Nonlinear Vibrations of Multiwalled Carbon Nanotubes under Various Boundary Conditions

Abstract

The present work deals with applying the homotopy perturbation method to the problem of the nonlinear oscillations of multiwalled carbon nanotubes embedded in an elastic medium under various boundary conditions. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals interlayer forces. The amplitude-frequency curves for large-amplitude vibrations of single-walled, double-walled, and triple-walled carbon nanotubes are obtained. The influences of some commonly used boundary conditions, changes in material constant of the surrounding elastic medium, and variations of the nanotubes geometrical parameters on the vibration characteristics of multiwalled carbon nanotubes are discussed. The comparison of the generated results with those from the open literature illustrates that the solutions obtained are of very high accuracy and clarifies the capability and the simplicity of the present method. It is worthwhile to say that the results generated are new and can be served as a benchmark for future works.

1. Introduction

In recent years, considerable effort has been devoted to the subject area of nanotechnology. Since nanomaterials possess unique mechanical, physical, and chemical properties, they occupy the chief topic of research in many scientific fields. Nowadays, they are being used as the substantial parts of nanoelectronics, nanodevices, and nanocomposites. Amongst the materials at the scale of nanometers, carbon nanotubes (CNTs) discovered by Iijima [1] in 1991 have attracted intense interest to many research workers. In spite of being too small and having light weight, CNTs have a very large Young's modulus up to 1 TPa in the axial direction. A good review on their properties and industrial applications can be found detailed in [2]. The application of analytical and numerical approaches to the study of CNTs is suggested recently [3–5]. Since the vibrations of CNTs are of considerable importance in a number of nanomechanical devices such as oscillators, charge detectors, field emission devices, and sensors, many researches have been so far devoted to the problem of the vibration of these nanomaterials [6–11]. However, most of the investigations conducted on the vibration of CNTs have been restricted to the linear regime. Fu et al. [12] studied the nonlinear vibrations of embedded nanotubes using the incremental harmonic balanced method (IHBM). In that work, single-walled nanotubes (SWNTs) and double-walled nanotubes (DWNTs) with simply supported end conditions were considered for the study. Recently, He [5] applied the homotopy perturbation method (HPM) to investigate the nonlinear vibrations of MWNTs using the same beam model as the one used in [12]. In that paper, they also extended Fu's work to the problem of triple-walled nanotubes (TWNTs) and gave the nonlinear amplitude-frequency curves. The present work is an extension of the authors' previous work for free nonlinear oscillations of MWNTs with arbitrary boundary conditions. The HPM as a powerful analytical approach was first introduced by He [13–20] and has been used by many mathematicians and engineers to solve various functional equations. In this method, the solution is considered as the summation of an infinite series which converges rapidly to the exact solution. Some applications of this method are summarized in a review article by He [21]. This simple method has been applied to solve linear and nonlinear equations of heat transfer [22–24], fluid mechanics [25], nonlinear Schrödinger equations [26], nonlinear oscillations [27, 28], some boundary value problems, and other subjects in different disciplines [29]. The recent developments of this perturbation-based method can be found in [30]. The convergence study of HPM for partial differential equations has been recently reported in [31]. The aim of this investigation is to show the effectiveness of HPM and the capability of this simple method for handling the nonlinear oscillations of many vibrating systems. It is shown that the first approximate solution of New HPM admits a remarkable accuracy in comparison with the results obtained from the incremental harmonic balanced method for the amplitude-frequency curves.

2. Basic Equations

Consider a CNT of length 𝐿, Young's modulus 𝐸, density 𝜌, cross-sectional area 𝐴, and cross-sectional inertia moment 𝐼, embedded in an elastic medium with constant 𝑘 determined by the material constants of the surrounding medium. This model is shown in Figure 1. Assume that 𝑢 and 𝑤 are the displacements of the nanotube along 𝑥 and 𝑧 directions respectively in terms of the spatial coordinate 𝑥 and the time variable 𝑡.

Figure 1 

Model of an embedded carbon nanotube.

The free vibration governing equation of this embedded beam-modeled CNT considering the geometric nonlinearity is [12] 𝜕𝜌𝐴²𝑤𝜕𝑡²𝜕+𝐸𝐼⁴𝑤𝜕𝑥⁴=𝐸𝐴2𝐿^𝐿₀𝜕𝑣𝜕𝑥²+𝜕𝑤𝜕𝑥²𝜕𝑑𝑥²𝑤𝜕𝑥²+𝑃, (2.1) where 𝑃 is the transverse load considered as the interaction pressure per unit axial length between the outermost tube and the surrounding medium, which can be described by the Winkler model [32, 33] as 𝑃(𝑥,𝑡)=−𝑘𝑤. (2.2)

Herein, the negative sign indicates that the pressure 𝑃 is opposite to the deflection of the outermost tube. Substituting (2.2) into (2.1) we get 𝜕𝜌𝐴²𝑤𝜕𝑡²𝜕+𝐸𝐼⁴𝑤𝜕𝑡⁴+𝑘𝑤=𝐸𝐴2𝐿^𝐿₀𝜕𝑤𝜕𝑥²𝜕𝑑𝑥²𝑤𝜕𝑥². (2.3) For a MWNT with 𝑁 layers, the pressure at any point between any two adjacent tubes, denoted as the van der Waals forces, can be expressed as 𝐹_𝑖=𝑐_𝑖𝑤_𝑖−𝑤_𝑖−1, (2.4) where 𝐹_𝑖 is the van der Waals force between the 𝑖th tube and the 𝑖−1th tube. 𝑐_𝑖 is the coefficient of the van der Waals force between the 𝑖th tube and the 𝑖−1th tube defined according to [34]. Therefore, imposing the effects of the van der Waals forces to (2.3) it results in the 𝑛 coupled nonlinear differential equations corresponding to the transverse nonlinear vibrations of an embedded MWNT with 𝑛 layers 𝐸𝐼₁𝜕⁴𝑤₁𝜕𝑥⁴+𝜌𝐴₁𝜕²𝑤₁𝜕𝑡²=𝐸𝐴₁2𝑙^𝐿₀𝜕𝑤₁𝜕𝑥²𝜕𝑑𝑥²𝑤₁𝜕𝑥²+𝑐₁𝑤₂−𝑤₁,𝐸𝐼₂𝜕⁴𝑤₂𝜕𝑥⁴+𝜌𝐴₂𝜕²𝑤₂𝜕𝑡²=𝐸𝐴₂2𝑙^𝐿₀𝜕𝑤₂𝜕𝑥²𝜕𝑑𝑥²𝑤₂𝜕𝑥²+𝑐₂𝑤₃−𝑤₂−𝑐₁𝑤₂−𝑤₁,⋮𝐸𝐼_𝑛𝜕⁴𝑤_𝑛𝜕𝑥⁴+𝜌𝐴_𝑛𝜕²𝑤_𝑛𝜕𝑡²=𝐸𝐴_𝑛2𝑙^𝐿₀𝜕𝑤_𝑛𝜕𝑥²𝜕𝑑𝑥²𝑤_𝑛𝜕𝑥²−𝑘𝑤_𝑛−𝑐_𝑛−1𝑤_𝑛−𝑤_𝑛−1. (2.5)

Assuming 𝑤_𝑖(𝑥,𝑡)=𝜑(𝑥)𝑊_𝑖(𝑡), 𝑖=1,2,…,𝑛, where 𝜑(𝑥) is the first eigenmode of the beam satisfying the kinematic boundary conditions [34] and 𝑊_𝑖(𝑡) is the time-dependent deflection parameter of the 𝑖th layer of the nanotube and applying the Galerkin method, the governing equations of motion are obtained as follows:𝐸𝐼₁𝑊₁𝛼₁+𝜌𝐴₁𝑑²𝑊₁𝑑𝑡²+𝑐₁𝑊₁−𝑐₁𝑊₂𝛼₂−𝐸𝐴₁𝑊2𝑙³¹𝛼₃=0,𝐸𝐼₂𝑊₂𝛼₁+𝜌𝐴₂𝑑²𝑊₂𝑑𝑡²+𝑐₁𝑊₂−𝑐₁𝑊₁+𝑐₂𝑊₂−𝑐₂𝑊₃𝛼₂−𝐸𝐴₂𝑊2𝑙³²𝛼₃⋮=0,𝐸𝐼_𝑛𝑊_𝑛𝛼₁+𝜌𝐴_𝑛𝜕²𝑊_𝑛𝜕𝑡²+𝑐_𝑛−1𝑊_𝑛−𝑊_𝑛−1+𝑘𝑊_𝑛−𝐸𝐴_𝑛𝑊2𝑙^3𝑛𝛼₃=0. (2.6)

The above equations are the differential equations of motion governing the nonlinear vibrations of CNTs subjected to the following initial conditions: 𝑊_𝑖(0)=𝑊_max,𝑑𝑊_𝑖(0)()𝑑𝑡=0,𝑖=1,2,…,𝑛, (2.7) where 𝑊_max denotes the maximum amplitude of oscillation. In (2.6), 𝛼₁, 𝛼₂, and 𝛼₃ are as follows:𝛼₁=^𝐿₀𝑑⁴𝜑(𝑥)𝑑𝑥⁴𝜑(𝑥)𝛼𝑑𝑥,₂=^𝐿₀𝜑²(𝑥)𝛼𝑑𝑥,₃=^𝐿₀𝑑²𝜑(𝑥)𝑑𝑥^2𝐿₀𝑑𝜑(𝑥)𝑑𝑥²𝑑𝑥𝜑(𝑥)𝑑𝑥. (2.8) The deflection of the nanotube is subjected to the following boundary conditions.

For a simply supported (S-S) nanotube,𝑤()𝜕0,𝑡=0,²𝑤(0,𝑡)𝜕𝑥²()𝜕=0,𝑤𝐿,𝑡=0,²𝑤(𝐿,𝑡)𝜕𝑥²=0. (2.9)

For a clamped-clamped (C-C) nanotube, 𝑤()()0,𝑡=0,𝜕𝑤0,𝑡()()𝜕𝑥=0,𝑤𝐿,𝑡=0,𝜕𝑤𝐿,𝑡𝜕𝑥=0. (2.10)

For a clamped-simply supported (C-S) nanotube, 𝑤()()0,𝑡=0,𝜕𝑤0,𝑡()𝜕𝜕𝑥=0,𝑤𝐿,𝑡=0,²𝑤(𝐿,𝑡)𝜕𝑥²=0. (2.11) The base functions corresponding to the above boundary conditions are given in Table 1.

Table 1 

Common boundary conditions for the transverse vibration of beams.

3. Application of HPM

3.1. Applying a New HPM for Nonlinear Vibrations of an SWNT

For an SWNT, the nonlinear vibration governing equation is given by (2.6) with 𝑛=1 as follows: 𝑑²𝑊𝑑𝑡²+𝐸𝐼𝛼𝜌𝐴𝛼¹₂+𝑘𝐸𝜌𝐴𝑊−𝛼2𝜌𝐿𝛼³₂𝑊³=0. (3.1)

Under the transformations 𝑟=𝐼/𝐴, 𝜏=𝜔𝑡, and 𝑎=𝑊/𝑟, the above equation can be transformed to the following nonlinear equation: 𝜔²𝑑²𝑎𝑑𝜏²+𝑓₁𝑎−𝑓₂𝑎³=0, (3.2) with 𝑓₁ and 𝑓₂ defined as 𝑓₁=𝐸𝐼𝛼𝜌𝐴𝛼¹₂+𝑘𝜌𝐴=𝜔²_𝑏,𝑓₂=𝐸𝐼𝛼2𝜌𝐴𝐿𝛼³₂. (3.3)

In (3.2), 𝑓₁=𝜔_𝑏 is the linear, free vibration frequency, and 𝜔 is an unknown angular frequency to be determined. Now, we apply a New HPM to seek the solution of (3.1). To this aim, we construct the following homotopy with 𝜔₀ as the initial approximation for the angular frequency:1−𝑝𝜔²₀𝑑²𝑎𝑑𝜏²+𝑎+𝑝𝜔²𝑑²𝑎𝑑𝜏²+𝑓₁𝑎−𝑓₂𝑎³=0. (3.4)

Here 𝑝 is an embedding parameter, 𝑎=𝑎(𝜏,𝑝), and 𝜔=𝜔(𝑝). Clearly, when 𝑝=0, (3.4) yields the linear harmonic 𝑑²𝑎𝑑𝜏²(0)(0)+𝑎=0,𝑎=𝑋,𝑑𝑎𝑑𝜏=0, (3.5) and, for 𝑝=1, it results the nonlinear equation (3.1). As embedding parameter 𝑝 travels from 0 to 1, the solutions 𝑎=𝑎(𝜏,𝑝) and 𝜔=𝜔(𝑝) of the homotopy (3.4) change from their initial approximations 𝑎₀(𝜏) and 𝜔₀ to the required solutions 𝑎(𝜏) and 𝜔 of (3.1).

Suppose the solution of (3.2) to be in the following form: 𝑎(𝜏)=𝑎₀(𝜏)+𝑝𝑎₁(𝜏)+𝑝²𝑎₂(𝜏)+⋯,𝜔=𝜔₀+𝑝𝜔₁+𝑝²𝜔₂+⋯. (3.6)

Substituting the above relations into the homotopy equation (3.4) and rearranging the coefficients of the terms with identical powers of 𝑝, we have a series of linear differential equations𝑝⁰∶𝑑²𝑎₀𝑑𝜏²+𝑎₀=0,𝑎₀(0)=𝑋,𝑑𝑎₀(0)𝑝𝑑𝜏=0,₁∶𝜔²₀𝑑²𝑎₁𝑑𝜏²+𝑎₁−𝜔²₀𝑎₀+𝑓₁𝑎₀−𝑓₂𝑎³₀=0,𝑎₁(0)=0,𝑑𝑎₁(0)𝑝𝑑𝜏=0,₂∶𝜔²₀𝑑²𝑎₂𝑑𝜏²+𝑎₂−𝜔²₀𝑎₁+2𝜔₀𝜔₁𝑑²𝑎₀𝑑𝜏²+𝑓₁𝑎₁−3𝑓₂𝑎²₀𝑎₁𝑎=0,₂(0)=0,𝑑𝑎₂(0)𝑑𝜏=0,⋮. (3.7)

The solution of the initial zeroth approximation is simply given by 𝑎₀(𝜏)=𝑋cos𝜏, (3.8) putting this into the first approximation equation, we get 𝜔²₀𝑑²𝑎₁𝑑𝜏²+𝜔²₀𝑎₁−𝜔²₀𝑋cos𝜏+𝑓₁𝑋cos𝜏−𝑓₂𝑋³cos³𝜏=0. (3.9)

Expanding the trigonometric function cos³𝜏 as cos³𝜏=(3/4)cos𝜏+(1/4)cos3𝜏, and letting the coefficient of cos𝜏 to be zero in order to eliminate the secular terms, we arrive at 𝜔₀=𝑓₁−34𝑓₂𝑋². (3.10)

Therefore, the ratio of the nonlinear frequency, 𝜔₀, to the linear frequency, 𝜔_𝑏, becomes Ψ=31−4𝑓₂𝑓₁𝑋². (3.11)

The solution of (3.9) can be easily achieved as 𝑎₁(𝜏)=𝑓₂𝑋³32𝑓₁−24𝑓₂𝑋²(cos𝜏−cos3𝜏). (3.12)

Thus, the first approximate solution of (3.2) can be written as follows: 𝑎(𝜏)=𝑎₀(𝜏)+𝑎₁(𝜏)𝑓=𝑋cos𝜏+²𝑋³32𝑓₁−24𝑓₂𝑋²(cos𝜏−cos3𝜏). (3.13)

3.2. Applying a New HPM for Nonlinear Vibrations of a DWNT

For a DWNT, the nonlinear vibration governing equation is given by (2.6) with 𝑛=2 as follows:𝑑²𝑊₁𝑑𝑡²+𝐸𝐼₁𝜌𝐴₁𝛼₁𝛼₂+𝑐₁𝜌𝐴₁𝑊₁−𝐸𝛼2𝜌𝐿𝛼³₂𝑊³₁−𝑐₁𝜌𝐴₁𝑊₂𝑑=0,₂𝑊₂𝑑𝑡²+𝐸𝐼₂𝜌𝐴₂𝛼₁𝛼₂+𝑘𝜌𝐴₂+𝑐₁𝜌A₂𝑊₂−𝐸𝛼2𝜌𝐿𝛼³₂𝑊³₂−𝑐₁𝜌𝐴₂𝑊₁=0. (3.14)

Substituting the following dimensionless parameters 𝑟=𝐼₁𝐴₁,𝑎₁=𝑊₁𝑟,𝑎₂=𝑊₂𝑟,𝜏=𝜔𝑡. (3.15)

Equation (3.14) can be transformed to the following dimensionless nonlinear system of equations:𝜔²𝑑²𝑎₁𝑑𝑡²+𝑓^′₁𝑎₁−𝑓^′₂𝑎³₁−𝑓^′₃𝑎₂𝜔=0,₂𝑑²𝑎₂𝑑𝑡²+𝑓^′₄𝑎₂−𝑓^′₅𝑎³₂−𝑓^′₆𝑎₁=0, (3.16) with 𝑓^′₁ to 𝑓^′₆ defined as𝑓^′₁=𝐸𝐼₁𝜌𝐴₁𝛼₁𝛼₂+𝑐₁𝜌𝐴₁,𝑓^′₁=𝐸𝐼₁2𝜌𝐴₁𝐿𝛼₃𝛼₂,𝑓₃=𝑐₁𝜌𝐴₁,𝑓^′₄=𝐸𝐼₂𝜌𝐴₂𝛼₁𝛼₂+𝑘𝜌𝐴₂+𝑐₁𝜌𝐴₂,𝑓^′₅=𝐸𝐼₁2𝜌𝐴₁𝐿𝛼₃𝛼₂,𝑓^′₆=𝑐₁𝜌𝐴₂. (3.17)

In a similar manner as above, one constructs a homotopy on (3.16) as follows:1−𝑝𝜔²₀𝑑²𝑎₁𝑑𝜏²+𝑎₁+𝑝𝜔²𝑑²𝑎₁𝑑𝜏²+𝑓^′₁𝑎₁−𝑓^′₂𝑎³₁−𝑓^′₃𝑎₂𝜔=0,1−𝑝²₀𝑑²𝑎₂𝑑𝜏²+𝑎₂+𝑝𝜔²𝑑²𝑎₂𝑑𝑡²+𝑓^′₄𝑎₂−𝑓^′₅𝑎³₂−𝑓^′₆𝑎₁=0. (3.18)

Here, the solutions 𝑎₁=𝑎₁(𝜏,𝑝), 𝑎₂=𝑎₂(𝜏,𝑝), and 𝜔=𝜔(𝑝) of the homotopy (3.18) change from their initial approximations 𝑎₁₀(𝜏), 𝑎₂₀(𝜏) and 𝜔₀ to the required solutions 𝑎₁(𝜏), 𝑎₂(𝜏) and 𝜔 of (3.16) as the embedding parameter 𝑝 travels from 0 to 1.

Suppose the solution of (3.18) to be in the following form 𝑎₁(𝜏)=𝑎₁₀(𝜏)+𝑝𝑎₁₁(𝜏)+𝑝²𝑎₁₂(𝜏)𝑎+⋯,₂(𝜏)=𝑎₂₀(𝜏)+𝑝𝑎₂₁(𝜏)+𝑝²𝑎₂₂(𝜏)+⋯,𝜔=𝜔₀+𝑝𝜔₁+𝑝²𝜔₂+⋯. (3.19)

Substituting the above relations into the homotopy equation (3.18) and rearranging the coefficients of the terms with identical powers of 𝑝, we have a series of linear differential equations 𝑝⁰∶𝑑²𝑎₁₀𝑑𝜏²+𝑎₁₀=0,𝑎₁₀(0)=𝑋₁,𝑑𝑎₁₀(0)𝑑𝑑𝜏=0,₂𝑎₂₀𝑑𝜏²+𝑎₂₀=0,𝑎₂₀(0)=𝑋₂,𝑑𝑎₂₀(0)𝑝𝑑𝜏=0,₁∶𝜔²₀𝑑²𝑎₁₁𝑑𝜏²+𝑎₁₁−𝜔²₀𝑎₁₀+𝑓^′₁𝑎₁₀−𝑓^′₂𝑎³₁₀−𝑓^′₃𝑎₂₀=0,𝑎₁₁(0)=0,𝑑𝑎₁₁(0)𝜔𝑑𝜏=0,₂₀𝑑²𝑎₂₁𝑑𝜏²+𝑎₂₁−𝜔²₀𝑎₂₀+𝑓^′₄𝑎₂₀−𝑓^′₅𝑎³₂₀−𝑓^′₆𝑎₁₀=0,𝑎₂₁(0)=0,𝑑𝑎₂₁(0)𝑑𝜏=0,⋮. (3.20)

The solution of the initial zeroth approximation is simply given by𝑎₁₀(𝜏)=𝑋₁𝑎cos𝜏,₂₀(𝜏)=𝑋₂cos𝜏. (3.21)

Substituting these into the first approximation together with eliminating the coefficient of cos𝜏 in the above system to avoid the secular terms it results in the following nonlinear system which can be easily solved using a simple mathematical algorithm such as Newton-Raphson technique: −𝑋₁𝜔²₀+𝑓^′₁𝑋₁−34𝑓^′₂𝑋³₁−𝑓^′₃𝑋₂=0,−𝑋₂𝜔²₀+𝑓^′₄𝑋₂−34𝑓^′₅𝑋³₂−𝑓^′₆𝑋₁=0. (3.22)

3.3. Applying a New HPM for Nonlinear Vibrations of a TWNT

For a TWNT, the nonlinear vibration governing equation is given by (2.6) with 𝑛=3 as follows: 𝑑²𝑊₁𝑑𝑡²+𝐸𝐼₁𝜌𝐴₁𝛼₁𝛼₂+𝑐₁𝜌𝐴₁𝑊₁−𝐸𝛼2𝜌𝐿𝛼³₂𝑊³₁−𝑐₁𝜌𝐴₁𝑊₂𝑑=0,₂𝑊₂𝑑𝑡²+𝐸𝐼₂𝜌𝐴₂𝛼₁𝛼₂+𝑐₁𝜌𝐴₂+𝑐₂𝜌𝐴₂𝑊₂−𝐸𝛼2𝜌𝐿𝛼³₂𝑊³₂−𝑐₁𝜌𝐴₂𝑐𝑊−²𝜌𝐴₂𝑊₂𝑑=0,₂𝑊₃𝑑𝑡²+𝐸𝐼₃𝜌𝐴₃𝛼₁𝛼₂+𝑐₂𝜌𝐴₃+𝑘𝜌𝐴₃𝑊₃−𝐸𝛼2𝜌𝐿𝛼³₂𝑊³₃−𝑐₂𝜌𝐴₃𝑊₂=0. (3.3)

Substituting the following dimensionless parameters: 𝑟=𝐼₁𝐴₁,𝑎₁=𝑊₁𝑟,𝑎₂=𝑊₂𝑟,𝑎₃=𝑊₃𝑟,𝜏=𝜔𝑡. (3.24)

Equation (3.3) can be transformed to the following dimensionless nonlinear system of equations: 𝜔²𝑑²𝑎₁𝑑𝑡²+𝑓₁^′′𝑎₁−𝑓₂^′′𝑎³₁−𝑓₃^′′𝑎₂𝜔=0,₂𝑑²𝑎₂𝑑𝑡²+𝑓₄^′′𝑎₂−𝑓₅^′′𝑎³₂−𝑓₆^′′𝑎₁−𝑓₇^′′𝑎₃𝜔=0,₂𝑑²𝑎₃𝑑𝑡²+𝑓₈^′′𝑎₃−𝑓₉^′′𝑎³₃−𝑓^′′₁₀𝑎₂=0, (3.25) with 𝑓₁^′′ to 𝑓^′′₁₀ defined as𝑓₁^′′=𝐸𝐼₁𝜌𝐴₁𝛼₁𝛼₂+𝑐₁𝜌𝐴₁,𝑓₂^′′=𝐸𝐼₁2𝜌𝐴₁𝐿𝛼₃𝛼₂,𝑓₃^′′=𝑐₁𝜌𝐴₁,𝑓₄^′′=𝐸𝐼₂𝜌𝐴₂𝛼₁𝛼₂+𝑐₁𝜌𝐴₂+𝑐₂𝜌𝐴₂,𝑓₅^′′=𝐸𝐼₁2𝜌𝐴₁𝐿𝛼₃𝛼₂,𝑓₆^′′=𝑐₁𝜌𝐴₂,𝑓₇^′′=𝑐₂𝜌𝐴₂,𝑓₈^′′=𝐸𝐼₃𝜌𝐴₃𝛼₁𝛼₂+𝑐₂𝜌𝐴₃+𝑘𝜌𝐴₃,𝑓₉^′′=𝐸𝐼₁2𝜌𝐴₁𝐿𝛼₃𝛼₂,𝑓^′′₁₀=𝑐₂𝜌𝐴₃. (3.26)

Likewise, we construct a homotopy on (3.25) as follows: 1−𝑝𝜔²₀𝑑²𝑎₁𝑑𝜏²+𝑎₁+𝑝𝜔²𝑑²𝑎₁𝑑𝜏²+𝑓₁^′′𝑎₁−𝑓₂^′′𝑎³₁−𝑓₃^′′𝑎₂𝜔=0,1−𝑝²₀𝑑²𝑎₂𝑑𝜏²+𝑎₂+𝑝𝜔²𝑑²𝑎₂𝑑𝑡²+𝑓₄^′′𝑎₂−𝑓₅^′′𝑎³₂−𝑓₆^′′𝑎₁−𝑓₇^′′𝑎₃𝜔=0,1−𝑝²₀𝑑²𝑎₃𝑑𝜏²+𝑎₃+𝑝𝜔²𝑑²𝑎₃𝑑𝑡²+𝑓₈^′′𝑎₃−𝑓₉^′′𝑎³₃−𝑓^′′₁₀𝑎₂=0. (3.27)

Suppose the solution of (3.27) to be in the following form: 𝑎₁(𝜏)=𝑎₁₀(𝜏)+𝑝𝑎₁₁(𝜏)+𝑝²𝑎₁₂(𝜏)𝑎+⋯,₂(𝜏)=𝑎₂₀(𝜏)+𝑝𝑎₂₁(𝜏)+𝑝²𝑎₂₂(𝜏)𝑎+⋯,₃(𝜏)=𝑎₃₀(𝜏)+𝑝𝑎₃₁(𝜏)+𝑝²𝑎₃₂(𝜏)+⋯,𝜔=𝜔₀+𝑝𝜔₁+𝑝²𝜔₂+⋯. (3.28)

Taking 𝑎₁₀=𝑋₁cos𝜏, 𝑎₂₀=𝑋₂cos𝜏, and 𝑎₃₀=𝑋₃cos𝜏, as the initial guesses for 𝑎₁, 𝑎₂, and 𝑎₃ then eliminating secular terms as above, the first approximate solution appears in the following nonlinear system:−𝑋₁𝜔²₀+𝑓₁^′′𝑋₁−34𝑓₂^′′𝑋³₁−𝑓₃^′′𝑋₂=0,−𝑋₂𝜔²₀+𝑓₄^′′𝑋₂−34𝑓₅^′′𝑋³₂−𝑓₆^′′𝑋₁−𝑓₇^′′𝑋₃=0,−𝑋₃𝜔²₀+𝑓₈^′′𝑋₃−34𝑓₉^′′𝑋³₂−𝑓^′′₁₀𝑋₂=0. (3.29)

Using the former solving process, the nonlinear amplitude frequency response curves can be obtained for TWNTs.

4. Discussion of Results

The material and geometric parameters of a CNT are taken as 𝐸=1.1 TPa, 𝜌=1300 kg/m³, 𝑙=45 nm, the outer diameter 𝑑₁=3 nm, and the thickness of the each layer 0.68 nm. The amplitude-frequency response curves for a TWNT for several spring constants 𝑘 are shown in Figure 2 in which Ψ is the ratio of the fundamental nonlinear frequency, 𝜔 to the fundamental linear frequency 𝜔_𝑏, and 𝑋 is the maximum vibration amplitude. The fundamental linear frequency can be simply calculated by setting 𝑊_𝑖=𝑊_max𝑖cos𝜔_𝑏𝑡, 𝑖=1,2,…,𝑛, into nonlinear vibration governing equations of MWNTs without considering its nonlinear terms. Then, by setting the determinant of the achieved matrix equal to zero, the frequency characteristic equation will be obtained. The fundamental linear vibration frequency of MWNT is the lowest root of the resulting characteristic equation as illustrated in [35]. As can be seen from Figure 2, in contrast to linear systems, the nonlinear frequency is a function of amplitude so that the larger the amplitude, the more pronounced the discrepancy between the linear and nonlinear frequencies becomes. It is also seen that as the spring constant 𝑘 increases, the nonlinear frequencies tend to approach the linear ones especially when 𝑘 exceeds the value 𝑘=10⁷N/m². It should be noted that the results are in an excellent agreement with those obtained via incremental harmonic balance method (IHBM) according to the formulations presented in [12]. Figure 3 shows the nonlinear to linear frequency ratio versus nondimensional amplitude ratio for three different boundary conditions and 𝑘=10⁷ n/m². Similar figures for DWNT and TWNT are illustrated in Figures 4 and 5. In Figures 4 and 5, the values of the coefficients of the van der Waals forces are 𝑐₁=𝑐₂=0.3×10¹² N/m² and 𝑘=10⁷ N/m².

Figure 2 

Effect of spring constant 𝑘 on nonlinear amplitude frequency response curves of TWNT.

Figure 3 

Nondimensional frequency ratio against nondimensional maximum amplitude of SWNTs under various boundary conditions (𝑘=107 N/m2).

Figure 4 

Nondimensional frequency ratio against nondimensional maximum amplitude of DWNTs under various boundary conditions (𝑘=107 N/m2).

Figure 5 

Nondimensional frequency ratio against nondimensional maximum amplitude of TWNTs under various boundary conditions (𝑘=107 N/m2).

Figure 6 illustrates the nonlinear frequency variation against length to radius ratio of a TWNT shell under different boundary conditions for 𝑘=10⁷ N/m² and 𝑋=2. It can be observed that with the increase of the aspect ratio of the nanotubes, the nonlinear vibration frequencies of MWNTs decrease. As is expected, the double clamped CNT has the highest natural frequency among the selected boundary conditions. A comparison between the nondimensional amplitude ratio of the first layer of a C-C SWNT and its linear counterpart is shown in Figure 7 for 𝑘=10⁷ N/m². To demonstrate the accuracy of the obtained analytical results, the authors also calculate the variation of nondimensional amplitude ratio versus the linear period of vibration for the CNT center using fourth-order Runge-Kutta method. As can be seen in the figure, the results obtained using the HPM have a good agreement with numerical results. Using (3.21) and Table 1, the amplitude-space-time diagram of an SWNT is plotted for S-S and C-C end conditions in Figures 8 and 9. The parameters used in these figures are 𝑘=10⁷ N/m² and 𝑋=2.

Figure 6 

Fundamental nonlinear frequency against length to the outermost diameter ratio of a TWNT under different boundary conditions (𝑘=107 N/m2 and 𝑋=2).

Figure 7 

Variation of the nondimensional amplitude ratio versus the linear period of a C-C SWNT (𝑘=107 N/m2).

Figure 8 

Amplitude-time-space diagram of an S-S SWNT (𝑘=107 N/m2 and 𝑋=2).

Figure 9 

Amplitude-time-space diagram of a C-C SWNT (𝑘=107 N/m2 and 𝑋=2).

5. Conclusions

The HPM has been successfully used to investigate the nonlinear vibration analysis of multiwalled carbon nanotubes with arbitrary end conditions. The generated results obtained have been compared with those available in open literature, and excellent correlation has been achieved. The significant dependency of this oscillation to the surrounding elastic medium is observed. The nonlinear vibration frequency of nanotubes rises rapidly with increasing the amplitude especially when the stiffness of the medium is relatively small. For larger values of the medium stiffnesses (say 𝑘>10⁸ N/m²), the nonlinear vibration tends to the linear regime. It has been shown that with the increase of the aspect ratio of the nanotubes, the nonlinear vibration frequencies of MWNTs decrease. Also, as one travels through the end conditions of S-S, to fully clamped, denoted by C-C, respectively, the influence of the boundary conditions is shown to increase the natural frequencies. This effect is more significant for lower values of length to outermost diameter ratios. However, for higher values, the effect of the changes in the boundary conditions diminishes so that the corresponding natural frequencies are found to coincide.

The proposed method can be readily extended to the multiwalled CNTs with the number of walls more than three. It is worthwhile to mention that HPM is straightforward and powerful, and it is a promising technique for solving strong nonlinear partial differential equations.