The vibration analysis, based on the Donnell thin shell theory, of single-walled carbon nanotubes (SWCNTs) has been investigated. The wave propagation approach in standard eigenvalue form has been employed in order to derive the characteristic frequency equation describing the natural frequencies of vibration in SWCNTs. The complex exponential functions, with the axial modal numbers that depend on the boundary conditions stated at edges of a carbon nanotube, have been used to compute the axial modal dependence. In our new investigations, the vibration frequency spectra are obtained and calculated for various physical parameters like length-to-diameter ratios for armchair and zigzag SWCNTs for different modes and in-plane rigidity and mass density per unit lateral area for armchair and zigzag SWCNTs on the vibration frequencies. The computer software MATLAB is used in order to compute these frequencies of the SWCNTs. The results obtained from wave propagation method are found to be in satisfactory agreement with that obtained through the previously known numerical molecular dynamics simulations.

## NOMENCLATURE

### List of symbols

*E*Young’s modulus

*h*Shell thickness

*L*Shell length

- $\nu $
Poisson’s ratio

- $\omega $
Natural angular frequency

*E*, $\nu $, $\rho $Effective material quantities

*L/d*Length-to-diameter ratio

*a*_{m},*b*_{m},*c*_{m}Vibration amplitude coefficients in axial, circumferential and radial direction

*n*Circumferential wave number

*Eh*In-plane rigidity

- $\rho h$
Mass density per unit lateral area

- $\theta $
Circumferential coordinate

*u(x*, $\theta $,*t)*Displacement functions in x direction

*v(x*, $\theta $,*t)*Displacement functions in$\theta $ direction

*w(x*, $\theta $,*t)*Displacement functions in

*z*direction- $\rho $
Mass density of shell material

*k*_{m}Axial wave number

*m*Half- axial wave number

*R*Shell radius

*x*Axial coordinate

*f*Natural frequency

### Abbreviations

- CS,s
Cylindrical shells

- PDE,s
Partial differential equation

- SWCNTs
Single walled carbon nanotubes

- MD
Molecular dynamics

- WPA
Wave propagation approach

- CSM
Cylindrical shell model

- CCSM
Continuum Cylindrical shell model

- ODEs
Ordinary differential equations

- FNFs
Fundamental natural frequencies

- CC
Clamped-clamped

- CF
Clamped-free

- BC’s
Boundary conditions

- THz
Tera Hertz

- GPa.nm
Giga Pascal nano meter

## I. INTRODUCTION

Understanding of the vibrational properties of the carbon nanotubes (CNTs) and their uses have been involved in various areas such as electronics, optical, medicine, charge detectors, sensors, field emission devices, aerospace, defense, construction and even fashion.^{1} Investigation of their remarkable properties, a bulk of research work was performed for their high springiness and characteristic ratio,^{2} a very effective Young modulus and tensile potency,^{3} well-bonding strength and superconductivity between carbon atoms.^{4} Vibrations of CNTs have been studied extensively in the last fifteen years and various cost effective continuum models such as thin shell,^{5} beam^{6} and ring^{7} as well as other continuum models^{3,8} have been proposed to capture the new physical phenomena and quantify the mechanical properties, and identify the major factors that affect the mechanical behavior of CNTs, which are difficult to observed through experimental and at atomistic simulation methods. Investigations of free vibration of CNTs have been examined with regard to their properties and material behavior. For practical applications, it needs more exploration to examine vibration characteristics of single-walled carbon nanotubes (SWCNTs). Moreover, vibration properties of CNTs have significant role in material strength analysis and have practical importance. A reliable knowledge of vibrational data is also important for an optimized design of processes and apparatus in various engineering and science fields (for instance electrical, biological sciences and chemical engineering). New innovative improvement and technologies such as nano-probes, emanation panel spectacle, nano-electronics, and chemical sensing and drug deliverance have been proposed. Vibration problems of SWCNTs can be investigated experimentally, theoretically and by simulation techniques.

This paper may in some aspects be regarded as an update of former reviews on different numerical simulations and experimental techniques to compute the frequencies of SWCNTs. Poncharal *et al.*^{9} and Treacy *et al.*^{10} performed experiment to calculate the resonance frequency of multi-walled CNTs (MWCNTs) for clamped-free excited by electrical loads or thermal process. Zhao *et al.*^{11} applied molecular dynamics (MD) simulations for the investigations of natural frequencies and to predict the Young’s modulus. The behaviors and material properties of CNTs have been reported in Ref. 12, by using either or continuum mechanics modeling or atomistic modeling. The results obtained through continuum mechanics modeling are well matched with MD simulations^{5} and this indicates current interest in the development of continuum mechanics model for researchers. A comprehensive MD study for the contraction and thermal expansion behaviors on different mode of vibration analysis is reported in Ref. 8. Lordi and Yao ^{13} performed MD simulations to determine the Young’s modulus and thermal vibration frequencies of SWCNTs using the universal force field with various clamped-free conditions based on the Euler beam theory. The calculations performed with cylindrical shell model are in somewhat covenant with those results which obtained through MD simulations.

Moreover, Hsu *et al.*^{6} reported the resonant frequency for CNTs model of chiral SWCNTs and these tubes are observed under a thermal vibration. The Timoshenko beam model (TBM) has been used for implicating the shear deformation and rotatory inertia of CNTs and nonlocal theory (NLT) of elasticity is employed for the vibration analysis of SWCNTs.^{14,15} The vibrational analysis of SWCNTs are estimated by using TBM and NLT.^{16,17} Galerkin’s method is applied for the computations of frequencies and end conditions are confirmed for SWCNTs. Furthermore, MD simulations of Zhang *et al.*^{18} are carried out for effectiveness and applicability of vibrational behavior of CNTs. The effects of length-to-diameter ratio on the vibration frequencies of SWCNTs are examined through TBM and MD simulations, for different modes and boundary conditions (clamped-clamped and clamped-free). These models predict satisfactory results and these models also supported the results obtained in Refs. 19 and 20. Molecular structural mechanics (MSM) method of Li and Chou^{21} has been employed in order to understand the feasibility of SWCNTs as a nanoresonator. The measured fundamental frequencies were perceptive to dimensions such as diameter and length along with boundary conditions (clamped-free or clamped) of SWCNTs. The nonlocal Donnell shell theory of Ansari *et al.*^{22} has been successfully used for vibration and buckling behaviors of CNTs. Shahrokh Hosseini-Hashemi *et al.*^{23} demonstrated the validity of Donnell shell theory for exact vibrations at small diameters/ aspect ratio (*L/d* ≥1). In addition, satisfactory results are shown for short aspect ratios against critical buckling strain by using Donnell shell theory.^{24,25} Keeping in view of these literature survey and there references herein, we have employed the improved Donnell shell theory for SWCNTs, which is capable to capture the length for CNT with appropriate aspect ratios (L/d > 4.68, present case) against the fundamental frequencies and hence also this improved Donnell shell theory has satisfactory results in modeling small aspect ratios.

A schema of graphene sheet and single-walled carbon nanotube are shown in Fig. 1 and it is shown that the CNT looks like a hollow structure tube. The static and dynamics properties of CNTs are computed by using MSM method successfully^{26} and the natural frequencies of SWCNTs are investigated in Refs. 21 and 26. Gibson *et al.*^{27} investigated high fundamental frequencies of 10∼300GHz (and 100∼1500 GHz) at length-to-diameter of 6∼20 nm (0.4∼0.8 nm), respectively, for CNTs. Chirality of nanotubes does not have a momentous effect on the fundamental frequency, however, CNTs have higher values of fundamental frequency at the lower values of diameter. Another research group with armchair and zigzag CNTs investigated the fundamental frequency with different values of length-to-diameter ratio^{28} and concluded that zigzag CNTs have higher frequencies than armchair CNTs. Moreover, continuum mechanics models depicted that thickness and Young’s modulus of CNTs plays a significant role for vibrational analysis of nanotubes. A molecular simulation technique has been used to calculate the ambiguity analysis of CNTs.^{29} Rafiee and Moghadam^{30} stimulated the surrounding polymer at microscale and performed the analysis of impact of carbon nanotube reinforced polymer (CNTRP) using multi-scale finite element modeling. In addition, Fereidoon *et al.*^{31} performed the analysis of a (CNT) reinforced polymer using a 3D finite-element model and constructed a non-bonded interphase region and the surrounding polymer with multiscale finite-element model. Moghadam *et al.*^{32} studied the natural frequencies of various parameters of defective CNT and observed the orientation of C–C bond concentrated at Stone–Thrower–Wales (STW) defect by using the modal finite element analysis. Fig. 2 represents the single walled carbon nanotubes armchair and zigzag with side lateral views.

Increasingly, computer computations have been used to investigate vibrational properties of CNTs (and or SWCNTs) through different models basis.^{3–6,8,9,17,28,33–37} The method of choice to study systems on nano level is cylindrical shell model (CSM) based wave propagation approach (WPA), which allows for the study of fundamental frequencies of SWCNTs over various combinations of parameters. There are many numerical techniques that have been used for vibration problems of CNTs such as classical molecular dynamics,^{8} continuum models,^{3,5,6,33} Galerkin’s method,^{17,34} finite elements method (FEM),^{4,28} wave propagation approach,^{35–37} and ab-initio quantum mechanical simulations.^{9} The beam model (BM)^{38} was employed to calculate the frequencies and associated mode shapes of MWCNTs, and Timoshenko BM (TBM)^{39} was used to investigate the motion of inner and outer tubes in a double-walled carbon nanotube (DWCNTs). Despite of its conceptual simplicity, the TBM computations are subject to several computational problems which have to be addressed. The TBM cannot calculate the deformations in the cross section of the CNTs because of inherent characteristics of beams and the mathematical rigor of TBM limits them to yield a large number of accurate natural frequencies and mode shapes.^{40} On the other hand, CSM based WPA was found to be a very popular tool to compute the vibrational properties of SWCNTs. Ghavanloo and Fazelzadeh^{41} and Yan *et al.*^{42} were performed elastic CSM to calculate the vibrational properties of both SWCNTs and DWCNTs. The BM and CSM based on WPA^{36} have been used on a variety of problems such as for the study of dispersion relations between the group velocity and wave number of CNTs. Moreover, the novel CSM based on WPA for estimating the natural frequencies of SWCNTs has been demonstrated to converge faster than the BM and other approaches. The CSM based on WPA is, therefore, another choice of powerful research technique of CNTs whose results are applicable in the limit of acceptable statistical errors than the earlier used BM and TBM approaches.^{35–39} In this work, the CSM based on WPA for computing the fundamental frequencies is extended to the SWCNTs, which is our particular motivation. Furthermore, up to now little is known about the vibration analyses of zigzag and armchair SWCNTs, and moreover, the effects of the *Eh* (in-plane rigidity) and an increase of $\rho h$ (mass density per unit lateral area) of SWCNTs have not been investigated, by using CSM based on WPA. This is also our motivation for carrying out the present work.

The main objective of this work is to study the vibration characteristics through an analytical investigation of SWCNTs, based on the Donnell thin shell theory. The extensive cylindrical shell model is used to study the free vibrations of SWCNTs and compared the results obtained to those through MD simulations. In our case, the wave propagation technique is applied to solve the presented shell dynamics equations. We have formulated the shell frequency equation in the eigen value form. We are investigating the influence of length-to-diameter ratio for different modes, in-plane rigidity and mass density per unit lateral area for armchair and zigzag SWCNTs on the vibration frequencies and results are obtained for various material parameters. The computer software MATLAB has been employed for the computation of presented frequencies and tested the results obtained in order to assess the accuracy and validity of the cylindrical shell model for predicting the vibration frequencies of SWCNTs. Moreover, very recently, we have been reported the numerical results in the literature,^{43} where we have predicted some preliminary outcomes for the fundamental frequencies of SWCNTs. In this paper, we present our numerical results for the predictions of fundamental frequency of SWCNTs for a wide range of material parameters (*L/d*, *m*, *n*) and varying higher in-plan rigidity (*Eh*) and mass density per unit lateral area ($\rho h$).

## II. THEORETICAL FORMATION

### A. Cylindrical shell model for the vibration of SWCNT

Carbon nanotubes have two kinds, which are single walled carbon nanotubes and multi-walled carbon nanotubes. Actually multi-walled carbon nanotubes are singled walled carbon nanotubes that are coaxially interposed with different radii. When a graphene sheet rolled up into one time, then it becomes a SWCNTs to produce a hollow cylinder but with end caps.

The structure of single-walled carbon nanotubes is similar to the circular cylinders with regard to geometrical shapes as depicted in Fig. 3. So, the motion equations for cylindrical shells are utilized for studying the free vibrations of SWCNTs. According to the Donnell thin shell theory (He *et al.*),^{44} the governing equation of motion for free vibration of a cylindrical shell is given by (1). Where *u*, *v* and *w* are the longitudinal, circumferential, and radial displacements of the shell, *R* is the radius of the shell, *Eh* is the in-plane rigidity, $\rho h$ is the mass density per unit lateral area, *t* is the time and $\nu $ is the Poisson ratio. It is assumed that for the representation of the modal deformation displacement functions in the axial, circumferential and radial directions are *u(x*, *θ*, *t)*, *v(x*, *θ*, *t)* and *w(x*, *θ*, *t)* correspondingly. The three unknown displacement functions for SWCNTs executing vibration, a system of PDE is given as:

where $D=Eh312(1\u2212v2)$ denotes the effective bending stiffness.

### B. Applications of the wave propagation approach

There are different techniques for the solution of system of differential equations. Since the present carbon nanotubes problem is in the form of differential equations. Galerkin’s method has been used to compute the fundamental frequencies under various boundary conditions for SWCNTs and MWCNTs.^{17,34} In the case of the application of the Galerkin’s method for solving the shell equations, the axial modal dependence is approximated by the characteristic beam functions. In this technique, there is involvement of many integrals of these functions. To solve these integral requires a very lengthy process. A simple approach which is termed as wave propagation approach was developed by Zhang *et al.*^{45} So an efficient and a simple technique which is corporate as wave propagation approach is employed for the solution of CNT problem in the form of differential equation. Before this, present method has been successively used for the study of shell vibrations^{46} and carbon nanotubes vibrations.^{35–37} Selection for the modal deformation displacement functions of three modal displacement expressions separate the independent space and time variables. The axial coordinate and time variable are denoted by *x*, *t* correspondingly and the circumferential coordinate signifies by *θ*. The functions *u(x*, *θ*, *t)*, *v(x*, *θ*, *t)* and *w(x*, *θ*, *t)* are used to designate their respective displacement deformation function. So for modal deformation displacements are written in the assumed expression as:

where *a*_{m}, *b*_{m} and *c*_{m} stand for three vibration amplitude coefficients in the axial, circumferential and radial directions. The axial half and the circumferential wave numbers are denoted by *m* and *n* respectively and angular frequency is designated by $\omega $. The formula for fundamental frequency *f* which is written as: *f* = $\omega /2\pi $. Where *k*_{m} is the axial wave number related with an end condition.

Using the expressions for *u(x*, *θ*, *t)*, *v(x*, *θ*, *t)*, *w(x*, *θ*, *t)* and their partial derivatives in applying the product method by substituting the modal displacement functions for partial differential equations, the space and time variable are split. Now the expressions for *u*, *v* and *w* given in (2a)–(2c) are substituted into (1a)–(1c) along with their partial derivatives and the following simplified equation is achieved.

The above system of equations is transmuted in matrix representation after the arrangement of terms, and to designate the vibration frequency equation for SWCNTs an eigenvalue problem is formed.

The expressions for the terms $Aij$’*s* are given in Appendix. The form of non-zero solution of (*a*_{m}, *b*_{m}, *c*_{m}) yields the vibration frequency and associated modes for SWCNTs. where the roots of the equation furnish the frequencies. The lowest root corresponds to the fundamental frequency of vibration. It is clear that the frequency should be minimized with respect to the wave numbers *m*, *n* in order to obtain the fundamental frequency of vibration.

## III. RESULTS AND DISCUSSION

In this section, the fundamental frequencies *f* (THz) obtained for SWCNTs that we have been evaluated from Donnell thin cylindrical shell theory using (4) are presented. The results obtained (variations of the frequencies) are compared and discussed with earlier simulation and theoretical results at nearly the same sets of material parameters of SWCNTs in the clamped-clamped (CC), clamped-free (CF) boundary conditions. In addition, our results of SWCNTs in the CC, CF armchair and CC, CF zigzag SWCNTs are also calculated and presented. The earlier investigations of *f* (THz) have been restricted to the lower *L/d* and screening values. The presented CSM based WPA to SWCNTs makes it possible to study all the feasible ranges of material data points (*L/d*, *m*, *n*). It is remarkable that the present CSM based on WPA provides satisfactory calculations for a wide domain of material parameters (*L/d*, *m*, *n*) than those used earlier, based on different numerical techniques,^{18,47} for the SWCNTs. It is interesting that considerable accurate data have been obtained for the *f* (THz) of SWCNTs through newly developed CSM based on WPA computations at nearly the same sets of data points, to show that the presented new numerical data of *f* (THz) are well matched with the earlier available theoretical and MD simulation results.

Comparisons of our computations in terms of the frequency spectra use parameters of the referenced MD simulations with Timoshenko beam model of Ref. 18. Notice that for parameters of mentioned simulation with Timoshenko beam model^{18} (the in-plane stiffness or rigidity *Eh* = 278.25 GPa⋅nm, the ratio of Young’s modulus and mass density, *E / $\rho $* = 3.6481 × 10^{8} m^{2} / s^{2} Poisson’s ratio $\nu $ = 0.2), the calculated values of *f* (THz) is in the range between 1.23445 and 0.17360 for clamped-clamped BC’s and between 0.17074 and 0.02051 for clamped-free BC’s. We calculate the fundamental natural frequencies of vibration using wave propagation approach in SWCNTs for a wide range of material parameters. In the present case, the range of thickness and Poisson’s ratio are varied from *h* = 0.0612 nm to 0.69 nm and $\nu $ = 0.14 to 0.34, respectively,^{5,7,48} and the value of used diameter is taken as *d* = 6.86645×10^{−10} m. On the basis of parameters mentioned in these MD simulation and continuum shell model, the calculated value of fundamental natural frequency is in the range 0.17074 and 0.02051 for clamped-free, which is satisfactory agreement the numerical data where the values of (0.05566-0.03777) were investigated.

Tables I and II show the natural frequency (THz) as a function of *L/d* (length-to-diameter ratio) for cases of CC and CF at first vibration mode (*m*, *n*) = (1, 1). These tables show the main results and also include the numerical results taken from MD simulations based on Timoshenko beam model of Zhang *et al.*^{18} and continuum cylindrical shell model (CCSM).^{12} For the case of *f* (THz) of CC SWCNTs (see Table I), the calculations are performed with five different lower settings of *L/d* (= 4.68, 6.67, 8.47, 10.26 and 13.89) at constant referenced parameters, mentioned above (*Eh* = 278.25 Gpa⋅nm, *E* / $\rho $ = 3.6481 × 10^{8} m^{2} / s^{2}, $\nu $ = 0.2). The presented calculated results are generally in good agreement for the whole length-to-diameter range and tables show overall the same trends as in the previous numerical investigations based on different numerical methods of CC and CF SWCNTs. Most of present data points for *L/d* = (4.68, 13.89) lie between the MD simulation data of Ref. 18 and CCSM results of Ref. 12. Moreover, for higher length-to-diameter ratio (*L/d* ≥ 10.26), it is observed that the present results obtained from CSM are well matched with earlier MD simulations.^{18} The deviation of the present data from the earlier MD set of reference data is still acceptable and nearly all the data points fall within a average percentage error of ±5% range around the reference data. For the case of *f* (THz) of CF SWCNTs (see Table II), computations are carried out for six various *L/d* (= 4.67, 6.47, 7.55, 8.27, 10.07, 13.69, 17.30, 20.89, 24.50, 28.07, 31.64 and 35.34) and taking rest of referenced parameters are constant. The computed results are in fair agreement with part of MD data in the literature with numerical values generally under predicting the *f* (THz). Comparison of computed fundamental frequency shows that the difference between the different sets of data is present at some *L/d* points. It is remarkable that the presented novel CSM based on WPA calculates the average percentage error of 3.6% for *f* (THz) of CC SWCNTs, which is less than that reported (9.3%) in the previous MD simulations.

. | Frequencies (THz) . | . | |||
---|---|---|---|---|---|

L/d
. | Present . | MD Simulation^{18}
. | Percentage Error . | Shell Model^{12}
. | Percentage Error . |

4.68 | 1.23445 | 1.06812 | 15.57 | 1.17470 | 5.09 |

6.67 | 0.67832 | 0.64697 | 4.85 | 0.59090 | 14.79 |

8.47 | 0.44146 | 0.43335 | 1.87 | 0.46250 | -2.10 |

10.26 | 0.30922 | 0.30518 | 1.32 | 0.34940 | -11.50 |

13.89 | 0.17360 | 0.18311 | -5.19 | 0.19380 | -10.42 |

. | Frequencies (THz) . | . | |||
---|---|---|---|---|---|

L/d
. | Present . | MD Simulation^{18}
. | Percentage Error . | Shell Model^{12}
. | Percentage Error . |

4.68 | 1.23445 | 1.06812 | 15.57 | 1.17470 | 5.09 |

6.67 | 0.67832 | 0.64697 | 4.85 | 0.59090 | 14.79 |

8.47 | 0.44146 | 0.43335 | 1.87 | 0.46250 | -2.10 |

10.26 | 0.30922 | 0.30518 | 1.32 | 0.34940 | -11.50 |

13.89 | 0.17360 | 0.18311 | -5.19 | 0.19380 | -10.42 |

. | Frequencies (THz) . | ||
---|---|---|---|

L/d
. | Present . | MD Simulation^{18}
. | Percentage error . |

4.67 | 0.17074 | 0.23193 | -26.38 |

6.47 | 0.09048 | 0.12872 | -29.70 |

7.55 | 0.06678 | 0.1000 | -31.61 |

8.28 | 0.05566 | 0.07935 | -29.85 |

10.07 | 0.03777 | 0.05493 | -31.23 |

13.69 | 0.02051 | 0.03052 | -32.79 |

17.30 | 0.01288 | 0.01831 | -29.65 |

20.89 | 0.00883 | 0.01381 | -27.68 |

24.50 | 0.00642 | 0.00916 | -29.13 |

28.07 | 0.00489 | 0.00690 | -36.13 |

31.64 | 0.00385 | 0.00610 | -36.88 |

35.34 | 0.00309 | 0.00458 | -32.53 |

. | Frequencies (THz) . | ||
---|---|---|---|

L/d
. | Present . | MD Simulation^{18}
. | Percentage error . |

4.67 | 0.17074 | 0.23193 | -26.38 |

6.47 | 0.09048 | 0.12872 | -29.70 |

7.55 | 0.06678 | 0.1000 | -31.61 |

8.28 | 0.05566 | 0.07935 | -29.85 |

10.07 | 0.03777 | 0.05493 | -31.23 |

13.69 | 0.02051 | 0.03052 | -32.79 |

17.30 | 0.01288 | 0.01831 | -29.65 |

20.89 | 0.00883 | 0.01381 | -27.68 |

24.50 | 0.00642 | 0.00916 | -29.13 |

28.07 | 0.00489 | 0.00690 | -36.13 |

31.64 | 0.00385 | 0.00610 | -36.88 |

35.34 | 0.00309 | 0.00458 | -32.53 |

The main results of CSM based on WPA computations are shown in Figs. 5–9, for CC and CF at different modes. The CC and CF SWCNTs configuration are principally applied to micro-oscillators as well as nano-strain sensors. The present results trends obtained at referenced parameters have a generally good agreement with the earlier MD simulations based on Timoshenko beam model and MD simulations of Zhang *et al.*^{18} Fig. 4, shows the schema for the boundary conditions (clamped-clamped, clamped-free) applied.

In order to classify boundary conditions as CC and or CF, a series of computations is carried out for the FNFs. For a better understanding of FNF of SWCNTs, the referenced parameters have been employed in our calculations. The CC and CF *f* (THz) data verses ratio of *L/d* are plotted in Figs. 5 and 6, for various modes of vibration. It is to be noted that the values of *L/d* are varied from 4.67 to 35.34, for both cases. It is examined from Fig. 5, that the overall FNF trends are found and depend on ratio of *L/d* in the SWCNTs for 1^{st} (1, 1) and 3^{rd} (3, 1) modes, and these trends agree well with the earlier results reported in the MD simulations based on different numerical models.^{18,47} In addition, for first four modes of vibration (at *L/d* = 6.71), the CC FNFs of SWCNTs are 0.671, 1.565, 2.552, and 3.523 respectively. These values are well matched as reported by Duan *et al.*^{33} (0.681, 1.535, 2.536, and 3.588, respectively) of CC FNFs. The CC SWCNTs have been studied both by experimentally^{5,6,49} and simulations techniques.^{18,20,21,47} Variations of CF FNF with ratio of *L/d* of SWCNTs are plotted in Fig. 6, for five different consecutive modes of (1, 1), (2, 1), (3, 1), (4, 1) and (5, 1). We have used the present developed CSM based on WPA, which has an excellent performance; its accuracy is comparable to that of the MD results of Ref. 18. The present CSM measurements slightly lie lower as compared the MD predictions measured by the local and nonlocal theory approaches.^{18} It is observed that presented fundamental frequency of CF SWCNTs is excellent well matched for 1^{st}, 2^{nd} and 3^{rd} modes (1, 1), (2, 1), and (3, 1), respectively. Moreover, for first four modes of vibration, the CF FNFs of SWCNTs reported by Duan *et al.*^{33} and Zhang *et al.*^{18} were 0.136, 0.716, 1.676 and 2.736 THz and 0.129, 0.684, 1.630, and 2.704 THz, respectively. These values are well matched with the present values (0.090, 0.678, 1.655, and 2.680 respectively) of CF FNFs. A first conclusion from these figures is that the FNF depends on the ratio of length-to-diameter and modes of vibration in the SWCNTs, clearly indicating CC and CF FNF confirming earlier simulations.^{18,33,47} Secondly, it can be seen that the values of FNF increases with increasing ratio of *L/d* and rank of mode of vibrations. Third, it is observed from both boundary conditions that the values CC FNS are higher as compared to the CF FNF^{33} that is important for the development of carbon nanotubes based micro-oscillators.

We now turn our attention to the FNF results obtained through the CSM based on WPA computations for CC and CF armchair and zigzag SWCNTs structures. These structures have different properties and exhibit the same vibrational modes due to the doubly clamped nanotubes and their applications in the perpendicular direction to the nanotube axis are perverted. In case of nanotube is not fully clamped, then, the vibrational behavior of SWCNTs can be different. The presented wave propagation approach to SWCNTs makes it possible to study all the feasible boundary conditions that are armchair and zigzag nanotubes that definitely vibrate differently. It is remarkable that the present approach provides satisfactory investigations for a wide range of material parameters (*L/d*, *m*, *n*) for CC and CF armchair and zigzag nanotubes. The computational data of this work are that the FNF of CC and CF armchair and zigzag SWCNTs can be obtained with satisfactory statistics by WPA. Variations of the fundamental natural frequencies are attained for length-to-diameter ratio. Moreover, the lattice interpretation indices (*m*, *n*) for armchair and zigzag nanotubes can be denoted as (*m*, *m)* for *n* = *m* and (*m*, 0) for *n* = 0 respectively. Due to these indices, a large number of likely parameter arrangements can occur.^{15} In our present case, we have performed calculations, using CSM based on WPA, for CC and CF eigen-frequencies of different indices (10, 10), (15, 15), (20, 20) and (7, 0), (9, 0), (12, 0) for armchair and zigzag SWCNTs, respectively.

Fig. 7, shows the FNF measured here from WPA for CC and CF eigen-frequency against *L/d* for different armchair and zigzag SWCNTs, respectively. For armchair case, a sequence of six different CC and CF computations is carried out using WPA with index varying from 10 to 20 and *L/d* = 4.86 to 35.53. The computationally traceable FNFs of *f* (THz) are found to be 0.97293 to 0.011812 and 0.76587 to 0.01640 for CC and CF conditions, respectively, as shown in Fig. 7 (a). It is examined that the natural frequency decreases with an increase of *L/d* ratio and the values of CC FNF is slightly higher than that of CF FNF values. For zigzag case, we have performed six various CC and CF measurements with index changes from 7 to 12 and *L/d* = 4.86 to 35.53. The CC (CF) FNFs data points are obtained from 13.855 to 3.1585THz (2.007 to 2.9058 THz) by using the presented CSM based on WPA. From Fig. 7 (b), it is clear that the results obtained for CC FNF are significantly higher as compared to CF FNF. It is interested that the value of FNF decreases with increasing *L/d* same as trend noted for CC and CF armchair SWCNTs. It is remarkable from both panels of Fig. 7, that the FNF values of CC and CF armchair are definitely higher compared to CC and CF zigzag values of SWCNTs. The possible reason for these differences (frequencies of armchair and zigzag values) that may arise from translation indices of SWCNTs and it explains the results. The symmetry of carbon nanotubes is described by the so-called line groups.^{50} Every nanotube with a particular arrangement (*m*_{1}, *m*_{2}) belongs to a different line group.^{51} Only armchair and zigzag tubes with the same *m* belong to the same symmetry group. Moreover, by starting with a single carbon atom and successively applying all symmetry operations of the group, the whole tube is constructed. Because the relation between the carbon atoms and the symmetry operations is one-to-one, the nanotubes in fact are the line groups. There is a considerable difference between the natural frequencies between the natural frequencies between armchair and zigzag type in the small aspect ratio.^{52–55,28} In the literature, the translation indices for the SWCNT is denoted by (*n*, *m*), for armchair and zigzag indicated as (*n*, *n*) and (*n*, 0). Now in the present study, the indices are denoted as (*m*, *n*), for armchair and zigzag can be written as (*m*, *m*) and (*m, 0*). Due to this reason, frequencies effects in the case of zigzag.

Two sets of figures are presented to show the natural frequency behaviors of the computed SWCNT systems at in-plane rigidity (*Eh*) and mass density per unit lateral area ($\rho h$) parameters. Both sets refer to the cases when the *Eh* varying from 278.25GPa⋅nm to 425GPa⋅nm^{55} and $\rho h$ changes from 740.52nm to 820.80nm. Fig. 8, shows the FNF, computed by the CSM based on WPA, with (7, 7) and (9, 0) armchair and zigzag SWCNT, respectively, and under different boundary conditions of CC and CF. It is can be seen from both panels of Fig. 8, that the FNF values of CC = (7, 7) and CF = (7, 7) for armchair are significantly lower than that of CC = (9, 0) and CF = (9, 0) for zigzag. It is observed that the FNF increases with an increase in Eh (in-plane rigidity) and its value decreases with increasing *L/d*,^{56} as CC = (9, 0) *f* (THz) ∼ 18.963 [CF (9, 0) *f* (THz] ∼ 16.967) at *L/d* = 4.67 and *Eh* = 300GPa.nm increases to CC = (9, 0) *f* (THz) ∼ 22.570 [CF (9, 0) *f* (THz] ∼ 20.185] at *L/d* = 4.67 and *Eh* = 425GPa.nm for zigzag SWCNTs, and CC = (7, 7) *f* (THz) ∼ 1.04524 [CF (7, 7) *f* (THz) ∼ 0.80828 (*L/d* = 4.67) at *Eh* = 300GPa.nm increases to CC = (7, 7) *f* (THz) ∼ 1.24408 [CF (7, 7) *f* (THz) ∼ 0.9620 (*L/d* = 4.67) at *Eh* = 425GPa.nm for armchair SWCNTs. Furthermore, for the zigzag case, CC = (9, 0) *f* (THz) ∼ 2.5059 [CF (9, 0) *f* (THz] ∼ 2.2421) at *L/d* = 35.34 and Eh = 300GPa.nm slightly changes to CC = (9, 0) *f* (THz) ∼ 2.9826 [CF (9, 0) *f* (THz] ∼ 2.6686] at *L/d* = 35.34 and *Eh* = 425GPa.nm, and for the armchair SWCNTs case, CC = (7, 7) *f* (THz) ∼ 0.020594 [CF (7, 7) *f* (THz) ∼ 0.01547 (*L/d* = 35.34) at *Eh* = 300GPa.nm slightly changes to CC = (7, 7) *f* (THz) ∼ 0.02451 [CF (7, 7) *f* (THz) ∼ 0.01842 (*L/d* = 35.34) at *Eh* = 425GPa.nm. The computer results for *f* (THz) as a function of L/d for $\rho h$ = 740.52nm, 800.640nm and 820.80nm are plotted in Fig. 9. We have performed our measurements of FNFs for different CC (= 12, 12) armchair and CF (= 14, 0) zigzag SWCNTs over three values of mass density per unit lateral area ($\rho h$). It can be noted from Fig. 9, that the FNF decreases with increasing in $\rho h$ (mass density per unit lateral area) and *L/d* (length-to-diameter ratio) of the SWCNTs, as CC = (12, 12) *f* (THz) ∼ 0.53352 [CF (12, 12) *f* (THz] ∼ 0.45575) at $\rho h$ = 740.52nm decreases to CC = (12, 12) *f* (THz) ∼ 0.50676 [CF (12, 12) *f* (THz] ∼ 0.43289] at $\rho h$ = 820.80nm for armchair SWCNTs and *L/d* = 6.5, and CC = (14, 0) *f* (THz) ∼ 20.325 [CF (14, 0) *f* (THz] ∼ 18.923) at $\rho h$ = 740.52nm decreases to CC = (14, 0) *f* (THz) ∼ 19.3057 [CF (14, 0) *f* (THz] ∼ 17.974] at $\rho h$ = 820.80nm for zigzag SWCNTs and *L/d* = 6.5. It is important to note that the FNF of CC = (14, 0) and CF = (14, 0) is normally overpredicting CC = (12, 12) and CF = (12, 12). At higher $\rho h$ and *Eh* values, instead, a constant *f* (THz) is observed for both armchair and zigzag which is decreasing with slightly increasing *L/d*.

## IV. CONCLUSIONS

A newly developed CSM based on wave propagation approach has been employed for the SWCNTs and provide an alternative method to investigate the vibrational behaviors of CC and CF armchair and zigzag SWCNTs. The FNF analysis of SWCNTs has been reported and compared with earlier simulation results, based on different numerical methods, over a wide range of material parameters of length-to-diameter ratio (*L/d*) and mode of vibrations (*m*, *n*) with different combinations of in-plane rigidity and mass density per unit lateral area of armchair and zigzag CNTs. It has been shown that the FNF is dependent on both the material parameters (*L/d*, *m*, *n*) and it is demonstrated that the FNF decreases with an increase in *L/d* for CC and CF zigzag and armchair SWCNTs, showing that the value of CC and CF armchair is dominated over CC and CF zigzag values throughout computations. Moreover, it has been shown that the FNF increases and decreases with increasing *Eh* (in-plane rigidity) and $\rho h$ (mass density per unit lateral area), respectively. However, the FNF decreases with increasing *L/d* (length-to-diameter ratio) of the SWCNTs. The fundamental frequency data obtained through an improved CSM based on WPA and those obtained numerically for single walled carbon nanotubes are found to be in suitable agreement, generally overpredicted with statistical limits, depending on the material parameters (*L/d*, *m*, *n*) and (*Eh*, $\rho h$). The presented CSM based on wave propagation approach has an excellent performance that its accuracy and consistency are comparable to those of the earlier MD simulations based on different models. Finally, it is concluded that the presented CSM method is an alternative best choice to examine the overall vibration behavior of SWCNTs. In future work, a better cylindrical shell model is needed to furnish more accurate prediction of the vibration frequencies of SWCNTs such as the nonlocal shell theory that incorporates the effect of small length scale behavior. This future model may be helpful in studies of multi-walled CNT vibrations that take into the account of facet of mechanical sensors and high frequency oscillators.