Dynamic stability of a nonlinear multiplenanobeam system
 498 Downloads
Abstract
We use the incremental harmonic balance (IHB) method to analyse the dynamic stability problem of a nonlinear multiplenanobeam system (MNBS) within the framework of Eringen’s nonlocal elasticity theory. The nonlinear dynamic system under consideration includes MNBS embedded in a viscoelastic medium as clamped chain system, where every nanobeam in the system is subjected to timedependent axial loads. By assuming the von Karman type of geometric nonlinearity, a system of m nonlinear partial differential equations of motion is derived based on the Euler–Bernoulli beam theory and D’ Alembert’s principle. All nanobeams in MNBS are considered with simply supported boundary conditions. Semianalytical solutions for time response functions of the nonlinear MNBS are obtained by using the singlemode Galerkin discretization and IHB method, which are then validated by using the numerical integration method. Moreover, Floquet theory is employed to determine the stability of obtained periodic solutions for different configurations of the nonlinear MNBS. Using the IHB method, we obtain an incremental relationship with the frequency and amplitude of timevarying axial load, which defines stability boundaries. Numerical examples show the effects of different physical and material parameters such as the nonlocal parameter, stiffness of viscoelastic medium and number of nanobeams on Floquet multipliers, instability regions and nonlinear amplitude–frequency response curves of MNBS. The presented results can be useful as a first step in the study and design of complex micro/nanoelectromechanical systems.
Keywords
Multiplenanobeam system Geometric nonlinearity Nonlocal elasticity Instability regions IHB method Floquet theory1 Introduction
Most dynamic stability studies in the literature are considering single or doublemicro/nanostructurebased systems [1, 2, 3, 4]. Investigation of the nonlinear dynamic behaviour of such systems has been an exciting perspective in the last decade due to possible applications in design procedures of different micro/nanoengineering systems [5], especially when these systems are aimed to be exploited as vibrating nanodevices such as resonators, nanosensors, or other nanoelectromechanical systems. However, it has been shown that organized nanostructure architectures made of vertically aligned forests, yarns, and sheets of carbon nanotubes (CNTs) give an exciting perspective to scale up the properties of individual CNTs and realize new functionalities [6]. The lack of reliable nonlinear dynamic models of multiplenanostructurebased systems makes a future investigation in this field as an attractive task for researchers. A seminal idea of this work is to fill this gap in the literature and propose new models and procedures of a solution to investigate the dynamic stability of the geometrically nonlinear multiplenanobeam system (MNBS). The presented model of MNBS consists of multiple individual simply supported nanobeams that are parallel to each other and placed into a viscoelastic medium. In general, such model can represent some nanocomposite material composed of vertically aligned CNTs array placed in some polymer. Therefore, the presented model of the nonlinear MNBS might be important to investigate the aligned arrays of CNTs [7] and CNT/polymer composites [8], especially stability of CNTs embedded in the polymer matrix. In addition, due to their remarkable physical and chemical characteristics, various applications and mathematical models of CNTs are suggested in the literature [9]. Based on experimental results, it is shown that smallscale effects play a significant role in the static and dynamic behaviour of micro/nanostructures. Since models based on classical continuum theory are scalefree, they need to be modified introducing some new assumptions in order to take into account size effects that are present at small scales. Eringen [10, 11] extended the classical continuum theory by introducing integral and differential forms of constitutive equations with a single material parameter, such that it takes into account forces between atoms and internal length scale. Moreover, it has been shown that the results obtained by using the nonlocal continuum models are in good agreement with the results obtained via molecular dynamics simulations. Many authors have analysed the mechanical behaviour of different types of nanostructures within the framework of nonlocal continuum mechanics.
A pioneering study where nonlocal elasticity theory is proposed to model micro/nanostructures is the work by Pedinson et al. [12]. They analysed the static and dynamic behaviour of a simple micro/nanobeam and proposed a possible application to real MEMS and NEMS. Reddy [13, 14, 15] has derived new equations of motion for Euler–Bernoulli, Timoshenko, Reddy and Levinson beam theories based on the Eringen’s nonlocal elasticity theory and then obtained analytical solutions for bending, vibrations, and buckling response of beams for simply supported boundary conditions. The new shear deformation beam theory proposed by HuuTai [16] is used to analyse the free vibration and stability of short nanobeams using Eringen’s nonlocal elasticity theory. Aydogdu [17] employed the Euler–Bernoulli, Timoshenko, Levinson, Reddy, and Aydogdu beam theories to analyse the bending, buckling, and vibration of nanobeams in an analytical manner. Ansari et al. [18, 19] compared results obtained from nonlocal continuum mechanics with the results obtained by molecular dynamic simulations for simple models of nanostructures and concluded that the results obtained from both theories are in good agreement. More complex nanoscale systems consist of two and more organized nanostructures such as rods, beams, and plates, which are usually coupled through some medium. The dynamic behaviour of such systems is interesting to observe and it is still not fully explored in the literature. A doublenanostructurebased system is the simplest model of a coupled multiplenanostructure systems, which can be composed of two nanobeams, nanorods or nanoplates coupled through a medium with elastic or viscoelastic properties. Murmu and Adhikari conducted detailed vibration studies of a doublenanorod, nanobeam, and nanoplate systems [20, 21, 22, 23], where partial differential equations of motion are obtained based on the D’ Alembert’s principle and nonlocal elasticity theory and solved by using analytical methods. They investigated the influence of smallscale effects and other physical parameters on natural frequencies and critical buckling loads and compared analytical results with results obtained by molecular dynamics simulations. One of the first dynamic behaviour studies of multilayered graphene sheet system is the paper by He et al. [24] and Liew et al. [25], where the authors derived an explicit formula for natural frequencies by utilizing the classical elasticity model. Using the methodology that was proposed for chainlike mechanical systems by Rašković [26] and multiple coupled structural elements by Hedrih [27], Karličić et al. [28] presented a straightforward method to obtain analytical solutions for natural frequencies and critical buckling loads of multiple nanorods, nanobeams, and nanoplates systems based on the Eringen’s nonlocal elasticity theory and trigonometric method. To this time, the mechanical behaviour of different nanostructures such as the longitudinal vibration of nanorods [29], transverse vibration of nanobeams [30, 31, 32], nanoplates [33] and nanoshells [34] with elastic properties are given in the literature.
Forced and parametric vibration problems of complex coupled nanostructurebased systems such as coupled nanobeams, nanorods, nanoplates have attracted much attention of the scientific community. Simsek [35] investigated the influence of a moving load on different types of singlewalled carbon nanotubes. Karaoglu and Aydogdu [36] investigated the forced vibration of carbon nanotubes via nonlocal Euler–Bernoulli beam model. Ansari et al. [37, 38] analysed the forced vibration of nanobeams as mechanical models of carbon nanotubes by using different numerical technics and compared the results with those obtained by the molecular dynamic simulations. In addition, Kiani [39, 40, 41, 42] has examined the influence of different physical fields and shear effects on the free and forced vibration response of nanoscale systems composed of multiple nanobeams coupled in membrane and forest configurations. Also, Kiani [43, 44, 45] conducted several studies on the influence of moving loads on doublecarbon nanotube system based on the nonlocal elasticity theory. Arani et al. [46] investigated the nonlinear dynamic stability of a doublegraphene sheet with integrated actuators and sensors based on the Gurtin–Murdoch elasticity theory. Further, in series of papers Wang et al. [47, 48] investigated the nonlinear behaviours of doublenanoplate systems for forced and parametric excitations. Recently, Pavlović et al. [49] observed the stochastic stability problem by determining the stability boundaries and regions of almost sure stability of a linear multinanobeam system embedded in a viscoelastic medium by using the Monte Carlo simulation method.
In general, dynamic stability analysis of nanobeamlike structures can play a significant role in design procedures of future nanodevices. For axially loaded nanobeams, where loads are timedependent harmonic functions, a failure due to dynamic instability might occur for much smaller amplitudes of load than the failure induced by static buckling. These instability conditions usually lead to the failure of micro/nanodevices. Based on that fact, authors usually intend to analyse stability regions caused by primary parametric resonance, where the frequency of excitation is two times larger than the first natural frequencies of MNBS [50, p. 23]. Therefore, the main aim of this paper is to extend the previous studies by investigating the dynamic stability of more complex systems such as the nonlinear MNBS and to explore its primary resonance state. In addition, investigation of the influence of smallscale parameter on the stability of periodic solutions and instability regions is a very important task for the design of different types of micro/nanodevices.
Up to this point, no investigation of the dynamic stability of the nonlinear MNBS has been carried out within the framework of nonlocal elasticity theory. In the present paper, we carry out the semianalytical procedure based on the incremental harmonic balance (IHB) method to find periodic solutions and instability regions of the axially loaded system of m coupled nonlinear nanobeams embedded in the viscoelastic medium. We assume that nanobeams are having the same material and geometric properties as well as boundary conditions. By considering the Euler–Bernoulli beam theory, nonlocal constitutive relation and von Karman nonlinear strains, we obtain a system of m nonlinear partial differential equations of motion. Singlemode Galerkin discretization will be employed to obtain a system of m nonlinear differential equations that will be solved using the IHB method in order to obtain semianalytical periodic solutions of the nonlinear MNBS for different configurations. Moreover, the stability of periodic solutions will be examined by introducing the Floquet theory. The parametric study will be conducted to study the influence of different parameters on the regions of instability, nonlinear amplitude–frequency response curve and Floquet multipliers. This study can be a starting point for investigation of more complex behaviour such as chaos in nonlinear MNBS.
2 Problem formulation
2.1 Formulation of the dynamic equations of motion
Let us consider the system formed by the set of straight and parallel nanobeams with geometric nonlinearity, which are embedded in a viscoelastic medium and subjected to timedependent axial compressive loads \(F_1 \left( t \right) =F_2 \left( t \right) =\cdots =F_m \left( t \right) =F \left( t \right) \), Fig. 1. All nanobeams in MNBS are having the same material and geometric properties such as elastic modulus E, mass density \(\rho \), uniform cross sections of area A and moments of inertia I. Nanobeams in MNBS are referred to as nanobeam 1, nanobeam 2 and so on until the mth nanobeam. The transverse displacement of the ith nanobeam is \(w_i \left( {x,t} \right) , i=1,2,3\ldots m\). This study is limited to the case of the EulerBernoulli’s beam model with simply supported boundary conditions (see Fig. 1). It considers only MNBS coupled in the clamped chain system, where the first and the last nanobeam in the system are coupled with a fixed base through a viscoelastic medium of stiffness k and viscosity b. Other nanobeams in MNBS are also coupled through the viscoelastic medium of stiffness per length k and viscosity b.
The most common method for determination of the nonlocal parameter for simple nanostructures is the fitting procedure based on molecular dynamics simulations. As mentioned previously, the nonlocal parameter \( \mu =\left( {e_0 a} \right) ^{2}\) is determined by the internal characteristic length a (lattice parameter, granular size or distance between C–C bounds that is for SWCNT a =1.42 (Å), e.g. see [55]) and constant \(e_0 \), whose value is different for each material. Eringen [10] obtained the value of nonlocal parameter by comparing the dispersion curves of plane waves from nonlocal model with those obtained by the atomic Born–Karman model of lattice dynamics. The author has found that the maximum difference of 6% is obtained for the value of constant \(e_0 = 0.39\). Furthermore, the value of key parameter \(e_0 \) can be found by comparing the results for frequencies found by the nonlocal elasticity model with those from the MD simulations. In order to analyse the multiwall carbon nanotubebased system, Sudak [56] proposed that \(e_0 = 112.7\), for the case when the, nonlocal parameter is \(\left( {e_0 a} \right) = 1.50 \times 10^{8}\) (cm) and \(a =1.42\) (Å). In the paper proposed by Zhang et al. [57], the authors have estimated the value of material constant \(e_0 \approx 0.82\) by matching the results from nonlocal continuum model with the results from molecular dynamics simulations given in Sears and Batra [58]. Ansari et al. [19, 59, 60] published a series of papers where values of nonlocal parameter are calibrated based on molecular dynamic simulations. The authors have employed the NanoHive simulator to perform quasistatic molecular dynamics simulations on SWCNTs with different chirality, aspect ratios, and boundary conditions. By using the nonlinear leastsquare fitting procedure, fundamental frequencies are obtained for different nonlocal beam theories and matched with those calculated from MD simulations, where \(\left( {e_0 a} \right) \) is set as the optimization variable for each nonlocal beam model. They obtained the values of nonlocal parameter \(\left( {e_0 a} \right) \) from the fitting procedure for both, armchair and zigzag SWCNTs, taking into account different boundary conditions. It is shown that values of nonlocal parameter are ranging from 0.13 to 1.42 for different nanobeam theories and boundary conditions. Based on previous results, in this study we adopted that the values of nonlocal parameter goes in the range \(\left( {e_0 a} \right) =02 \left( {\mathrm{nm}} \right) \).
3 Galerkin discretization method
4 Instability regions and periodic solution: IHB method
Since the system of Eq. (17) has two more unknowns than the number of equations in the system, we need to set one value of Fourier coefficients in Eq. (15) to be constant and corresponding increment equal to zero. In the \(\tilde{\Lambda } \)incrementation process, the \({\Delta }\tilde{\Lambda } \) is an active increment whereby giving the initial values of \({{\varvec{A}}}_i\), \({\Omega }_0 \) and \(\tilde{\Lambda } _0 \) one can obtain principal instability regions that are bounded by two boundaries. When using the Newton–Raphson method for solving the system (17), we need to do the following two steps. First, we determine the initial values of \({{\varvec{A}}}_i\), \({\Omega }_0 \) and \(\tilde{\Lambda } _0\), set \(a_{i1} =\mathrm{Const.}\), \(b_{i1} = \mathrm{Const.}, \quad \Delta a_{i1} =0\), \(\Delta b_{i1} =0\) and consider \({\Delta }\tilde{\Lambda } \) as an active increment during simulation to obtain the same number of equations and unknowns in Eq. (17). Second, by using the iterative Newton–Raphson method, we solve the incremental relation (17), where the results for \(\Delta {{\varvec{A}}}_i \) and \(\Delta {\Omega }\) are obtained iteratively until the residue Euclidian norm \({\vert }\left[ {{\varvec{R}}} \right] \left\{ {{\varvec{A}}} \right\} {\vert }\) is smaller than a preset tolerance for \(\left\{ {{{\varvec{A}}}_i } \right\} ^{p+1}=\left\{ {{{\varvec{A}}}_i } \right\} ^{p}+\left\{ {\Delta {{\varvec{A}}}_i } \right\} ^{p+1}\) and \({\Omega }_0^{p+1} ={\Omega }_0^p +{\Omega }^{p+1}\). In the next step, the initial value of \(\tilde{\Lambda } \)—incrementation procedure is \(\tilde{\Lambda } _0 +{ \Delta }\tilde{\Lambda } \), and we repeat an iterative procedure until the residue Euclidian norm \({\vert }\left[ {{\varvec{R}}} \right] \left\{ {{\varvec{A}}} \right\} {\vert }\) is smaller then a preset tolerance. The process is repeated until \(\tilde{\Lambda } _0 \) reaches the set value.
5 Stability of the periodic solution: Floquet theory
Based on the Floquet theory [68, 69, 70], the stability criteria for determined periodic solutions of MNBS is related to eigenvalues of the matrix \({{\varvec{M}}}\) (Floquet multipliers), which is the transition matrix and the real parts of characteristic exponents (Floquet exponents). In the case when all values of Floquet multipliers are located inside the unit circle centered at the origin of the complex plane, the periodic solutions are stable or asymptotically stable. However, when the values of Floquet multipliers are out of the unit circle, then the periodic solutions are unstable [68, 69, 70].
6 Numerical results
This section is divided into four parts. First, we show a comparative study where results for periodic solutions of the system of nonlinear differential equations obtained by employing the IHB method are validated with the results from the Runge–Kutta method (ode45Matlab). Afterwards, we analyse the stability of periodic motions of the nonlinear MNBS by employing the Floquet stability theory. In that context, we show Floquet multipliers as functions of different physical parameters. In the second part of this section, we show the stability boundaries and instability regions determined by the IHB method for different configurations of the nonlinear MNBS. The free nonlinear vibrations of MNBS are analysed in the third part, where frequency response curves are shown for different numbers of nanobeams in the system. The validation study is presented in the last subsection.
6.1 Periodic solutions and stability
In this study, we consider several different cases of MNBS with \(m=1, 3, 5\) nanobeams and for different initial conditions. We adopted the material and geometric properties for nanobeams in MNBS corresponding to the properties of singlewalled carbon nanotubes as: Young’s modulus \(E=1.1 \hbox { TPa}\), density \(\rho =1300 \frac{\mathrm{kg}}{\mathrm{m}^{3}} \), length \(L=45 \hbox { nm}\), inner \(d_1 =3\,\hbox { nm}\) and outer \(d_0 =2.32 \hbox { nm}\) diameters of the nanotube. For the numerical purposes, we use the value of elastic coefficient of the medium as \(k=10^{8}\,({\hbox {N}/\hbox {m}^{2}})\), while influence of damping coefficient is almost neglected and small value of \(b=10^{6}\,({\hbox {Ns}/\hbox {m}^{2}})\) is adopted.
In order to illustrate the accuracy of the proposed IHB methodology, we compare the obtained results with the results from the Runge–Kutta method in Figs. 2, 3 and 4, where a fine agreement of the results is achieved. These figures show time response functions of the first nanobeam and periodic orbits of each nanobeam for different configurations of the nonlinear MNBS. Figure 2 shows periodic orbits of the special case of the nonlinear MNBS consisting of a single nanobeam, which is the case very often analysed in the literature. However, in Figs. 3 and 4, one can observe periodic orbits of the nonlinear MNBS consisting of three and fivecoupled nanobeams. We can notice that the periodic solutions strongly depend on initial conditions.
The influence of axial load amplitude on the Floquet multipliers for the special case of MNBS consisting of a single nanobeam is shown in Fig. 5. Moreover, we get Floquet multipliers as complex numbers, where real parts are shown in Fig. 5a and imaginary parts in Fig. 5b. By increasing the value of axial load \(\tilde{\Lambda } \), both parts of the Floquet multipliers remain within the unit circle of the complex plane, as shown in Fig. 5c. Based on that fact, we can conclude that obtained periodic solutions for the nonlinear system are stable for both values of the nonlocal parameter. It should be noted that the real parts of Floquet multipliers decrease for an increase in the nonlocal parameter, while the imaginary parts increases.
Figure 6 shows three values of Floquet multipliers as functions of the nonlocal parameter and amplitude of axial load for the nonlinear MNBS consisting of three nanobeams. From obtained curves, it can be observed that the influence of the amplitude of axial load on the Floquet multipliers is not a smooth function coming out of the unit circle in the complex plane, as shown in Fig. 6g, i. Based on that fact, we can conclude that the periodic solution of the third nanobeam in MNBS is unstable. For the Floquet multipliers regarding the first and the second nanobeam, we can observe that obtained periodic solutions are stable, as shown in Fig. 6c, f. It should be noted that the nonlocal parameter has a significant influence on the first two Floquet multipliers while for the third one it almost vanishes, as shown in Fig. 6i.
The Floquet multipliers determining the stability of the nonlinear MNBS composed of five nanobeams are shown in Fig. 7. We can observe a very interesting behaviour of Floquet multipliers with a significant dependence on the nonlocal parameter. From obtained results, we can notice that only the fourth and the fifth nanobeam in MNBS are having stable periodic solutions with the values of Floquet multipliers remaining within the unit circle of the complex plane, as shown in Fig. 7l, p. However, other periodic solutions of MNBS are unstable, where the values of the Floquet multipliers are out of the unit circle, as shown in Fig. 7c, f, i. From the physical point of view, we can conclude that the number of nanobeams decreases the overall stiffness of the MNBS and this fact lead to the reduction of the system stability.
6.2 Instability regions
Figure 8 shows the influence of stiffness coefficient k and the number of nanobeams on the instability regions of the linear and nonlinear cases. It can be concluded that an increase in stiffness coefficient k leads to narrowing of instability regions while an increase in the number of nanobeams in MNBS leads to an increase in instability regions. From the physical point of view, an increase in the number of nanobeams decreases the overall stiffness of the system and dynamic system is becoming more unstable. From the nonlinear case (Fig. 8b), we can notice that different configurations of MNBS shifts the instability regions, which is the consequence of the system’s nonlinearity.
The last Fig. 10 illustrates the effects of the amplitude of static axial load on instability regions of MNBS. An increase in the amplitude of static axial load increases the instability regions. This increase is smaller for the nonlinear case than for the linear one. In addition, this effect occurs as a result of an increase in the overall stiffness of the system. For all the presented cases, an increase in the amplitude of axial load \(\tilde{\Lambda } \) leads to an increase in instability regions and the system becomes more unstable.
In a general case, we can conclude that change of a number of nanobeams in MNBS affects its instability regions in such a manner that these regions are expanding for an increase in the parameter m and small initial amplitudes. On the other side, for larger initial amplitudes, we can notice that the effect of a number of nanobeams goes in two directions. First, the instability regions are shifted relative to the case of small initial amplitudes and second, instability regions are increasing for a change of a number of nanobeams in the system. From the mechanical point of view, we can conclude that the overall stiffness of MNBS is reduced when increasing the number of nanobeams, which consequently results in reduced stability of the system.
6.3 The free nonlinear vibration of MNBS
In order to analyse the free nonlinear vibration of MNBS, amplitude–frequency responses for different configurations of MNBS obtained by using the IHB method are presented in Figs. 11 and 12. It should be noted that nonlinear amplitude–frequency curves are determined by using the \({\Delta }b_1 \)incrementation, where the value of the increment is \({\Delta }b_1 =0.001\), using the five odd cosine terms [67]. In this subsection, we will neglect the influence of the axial load and damping coefficient in Eq. (5) to obtain the governing equation for the free vibration of the nonlinear MNBS embedded in the elastic medium.
6.4 Validation study
In this work, results are obtained for MNBS embedded in the viscoelastic medium and the authors did not find a similar mechanical model in the literature that can be used for the validation study. However, for the special case of MNBS, i.e. the case with a single nanobeam on elastic foundation, we use the paper by Fu et al. [67]. Figure 13 shows the nonlinear amplitude–frequency response of MNBS consisting of one, three and five nanobeams and the amplitude–frequency response of the system with a single nanobeam resting on the elastic foundation presented in Fu et al. [67]. It should be noted that our model of MNBS is coupled in clamped chain system, i.e. for the case of a single nanobeam it is coupled with the fixed base through the elastic medium on both sides. This leads to higher values of the overall stiffness and such system is more constrained than the model proposed by Fu et al. [67].
7 Scope and limitations of the presented model
The given nonlinear MNBS is based on the nonlocal Euler–Bernoulli beam model and von Karman nonlinear deformations and does not take into account shear deformations. However, in order to consider this effect, one should consider Timoshenko or higherorder shear deformation beam theories. From the literature [18], we can see that the effect of transverse shear deformation leads to lower vibration frequencies comparing to the case with Euler–Bernoulli beam. This effect is more prominent in higher vibration modes. Therefore, the consideration of transverse shear deformation leads to a decrease in stability of the presented system, i.e. it leads to widening of instability regions [72]. It should be noted that applied axial loads are smaller than the critical buckling load, and MNBS is not buckled. However, the nonlinear vibration behaviour of buckled MNBS is also a very important issue and should be studied in some future works.
The second limitation refers to the fact that presented MNBS model consider amplitudes of nanobeams that are bounded by the surrounding viscoelastic medium. Based on that, we analysed only small nonlinear deformations by using the von Karman theory. It should be emphasized that we employed the phenomenological viscoelastic model based on simple mechanical analogy to describe the rheological behaviour of the viscoelastic medium in MNBS. Since the applied force–displacement relation is a phenomenological model of Kelvin–Voight’s viscoelastic stress–strain constitutive relation and reaction force of viscoelastic medium is distributed over nanobeam’s length, it can be represented by parallelly connected springs and dashpots with corresponding elasticity and viscosity coefficients. An interested reader can find more on experimental works and dynamic behaviour of CNT/polymer nanocomposites in [51, 52, 53, 75].
In the literature, the authors have found a series of papers [61, 67, 73, 74] where linear mode shape functions are used for a nonlinear vibration analysis of classical and scaledependent beam models. However, we should mention two interesting papers by ElBorgi et al. [73] and Nayfeh and Lacarbonara [74], where the authors concluded that the Galerkin singlemode method is suitable for the discretization of governing equations of nonlinear vibration of simply supported beams, especially for the primary resonance state and the first vibration mode. According to this, the authors of the present paper employed only a singlemode Galerkin method for discretization of the governing equations of given MNBS. One should know that each nanobeam in MNBS has the same simply supported boundary conditions and therefore, same linear mode shape functions are considered in the assumed solutions of the system of governing equations. Moreover, in the absence of internal resonances, the steadystate response contains only the directly excited modes. The mathematical model of MNBS with considered interactions between modes, internal and combined resonances, is interesting to be analysed in some future study.
8 Conclusions

The Floquet multipliers are obtained as functions of different parameters of the stiffness coefficient, amplitude of the axial load and number of nanobeams.

The stability boundaries are obtained for different configurations of the nonlinear MNBS.

Coupled influence of nonlocal and nonlinear effect in the context of the stability behaviour of the nonlinear MNBS.

Semianalytical periodic solutions are obtained for nonlinear MNBS by using the IHB method, without introducing small parameter.
Notes
Acknowledgements
This research was supported by the project of the Ministry of Education, Science and Technological Development of the Republic of Serbia under the Grant No. OI 174001.
References
 1.Pourkiaee, S.M., Khadem, S.E., Shahgholi, M., Bab, S.: Nonlinear modal interactions and bifurcations of a piezoelectric nanoresonator with threetoone internal resonances incorporating surface effects and van der Waals dissipation forces. Nonlinear Dynam. 3(88), 1785–1816 (2017)CrossRefGoogle Scholar
 2.Pourkiaee, S.M., Khadem, S.E., Shahgholi, M.: Parametric resonances of an electrically actuated piezoelectric nanobeam resonator considering surface effects and intermolecular interactions. Nonlinear Dyn. 4(84), 1943–1960 (2016)CrossRefzbMATHGoogle Scholar
 3.Wang, Y., Li, F.M., Wang, Y.Z.: Nonlocal effect on the nonlinear dynamic characteristics of buckled parametric doublelayered nanoplates. Nonlinear Dyn. 85(3), 1719–1733 (2016)CrossRefGoogle Scholar
 4.Ghayesh, M.H., Amabili, M., Farokhi, H.: Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. Sci. 63, 52–60 (2013)MathSciNetCrossRefGoogle Scholar
 5.Belardinelli, P., Ghatkesar, M.K., Staufer, U., Alijani, F.: Linear and nonlinear vibrations of fluidfilled hollow microcantilevers interacting with small particles. Int. J. Nonlinear Mech. 93, 30–40 (2017)CrossRefGoogle Scholar
 6.Hinds, B.J., Chopra, N., Rantell, T., Andrews, R., Gavalas, V., Bachas, L.G.: Aligned multiwalled carbon nanotube membranes. Science 303(5654), 62–65 (2004)CrossRefGoogle Scholar
 7.Kang, S.J., Kocabas, C., Ozel, T., Shim, M., Pimparkar, N., Alam, M.A., Rotkin, S.V., Rogers, J.A.: Highperformance electronics using dense, perfectly aligned arrays of singlewalled carbon nanotubes. Nature Nanotechnol. 2(4), 230 (2007)CrossRefGoogle Scholar
 8.Wang, S., Liang, R., Wang, B., Zhang, C.: Loadtransfer in functionalized carbon nanotubes/polymer composites. Chem. Phys. Lett. 457(4), 371–375 (2008)CrossRefGoogle Scholar
 9.Kacem, N., Arcamone, J., PerezMurano, F., Hentz, S.: Dynamic range enhancement of nonlinear nanomechanical resonant cantilevers for highly sensitive NEMS gas/mass sensor applications. J. Micromech. Microeng. 20(4), 045023 (2010)CrossRefGoogle Scholar
 10.Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54(9), 4703–4710 (1983)CrossRefGoogle Scholar
 11.Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, Berlin (2002)zbMATHGoogle Scholar
 12.Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41(3), 305–312 (2003)CrossRefGoogle Scholar
 13.Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2), 288–307 (2007)CrossRefzbMATHGoogle Scholar
 14.Reddy, J.N.: Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48(11), 1507–1518 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
 15.Reddy, J.N., ElBorgi, S.: Eringen’s nonlocal theories of beams accounting for moderate rotations. Int. J. Eng. Sci. 82, 159–177 (2014)MathSciNetCrossRefGoogle Scholar
 16.Thai, H.T.: A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 52, 56–64 (2012)MathSciNetCrossRefGoogle Scholar
 17.Aydogdu, M.: A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys. E Low Dimens. Syst. Nanostruct. 41(9), 1651–1655 (2009)CrossRefGoogle Scholar
 18.Ansari, R., Gholami, R., Rouhi, H.: Vibration analysis of singlewalled carbon nanotubes using different gradient elasticity theories. Compos. Part B Eng. 43(8), 2985–2989 (2012)CrossRefGoogle Scholar
 19.Ansari, R., Sahmani, S.: Small scale effect on vibrational response of singlewalled carbon nanotubes with different boundary conditions based on nonlocal beam models. Commun. Nonlinear Sci. Numer. Simul. 17(4), 1965–1979 (2012)MathSciNetCrossRefGoogle Scholar
 20.Murmu, T., Adhikari, S.: Nonlocal transverse vibration of doublenanobeamsystems. J. Appl. Phys. 108(8), 083514 (2010)CrossRefGoogle Scholar
 21.Murmu, T., Adhikari, S.: Nonlocal effects in the longitudinal vibration of doublenanorod systems. Phys. E Low Dimens. Syst. Nanostruct. 43(1), 415–422 (2010)CrossRefGoogle Scholar
 22.Murmu, T., Sienz, J., Adhikari, S., Arnold, C.: Nonlocal buckling of doublenanoplatesystems under biaxial compression. Compos. Part B Eng. 44(1), 84–94 (2013)CrossRefGoogle Scholar
 23.Murmu, T., Adhikari, S.: Axial instability of doublenanobeamsystems. Phys. Lett. A 375(3), 601–608 (2011)CrossRefGoogle Scholar
 24.He, X.Q., Kitipornchai, S., Liew, K.M.: Resonance analysis of multilayered graphene sheets used as nanoscale resonators. Nanotechnology 16(10), 2086 (2005)CrossRefGoogle Scholar
 25.Liew, K.M., He, X.Q., Kitipornchai, S.: Predicting nanovibration of multilayered graphene sheets embedded in an elastic matrix. Acta Mater. 54(16), 4229–4236 (2006)CrossRefGoogle Scholar
 26.Rašković, D.: On some characteristics of the frequency equation of torsional vibrations of light shafts with several disks. Publ. l’Inst. Math. 5(11), 155–164 (1953)zbMATHGoogle Scholar
 27.Hedrih, K.S.: Dynamics of coupled systems. Nonlinear Anal. Hybrid Syst. 2(2), 310–334 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 28.Karličić, D., Murmu, T., Adhikari, S., McCarthy, M.: Nonlocal Structural Mechanics. Wiley, London (2015)zbMATHGoogle Scholar
 29.Li, X.F., Tang, G.J., Shen, Z.B., Lee, K.Y.: Sizedependent resonance frequencies of longitudinal vibration of a nonlocal Love nanobar with a tip nanoparticle. Math. Mech. Solids 226, 1529 (2016)MathSciNetzbMATHGoogle Scholar
 30.Kiani, K.: Nonlocal discrete and continuous modeling of free vibration of stocky ensembles of vertically aligned singlewalled carbon nanotubes. Curr. Appl. Phys. 14(8), 1116–1139 (2014)CrossRefGoogle Scholar
 31.Kiani, K.: Nonlinear vibrations of a singlewalled carbon nanotube for delivering of nanoparticles. Nonlinear Dyn. 76(4), 1885–1903 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 32.Kiani, K.: Longitudinally varying magnetic field influenced transverse vibration of embedded doublewalled carbon nanotubes. Int. J. Mech. Sci. 87, 179–199 (2014)CrossRefGoogle Scholar
 33.Mohammadi, M., Ghayour, M., Farajpour, A.: Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model. Compos. Part B Eng. 45(1), 32–42 (2013)CrossRefGoogle Scholar
 34.Hu, Y.G., Liew, K.M., Wang, Q., He, X.Q., Yakobson, B.I.: Nonlocal shell model for elastic wave propagation in singleand doublewalled carbon nanotubes. J. Mech. Phys. Solids 56(12), 3475–3485 (2008)CrossRefzbMATHGoogle Scholar
 35.Şimşek, M.: Vibration analysis of a singlewalled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory. Phys. E Low Dimens. Syst. Nanostruct. 43(1), 182–191 (2010)CrossRefGoogle Scholar
 36.Karaoglu, P., Aydogdu, M.: On the forced vibration of carbon nanotubes via a nonlocal Euler–Bernoulli beam model. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 224(2), 497–503 (2010)CrossRefGoogle Scholar
 37.Ansari, R., Mohammadi, V., Shojaei, M.F., Gholami, R., Sahmani, S.: On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory. Compos. Part B Eng. 60, 158–166 (2014)CrossRefGoogle Scholar
 38.Ansari, R., Ramezannezhad, H., Gholami, R.: Nonlocal beam theory for nonlinear vibrations of embedded multiwalled carbon nanotubes in thermal environment. Nonlinear Dyn. 67(3), 2241–2254 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 39.Kiani, K.: Wave characteristics in aligned forests of singlewalled carbon nanotubes using nonlocal discrete and continuous theories. Int. J. Mech. Sci. 90, 278–309 (2015)CrossRefGoogle Scholar
 40.Kiani, K.: Nonlocal continuous models for forced vibration analysis of twoand threedimensional ensembles of singlewalled carbon nanotubes. Phys. E Low Dimens. Syst. Nanostruct. 60, 229–245 (2014)CrossRefGoogle Scholar
 41.Kiani, K.: Inand outofplane dynamic flexural behaviours of twodimensional ensembles of vertically aligned singlewalled carbon nanotubes. Phys. B Condens. Matter 449, 164–180 (2014)CrossRefGoogle Scholar
 42.Kiani, K.: Free vibration of inplanealigned membranes of singlewalled carbon nanotubes in the presence of inplaneunidirectional magnetic fields. J. Vib. Control 22(17), 3736–3766 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
 43.Kiani, K.: Forced vibrations of a currentcarrying nanowire in a longitudinal magnetic field accounting for both surface energy and size effects. Phys. E Low Dimens. Syst. Nanostruct. 63, 27–35 (2014)CrossRefGoogle Scholar
 44.Kiani, K.: Application of nonlocal beam models to doublewalled carbon nanotubes under a moving nanoparticle. Part I: theoretical formulations. Acta Mech. 216(1), 165–195 (2011)CrossRefzbMATHGoogle Scholar
 45.Kiani, K.: Application of nonlocal beam models to doublewalled carbon nanotubes under a moving nanoparticle. Part II: parametric study. Acta Mech. 216(1), 197–206 (2011)CrossRefzbMATHGoogle Scholar
 46.Arani, A.G., Kolahchi, R., Zarei, M.S.: Viscosurfacenonlocal piezoelasticity effects on nonlinear dynamic stability of graphene sheets integrated with ZnO sensors and actuators using refined zigzag theory. Compos. Struct. 132, 506–526 (2015)CrossRefGoogle Scholar
 47.Wang, Y., Li, F.M., Wang, Y.Z.: Nonlocal effect on the nonlinear dynamic characteristics of buckled parametric doublelayered nanoplates. Nonlinear Dyn. 85(3), 1719–1733 (2016)CrossRefGoogle Scholar
 48.Wang, Y., Li, F.M., Wang, Y.Z.: Homoclinic behaviours and chaotic motions of double layered viscoelastic nanoplates based on nonlocal theory and extended Melnikov method. Chaos: An Interdisciplinary. J. Nonlinear Sci. 25(6), 063108 (2015)MathSciNetGoogle Scholar
 49.Pavlović, I.R., Karličić, D., Pavlović, R., Janevski, G., Ćirić, I.: Stochastic stability of multinanobeam systems. Int. J. Eng. Sci. 109, 88–105 (2016)MathSciNetCrossRefGoogle Scholar
 50.Bolotin, V.V.: The Dynamic Stability of Elastic Systems (translated from Russian). HoldenDay, San Francisco (1964)Google Scholar
 51.Suhr, J., Koratkar, N., Keblinski, P., Ajayan, P.: Viscoelasticity in carbon nanotube composites. Nature Mater. 4(2), 134–137 (2005)CrossRefGoogle Scholar
 52.Moniruzzaman, M., Winey, K.I.: Polymer nanocomposites containing carbon nanotubes. Macromolecules 39(16), 5194–5205 (2006)CrossRefGoogle Scholar
 53.Ahir, S.V., Huang, Y.Y., Terentjev, E.M.: Polymers with aligned carbon nanotubes: active composite materials. Polymer 49(18), 3841–3854 (2008)CrossRefGoogle Scholar
 54.Atanackovic, T.M., Pilipovic, S., Stankovic, B., Zorica, D.: Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. Wiley, London (2014)CrossRefzbMATHGoogle Scholar
 55.Lu, J.P.: Elastic properties of carbon nanotubes and nanoropes. Phys. Rev. Lett. 79(7), 1297 (1997)CrossRefGoogle Scholar
 56.Sudak, L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94(11), 7281–7287 (2003)CrossRefGoogle Scholar
 57.Zhang, Y.Q., Liu, G.R., Xie, X.Y.: Free transverse vibrations of doublewalled carbon nanotubes using a theory of nonlocal elasticity. Phys. Rev. B 71(19), 195404 (2005)CrossRefGoogle Scholar
 58.Sears, A., Batra, R.C.: Macroscopic properties of carbon nanotubes from molecularmechanics simulations. Phys. Rev. B 69(23), 235406 (2004)CrossRefGoogle Scholar
 59.Ansari, R., Rouhi, H., Sahmani, S.: Calibration of the analytical nonlocal shell model for vibrations of doublewalled carbon nanotubes with arbitrary boundary conditions using molecular dynamics. Int. J. Mech. Sci. 53(9), 786–792 (2011)CrossRefGoogle Scholar
 60.Ansari, R., Ajori, S., Arash, B.: Vibrations of singleand doublewalled carbon nanotubes with layerwise boundary conditions: a molecular dynamics study. Curr. Appl. Phys. 12(3), 707–711 (2012)CrossRefGoogle Scholar
 61.Lau, S.L., Cheung, Y.K., Wu, S.Y.: A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems. ASME J. Appl. Mech. 49(4), 849–853 (1982)CrossRefzbMATHGoogle Scholar
 62.Cheung, Y.K., Lau, S.L.: Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems. J. Appl. Mech. 50, 871–876 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
 63.Shen, Y.J., Wen, S.F., Li, X.H., Yang, S.P., Xing, H.J.: Dynamic analysis of fractionalorder nonlinear oscillator by incremental harmonic balance method. Nonlinear Dyn. 85(3), 1457–1467 (2016)CrossRefGoogle Scholar
 64.Sze, K.Y., Chen, S.H., Huang, J.L.: The incremental harmonic balance method for nonlinear vibration of axially moving beams. J. Sound Vib. 281(3), 611–626 (2005)CrossRefGoogle Scholar
 65.Yuanping, L., Siyu, C.: Periodic solution and bifurcation of a suspension vibration system by incremental harmonic balance and continuation method. Nonlinear Dyn. 83(1–2), 941–950 (2016)MathSciNetCrossRefGoogle Scholar
 66.Azizi, Y., Bajaj, A.K., Davies, P., Sundaram, V.: Prediction and verification of the periodic response of a singledegreeoffreedom foammass system by using incremental harmonic balance. Nonlinear Dyn. 82(4), 1933–1951 (2015)CrossRefGoogle Scholar
 67.Fu, Y.M., Hong, J.W., Wang, X.Q.: Analysis of nonlinear vibration for embedded carbon nanotubes. J. Sound Vib. 296(4), 746–756 (2006)CrossRefGoogle Scholar
 68.Simić, S.: Analytical Mechanics: Dynamics, Stability, Bifurcation. FTN Novi Sad, Serbia (2009)Google Scholar
 69.Hsu, C.S.: On approximating a general linear periodic system. J. Math. Anal. Appl. 45(1), 234–251 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
 70.Friedmann, P., Hammond, C.E., Woo, T.H.: Efficient numerical treatment of periodic systems with application to stability problems. Int. J. Numer. Methods Eng. 11(7), 1117–1136 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
 71.Shen, J.H., Lin, K.C., Chen, S.H., Sze, K.Y.: Bifurcation and routetochaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method. Nonlinear Dyn. 52(4), 403–414 (2008)CrossRefzbMATHGoogle Scholar
 72.Hagedorn, P., Koval, L.R.: On the parametric stability of a Timoshenko beam subjected to a periodic axial load. Arch. Appl. Mech. 40(3), 211–220 (1971)zbMATHGoogle Scholar
 73.ElBorgi, S., Fernandes, R., Reddy, J.N.: Nonlocal free and forced vibrations of graded nanobeams resting on a nonlinear elastic foundation. Int. J. Nonlinear Mech. 77, 348–363 (2015)CrossRefGoogle Scholar
 74.Nayfeh, A.H., Lacarbonara, W.: On the discretization of distributedparameter systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 13(3), 203–220 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
 75.Lu, H., Huang, G., Wang, B., Mamedov, A., Gupta, S.: Characterization of the linear viscoelastic behaviour of singlewall carbon nanotube/polyelectrolyte multilayer nanocomposite film using nanoindentation. Thin Solid Films 500(1), 197–202 (2006)Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.