This paper presents an investigation on the dynamical properties of singlewalled carbon nanotubes (SWCNTs), and nonlinear modal interaction and energy exchange are analysed in detail. Resonance interactions between two conjugate circumferential flexural modes (CFMs) are investigated. The nanotubes are analysed through a continuous shell model, and a thin shell theory is used to model the dynamics of the system; freefree boundary conditions are considered. The Rayleigh–Ritz method is applied to approximate linear eigenfunctions of the partial differential equations that govern the shell dynamics. An energy approach, based on Lagrange equations and series expansion of the displacements, is considered to reduce the initial partial differential equations to a set of nonlinear ordinary differential equations of motion. The model is validated in linear field (natural frequencies) by means of comparisons with literature. A convergence analysis is carried out in order to obtain the smallest modal expansion able to simulate the nonlinear regimes. The time evolution of the nonlinear energy distribution over the SWCNT surface is studied. The nonlinear dynamics of the system is analysed by means of phase portraits. The resonance interaction and energy transfer between the conjugate CFMs are investigated. A travelling wave moving along the circumferential direction of the SWCNT is observed.
Carbon nanotubes (CNTs) were first discovered in 1991 by Iijima in the laboratories of the NEC Corporation in Japan. These “helical microtubules of graphitic carbon” were observed in the form of needles comprising several coaxial tubes of graphitic sheets by means of a transmission electron microscopy (TEM). They were grown on the negative end of the carbon electrode used in the arcdischarge synthesis of fullerenes and were detected by highresolution electron micrographs [
Starting from their discovery, CNTs have attracted increasing interest from researchers all over the world because of their outstanding mechanical properties, in particular, very high Young’s modulus (of the order of 12 TPa) and tensile strength (up to 100 GPa) [
The combination of the two previous mechanical properties allows CNTs to reach natural vibration frequencies of the order of THz: these ultrahigh frequencies have led to experiment the application of CNTs and their composites as ultrahigh sensitivity resonators within nanoelectromechanicalsystems (NEMS), such as oscillators, sensors, and charge detectors [
Several methodologies have been adopted in order to better investigate the mechanical properties of CNTs and their application in many industrial fields. Experimental tests, numerical simulations, and continuum mechanics studies have been performed. Singlewalled and multi‐walled carbon nanotubes (MWCNTs) have been considered.
Experiments based on the resonant Raman spectroscopy (RRS) technology have been carried out to obtain atomic structure and chirality of SWCNTs considering the frequency spectrum of the CNT and applying the theory of the resonant transitions. This method starts from the measurement of the SWCNT diameter obtained by means of atomic force microscopy (AFM) and investigates natural frequency and energy of the SWCNT radial breathing mode (RBM) [
However, owing to their high technological complexity and costs, experimental methods cannot be considered as efficient approaches to investigate the mechanical behaviour of CNTs.
Molecular dynamics simulations (MDS) have been performed to get numerically the time evolution of SWCNT as a system of atoms treated as pointlike masses interacting with one another according to an assumed potential energy which describes the atomic bond interactions along the CNT. The vibrations of the free atoms of the SWCNT are recorded for a certain duration at fixed temperature and the corresponding natural frequencies are computed by the Discrete Fourier Transform (DFT) [
Theoretical models based on continuum mechanics are computationally more efficient than MDS. In such kind of modelling, CNTs are treated as continuous homogeneous structures, ignoring their intrinsic atomic nature; this allows the achievement of a strong reduction of the number of degrees of freedom. In order to investigate the vibration properties of SWCNTs and MWCNTs, both beamlike [
In particular, the analogy between circular cylindrical shell and CNT nanostructures led to extensive application of continuous shell models for the CNT vibration analysis. To deepen the knowledge on shells, the fundamental books of Leissa [
In order to study the CNT dynamics through continuous models, equivalent mechanical parameters (i.e., Young’s modulus, Poisson’s ratio, and mass density) must be applied. Indeed, we are going to simulate the intrinsically discrete CNT with its continuous twin: a circular cylindrical shell; in order to be dynamically equivalent all shell parameters must be suitably set, this was achieved comparing the equations of the classical shell theory for tensile and bending rigidity with the MDS results [
One of the most controversial topics in modelling the discrete CNTs as continuous cylindrical shells is denoted by the inclusion of the size effects, i.e., surface stresses, strain gradients, and nonlocalities, into the elastic shell theory considered. It was found in the literature [
Both theoretical and experimental aspects of nonlinear dynamics and stability of shells are widely explored in the literature [
It must be stressed that the application of the CNTs and their composites as ultrahighsensitivity resonators within NEMS requires deep investigation of the nonstationary dynamics phenomena, i.e., energy transfers, internal resonances, and propagating waves, in the CNTs, considered as strongly nonlinear systems. To this aim, the nonlinear resonance interaction and energy exchange between vibration modes in SWCNTs were extensively studied in the literature [
An interesting nonstationary dynamic phenomenon, a travelling wave response, was observed both numerically and experimentally in circular cylindrical shells, with and without fluidstructure interaction [
In this paper, the nonlinear oscillations and energy distribution of SWCNTs are investigated. The resonance interaction between two CFMs having the same natural frequency and with mode shapes shifted of
In this work, the SandersKoiter elastic thin shell theory is used to model the SWCNT dynamics.
It must be stressed that in the present paper, the contribution of the size effects (surface stresses, strain gradients, and nonlocalities) is not taken into account in the stressstrain relationships.
Since size effects have a relevant influence in the higher region of the SWCNT frequency spectrum (vibration modes with relatively high number of longitudinal halfwaves and resonant frequencies), then they can be neglected in this manuscript, which is focused on the lower region of the SWCNT frequency spectrum (conjugate CFMs with relatively low number of longitudinal halfwaves and resonant frequencies). Therefore, in the present work, the local continuum mechanics neglecting the size effects can be applied correctly without loss of accuracy (see ref. [
In Figure
Equivalent continuum structure of a SWCNT. (a) Discrete SWCNT [
In Figures
Since one of the most important issues of CNT continuum modelling is given by the infinitesimal dimensions, which cause very fast time scales and therefore induce hard computational troubles during numerical integration, then, in this work, the governing parameters, i.e., shell displacements and time, are transformed into dimensionless form.
The dimensionless displacement field
The SandersKoiter shell theory is based on Love’s first approximation, which considers the following five hypotheses [
The consequences of the last two assumptions are that the transverse shear the deformations of the shell are neglected (
The dimensionless middle surface strains of the shell
The separation of the linear and nonlinear terms in equations (
The dimensionless middle surface changes in curvature and torsion of the shell
The dimensionless force (
In the case of homogeneous and isotropic material, the dimensionless elastic strain energy
Another relevant issue of the thinshell modelling of SWCNTs is given by the choice of an isotropic or anisotropic model; indeed, CNTs are discrete systems, i.e., they are intrinsically nonisotropic. However, it was proven in the past that the use of isotropic models for SWCNTs does not lead to significant errors. Therefore, in the present work, an isotropic elastic shell model is considered.
The dimensionless time variable
The dimensionless velocity field
The dimensionless kinetic energy
The dimensionless total energy
In order to study the linear and nonlinear vibrations of the shell, a twostep energy based procedure is adopted: (i) in the linear analysis, the three displacements are expanded by using a double mixed series, the potential and kinetic energies are developed in terms of the series free parameters, and the Rayleigh–Ritz method is considered to obtain approximated eigenfunctions; (ii) in the nonlinear analysis, the three displacements are re‐expanded by using the approximated eigenfunctions of the linear analysis, the potential and kinetic energies are developed in terms of modal coordinates and the Lagrange equations are applied to obtain a system of nonlinear ordinary differential equations of motion, which is solved numerically.
The linear vibration analysis is carried out by considering only the quadratic terms in the equation of the dimensionless elastic strain energy (
A modal vibration, i.e., a synchronous motion, can be formally written in the form [
The dimensionless mode shape
It must be noted that
Due to the axial symmetry and the isotropy of the system, in the absence of imperfections and in the case of axisymmetric boundary conditions (e.g., simply, clamped, and free), the harmonic functions are orthogonal with respect to the linear operator of the shell in the circumferential direction
The dimensionless mode shape
From expansions (
Freefree boundary conditions are given by the following equations [
It can be observed from equations (
In the present paper, focused on freefree SWCNTs, no boundary equations are imposed and no constraints are applied to the unknown coefficients of expansions (
The maximum number of variables needed for describing a vibration mode with
In this paper, since no boundary equations are imposed for the freefree SWCNTs,
For a multi‐mode analysis including different values of nodal diameters
Equations (
After imposing the stationarity to the Rayleigh quotient, the eigenvalue problem is obtained [
The approximated mode shape of the
The approximation of the
The
The previous normalization has the following simple physical meaning. Suppose that the dominant direction of vibration of the
In the nonlinear vibration analysis, the full expression of the elastic strain energy (
The three dimensionless displacements
From expansions (
The expansions (
The dimensionless Lagrangian coordinates
By substituting the dimensionless vector functions
Introducing the vector function
The set of nonlinear ordinary differential equations (
The numerical analyses carried out in this section are obtained considering the equivalent mechanical parameters listed in Table
Equivalent mechanical parameters of the freefree SWCNT [
Young’s modulus 
5.5 
Poisson’s ratio 
0.19 
Mass density 
11700 
Thickness 
0.066 
In Tables
Natural frequencies of the radial breathing mode (
Radius  Aspect ratio  Natural frequency (THz)  Difference (%)  



SKT (present model)  RRS [  
0.509  10  6.905  7.165  3.63 
0.518  10  6.785  7.105  4.50 
0.527  10  6.669  6.865  2.85 
0.569  10  6.177  6.295  1.87 
0.705  10  5.025  5.276  4.76 
0.708  10  4.964  5.216  4.83 
0.718  10  4.895  5.066  3.37 
0.733  10  4.788  4.947  3.21 
0.746  10  4.711  4.917  4.19 
0.764  10  4.594  4.797  4.23 
0.770  10  4.559  4.737  3.76 
0.782  10  4.494  4.677  3.91 
0.799  10  4.393  4.617  4.85 
0.814  10  4.318  4.527  4.62 
0.822  10  4.271  4.437  3.74 
0.847  10  4.150  4.317  4.04 
Natural frequencies of the beamlike mode (
Radius  Aspect ratio  Natural frequency (THz)  Difference (%)  



SKT (present model)  MDS [  
0.339  5.26  0.217  0.212  2.36 
0.339  5.62  0.191  0.188  1.60 
0.339  5.99  0.169  0.167  1.20 
0.339  6.35  0.151  0.150  0.67 
0.339  6.71  0.136  0.136  ≤0.01 
0.339  7.07  0.123  0.123  ≤0.01 
0.339  7.44  0.112  0.112  ≤0.01 
0.339  7.80  0.102  0.102  ≤0.01 
0.339  8.16  0.094  0.094  ≤0.01 
0.339  8.52  0.086  0.086  ≤0.01 
0.339  8.89  0.080  0.080  ≤0.01 
0.339  9.25  0.074  0.074  ≤0.01 
0.339  9.61  0.069  0.069  ≤0.01 
0.339  9.98  0.064  0.064  ≤0.01 
0.339  10.34  0.060  0.060  ≤0.01 
0.339  10.70  0.056  0.056  ≤0.01 
0.339  11.06  0.053  0.053  ≤0.01 
It must be underlined that this validation in linear field confirms the correctness of the continuous shell model proposed in the present paper with regard to the choice of the values of the equivalent mechanical parameters and to the previously assumed hypotheses (no size effects, isotropic shell).
In Table
Natural frequencies of the freefree SWCNT of Table
Vibration mode ( 
Natural frequency (THz) 

(0,2)  1.17609 
(1,2)  1.21558 
(2,2)  1.52195 
(3,2)  2.32386 
(0,4)  6.37264 
(1,4)  6.42757 
(2,4)  6.61777 
(3,4)  6.95070 
(0,0)  9.04466 
(1,0)  9.28746 
(2,0)  8.60968 
(3,0)  8.79534 
Mode shapes of the freefree SWCNT of Table
Mode shapes of the freefree SWCNT of Table
Mode shapes of the freefree SWCNT of Table
In the following, a further index is used for defining a mode shape and its longitudinal feature, i.e., the index
In Figures
Mode shape comparisons of the freefree SWCNT of Table
Mode shape comparisons of the freefree SWCNT of Table
In order to better understand the nonlinear effect of the resonance interaction between the conjugate CFMs considered, an initial linear study is carried out by taking into account only the linear terms of the dimensionless middle surface strains (
The three displacements
The modal initial conditions
It should be highlighted that, in the present analysis, the initial velocities are taken equal to zero (the modal initial conditions are imposed only on the displacements).
The system of linear ordinary differential equations of motion (
The natural frequency and dimensionless time period of the two conjugate modes (1,2), (1,2,
The time histories of the two conjugate modes (1,2), (1,2,
Time histories (a) and frequency spectra (b) of the two conjugate modes (1,2), (1,2,
The corresponding frequency spectra are shown in Figure
It must be stressed that these analyses were performed in order to compare the results of the linear vibrations with the following results of the nonlinear vibrations.
The twodimensional phase portrait of the mode (1,2) in the plane displacementvelocity is shown in Figure
Phase portraits of the conjugate modes (1,2) (a) and (1,2,
Similarly, the twodimensional phase portrait of the conjugate vibration mode (1,2,
The evolution in time of the total energy distribution
Contour plot (a) and 3D plot (b) of the total energy distribution
The total energy distribution
Contour plot of the total energy distribution
3D plot of the total energy distribution
After integration along the longitudinal
Evolution in time of the energy of the conjugate modes (1,2), (1,2,
In the following, the previous analyses carried out for the conjugate CFMs (1,2), (1,2,
In the nonlinear vibration analysis, the full expression of the dimensionless elastic strain energy (
The first step of the nonlinear study is the convergence analysis in terms of the modal expansion (
In particular, the aim of this convergence analysis is to find a modal expansion with the minimum number of degrees of freedom that allows the nonlinear behaviour of the sum of the time histories of the two conjugate modes (1,2), (1,2,
An initial twomode approximation involving only the conjugate modes (1,2) and (1,2,
Nonlinear convergence analysis of the sum of the time histories of the two conjugate modes (1,2), (1,2,
( 
(1,2)  (1,2, 
(1,4)  (1,4, 
(3,2)  (3,2, 
(1,0)  (3,0)  ERROR_{RMS}% 

6 dof model 


—  —  —  —  —  —  12.8 
8 dof model 


—  —  —  — 

—  8.63 
14 dof model 




—  — 

—  4.54 
20 dof model 







—  2.23 
22 dof model 








— 
The modal initial conditions
The convergence is reached by means of a 22 dof model, described in Table
In Figure
Nonlinear convergence analysis of the sum of the time histories of the two conjugate modes (1,2), (1,2,
From this convergence analysis, it can be found that the smallest model able to predict the nonlinear dynamics of the system with an acceptable accuracy is the 14 dof model (ERROR_{RMS}% = 4 ÷ 5). The main weakness of the 6–8 dof models is the insufficient number of asymmetric and axisymmetric modes, which are very important for properly modelling bending deformation (asymmetric modes) and circumferential stretching (axisymmetric modes) during the modal vibration, see [
Since the 14 dof model provides accurate results with the minimal computational effort, then, in the following, this model will be used for studying the nonlinear oscillations of the two conjugate modes (1,2), (1,2,
modes (1,2), (1,2,
modes (1,2), (1,2,
modes (1,2), (1,2,
In this section, the minimum value of the modal initial conditions (
Modal initial conditions imposed on the conjugate modes (1,2), (1,2,
Case  Modal initial conditions on the modes (1,2), (1,2, 
Modal initial condition ratio  

A 



B 



C 



D 



E 



F 



In Figure
Nonlinear time histories of the conjugate modes (1,2), (1,2,
From Table
The time histories of the conjugate modes (1,2), (1,2,
Time histories (a) and frequency spectra (b) of the two conjugate modes (1,2), (1,2,
Phase portraits of the mode (1,2) for different times are shown in Figure
2D and 3D phase portraits of the mode (1,2). Freefree SWCNT of Table
Similarly, phase portraits of the conjugate mode (1,2,
2D and 3D phase portraits of the conjugate mode (1,2,
From Figures
The evolution in time of the total energy distribution
Contour plot (a) and 3D plot (b) of the total energy distribution
In the present section, the energy exchange between the conjugate CFMs (1,2), (1,2,
Time histories of the two conjugate modes (1,2), (1,2,
The evolution of the total energy distribution
Contour plot of the total energy distribution
Contour plot of the total energy distribution
Contour plot of the total energy distribution
3D plot of the total energy distribution
3D plot of the total energy distribution
3D plot of the total energy distribution
From Figures
It is reported in the literature for the circular cylindrical shells that the presence of one couple of modes having the same mode shape but different angular orientation (such as the conjugate modes (1,2), (1,2,
In the present paper, the nonlinear resonance interaction between two conjugate CFMs of a SWCNT with freefree boundary conditions is analysed. The SandersKoiter shell theory is used to model the dynamics of the shell. The Rayleigh–Ritz method is applied to obtain approximated eigenfunctions. The Lagrange equations are used to get the system of nonlinear ordinary differential equations of motion, which is numerically solved by means of the implicit Runge–Kutta iterative method.
Numerical results from the linear analysis
The model proposed in this paper is validated in linear field by comparing the natural frequencies obtained with data retrieved from the pertinent literature: from these comparisons, it can be observed that the present model is in very good agreement with the results of experimental analyses (RRS) and structural simulations (MDS).
The time histories, frequency spectra, and phase portraits obtained demonstrate a periodic vibration of the SWCNT in linear field.
The total energy distribution over the carbon nanotube surface is periodic along the circumferential direction and symmetric along the longitudinal direction, where the initial energy imposed on the edge sections of the SWCNT is preserved in amplitude throughout the time; the total energy conservation of the considered undamped system is verified.
Numerical results from the nonlinear analysis
A convergence analysis is carried out to select the vibration modes to be added to the two directly excited CFMs providing an accurate description of the actual nonlinear behaviour of the SWCNT.
The modal expansion obtained involves, in addition to the conjugate modes (1,2) and (1,2,
The modal initial conditions activating the nonlinear response of the mode (1,2,
The time histories, frequency spectra, and phase portraits obtained demonstrate a chaotic vibration of the SWCNT in nonlinear field.
The total energy distribution over the nanotube surface evolves in a complex pattern, maintaining symmetry along the longitudinal direction and periodicity along the circumferential direction, where the initial energy on the edge sections of the CNT is no longer preserved in amplitude in the time.
The energy exchange between the two conjugate CFMs generates a pure travelling wave moving circumferentially around the SWCNT.
The authors declare that the data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.