Comparison of nondimensional frequencies (L/R = 1, n = 1).
This chapter deals with the vibrational properties of single-walled carbon nanotubes (SWCNTs), based on nonlocalized theory of elasticity (NLT). The nanotube pilot control with nonlinear parameters was derived from Euler’s beam theory. The wave propagation (WPA) approach was used to derive the frequency equation describing the natural frequencies of vibration in SWCNTs. Complex exponentials depend on the boundary conditions given at the edges of the carbon nanotubes used. Vibration frequency spectra were obtained and evaluated for different physical parameters such as diameter ratio for single chiral carbon nanotubes and flexural strength for chiral SWCNT. The results show that the natural frequencies are significantly reduced by increasing the nonlocal parameters, but by increasing the ratio of the diameter length (aspect ratio), the natural frequency increases. The frequency of SWCNTs is calculated with the help of MATLAB computer software. These results are compared to previously known numerical simulations.
- wave propagation approach
Vibrational properties of CNTs play important critical roles in controlling the performance of various scientific and engineering fields and stability of CNT-based devices, superconductivity, and material strength analysis. New technologies and innovative improvements such as nano-probe, wood mirror, nano-electronic devices, chemical release, and drug release have been proposed. The important application of the current investigation of rotating FG-CNT is in nano-engineering structure as nano-components like sensors and actuators. The use of carbon nanotubes (CNTs) has been practiced in a variety of fields such as field emission, construction, electronics, and fashion . CNT’s free vibration surveys have been tested in relation to their physical properties and behavior. Much of the work was done with high rate of elasticity and characterization, a very effective Young module , and the bond strength between carbon atoms . In the past 15 years, researchers have used different models such as the ring , the beam , the shell , and other continuous models [2, 7] to capture the object anew. Due to their attractive applications, dynamic features such as buckling, stability, and vibration are explored in a theoretical way and avoid potential risks for future use. Therefore, a new model is needed to capture the nanoscale structure. Researchers [8, 9, 10, 11, 12] have conducted investigations of higher-order elasticity theories. Other nonclassical theories of elasticity have attracted the attention of researchers such as the theory of stress [13, 14], theories of stress [15, 16], and nonlocal theory [17, 18].
In NLT, the pressure applied at a certain point depends on the stress at all points, which is quite different from conventional theory. Wang et al.  and Yang et al.  presented a survey of SWCNT based on nonlocal Timoshenko beam theory (TBM). CNT analysis has been explained by some researchers [21, 22, 23, 24, 25]. Bocko and Lengvarský studied the vibration frequencies of CNTs for different termination conditions and mode shapes using nonlocal elastic theory. Chawis et al.  analyzes vibrational behavior of SWCNTs with small-scale effects using nonlocal theory. Recently, some researchers have investigated the vibrational behavior of SWCNTs [27, 28, 29].
In addition, many researchers investigated the vibrational behavior of the above structures using different types of theories. The method chosen for studying nanoscale systems is the NLT (WPA) wave propagation method, which allows the study of fundamental frequencies of SWCNTs through combinations of different parameters. The beam model (BM)  is used to calculate the associated frequencies and shapes of MWCNT. Bocko and Lengvarský  investigate the bending free vibration of SWCNTs with four different boundary conditions. A continuum approach is used for the computation of natural frequency based on nonlocal theory of bending beam. The natural frequencies are given for several nano-parameters with two different diameters of nanotubes and continuously changed length. It is concluded that when tube length increases, the frequency decreases the nonlocal parameter and diameter. Chawis et al.  reported SWCNT vibration based on nonlocal theory to access the scale length. With the addition of nonlocal parameter in Euler beam theory, the governing equation is derived.
NLT-based are another option of robust research techniques of CNTs within the acceptable error range compared to previously used BMs and the subsequent other approaches [2, 3, 4, 5, 21, 25, 32, 33, 34]. For solving ordinary differential equations (ODEs), Fourier variable separation method is used. According to the current model, the basic natural frequencies of chiral SWCNTs are calculated and obtained for various physical parameters such as the aspect ratio (length-to-diameter ratios) of SWCNTs with different nonlocal parameters and the effect of bending stiffness (rigidity) on the vibration frequencies of SWCNT. This is also our motivation for doing the present work.
2. Governing equation of motion
In classical theory, the physical mass acts as a local action. In conventional theory, the stress produced at a point is influenced by stress at the point. According to Eringen , in nonlocal theory, the stress applied at a given point depends on the stress at all points , which is quite different from conventional theory. Under this assumption, the nonlocal relationship based on homogeneous isotropic beams can be expressed as
The factor is termed as the small-scale effect, where and are defined as the material constant and lattice spacing length or internal characteristic length.
Equation (1) can be written as
where , , and E are, respectively, the normal pressure, normal stress, and modulus of the child. In general, the parameter is called a nonlocal parameter, and in classical theory, this parameter is used to investigate the vibration, pinching, and bending problems of the beam [36, 37]. According to Euler’s theory  equation can be written as
where and J are the mass per unit length and moment of inertia of CNT, respectively. By using the Fourier method of variable separation, the two leading differential equation systems are normalized (ODE). In this system, an equation involving the space variable x and other equations is connected to the time variable
For harmonic response
Here denotes the mode shape (eigenshape).
where the general solution of fourth-order ODEs is
where and are the unknown constants.
Equation (11) becomes
3. Application of WPA
For the solution of CNT problem, an analytical technique wave propagation approach is evoked. A simple approach called the propagation waveform (WPA) was developed by Zhang et al. . Thus, a simple and effective technique is applied as the wave propagation method [34, 39, 40] used for problem-solving in the form of differential equations. Prior to this, many techniques have been sequentially used to study the vibration of CNTs [41, 42, 43]. Previously, the current approach was used continuously to study the vibration of carbon nanotubes [27, 29, 44, 45, 46, 47].
where denoted the wave number in axial direction and used for support conditions of SWCNTs . ω = 2π f is the angular frequency.
After putting these values in Eq. (14), we get
By using Eq. (12), we can write as
4. Nonlocal boundary conditions
In the article, the vibration of chiral SWCNTs with CC boundary conditions was investigated. In addition, the network interpretive directives (m, n) for chiral CNTs can be expressed as (m, n) for n m correspondingly as shown in Figure 1. General boundary conditions C▬C and C▬F are considered for the system; then it is used in order to find the frequency equation of SWCNTs and eigenfrequencies of different indices for chiral SWCNTs such as (12, 5), (22, 7), and (25, 10).
Support conditions in the form of frequency.
From Eq. (19)
where (CC boundary condition).
where (CF boundary condition).
5. Results and discussion
The fundamental natural frequencies (FNF) f (Hz) of SWCNTs obtained from nonlocal theory (NLT) based on wave propagation approach (WPA) with C-C and C-F boundary conditions are presented. A comparison of nondimensionalized natural frequencies Δ =of SWCNT is presented in Table 1. It is noted that from Table 1, the frequency value of present model have the small values as the values followed by the Alibeigloo and Shaban  shows a frequency difference between these studies. It can be seen that the error percentage is negligible, hence showing high rate of convergence. The results of nondimensional frequency are computed for two different values of n = 1, 2 with circumferential wave number (m = 0, 1, 2, 3, 4, 5) as shown in Table 1.
In order to analyze the effect of nonlocal parameter and bending rigidity on the vibration of chiral SWCNTs for different scales, the effects of nonlocal parameters on natural frequencies are illustrated in Figures 2–5. The natural frequencies are reduced by increasing nonlocal parameters (p = 0.5, 1, 1.5, 2). The results for the chiral clamped SWCNT with the indices (12, 5), (22, 7), and (25, 10) are shown in Figure 2. For the effect of nonlocal parameters with chiral index C▬C(=12, 5) at L/d(=1∼10), the first 10 frequencies at p = 0.5 are 0.2773, 1.1094, 2.4961, 4.4376, 6.9337, 9.9846, 13.501, 17.7503, 22.4653, and 27.7349. When p = 2, then the frequency peaks are 0.1387, 0.5547, 1.2481, 2.2188, 3.4669, 4.9923, 6.7951, 8.8752, 11.2326, and 13.8675. Now, for C▬C(=22, 7), with the same parameters, the first 10 frequencies at p = 0.5 and 2 are 2.6958, 10.7833, 26.2624, 43.1331, 67.3955, 97.0496, 132.0953, 172.5326, 218.3616, and 269.5822 and 1.3472, 5.3886, 12.1244, 21.5566, 33.6788, 48.4975, 66.0105, 86.2178, 109.1808, and 134.7152, respectively. Now, for C▬C(=25, 10), with the same parameters, the first 10 frequencies at p = 0.5 and 2 are 4.9142, 19.6569, 44.2280, 78.6276, 122.8556, 176.9120, 240.7969, 314.5103, 398.0521, and 491.4223 and 2.4571, 9.8284, 22.1140, 39.3138, 61.4278, 88.4560, 120.3985, 15.2551, 199.0260, and 245.7112, respectively. It can be seen that FNF is reduced by increasing nonlocal parameters (p = 0.5, 1, 1.5, 2). To illustrate the effect of different nonlocal parameters on natural frequencies for chiral SWCNTs with indices (12, 5), (22, 7), and (25, 10) based on NLT as shown in Figures 2 and 3. It is remarkable that from Figures 2–5, the FNF values of the chiral CC tubes are certainly higher than the chiral CF values of the SWCNTs.
Figures 4 and 5 show the FNFs against aspect ratio with varying bending strength index (EI). They refer to instances when EI changes from 5.1122e−9 to 7.2629e−9nm with nonlocal parameters p = 1. These figures show the natural frequency behavior of the calculated SWCNT system under bending rigidity (EI) parameters. It is considered that with the increase of the bending rigidity (EI = 5.1122e−9 to 7.2617e−9 nm), the fundamental natural frequencies increase, and with the increase of aspect ratio, the frequencies also increase as C▬C = (12, 5) f (Hz): 0.1961–0.2337 [C▬F (12, 5) f (Hz): 0.1660–0.1978)] and C▬C = (22, 7) f (Hz): 1.9062–2.2719 [CF (22, 7) f (Hz):1.7406–2.0745)] and C▬C = (25, 10) f (Hz): 3.4749–4.1415 [C▬F (25, 10) f (Hz):3.2077–3.8230)] at L/d = 1. The fundamental natural frequencies at L/d = 10 are as C▬C = (12, 5) f (Hz): 27.7349–23.3737 [C▬F (12, 5) f (Hz:16.5992–19.7835)] and CC = (22, 7) f (Hz): 190.6234–227.1912 [C▬F (22, 7) f (Hz):174.0556–207.4452)] and C▬C = (25, 10) f (Hz): 347.4881–414.1476 [C▬F (25, 10) f (Hz): 320.7685–382.3023)]. These relate to the case where EI varies from 5.1122e−9 to 7.2617e−9 nm and the nonlocal parameter p = 1. A trend of increasing frequencies of indices with bending rigidity is as (25, 10) > (22, 7) > (12, 5).
The tendency to increase the frequency of indices with bending stiffness is (25, 10) > (22, 7) > (12, 5). Figure 5 shows that FNF, calculated by NLT, is based on WPA, with (12, 5), (22, 7), and (25, 10) CF chiral SWCNT, respectively. It was observed that FNF increased with increasing EI (hardness) and its value increased with increasing L/d. From our results, we can easily conclude that the climbing frequencies for bending the hardness of the curves (12, 5), (22, 7), and (25, 10) are as follows: (12, 5) < (22, 7) < (25, 10).
In this study, the influence of boundary conditions on the vibration of single-walled carbon nanotubes was analyzed in chiral fashion with indices (12, 5), (22, 7), and (25, 10), respectively. An attempt of nonlocal elasticity theory models has been employed to study the vibration characteristics of SWCNTs analytically, and the WPA is exploited to develop the ODE of the vibrations of the SWCNTs. The influences of different boundary conditions and bending rigidity of chiral SWCNTs against aspect ratio have investigated. As can be seen from these, by increasing the aspect ratio of the carbon nanotube, fundamental natural frequency increases. In addition, as can be seen, increasing the bending rigidity results in the increase of the fundamental frequencies. The frequencies of CC end condition are higher than CF end condition for all computations in this chapter.