of micro/nanodevices.Based on that fact, authors usually intend to analyse sta-bility regions caused by primary parametric resonance,where the frequency of excitation is two times largerthan the first natural frequencies of MNBS [50, p. 23].Therefore, the main aim of this paper is to extend theprevious studies by investigating the dynamic stabilityof more complex systems such as the nonlinear MNBS

and to explore its primary resonance state. In addition,investigation of the influence of small-scale parame-ter on the stability of periodic solutions and instabilityregions is a very important task for the design of dif-ferent types of micro/nanodevices.

It is well known that CNT/polymer composites canbe observed as materials with viscoelastic properties[51–53]. Based on experiments and complex modu-lus characterization results [52], one can obtain uniqueproperties of composite materials with aligned carbonnanotubes embedded in the polymer matrix. Knowingstorage and loss modules is the first step in the identi-fication of parameters of chosen viscoelasticity modelto represent such materials. In addition, it is confirmedthat in CNT/polymer composites, relaxation time andviscosity of polymers are temperature-dependent quan-tities [53], which can be easily obtained from complexmodulus under the assumption of isothermal deforma-tion [54]. On the other side, a model of MNBS embed-ded in the elastic medium is suitable to describe the vander Waals interactions between adjacent nanobeams,where values of the stiffness coefficient of the elasticmedium can be determined from van der Waals forcesusing the methodology from the literature [39–42].

Up to this point, no investigation of the dynamicstability of the nonlinear MNBS has been carried outwithin the framework of nonlocal elasticity theory. Inthe present paper, we carry out the semi-analyticalprocedure based on the incremental harmonic balance(IHB) method to find periodic solutions and instabil-ity regions of the axially loaded system of m cou-pled nonlinear nanobeams embedded in the viscoelas-tic medium. We assume that nanobeams are havingthe same material and geometric properties as wellas boundary conditions. By considering the Euler–Bernoulli beam theory, nonlocal constitutive relationand von Karman nonlinear strains, we obtain a sys-tem of m nonlinear partial differential equations ofmotion. Single-mode Galerkin discretization will beemployed to obtain a system of m nonlinear differen-tial equations that will be solved using the IHBmethodin order to obtain semi-analytical periodic solutionsof the nonlinear MNBS for different configurations.Moreover, the stability of periodic solutions will beexamined by introducing the Floquet theory. The para-metric study will be conducted to study the influence ofdifferent parameters on the regions of instability, non-linear amplitude–frequency response curve and Flo-quet multipliers. This study can be a starting point for

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Fig. 1 The mechanicalmodel of MNBS

investigation of more complex behaviour such as chaosin nonlinear MNBS.

2 Problem formulation

2.1 Formulation of the dynamic equations of motion

Let us consider the system formed by the set of straightand parallel nanobeams with geometric nonlinearity,which are embedded in a viscoelastic medium andsubjected to time-dependent axial compressive loadsF1 (t) = F2 (t) = · · · = Fm (t) = F (t), Fig. 1.All nanobeams in MNBS are having the same mate-

rial and geometric properties such as elastic modulusE , mass density ρ, uniform cross sections of area Aand moments of inertia I . Nanobeams in MNBS arereferred to as nanobeam 1, nanobeam 2 and so on untilthe m-th nanobeam. The transverse displacement ofthe i-th nanobeam is wi (x, t) , i = 1, 2, 3 . . .m. Thisstudy is limited to the case of the Euler-Bernoulli’sbeam model with simply supported boundary condi-tions (see Fig. 1). It considers only MNBS coupled inthe clamped chain system, where the first and the lastnanobeam in the system are coupled with a fixed basethrough a viscoelastic medium of stiffness k and vis-cosity b. Other nanobeams in MNBS are also coupled

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through the viscoelastic medium of stiffness per lengthk and viscosity b.

In order to explore thedynamic stability of nanoscalestructures, we first need to consider size effects thatmight significantly change dynamic properties of non-linear dynamic systems. For this purposes, modifiedcontinuum theories shown to be a reliable tool for mod-elling of nanoscale systems [13–15]. Eringen [10,11]derived a new constitutive relation in the integral formand later on in the differential form, based on theassumption that the stress at some point of an elas-tic body is a function of strains at all other points ofthat body. The aforementioned differential form of thenonlocal constitutive relation for one-dimensional caseis given as

σxx − μ∂2σxx

∂x2= Eεxx , (1)

where E is Young’s modulus of the nanobeam, μ =(e0a)2 is the nonlocal parameter (length scale) with adenoting the internal characteristic length and e0 denot-ing the dimensionless material constant. Further, σxxis the normal nonlocal stress, whereas the von Karmannonlinear strain–displacements relation is given as fol-lows:

εxx = ∂u

∂x− z

∂2w

∂x2+ 1

2

(∂w

∂x

)2

. (2)

In the literature, it is shown that the exact value of thenonlocal parameter is scattered and depends on appliedboundary conditions,material andgeometric propertiesof nanostructures.

The most common method for determination of thenonlocal parameter for simple nanostructures is the fit-ting procedure based on molecular dynamics simula-tions. Asmentioned previously, the nonlocal parameterμ = (e0a)2 is determined by the internal characteris-tic length a (lattice parameter, granular size or distancebetween C–C bounds that is for SWCNT a =1.42 (Å),e.g. see [55]) and constant e0, whose value is differ-ent for each material. Eringen [10] obtained the valueof nonlocal parameter by comparing the dispersioncurves of plane waves from nonlocal model with thoseobtained by the atomic Born–Karman model of lat-tice dynamics. The author has found that the maximumdifference of 6% is obtained for the value of constante0 = 0.39. Furthermore, the value of key parametere0 can be found by comparing the results for frequen-

cies found by the nonlocal elasticity model with thosefrom theMD simulations. In order to analyse themulti-wall carbon nanotube-based system, Sudak [56] pro-posed that e0 = 112.7, for the case when the, nonlocalparameter is (e0a) = 1.50 × 10−8 (cm) and a = 1.42(Å). In the paper proposed by Zhang et al. [57], theauthors have estimated the value of material constante0 ≈ 0.82 by matching the results from nonlocal con-tinuum model with the results from molecular dynam-ics simulations given in Sears and Batra [58]. Ansari etal. [19,59,60] published a series of paperswhere valuesof nonlocal parameter are calibrated based on molec-ular dynamic simulations. The authors have employedthe NanoHive simulator to perform quasi-static molec-ular dynamics simulations on SWCNTs with differentchirality, aspect ratios, and boundary conditions. Byusing the nonlinear least-square fitting procedure, fun-damental frequencies are obtained for different nonlo-cal beam theories and matched with those calculatedfrom MD simulations, where (e0a) is set as the opti-mization variable for each nonlocal beam model. Theyobtained the values of nonlocal parameter (e0a) fromthe fitting procedure for both, armchair and zigzagSWCNTs, taking into account different boundary con-ditions. It is shown that values of nonlocal parameterare ranging from 0.13 to 1.42 for different nanobeamtheories and boundary conditions. Based on previousresults, in this study we adopted that the values of non-local parameter goes in the range (e0a) = 0 − 2 (nm).

The equations ofmotion can be expressed in terms ofdisplacements wi (x, t), where by taking into accountnonlocal constitutive relation (1) and von Karman non-linear strain–displacements relation (2) in theD’Alem-bert’s principle, we obtain the following equation ofmotion for the i-th nanobeam [59],(1 − μ

∂2

∂x2

)[ρA

∂2wi

∂t2+ (F0 + F1 cos 2�t)

∂2wi

∂x2

− E A

2L

∂2wi

∂x2

∫ L

0

(∂wi

∂x

)2

dx + k (wi − wi+1)

+ k (wi − wi−1) + b

(∂wi

∂t− ∂wi+1

∂t

)

+ b

(∂wi

∂t− ∂wi−1

∂t

)]

+ E I∂4wi

∂x4= 0, i = 1, 2, . . .m (3)

and clamped chain conditions as follows:

w0 (x, t) = 0 and wm+1 (x, t) = 0 (4)

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where the axial load is F = F0 + F1 cos 2�t withamplitudes of static F0 and dynamic part F1 of axialload, while � is the frequency of the dynamic part ofaxial load.

FromEqs. (3) and (4),we obtain equations ofmotionof the clamped chain system, shown in Fig. 1, as:(1 − μ

∂2

∂x2

)[ρA

∂2w1

∂t2+ (F0 + F1 cos 2�t)

∂2w1

∂x2

− E A

2L

∂2w1

∂x2

∫ L

0

(∂w1

∂x

)2

dx

+ 2kw1 + 2b∂w1

∂t− kw2 − bw2

]

+ E I∂4w1

∂x4= 0, i = 1, (5a)

(1 − μ

∂2

∂x2

)[ρA

∂2wi

∂t2+ (F0 + F1 cos 2�t)

∂2wi

∂x2

− E A

2L

∂2wi

∂x2

∫ L

0

(∂wi

∂x

)2

dx + ki (wi − wi+1)

+ ki−1 (wi − wi−1)

+ bi

(∂wi

∂t− ∂wi+1

∂t

)+ bi−1

(∂wi

∂t− ∂wi−1

∂t

)]

+ E I∂4wi

∂x4= 0, i = 2, . . .m − 1 (5b)

(1 − μ

∂2

∂x2

)[ρA

∂2wm

∂t2+ (F0 + F1 cos 2�t)

∂2wm

∂x2

− E A

2L

∂2wm

∂x2

∫ L

0

(∂wm

∂x

)2

dx

+ 2kwm + 2b∂wm

∂t− kwm−1 − bwm−1

]

+ E I∂4w1

∂x4= 0, i = m. (5c)

Mathematical expressions for initial and boundary con-ditions of simply supported Euler-Bernoullinanobeams, of the same length L , are given as follows:*initial conditions

wi (x, 0) = wi0 (x) ,∂wi (x, 0)

∂t= vi0 (x) , (6a)

*boundary conditions for simply supported (S–S)nanobeams

wi (0, t) = wi (L , t) = 0, M f i (0, t)

= M f i (L , t) = 0, i = 1, 2, 3, ..,m. (6b)

3 Galerkin discretization method

Now, we can reduce the system of m nonlinear partialdifferential equations of motion (5) to the system of

m nonlinear ordinary differential equations by usingthe Galerkin discretization, where obtained equationsrepresent only the time-varying part of the solution.The first mode of approximation is assumed to be inthe following form:

wi (x, t) = r fi (t) φ (x) , (7)

in which r =√

IA is the radius of gyration of the

cross section, fi (t) is the corresponding time functionand φ (x) is the linear mode shape function determinedfrom the boundary conditions. The mode shape func-tion for corresponding boundary conditions of MNBSis given as follows:

φ (x) = sin (αx) , α = π

L, (8)

Inserting Eq. (7) into (5), multiplying the results by cor-responding linearmode shape functionEq. (8), and thenintegrating them over the length of the nanobeam, weobtain a set of m nonlinear ordinary differential equa-tions expressed in terms of corresponding time func-tions fi (t) as follows:

f1 +(ω2F + 2ω2

K + ω2L + 2� cos 2�t

)f1

+ 2υ f1 − γ f 31 − ω2K f2 − υ f2 = 0,

f2 +(ω2F + 2ω2

K + ω2L + 2� cos 2�t

)f2

+ 2υ f2 − γ f 32 − ω2K f1 − υ f1 − ω2

K f3 − υ f3 = 0,

f3 +(ω2F + 2ω2

K + ω2L + 2� cos 2�t

)f3

+ 2υ f3 − γ f 33 − ω2K f2 − υ f2 − ω2

K f4 − υ f4 = 0,

...

fi +(ω2F + 2ω2

K + ω2L + 2� cos 2�t

)fi

+ 2υ fi − γ f 3i − ω2K fi−1 − υ fi−1

−ω2K fi+1 − υ fi+1 = 0,

...

fm−1 +(ω2F + 2ω2

K + ω2L + 2� cos 2�t

)fm−1

+ 2υ fm−1 − γ f 3m−1 − ω2K fm−2 − υ fm−2

−ω2K fm − υ fm = 0,

fm +(ω2F + 2ω2

K + ω2L + 2� cos 2�t

)fm

+ 2υ fm − γ f 3m − ω2K fm−1 − υ fm−1 = 0,

(9)

whereωL is the natural frequency of corresponding lin-ear system,υ is the damping ratio and γ is the nonlinearstiffness

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ω2L = E Ia3

ρA (a1 − μa2), υ = b

ρA,

γ = E Ar2a4 (a2 − μa3)

2LρA (a1 − μa2), (10a)

and ω2K , ω2

F and � are reduced constants given as fol-lows:

ω2F = F0 (a2 − μa3)

ρA (a1 − μa2), ω2

K = k

ρA,

� = F1 (a2 − μa3)

2ρA (a1 − μa2), (10b)

{a1, a2, a3, a4} =∫ L

0

{φ2, φ · φ′′, φ · φ I V , (φ′)2

}dx,

(10c)

In the following, we introduce the IHBmethod to deter-mine the periodic solutions of the presented system ofm nonlinear ordinary differential equations, instabilityregions of the nonlinear MNBS and Floquet multipli-ers. It should be noted that stability of determined peri-odic solutions of the nonlinear MNBS is studied byusing the Floquet stability theory.

4 Instability regions and periodic solution: IHBmethod

Lau et al. [61] developed the methodology for the IHBmethod which is later on applied to different problems[62–66]. In general, IHB is the semi-analytical methodthat can be used for finding the solutions of differentialequations appearing inmathematical models of variousproblems in mechanics. In order to obtain instabilityregions and periodic solutions of the system of non-linear ordinary differential equations (9), we use theIHB method to obtain iterative relationships with fre-quency, response amplitudes and excitation amplitudeof time-varying axial load. First, one should introducea new time scale τ = �t into Eq. (9) to obtain the sys-tem of nonlinear ordinary differential equations in thefollowing form:

�2 d2 fidτ 2

+(ω2F + 2ω2

K + ω2L + 2� cos 2�t

)fi

+ 2υ�d fidτ

− γ f 3i

−ω2K fi−1 − υ�

d fi−1

dτ− ω2

K fi+1

−υ�d fi+1

dτ= 0, i = 1, 2, . . . ,m, (11)

Dividing Eq. (15) with ω2L , one can obtain a new form

of Eq. (11) as follows:(�

ωL

)2 d2 fidτ 2

+(

ω2F + 2ω2

K

ω2L

+ 1 + 2 � cos 2τ

)fi

+ 2υ�

ω2L

d fidτ

− γ

ω2L

f 3i

−(

ωK

ωL

)2

ω2K fi−1 − υ

�

ω2L

d fi−1

dτ−

(ωK

ωL

)2

fi+1

−υ�

ω2L

d fi+1

dτ= 0, i = 1, 2, . . . ,m, (12)

where the values of the frequency � depends onthe configuration of the nonlinear MNBS and � =�

ω2L. Now, we introduce the incremental relations for

fi (τ ) ,�, and �, where fi0 (τ ), �0 and �0 denotes aknown vibration state, and � fi (τ ), �� and �� arecorresponding increments of amplitude, frequency andamplitude of excitation load, respectively, for neigh-bouring vibration state, which is given as follows:

fi (τ ) = fi0 (τ ) + � fi (τ ) , � = �0 + ��,

� = �0 + ��, i = 1, 2, . . . ,m. (13)

Substituting relations (13) into Eq. (12) and neglectinghigher-order terms, a linearized incremental relationcan be expressed as

−[(

ωK

ωL

)2

� fi−1 + υ�0

ω2L

d� fi−1

dτ

]

−[(

ωK

ωL

)2

� fi+1 + υ�0

ω2L

d� fi+1

dτ

]

+[(

�0

ωL

)2 d2� fidτ 2

+(

ω2F + 2ω2

K

ω2L

+ 1

+ 2�0 cos 2τ)

� fi + 2υ�0

ω2L

d� fidτ

− 3γ

ω2L

f 2i0� fi

]

= −[(

�0

ωL

)2 d2 fi0dτ 2

+(

ω2F + 2ω2

K

ω2L

+ 1

+ 2�0 cos 2τ)fi0 + 2υ

�0

ω2L

d fi0dτ

− γ

ω2L

f 3i0

]

+[(

ωK

ωL

)2

fi−10 + υ�0

ω2L

d fi−10

dτ

]

+[(

ωK

ωL

)2

fi+10 + υ�0

ω2L

d fi+10

dτ

]

−(2�0

ω2L

d2 fi0dτ 2

+ 2υ

ω2L

d fi0dτ

)��

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− 2 fi0 cos 2τ �� + υ

ω2L

d fi−10

dτ��

+ υ

ω2L

d fi+10

dτ��, i = 1, 2, . . . ,m. (14)

Assuming the periodic solutions of Eq. (14), the func-tions fi0 (τ ) and � fi (τ ) can be taken as Fourier seriesin the following forms:

fi0 (τ ) =Nh∑k=1

{aik cos (kτ) + bik sin (kτ)}

= C · Ai , i = 1, 2, . . . ,m,

� fi (τ ) =Nh∑k=1

{�aik cos (kτ) + �bik sin (kτ)}

= C · �Ai , i = 1, 2, . . . ,m, (15)

where

C = [ cos τ cos2τ cos3τ . . . cos Nhτ

sin τ sin2τ sin 3τ . . . sin Nhτ ] ,

Ai = [ai1 ai2 ai3 . . . aiNh bi1 bi2 bi3 . . . biNh

]T,

�Ai = [�ai1 �ai2 �ai3 . . .

�aiNh �bi1 �bi2 �bi3 . . . �biNh

]T. (16)

Inserting Eq. (15) into the systemof Eq. (14) and apply-ing the Galerkin procedure presented in [61–68], weobtain the system of linear algebraic equations in termsof unknown increments of amplitudes �Ai in the fol-lowing form:

⎡⎢⎢⎢⎢⎢⎣

K11 K12 0 0 0 0K21 K22 K23 0 0 00 K32 K33 K34 . . . 0

0 0 0 . . .. . . 0

0 0 0 0 Km−1m Kmm

⎤⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

�A1�A2�A3

...

�Am−1�Am

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎣

R11 R12 0 0 0 0R21 R22 R23 0 0 00 R32 R33 R34 . . . 0

0 0 0 . . .. . . 0

0 0 0 0 Rm−1m Rmm

⎤⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

A1A2A3...

Am−1Am

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

+

⎡⎢⎢⎢⎢⎢⎣

P11 P12 0 0 0 0P21 P22 P23 0 0 00 P32 P33 P34 . . . 0

0 0 0 . . .. . . 0

0 0 0 0 Pm−1m Pmm

⎤⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

A1A2A3...

Am−1Am

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

��

+

⎡⎢⎢⎢⎢⎢⎣

Q11 0 0 0 0 00 Q22 0 0 0 00 0 Q33 0 . . . 0

0 0 0 . . .. . . 0

0 0 0 0 0 Qmm

⎤⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

A1A2A3...

Am−1Am

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

��

(17)

where

[Kii] = 1

π

∫ 2π

0

[(�0

ωL

)2

CT d2Cdτ 2

+(

ω2F + 2ω2

K

ω2L

+ 1 + 2�0 cos 2τ

)CTC

+ 2υ�0

ω2L

CT dCdτ

− 3γ

ω2L

f 2i0CTC

]dτ,

[Kij

] = − 1

π

∫ 2π

0

[(ωK

ωL

)2

CTC + υ�0

ω2L

CT dCdτ

]dτ,

[Rii] = − 1

π

∫ 2π

0

[(�0

ωL

)2

CT d2Cdτ 2

+(

ω2F + 2ω2

K

ω2L

+ 1

+2�0 cos 2τ)CTC + 2υ

�0

ω2L

CT dCdτ

− γ

ω2L

f 2i0CTC

]dτ,

[Rij

] = 1

π

∫ 2π

0

[(ωK

ωL

)2

CTC + υ�0

ω2L

CT dCdτ

]dτ,

[Pii] = − 1

π

∫ 2π

0

[2�0

ω2L

CT d2Cdτ 2

+ 2υ

ω2L

CT dCdτ

]dτ,

[Pij

] = 1

π

∫ 2π

0

[υ

ω2L

CT dCdτ

]dτ,

[Qij

] = − 1

π

∫ 2π

0

[2 cos 2τCTC

]dτ,

(i, j) = 1, 2, . . . ,m, . (18)

In Eq. (17), corrective vector term [R] {A} tends tozero when the values of fi0 (τ ) ,�0 and �0 tendsto exact solutions. Set of linearized algebraic equa-tions presented in Eq. (17) can be solved incremen-tally by starting from any known linear solution using�-incrementation procedure.

Since the systemof Eq. (17) has twomore unknownsthan the number of equations in the system, we needto set one value of Fourier coefficients in Eq. (15)to be constant and corresponding increment equal tozero. In the �-incrementation process, the �� is anactive increment whereby giving the initial values of

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Ai , �0 and �0 one can obtain principal instabilityregions that are bounded by two boundaries. Whenusing the Newton–Raphson method for solving thesystem (17), we need to do the following two steps.First, we determine the initial values of Ai , �0 and�0, set ai1 = Const., bi1 = Const., �ai1 = 0,�bi1 = 0 and consider �� as an active incrementduring simulation to obtain the same number of equa-tions and unknowns in Eq. (17). Second, by using theiterative Newton–Raphson method, we solve the incre-mental relation (17), where the results for �Ai and�� are obtained iteratively until the residue Euclid-ian norm | [R] {A} | is smaller than a preset toler-ance for {Ai }p+1 = {Ai }p + {�Ai }p+1 and �

p+10 =

�p0 + �p+1. In the next step, the initial value of �—

incrementation procedure is �0 + ��, and we repeatan iterative procedure until the residue Euclidian norm| [R] {A} | is smaller then a preset tolerance. The pro-cess is repeated until �0 reaches the set value.

In order to obtain periodic solutions of the systemof nonlinear ordinary differential equations (12), wereduce the incremental relations to

fi (τ ) = fi0 (τ ) + � fi (τ ) , i = 1, 2, . . . ,m,

(19a)

and for a periodic solution we obtain

fi0 (τ ) =Nh∑k=0

{aik cos (kτ) + bik sin (kτ)}

= C · Ai , i = 1, 2, . . . ,m,

� fi (τ ) =Nh∑k=0

{�aik cos (kτ) + �bik sin (kτ)}

= C · �Ai , i = 1, 2, . . . ,m, (19b)

where

C = [1 cos τ cos2τ cos3τ . . . cos Nhτ

sin τ sin2τ sin 3τ . . . sin Nhτ ] ,

Ai = [ai0 ai1 ai2 ai3 . . . aiNh bi1 bi2 bi3 . . . biNh

]T,

�Ai = [�ai0 �ai1 �ai2 �ai3 . . . �aiNh

�bi1 �bi2 �bi3 . . . �biNh

]T (19c)

Then, the incremental relation (17) becomes

⎡⎢⎢⎢⎢⎢⎣

K11 K12 0 0 0 0K21 K22 K23 0 0 00 K32 K33 K34 . . . 0

0 0 0 . . .. . . 0

0 0 0 0 Km−1m Kmm

⎤⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

�A1

�A2

�A3...

�Am−1

�Am

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎣

R11 R12 0 0 0 0R21 R22 R23 0 0 00 R32 R33 R34 . . . 0

0 0 0 . . .. . . 0

0 0 0 0 Rm−1m Rmm

⎤⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

A1

A2

A3...

Am−1

Am

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

.

(20)

5 Stability of the periodic solution: Floquet theory

In order to analyse the stability of periodic solutionsof the system of nonlinear ordinary differential equa-tions (9), we consider the Floquet theory in R

N . Now,we write the system of nonlinear ordinary differentialequations for nonlinear MNBS in general form as fol-lows:F

(f ′′, f ′, f ,�, τ

) = 0, (21)

where f = [ f1 (τ ) , f2 (τ ) , . . . fN (τ )] is N -dimen-sional displacement vector and f ′ = df /dτ .

Introducing small perturbations �f (τ ) in a neigh-bourhood of the periodic solution f 0 (τ ), i.e. byletting

f (τ ) = f 0 (τ ) + �f (τ ) , (22)

we can analyse the stability of periodic solutions byanalysing the system of linear equations with variablecoefficients in terms of small perturbations �f (τ ).Introducing Eq. (22) into the system of Eq. (21) andafter linearization, we obtain the system of linear dif-ferential equations as follows:(

∂F∂f ′′

)0�f ′′ +

(∂F∂f ′

)0�f ′+

(∂F∂f

)0�f = 0, (23)

where f 0 (τ ) is the periodic solution determined byconsidering the IHB method. It should be noted thatEq. (23) represents the system of perturbed equationsaround known solutions f 0 (τ ). The stability charac-teristics of known periodic solutions are found throughthe multi-variable Floquet theory.

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We introduce state-space variables where

dYdτ

= G (τ )Y, (24)

for Y (τ ) = [�f ,�f ′]T and G (τ ) denoting the peri-

odic matrix.Based on the Floquet theory [68–70], the stability

criteria for determined periodic solutions of MNBS isrelated to eigenvalues of the matrix M (Floquet multi-pliers), which is the transition matrix and the real partsof characteristic exponents (Floquet exponents). In thecase when all values of Floquet multipliers are located

inside the unit circle centered at the origin of the com-plex plane, the periodic solutions are stable or asymp-totically stable. However, when the values of Floquetmultipliers are out of the unit circle, then the periodicsolutions are unstable [68–70].

In order to numerically determine the Floquet mul-tipliers, we assume that period T = 2π of f 0 (τ ) isdivided into Nk subintervals, in which the k-th intervalis�k = τk −τk−1 for τk = kT/Nk . Moreover,G (τ ) isthe continuous matrix with respect to τ , so in the k-thinterval it can be replaced by the constant matrix pro-vided in the case when Nk is chosen to be sufficientlylarge

Gk = 1

�k

∫ τk

τk−1

G (τ ) dτ. (25)

Now,we canwrite the transitionmatrix in the followingform:

M =Nk∏i=1

eAi�i =Nk∏i=1

⎛⎝I +

N j∑j=1

(Gi�i )j

j !

⎞⎠ , (26)

where N j denotes the number of terms in the approx-imation of the constant matrix Gk . From the transitionmatrixM, one can obtain Floquet multipliers as eigen-values of Eq. (26) in the following form:

det (M − σ I) = 0. (27)

For the nonlinear MNBS consisting of m nanobeams,the periodic matrix G (τ ) has the following form:

G (τ ) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 0 0 0 0 0 0 0−P1 −2R T R 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0T R −P2 −2R T R 0 0 0 0

0 0 0 0 0. . . . . . 0 0 0

0 0...

.... . .

. . . . . . R 0 00 0 0 0 0 0 0 1 0 00 0 0 0 T R −Pm−1 −2R T R0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 T R −Pm −2R

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦2m × 2m

, (28)

where Pi , (i = 1, 2, . . .m), R and T are defined as

Pi =(ωL

�

)2 (ω2F + 2ω2

K

ω2L

+ 1 + 2�

ω2L

cos 2τ

− 3γ

ω2L

f 2i0

), R = υ

�, T =

(ωK

�

)2,

fi0 (τ ) =Nh∑k=0

{aik cos (kτ)

+ bik sin (kτ)} , i = 1, 2, . . . ,m. (29)

From the relation (28), it is easy to obtain a systemmatrix for a specific configuration of the nonlinearMNBS.

6 Numerical results

This section is divided into four parts. First, we showa comparative study where results for periodic solu-tions of the system of nonlinear differential equationsobtained by employing the IHB method are validated

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Fig. 2 Periodic solution of a single nanobeam system as a special case of MNBS. a time functions of the nanobeam, b periodic orbitof the nanobeam

with the results from the Runge–Kutta method (ode45-Matlab). Afterwards, we analyse the stability of peri-odic motions of the nonlinear MNBS by employingthe Floquet stability theory. In that context, we showFloquet multipliers as functions of different physicalparameters. In the second part of this section, we showthe stability boundaries and instability regions deter-mined by the IHB method for different configurationsof the nonlinear MNBS. The free nonlinear vibrationsof MNBS are analysed in the third part, where fre-quency response curves are shown for different num-bers of nanobeams in the system. The validation studyis presented in the last subsection.

The system of m nonlinear differential equations(9) represents a mathematical model of the nonlinearMNBS that is vibrating under the influence of time-varying axial loads with considered initial conditions.In the case of larger systems, such as the nonlinearMNBS, it is difficult to find system responses usingonly analytical methods. In addition, in a large num-ber of nanoengineering applications, the amplitude ofvibration is not small and for such systems, we can onlyuse numerical or approximate technics. It should benoted that IHB method has several advantages in com-parison with classical approximated methods. First,there are no limitations such as small exciting param-eters or weak nonlinearities and second, it is easy forcomputer implementation. Because of these facts, wedetermine periodic solutions of the nonlinear MNBS

without introducing any limitations such as the book-keeping parameter.

6.1 Periodic solutions and stability

In this study, we consider several different cases ofMNBS with m = 1, 3, 5 nanobeams and for differ-ent initial conditions. We adopted the material andgeometric properties for nanobeams in MNBS cor-responding to the properties of single-walled carbonnanotubes as: Young’s modulus E = 1.1 TPa, densityρ = 1300 kg

m3 , length L = 45 nm, inner d1 = 3 nmand outer d0 = 2.32 nm diameters of the nanotube.For the numerical purposes, we use the value of elasticcoefficient of the medium as k = 108 (N/m2), whileinfluence of damping coefficient is almost neglectedand small value of b = 10−6 (Ns/m2) is adopted.

In order to illustrate the accuracy of the proposedIHB methodology, we compare the obtained resultswith the results from the Runge–Kutta method inFigs. 2, 3 and 4, where a fine agreement of the resultsis achieved. These figures show time response func-tions of the first nanobeam and periodic orbits of eachnanobeam for different configurations of the nonlin-ear MNBS. Figure 2 shows periodic orbits of the spe-cial case of the nonlinear MNBS consisting of a singlenanobeam, which is the case very often analysed in theliterature. However, in Figs. 3 and 4, one can observeperiodic orbits of the nonlinear MNBS consisting ofthree- and five-coupled nanobeams. We can notice that

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Fig. 3 Periodic solutions ofMNBSwith three nanobeams, a time functions of the first nanobeam, b periodic orbit of the first nanobeam,c periodic orbit of the second nanobeam and d periodic orbit of the third nanobeam in MNBS

the periodic solutions strongly depend on initial condi-tions.

Now, we can investigate the stability of determinedperiodic solutions of the nonlinear MNBS subjected toaxial time-dependent loads (Figs. 5, 6, 7). By consider-ing the Floquet theory, we can obtain Floquet multipli-ers for different configurations of the nonlinear MNBSconsisting of one, three and five nanobeams in the sys-tem. By calculating the Floquet multipliers and trac-ing their evaluation, we predict the stability or identifybifurcations of the presented system. For stability ofthe periodic solution, Floquet multipliers must be cen-tred within the unit circle with the origin in the centreof the complex plane. However, if the values of Flo-quet multipliers leave the unit circle, we can predict

bifurcation according to [71]. For particular values ofmaterial parameters and initial conditions, we considerthe iterative equation (20) and obtain the following sta-ble periodic solutions wherein the stability of the tra-jectory is confirmed by using the Floquet theory andabove-described iterative procedure [69,70].We set thevalues of Nk = 104 and N j = 5 in the relation (26)showing Floquet multipliers as functions of amplitudesof axial loads and nonlocal parameter, Figs. 5, 6 and 7,for different configurations of the nonlinear MNBS.

The influence of axial load amplitude on the Floquetmultipliers for the special case of MNBS consisting ofa single nanobeam is shown in Fig. 5. Moreover, weget Floquetmultipliers as complex numbers, where realparts are shown inFig. 5a and imaginaryparts inFig. 5b.

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Fig. 4 Periodic solutions for five nanobeams in MNBS, a timefunctions of the first nanobeam, b periodic orbit of the firstnanobeam, c periodic orbit of the second nanobeam, d peri-

odic orbit of the third nanobeam, e periodic orbit of the fourthnanobeam and f periodic orbit of the fifth nanobeam in MNBS

123


Fig. 5 Floquet multiplier for one nanobeam in MNBS

Fig. 6 Floquet multipliers for three nanobeams in MNBS

123


Fig. 7 Floquet multipliers for five nanobeams in MNBS

123


By increasing the value of axial load �, both parts ofthe Floquet multipliers remain within the unit circle ofthe complex plane, as shown in Fig. 5c. Based on thatfact, we can conclude that obtained periodic solutionsfor the nonlinear system are stable for both values ofthe nonlocal parameter. It should be noted that the realparts of Floquet multipliers decrease for an increasein the nonlocal parameter, while the imaginary partsincreases.

Figure 6 shows three values of Floquetmultipliers asfunctions of the nonlocal parameter and amplitude ofaxial load for the nonlinear MNBS consisting of threenanobeams. From obtained curves, it can be observedthat the influence of the amplitude of axial load on theFloquet multipliers is not a smooth function comingout of the unit circle in the complex plane, as shown inFig. 6g, i. Based on that fact, we can conclude that theperiodic solution of the third nanobeam in MNBS isunstable. For the Floquet multipliers regarding the firstand the second nanobeam,we can observe that obtainedperiodic solutions are stable, as shown in Fig. 6c, f.It should be noted that the nonlocal parameter has asignificant influence on the first two Floquetmultiplierswhile for the third one it almost vanishes, as shown inFig. 6i.

The Floquet multipliers determining the stabilityof the nonlinear MNBS composed of five nanobeamsare shown in Fig. 7. We can observe a very interest-ing behaviour of Floquet multipliers with a significantdependence on the nonlocal parameter. From obtainedresults, we can notice that only the fourth and the fifthnanobeam in MNBS are having stable periodic solu-tions with the values of Floquet multipliers remainingwithin the unit circle of the complex plane, as shown inFig. 7l, p. However, other periodic solutions of MNBSare unstable, where the values of the Floquet multipli-ers are out of the unit circle, as shown in Fig. 7c, f, i.From the physical point of view, we can conclude thatthe number of nanobeams decreases the overall stiff-ness of the MNBS and this fact lead to the reduction ofthe system stability.

6.2 Instability regions

Figures 8, 9 and 10 show instability regions of thenonlinear MNBS obtained by using the IHB method.It should be noted that boundary curves of instabilityregions are determined by using the λ-incrementation,

where the value of the increment is �λ = 0.001, withfive cosine and five sine harmonic terms. The bound-ary lines leaning to the left are determined by usingthe sine terms, while boundary lines leaning to theright are defined by the cosine terms only. When con-sidering incremental relations to determine instabilityregions, we can examine cases with small and largeinitial amplitudes as well. Small initial amplitudes leadto the linear case while large values of initial ampli-tudes lead to the nonlinear case, for the same values ofthe excitation force amplitude. Both cases are explored,where amplitudes are shown as functions of the ampli-tude of axial load, a number of nanobeams and differ-ent physical parameters such as stiffness coefficients(Fig. 8), the nonlocal parameter (Fig. 9) and the ampli-tude of static axial load (Fig. 10). The values of initialamplitudes for the incremental equation (17) are givenas ai1 = 0.1 and bi1 = 0.1 for the linear case andai1 = 3 and bi1 = 3 for the nonlinear case. ApplyingtheNewton–Raphsonprocedure,where incrementationprocess starts from λ = 0 to λ = 1, we can investi-gate the instability region of the nonlinear MNBS. Forthe present analysis, we adopted the same material andgeometrical properties of nanobeams as in the previ-ous case. The instability regions are represented by theareas that are surrounded by two stability boundaries.

Figure 8 shows the influence of stiffness coefficient kand the number of nanobeams on the instability regionsof the linear and nonlinear cases. It can be concludedthat an increase in stiffness coefficient k leads to nar-rowing of instability regions while an increase in thenumber of nanobeams in MNBS leads to an increasein instability regions. From the physical point of view,an increase in the number of nanobeams decreases theoverall stiffness of the system and dynamic systemis becoming more unstable. From the nonlinear case(Fig. 8b), we can notice that different configurationsof MNBS shifts the instability regions, which is theconsequence of the system’s nonlinearity.

The influence of the nonlocal parameter on insta-bility regions of MNBS for different configurations isshown in Fig. 9. One can observe that the influence ofthe nonlocal parameter on instability regions is smallfor small values of the amplitude of axial load in thelinear case. However, in the nonlinear case, the non-local parameter has a significant influence on instabil-ity regions. As expected, we have shifts of instabilityregions of MNBS for a different number of nanobeamsin the system. The amplitudes of nonlinearity and over-

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Fig. 8 Instability regions of MNBS for different values of stiffness coefficients, a linear and b nonlinear case

Fig. 9 Instability regions of MNBS for different values of the nonlocal parameter, a linear and b nonlinear case

all stiffness of MNBS will dictate shifts of instabilityregions.

The last Fig. 10 illustrates the effects of the ampli-tude of static axial load on instability regions ofMNBS. An increase in the amplitude of static axialload increases the instability regions. This increase issmaller for the nonlinear case than for the linear one.In addition, this effect occurs as a result of an increasein the overall stiffness of the system. For all the pre-sented cases, an increase in the amplitude of axial load

� leads to an increase in instability regions and thesystem becomes more unstable.

In a general case, we can conclude that change ofa number of nanobeams in MNBS affects its insta-bility regions in such a manner that these regions areexpanding for an increase in the parameterm and smallinitial amplitudes. On the other side, for larger initialamplitudes, we can notice that the effect of a numberof nanobeams goes in two directions. First, the insta-bility regions are shifted relative to the case of smallinitial amplitudes and second, instability regions are

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Fig. 10 Instability regions of MNBS for different values of amplitude of the static part of axial load, a linear and b nonlinear case

increasing for a change of a number of nanobeams inthe system. From the mechanical point of view, we canconclude that the overall stiffness of MNBS is reducedwhen increasing the number of nanobeams, which con-sequently results in reduced stability of the system.

6.3 The free nonlinear vibration of MNBS

In order to analyse the free nonlinear vibration ofMNBS, amplitude–frequency responses for differentconfigurations of MNBS obtained by using the IHBmethod are presented in Figs. 11 and 12. It should benoted that nonlinear amplitude–frequency curves aredetermined by using the �b1-incrementation, wherethe value of the increment is �b1 = 0.001, using thefive odd cosine terms [67]. In this subsection, we willneglect the influence of the axial load and dampingcoefficient in Eq. (5) to obtain the governing equationfor the free vibration of the nonlinearMNBS embeddedin the elastic medium.

Figure 11 shows the influence of the stiffnesscoefficient of the elastic medium on the nonlinearamplitude–frequency response curve of MNBS con-sisting of one-, three- and five-coupled nanobeams.It can be observed that an increase in the stiffnesscoefficient tends to decrease nonlinear effects visibleon amplitude–frequency response curves, while theseeffects almost vanish for the highest values of stiff-ness coefficient. Moreover, in the case of the larger

stiffness of the elastic medium, nonlinear frequenciesare approaching the linear ones. On the other hand, anincrease in the number of nanobeams inMNBS leads toa decrease in overall stiffness and nonlinearity becomesmore prominent.

The effects of the nonlocal parameter and a numberof nanobeams on the nonlinear amplitude–frequencyresponse curves for each case of nanobeams in the sys-tem are shown in Fig. 12. It can be noticed that forgiven amplitudes and for an increase in the nonlocalparameter, the frequency ratio increases but the fre-quency decreases. This fact leads to the conclusion thatthe nonlocal parameter softens the system, i.e. reducesthe overall stiffness. Moreover, one can observe thatthe influence of nonlocal effects is larger for a highernumber of nanobeams and larger vibration amplitudes.

6.4 Validation study

In this work, results are obtained for MNBS embeddedin the viscoelastic medium and the authors did not finda similar mechanical model in the literature that can beused for the validation study. However, for the specialcase of MNBS, i.e. the case with a single nanobeam onelastic foundation, we use the paper by Fu et al. [67].Figure 13 shows the nonlinear amplitude–frequencyresponse of MNBS consisting of one, three and fivenanobeams and the amplitude–frequency response of

123


Fig. 11 Amplitude–frequency curves fordifferent values of stiffnesscoefficient of the elasticmedium, (e0a) = 1 nm

Fig. 12 Amplitude–frequency curve fordifferent values of thenonlocal parameterk = 107 (N/m2)

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Fig. 13 Amplitude–frequency responses ofpresent MNBS and Fu et al.[67] for (e0a) = 1 nm

the system with a single nanobeam resting on the elas-tic foundation presented in Fu et al. [67]. It should benoted that our model of MNBS is coupled in clampedchain system, i.e. for the case of a single nanobeam it iscoupled with the fixed base through the elastic mediumon both sides. This leads to higher values of the overallstiffness and such system is more constrained than themodel proposed by Fu et al. [67].

7 Scope and limitations of the presented model

The given nonlinear MNBS is based on the nonlocalEuler–Bernoulli beam model and von Karman nonlin-ear deformations and does not take into account sheardeformations. However, in order to consider this effect,one should consider Timoshenko or higher-order sheardeformation beam theories. From the literature [18],we can see that the effect of transverse shear defor-mation leads to lower vibration frequencies compar-ing to the case with Euler–Bernoulli beam. This effectis more prominent in higher vibration modes. There-fore, the consideration of transverse shear deformationleads to a decrease in stability of the presented system,i.e. it leads to widening of instability regions [72]. Itshould be noted that applied axial loads are smaller thanthe critical buckling load, and MNBS is not buckled.However, the nonlinear vibration behaviour of buckled

MNBS is also a very important issue and should bestudied in some future works.

The second limitation refers to the fact that presentedMNBS model consider amplitudes of nanobeams thatare bounded by the surrounding viscoelastic medium.Based on that, we analysed only small nonlinear defor-mations by using the von Karman theory. It should beemphasized that we employed the phenomenologicalviscoelastic model based on simple mechanical anal-ogy to describe the rheological behaviour of the vis-coelastic medium in MNBS. Since the applied force–displacement relation is a phenomenological modelof Kelvin–Voight’s viscoelastic stress–strain constitu-tive relation and reaction force of viscoelastic mediumis distributed over nanobeam’s length, it can be rep-resented by parallelly connected springs and dash-pots with corresponding elasticity and viscosity coef-ficients. An interested reader can find more on experi-mental works and dynamic behaviour of CNT/polymernanocomposites in [51–53,75].

In the literature, the authors have found a series ofpapers [61,67,73,74] where linear mode shape func-tions are used for a nonlinear vibration analysis ofclassical and scale-dependent beam models. However,we should mention two interesting papers by El-Borgiet al. [73] and Nayfeh and Lacarbonara [74], wherethe authors concluded that the Galerkin single-mode

123


method is suitable for the discretization of governingequations of nonlinear vibration of simply supportedbeams, especially for the primary resonance state andthe first vibration mode. According to this, the authorsof the present paper employed only a single-modeGalerkin method for discretization of the governingequations of given MNBS. One should know that eachnanobeam in MNBS has the same simply supportedboundary conditions and therefore, same linear modeshape functions are considered in the assumed solu-tions of the system of governing equations. Moreover,in the absence of internal resonances, the steady-stateresponse contains only the directly excited modes. Themathematical model of MNBS with considered inter-actions between modes, internal and combined reso-nances, is interesting to be analysed in some futurestudy.

8 Conclusions

The present analysis describes the dynamic stabil-ity problem of the nonlinear MNBS embedded in aviscoelastic medium, where each nanobeam in thesystem is simply supported and subjected to axialtime-dependent loads. A mathematical model of thenonlinear MNBS is based on the system of nonlin-ear partial differential equations of motion derivedby using the Eringen’s nonlocal elasticity theory,Euler–Bernoulli’s beam theory and nonlinear von Kar-man strain–displacement relation. By employing theGalerkin discretization technique and IHB method, wedetermined the stability boundaries and periodic solu-tions of the system of nonlinear ordinary differentialequations. For such determined periodic solutions, weintroduced a Floquet theory to analyse their stability.In the numerical study, we have shown the influenceof different material and geometric parameters on thestability behaviour of the presented nonlinear MNBS.A few novelties are proposed in this paper:

• The Floquet multipliers are obtained as functionsof different parameters of the stiffness coeffi-cient, amplitude of the axial load and number ofnanobeams.

• The stability boundaries are obtained for differentconfigurations of the nonlinear MNBS.

• Coupled influence of nonlocal and nonlinear effectin the context of the stability behaviour of the non-linear MNBS.

• Semi-analytical periodic solutions are obtained fornonlinearMNBSby using the IHBmethod,withoutintroducing small parameter.

It can be concluded that the analysis of periodicsolutions of the nonlinear MNBS without taking intoaccount nonlocal effectsmight lead to large errorswheninvestigating the dynamic behaviour of such complexsystems. It is well known that a very small variationin amplitudes of axial loads can cause structural col-lapse and the values of amplitudes should be cho-sen carefully. Information based on reliable modelsare important for the design of future nanostructure-based devices; therefore, this study besides theoreticalcould also have practical applications in nanoengineer-ing fields. In addition, this study might be importantfor future theoretical analyses of the nonlinear dynamicbehaviour of MNBS such as internal resonance, com-bined resonance conditions, and chaos.

Acknowledgements This research was supported by theproject of the Ministry of Education, Science and Technologi-cal Development of the Republic of Serbia under the Grant No.OI 174001.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/),whichpermits unrestricteduse, distribution, and reproduction in any medium, provided yougive appropriate credit to the original author(s) and the source,provide a link to the Creative Commons license, and indicate ifchanges were made.

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