1 Introduction

The novel nanostructure materials discovered by Iijima [1] have led to considerable number of studies on carbon nanotubes due to their promising applications in nanodevices, nanoelectronics, and nanocomposites. Also, the excellent mechanical, electrical, structural and thermal properties coupled with high strength to weight ratio property of carbon nanotubes have continuously and tremendously expanded their applications in various industrial, engineering, physical and natural sciences processes. In fact, the nanostructures have merits when applied to the functionability of transistors and diodes. However, carbon nanotubes (CNTs) are capable of undergoing large deformations within the elastic limit and vibrate at frequency in the order of GHz and THz. Consequently, logical investigations and analysis of carbon nanotube have been a subject of interest such as the vibrations of a micro-resonator that is excited by electrostatic and piezoelectric actuations. Various studies have been carried out on beams, carbon nanotube, nano-wires, nano-rods and nano-beam so as to specifically understand and achieve their area of best fit [2,3,4,5,6,7,8,9,10,11,12,13]. In achieving this, the well know beam models were employed and dynamic ranges were obtained in the scope of the structures. In such studies, Liew et al. [5], Pantano et al. [6, 7], Qian et al. [8] and Salvetat et al. [9] examined the mechanics of single and multiwalled carbon nanotubes. Sears and Batra [10] analyzed carbon nanotubes buckling under the influence of axial compression. Yoon et al. [11] and Wang and Cai [12] investigated the impacts of initial stress on multiwall carbon nanotube with a focus on non-coaxial resonance. Wang et al. [13] explored the dynamic response of multi-walled carbon nanotubes using Timoshenko beam model. Zhang et al. [14] scrutinized the influence of compressive axial load on the transverse dynamic behaviour of double-walled carbon nanotubes (DWCNT). Another work on the vibration of double-walled carbon nanotubes was presented by Elishakoff and Pentaras [15]. Also, studies on nonlinear vibration of nanomechanical resonator, nanotube and nanowire-based electromechanical systems have been carried out by Buks and Yurke [16] and Postma et al. [17] while Fu et al. [18] examined nonlinear vibration analysis of embedded carbon nanotubes. In the same year, Xu et al. [19] considered the dynamic response of a double-walled carbon nanotube under the influence of nonlinear intertube van der Waals forces. The vibration of carbon nanotube-based switches with focus on static and dynamic responses was analyzed by Dequesnes et al. [20]. Few years later, Ouakad and Younis [21] investigated the nonlinear vibration of electrically actuated carbon nanotube resonators. In an earlier work, Zamanian et al. [22] presented the non-linear vibrations analysis of a microresonator subjected to piezoelectric and electrostatic actuations. As a continuation of the tremendous work, Abdel-Rahman, Hawwa, Hajnayeb, and Belhadj [23,24,25,26] performed a vibration and instability studies of DWCNT using a nonlinear model and considering an electrostatic actuation as an external excitation agent. In their work, a DWCNT was situated and conditioned to a direct and alternating voltage and different behaviors of the nanotubes were recorded as the exciting agent is varied. They went further to determine the bifurcation point of the DWCNT and concluded that both walls have the same frequency of vibration under the two resonant conditions considered. Belhadj et al. [26] carried out the vibration analysis of a pinned–pinned supported SWCNT employing nonlocal theory of elasticity and obtained natural frequency up to third mode. The authors also put forward an explanation on the advantages of the high frequency obtained in their work to optical applications. Lei et al. [27] studied the dynamic behaviour of DWCNT by employing the well-known Timoshenko theory of beam. The nonlinear governing equations generated by Sharabiani and Yazdi [28] derived relations in the application to nanobeams that are graded and have surface roughness. Wang [29] generated a close form model for the aforementioned surface roughness effect for an unforced fluid conveying nanotubes and beams based on nonlocal theory of elasticity and ascertained the significance of the study for reasonably small thickness of the tube considered. Interesting foundation studies have been considered after modelling of CNTs as structures resting or embedded on elastic foundations such as Winkler, Pasternak and Visco-Pasternak medium [30,31,32,33,34,35]. Other interesting works through modelling and experiment have also been presented to justify the widespread application of SWCNTs [36,37,38,39,40,41].

The dynamic behaviour of SWCNTs and DWCNTs have been characterized and their dynamic behaviour have been investigated with the aids of experimental measurements, density functional theory, molecular dynamics simulations, and continuum mechanics. However, there are difficulties in performing experiment at the nanoscale level. Consequently, over the years, the classical continuum models (which do not consider the small-scale effects) have been widely applied to the small-scale structures as reviewed in the preceding section. The demerit of such classical continuum theories is witnessed in their scale-free models as they cannot incorporate the small-scale effects in their formulations. For the purpose of correcting the inadequacy in the classical continuum models, Eringen [42,43,44,45] developed nonlocal continuum mechanics based on nonlocal elasticity theory. The nonlocal elasticity theory considers the stress state at a given point to be a function of the strain field at all points in the body. Therefore, in this work, nonlocal elasticity theory is used to analyzed nonlinear vibrations of single- and double-walled carbon nanotubes resting on two-parameter foundation in a thermal and magnetic environment. With the aid of van der Waals interlayer interaction, the nested slender double-walled nanotubes are coupled with each other. Such study on the simultaneous influences of thermal and magnetic field, two-parameter foundation on the vibration of single- and double-walled carbon nanotubes using nonlocal elasticity theory has not been presented in literature. Additionally, the development of analytical expressions for the frequencies, frequency ratio and deflections of the double-walled carbon nanotubes is shown to be another novel idea of the present study. The analytical solutions are used to investigate the influences of elastic foundations, magnetic field, temperature rise, interlayer forces, small scale parameter and boundary conditions on the frequency ratio.

2 Problem description and the governing equations

In order to develop the governing equations of motion for the SWCNTs and DWCNTs, we first consider a SWCNT under the influence of stretching effects and resting on Winkler and Pasternak foundations in a thermal and magnetic environment as depicted in Fig. 1. With the aid of the Eringen’s nonlocal elasticity theory, Euler–Bernoulli beam theory and Hamilton’s principle, the governing equation of motion for the SWCNT is given by

$$\begin{aligned} & EI\frac{{\partial^{4} w}}{{\partial x^{4} }} + m_{c} \frac{{\partial^{2} w}}{{\partial t^{2} }} + k_{1} w + k_{3} w^{3} - \left( {\frac{EA}{2L}\int\limits_{0}^{L} {\left( {\frac{\partial w}{\partial x}} \right)}^{2} dx} \right)\frac{{\partial^{2} w}}{{\partial x^{2} }} \\ & \quad - EA\alpha_{x} T\frac{{\partial^{2} w}}{{\partial x^{2} }} - \eta AH_{x}^{2} \frac{{\partial^{2} w}}{{\partial x^{2} }} + \left( {e_{o} a} \right)^{2} \left( {m_{c} \frac{{\partial^{4} w}}{{\partial x^{2} \partial t^{2} }} + k_{1} \frac{{\partial^{2} w}}{{\partial x^{2} }}} \right. \\ & \quad + 6k_{3} w\left( {\frac{\partial w}{\partial x}} \right)^{2} + 3k_{3} w^{2} \left( {\frac{\partial w}{\partial x}} \right) - \left( {\frac{EA}{2L}\int\limits_{0}^{L} {\left( {\frac{\partial w}{\partial x}} \right)}^{2} dx} \right)\frac{{\partial^{4} w}}{{\partial x^{4} }} \\ & \quad \left. {-\,EA\alpha_{x} T\frac{{\partial^{4} w}}{{\partial x^{4} }} - \eta AH_{x}^{2} \frac{{\partial^{4} w}}{{\partial x^{4} }}} \right) = 0 \\ \end{aligned}$$
(1)

where w(x, t) is the bending deflection of the tube, t is the time coordinate, EI is the bending rigidity, mc is the mass of tube per unit length. The term \(EA\alpha_{x} T\) denotes the constant axial force due to thermal effects and the term \(\eta AH_{x}^{2}\) is the magnetic force per unit length due to Lorentz force exerted on the tube in z-direction. Also, A is the cross-sectional area of the tube, \(\alpha_{x}\) is the coefficient of thermal expansion and T is the change in temperature. Also, the term η is the magnetic field permeability and Hx is the magnetic field strength (Fig. 2).

Fig. 1
figure 1

The SWCNT on two-parameter elastic foundation in a thermal and magnetic field influnce

Full size image
Fig. 2
figure 2

The embedded DWCNT in a thermal and magnetic environment

Full size image

For the purpose of incorporating the interlayer interactions for the DWCNTs with two layers, it is established that the pressure at any point between any two adjacent tubes depends on the difference in their deflections at that point. Therefore, one can express the linearized form of the van der Waals forces as

$$F_{i} = c_{i} (w_{i} - w_{i - 1} ).$$
(2)

where Fi is the van der Waals force between the ith tube and the i − 1th tube, ci is the coefficient of the van der Waals force between the ith tube and the (i − 1)th tube. Assuming that the nested individual tubes of the DWCNT vibrate in the same plane, using the van der Waals forces in Eq. (2), the developed nonlinear governing equations of vibration for the embedded DWCNT in a thermal and magnetic environment with two layers are given as

$$\begin{aligned} & EI_{1} \frac{{\partial^{4} w_{1} }}{{\partial x^{4} }} + m_{c1} 1\frac{{\partial^{2} w_{1} }}{{\partial t^{2} }} - \left( {\frac{{EA_{1} }}{2L}\int\limits_{0}^{L} {\left( {\frac{{\partial w_{1} }}{\partial x}} \right)}^{2} dx} \right)\frac{{\partial^{2} w_{1} }}{{\partial x^{2} }} \\ & \quad - EA_{1} \alpha_{x} T\frac{{\partial^{2} w_{1} }}{{\partial x^{2} }} - \eta A_{1} H_{x}^{2} \frac{{\partial^{2} w_{1} }}{{\partial x^{2} }} - \left( {e_{o} a} \right)^{2} \left( {m_{c1} \frac{{\partial^{4} w_{1} }}{{\partial x^{2} \partial t^{2} }}} \right. \\ & \quad - \left( {\frac{{EA_{1} }}{2L}\int\limits_{0}^{L} {\left( {\frac{{\partial w_{1} }}{\partial x}} \right)}^{2} dx} \right)\frac{{\partial^{4} w_{1} }}{{\partial x^{4} }} - E_{1} A\alpha_{x} T\frac{{\partial^{4} w_{1} }}{{\partial x^{4} }} \\ & \quad \left. {-\,\eta_{1} AH_{x}^{2} \frac{{\partial^{4} w_{1} }}{{\partial x^{4} }} + c_{1} \left( {\frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{1} }}{{\partial x^{2} }}} \right)} \right) = c_{1} (w_{1} - w_{2} ) \\ \end{aligned}$$
(3)
$$\begin{aligned} & EI_{2} \frac{{\partial^{4} w_{N} }}{{\partial x^{4} }} + m_{c2} \frac{{\partial^{2} w_{2} }}{{\partial t^{2} }} + k_{1} w_{2} + k_{3} w_{2}^{3} \\ & \quad - \left( {\frac{{EA_{2} }}{2L}\int\limits_{0}^{L} {\left( {\frac{{\partial w_{2} }}{\partial x}} \right)}^{2} dx} \right)\frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} - EA_{2} \alpha_{x} T\frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} \\ & \quad - \eta A_{2} H_{x}^{2} \frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} - \left( {e_{o} a} \right)^{2} \left( {m_{c2} \frac{{\partial^{4} w_{2} }}{{\partial x^{2} \partial t^{2} }}} \right. \\ & \quad + k_{1} \frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} + 6k_{3} w_{2} \left( {\frac{{\partial w_{2} }}{\partial x}} \right)^{2} + 3k_{3} w_{2}^{2} \left( {\frac{{\partial w_{2} }}{\partial x}} \right) \\ & \quad - \left( {\frac{{EA_{2} }}{2L}\int\limits_{0}^{L} {\left( {\frac{{\partial w_{2} }}{\partial x}} \right)}^{2} dx} \right)\frac{{\partial^{4} w_{2} }}{{\partial x^{4} }} - E_{2} A\alpha_{x} T\frac{{\partial^{4} w_{2} }}{{\partial x^{4} }} \\ & \quad \left. {-\,\eta A_{2} H_{x}^{2} \frac{{\partial^{4} w_{2} }}{{\partial x^{4} }} + c_{1} \left( {\frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{1} }}{{\partial x^{2} }}} \right)} \right) = c_{1} (w_{2} - w_{1} ) \\ \end{aligned}$$
(4)

It should be noted that the k1 and k3 will not enter into the equations of the inner tubes since only the outer tube interacts with the elastic medium.

The displacements of the nanotubes are subjected to the following boundary conditions:

For simply supported (S–S) nanotube,

$$w_{i} (0,t) = 0,\quad \frac{{\partial^{2} w_{i} (0,t)}}{{\partial^{2} x}} = 0,\quad w_{i} (L,t) = 0,\quad \frac{{\partial^{2} w_{i} (L,t))}}{{\partial^{2} x}} = 0.$$
(5)

For clamped–clamped supported (C–C) nanotube,

$$w_{i} (0,t) = 0,\quad \frac{{\partial w_{i} (0,t)}}{\partial x} = 0,\quad w_{i} ((L,t) = 0,\quad \frac{{\partial w_{i} (L,t)}}{\partial x} = 0.$$
(6)

For a clamped–simply supported (C–S) nanotube,

$$w_{i} (0,t) = 0,\quad \frac{{\partial w_{i} (0,t)}}{\partial x} = 0,\quad w_{i} (L,t) = 0,\quad \frac{{\partial^{2} w_{i} ((L,t)}}{{\partial^{2} x}} = 0.$$
(7)

3 Solution methodology

Using the Galerkin’s decomposition procedure to separate the spatial and temporal parts of the lateral displacement functions,

$$w_{i} \left( {x,t} \right) = \phi (x)W_{i} (t)\quad i = 1,2.$$
(8)

where \(w_{i} \left( {x,t} \right)\) is the lateral displacement functions, \(W_{i} \left( t \right)\) is the time-dependent parameter or time-dependent maximum amplitude of oscillation of the i-th layer of the nanotube and \(\phi \left( x \right)\) is a trial/comparison function that will satisfy both the geometric and natural boundary conditions.

Applying one-parameter Galerkin to a generalized form of the Eqs. (3) and (4), we have

$$\int\limits_{0}^{L} R \left( {x,t} \right)\phi \left( x \right)dx = 0$$
(9)

where \(R_{N} \left( {x,t} \right)\) is the equation of motion for each wall. For the outer wall of double-walled carbon nanotubes,

$$\begin{aligned} R\left( {x,t} \right) & = EI_{2} \frac{{\partial^{4} w_{2} }}{{\partial x^{4} }} + m_{c2} \frac{{\partial^{2} w_{2} }}{{\partial t^{2} }} + k_{1} w_{2} \\ & \quad + k_{3} w_{2}^{3} - \left( {\frac{{EA_{2} }}{2L}\int\limits_{0}^{L} {\left( {\frac{{\partial w_{2} }}{\partial x}} \right)}^{2} dx} \right)\frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} \\ & \quad - EA_{2} \alpha_{x} T\frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} - \eta A_{2} H_{x}^{2} \frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} \\ & \quad - \left( {e_{o} a} \right)^{2} \left( {m_{c2} \frac{{\partial^{4} w_{2} }}{{\partial x^{2} \partial t^{2} }} + k_{1} \frac{{\partial^{2} w_{2} }}{{\partial x^{2} }}} \right. \\ & \quad + 6k_{3} w_{2} \left( {\frac{{\partial w_{2} }}{\partial x}} \right)^{2} + 3k_{3} w_{2}^{2} \left( {\frac{{\partial w_{2} }}{\partial x}} \right) \\ & \quad - \left( {\frac{{EA_{2} }}{2L}\int\limits_{0}^{L} {\left( {\frac{{\partial w_{2} }}{\partial x}} \right)}^{2} dx} \right)\frac{{\partial^{4} w_{2} }}{{\partial x^{4} }} \\ & \quad - E_{2} A\alpha_{x} T\frac{{\partial^{4} w_{2} }}{{\partial x^{4} }} - \eta A_{2} H_{x}^{2} \frac{{\partial^{4} w_{2} }}{{\partial x^{4} }} \\ & \quad \left. {+\,c_{1} \left( {\frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{1} }}{{\partial x^{2} }}} \right)} \right) = c_{1} \left( {w_{2} - w_{1} } \right) \\ \end{aligned}$$
(10)

One arrives at

$$\begin{aligned} & \alpha_{1} EI_{2} W_{2} + \alpha_{2} m_{C2} \frac{{d^{2} W_{2} }}{{dt^{2} }} + \alpha_{2} k_{1} W_{2} + \alpha_{3} k_{3} W_{2}^{3} \\ & \quad - \alpha_{4} \frac{{EA_{2} }}{2L}W_{N}^{3} - \left( {a_{0} a} \right)^{2} \alpha_{5} m_{c2} \frac{{d^{2} W_{2} }}{{dt^{2} }} \\ & \quad - \left( {a_{0} a} \right)^{2} \alpha_{5} k_{1} W_{2} - 6\alpha_{6} \left( {a_{0} a} \right)^{2} k_{3} W_{2}^{3} \\ & \quad - 3\alpha_{7} \left( {a_{0} a} \right)^{2} k_{3} W_{2}^{3} + \alpha_{8} \left( {a_{0} a} \right)^{2} \frac{{EA_{2} }}{2L}W_{2}^{3} \\ & \quad + \alpha_{{_{1} }} \left( {a_{0} a} \right)^{2} \left( {EA_{2} \alpha_{x} T_{t} + \eta A_{2} H_{x}^{2} } \right)W_{2} \\ & \quad - \alpha_{5} \left( {a_{0} a} \right)^{2} c_{1} (W_{2} - W_{1} ) - \alpha_{5} \left( {EA_{2} \alpha_{x} T_{t} } \right. \\ & \quad \left. {+\,\eta A_{2} H_{x}^{2} } \right)W_{2} + \alpha_{2} \left( {a_{0} a} \right)^{2} c_{1} (W_{2} - W_{1} ) = 0 \\ \end{aligned}$$
(11)

After collecting like terms, we have

$$\begin{aligned} & \frac{{d^{2} W_{2} }}{{dt^{2} }} + \left( {\frac{{\alpha_{1} EI_{1} + \alpha_{2} k_{1} - \alpha_{5} \mu k_{1} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA_{2} \alpha_{x} T_{t} + \eta A_{2} H_{x}^{2} } \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{2} }}} \right)W_{2} \\ & \quad + \left( {\frac{{(\alpha_{2} - \alpha_{5} \mu )c_{1} }}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{2} }}} \right)(W_{2} - W_{1} ) \\ & \quad + \left( {\frac{{\alpha_{3} k_{3} - \alpha_{4} \frac{{EA_{2} }}{2L} - 6\alpha_{6} \mu k_{3} - 3\alpha_{7} \mu k_{3} + \alpha_{8} \mu \frac{{EA_{2} }}{2L}}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{2} }}} \right)W_{2}^{3} = 0 \\ \end{aligned}$$
(12)

where

$$\begin{aligned} \alpha_{1} & = \int\limits_{0}^{L} {\phi \left( x \right)\frac{{d^{4} \phi \left( x \right)}}{{dx^{4} }}dx} ,\quad \alpha_{2} = \int\limits_{0}^{L} {\phi^{2} \left( x \right)dx} ,\quad \alpha_{3} = \int\limits_{0}^{L} {\phi^{4} \left( x \right)dx} , \\ \alpha_{4} & = \int\limits_{0}^{L} {\left( {\phi \left( x \right)\left( {\int\limits_{0}^{L} {\left( {\frac{d\phi \left( x \right)}{\partial x}} \right)}^{2} dx} \right)\frac{{d^{2} \phi \left( x \right)}}{{dx^{2} }}dx} \right)} , \\ \alpha_{5} & = \int\limits_{0}^{L} {\phi \left( x \right)\frac{{d^{2} \phi \left( x \right)}}{{dx^{2} }}dx} ,\quad \alpha_{6} = \int\limits_{0}^{L} {\phi^{2} \left( x \right)\left( {\frac{d\phi \left( x \right)}{dx}} \right)^{2} dx} \\ \alpha_{7} & = \int\limits_{0}^{L} {\phi^{3} \left( x \right)\frac{d\phi \left( x \right)}{dx}dx} ,\quad \alpha_{8} = \int\limits_{0}^{L} {\left( {\phi \left( x \right)\left( {\int\limits_{0}^{L} {\left( {\frac{d\phi \left( x \right)}{\partial x}} \right)}^{2} dx} \right)\frac{{d^{4} \phi \left( x \right)}}{{dx^{4} }}dx} \right)} , \\ \mu & = \left( {e_{0} a} \right)^{2} ,\quad m_{cN} = \rho A_{N} , \\ \end{aligned}$$
(13)

Similarly, the same procedure is applied to other inner walls appropriately.

Therefore, the governing equations of motion for nonlinear vibrations of embedded DWCNTs in a thermal and magnetic environment in ODE form is obtained as,

$$\begin{aligned} & \frac{{d^{2} W_{1} }}{{dt^{2} }} + \left( {\frac{{\alpha_{1} EI_{1} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA_{1} \alpha_{x} T_{t} + \eta A_{1} H_{x}^{2} } \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{1} }}} \right)W_{1} \\ & \quad - \frac{{c_{1} }}{{\rho A_{1} }}(W_{2} - W_{1} ) + \left( {\frac{{(\alpha_{8} \mu - \alpha_{4} )\frac{{EA_{1} }}{2L}}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{1} }}} \right)W_{1}^{3} = 0 \\ \end{aligned}$$
(14)
$$\begin{aligned} & \frac{{d^{2} W_{2} }}{{dt^{2} }} + \left( {\frac{{\alpha_{1} EI_{2} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA_{2} \alpha_{x} T_{t} + \eta A_{2} H_{x}^{2} } \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{2} }}} \right)W_{2} \\ & \quad + \frac{{c_{1} }}{{\rho A_{2} }}(W_{2} - W_{1} ) + \left( {\frac{{(\alpha_{8} \mu - \alpha_{4} )\frac{{EA_{2} }}{2L}}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{2} }}} \right)W_{2}^{3} = 0 \\ \end{aligned}$$
(15)

and the initial conditions are

$$\begin{aligned} W_{1} \left( 0 \right) & = X\quad {\text{and}}\quad \frac{{dW_{1} \left( 0 \right)}}{dt} = 0 \\ W_{2} \left( 0 \right) & = X\quad {\text{and}}\quad \frac{{dW_{2} \left( 0 \right)}}{dt} = 0 \\ \end{aligned}$$
(16)

3.1 Homotopy perturbation method

The nonlinear terms in Eqs. (14) and (15) make the development of exact analytical solution. Therefore, for the purpose of generating a symbolic solution for the nonlinear equations, we made a recourse homotopy perturbation method. The principle and the procedures of the method can be found in our previous works [46, 47].

3.1.1 Analysis of single-walled carbon nanotube

For SWCNT, the governing equation is given by

$$\begin{aligned} & \frac{{d^{2} W}}{{dt^{2} }} + \left( {\frac{{\alpha_{1} EI + \alpha_{2} k_{1} - \alpha_{5} \mu k_{1} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA\alpha_{x} T_{t} + \eta AH_{x}^{2} } \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{N} }}} \right)W \\ & \quad + \left( {\frac{{\alpha_{3} k_{3} - \alpha_{4} \frac{EA}{2L} - 6\alpha_{6} \mu k_{3} - 3\alpha_{7} \mu k_{3} + \alpha_{8} \mu \frac{EA}{2L}}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A}}} \right)W^{3} = 0 \\ \end{aligned}$$
(17)

Introducing the following dimensionless quantities,

$$r = \sqrt {\frac{I}{A}} ,\quad \tau = \omega_{0} t,\quad a = \frac{W}{r}$$
(18)

After applying the dimensionless parameters in Eq. (18), Eq. (17) is transformed to

$$\omega_{0}^{2} \frac{{d^{2} a}}{{d\tau^{2} }} + f_{1} a + f_{2} a^{3} = 0$$
(19)

where f1 and f2 are defined as

$$f_{1} = \frac{{\alpha_{1} EI + \alpha_{2} k_{1} - \alpha_{5} \mu k_{1} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA\alpha_{x} T_{t} + \eta AH_{x}^{2} } \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A}} = \omega^{2}$$
(20)
$$f_{2} = \left( {\frac{{\alpha_{3} k_{3} - \alpha_{4} \frac{EA}{2L} - 6\alpha_{6} \mu k_{3} - 3\alpha_{7} \mu k_{3} + \alpha_{8} \mu \frac{EA}{2L}}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A}}} \right) \cdot \frac{I}{A}$$
(21)

and the initial conditions are

$$a\left( 0 \right) = X\quad {\text{and}}\quad \frac{da\left( 0 \right)}{d\tau } = 0$$
(22)

It is shown from Eq. (19) that

$$\omega = \sqrt {f_{1} } = \sqrt {\frac{{\alpha_{1} EI + \alpha_{2} k_{1} - \alpha_{5} \mu k_{1} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA\alpha_{x} T_{t} + \eta AH_{x}^{2} } \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A}}}$$
(23)

It should be noted that \(\omega\) is the linear forced vibration frequency, and \(\omega_{0}\) is an unknown nonlinear angular frequency to be determined.

In order to solve Eq. (17), we construct the following homotopy with \(\omega_{0}\) as the initial approximation for the angular nonlinear frequency as

$$\left( {1 - p} \right)\left\{ {\omega_{0}^{2} \left( {\frac{{d^{2} a}}{{d\tau^{2} }} + a} \right)} \right\} + p\left( {\omega_{0}^{2} \frac{{d^{2} a}}{{d\tau^{2} }} + f_{1} a + f_{2} a^{3} } \right) = 0$$
(24)

It should be noted that the solutions of \(a = a(\tau ,p)\) and \(\upomega =\upomega(p)\) of the homotopy change from their initial approximations \(a_{0} = a(\tau )\) and \(\omega_{0}\) to the required solutions \(a(\tau )\) and \(\omega_{o}\) of (19) as the embedding parameter p travels from 0 to 1.

Assuming that the solution of Eq. (19) take the form of:

$$a\left( \tau \right) = a_{0} \left( \tau \right) + pa_{1} \left( \tau \right) + p^{2} a_{2} \left( \tau \right) + p^{3} a_{3} \left( \tau \right) + \cdots ,$$
(25a)
$$\omega_{0} = \omega_{0} + p\omega_{1} + p^{2} \omega_{2} + p^{3} \omega_{3} + \cdots ,$$
(25b)

After substituting Eq. (25) into the homotopy Eq. (24) and rearranging the coefficients of the terms with identical powers of p, we have a series of linear differential equations of the form

$$p^{0} :\omega_{0}^{2} \left( {\frac{{d^{2} a_{0} }}{{d\tau^{2} }} + a_{0} } \right) = 0,$$
(26a)

with initial conditions

$$\begin{aligned} & a_{0} (0) = X\quad {\text{and}}\quad \frac{{da_{0} (0)}}{d\tau } = 0, \\ & p^{1} :\omega_{0}^{2} \left[ {\frac{{d^{2} a_{1} }}{{d\tau^{2} }} + a_{1} - \left( {\frac{{d^{2} a_{0} }}{{d\tau^{2} }} + a_{0} } \right)} \right] \\ & \quad + \omega_{0}^{2} \frac{{d^{2} a_{0} }}{{d\tau^{2} }} + f_{1} a_{0} + f_{2} a_{0}^{3} = 0 \\ \end{aligned}$$
(26b)

the corresponding initial conditions are

$$\begin{aligned} & a_{1} (0) = 0\quad {\text{and}}\quad \frac{{da_{1} (0)}}{d\tau } = 0, \\ & p^{2} :\omega_{0}^{2} \left( {\frac{{d^{2} a_{2} }}{{d\tau^{2} }} + a_{2} } \right) - \omega_{0}^{2} a_{1} \\ & \quad + 2\omega_{0} \omega_{1} \frac{{d^{2} a_{0} }}{{d\tau^{2} }} + f_{1} a_{1} - f_{2} a_{0}^{2} a_{1} = 0 \\ \end{aligned}$$
(26c)

And the initial conditions are given as

$$a_{2} (0) = 0\quad {\text{and}}\quad \frac{{da_{0} (0)}}{d\tau } = 0,$$

Since \(\frac{{d^{2} a_{0} }}{{d\tau^{2} }} + a_{0} = 0\) and \(\frac{{d^{2} a_{0} (0)}}{{d\tau^{2} }} = -\,a_{0}\) from Eq. (26a), we can write Eq. (26b) as

$$\omega_{0}^{2} \left( {\frac{{d^{2} a_{1} }}{{d\tau^{2} }} + a_{1} } \right) - \omega_{0}^{2} a_{0} + f_{1} a_{0} + f_{2} a_{0}^{3} = 0,$$
(27)

The solution of the initial zeroth approximation is given by

$$a_{0} = Xcos\tau ,$$
(28)

On substituting Eq. (28) into the first approximation equation in Eq. (27), one arrives at

$$\omega_{0}^{2} \left( {\frac{{d^{2} a_{1} }}{{d\tau^{2} }} + a_{1} } \right) - \omega_{0}^{2} Xcos\tau + f_{1} Xcos\tau + f_{2} \left( {Xcos\tau } \right)^{3} = 0,$$
(29)

After the application of trigonometry identities to the fourth-term in the LHS of Eq. (29), we have

$$\omega_{0}^{2} \left( {\frac{{d^{2} a_{1} }}{{d\tau^{2} }} + a_{1} } \right) - \omega_{0}^{2} Xcos\tau + f_{1} Xcos\tau + \frac{3}{4}f_{2} X^{3} cos\tau + \frac{1}{4}f_{2} X^{3} cos3\tau = 0,$$
(30)

in order to eliminate the secular terms, we set the coefficient of \(cos\tau\) in Eq. (30) to zero

$$-\,\omega_{0}^{2} Xcos\tau + f_{1} Xcos\tau + \frac{3}{4}f_{2} X^{3} cos\tau = 0,$$
(31)

It gives the nonlinear natural frequency as,

$$\omega_{0} = \sqrt {f_{1} + \frac{3}{4}f_{2} X^{2} } ,$$
(32)

It should be noted that the frequency ratio is given as \(\psi = \frac{{\omega_{o} }}{\omega }\).

Therefore, from Eqs. (32) and (23), we have the frequency ratio as

$$\psi = \frac{{\sqrt {f_{1} + \frac{3}{4}f_{2} X^{2} } }}{{\sqrt {f_{1} } }} = \sqrt {1 + \frac{3}{4}\frac{{f_{2} }}{{f_{1} }}X^{2} }$$
(33)

On substituting Eqs. (20) and (21) into Eq. (33), we have

$$\psi = \sqrt {1 + \frac{3}{4}\left\{ {\frac{{\left( {\alpha_{3} k_{3} - \alpha_{4} \frac{EA}{2L} - 6\alpha_{6} \mu k_{3} - 3\alpha_{7} \mu k_{3} + \alpha_{8} \mu \frac{EA}{2L}} \right).\frac{I}{A}}}{{\alpha_{1} EI + \alpha_{2} k_{1} - \alpha_{5} \mu k_{1} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA\alpha_{x} T_{t} + \eta AH_{x}^{2} } \right)}}} \right\}X^{2} }$$
(34)

The solution of Eq. (30) produces

$$a_{1} (\tau ) = \frac{{f_{2} X^{3} }}{{32f_{1} + 24f_{2} X^{2} }}\left( {cos3\tau - \cos \tau } \right)$$
(35)

Hence, the first approximate solution of Eq. (19) can be written as

$$a(\tau ) = a_{0} (\tau ) + a_{1} (\tau ) = Xcos\tau + \frac{{f_{2} X^{3} }}{{32f_{1} + 24f_{2} X^{2} }}\left( {cos3\tau - \cos \tau } \right)$$
(36)

From Eqs. (8) and (18), we have

$$w\left( {x,t} \right) = \phi \left( x \right){\text{W}}\left( t \right),\quad W = a\left( \tau \right)\sqrt {\frac{I}{A}}$$
(37)

Therefore, the displacement of the nanotube to be expressed as

$$w\left( {x,t} \right) = \phi \left( x \right)a\left( \tau \right)\sqrt {\frac{I}{A}} ,$$
(38)

Substituting Eq. (36) and the shape functions in the Table 1 into Eq. (38), for simple simply support, we have

$$w\left( {x,t} \right) = \left[ {\left( {Xcos\tau + \frac{{f_{2} X^{3} }}{{32f_{1} + 24f_{2} X^{2} }}\left( {cos3\tau - \cos \tau } \right)} \right)} \right]\sqrt {\frac{I}{A}} sin\left( {\frac{n\pi x}{l}} \right)$$
(39)

while for clamped–clamped gives

$$\begin{aligned} w\left( {x,t} \right) & = \left[ {Xcos\tau + \frac{{f_{2} X^{3} }}{{32f_{1} + 24f_{2} X^{2} }}\left( {cos3\tau - \cos \tau } \right)} \right] \\ & \quad \sqrt {\frac{I}{A}} \left\{ {\left( {\cosh \left( {\frac{\beta x}{L}} \right) - \cos \left( {\frac{\beta x}{L}} \right)} \right)} \right. \\ & \quad \left. { - \left( {\frac{\sinh \beta + \sin \beta }{\cosh \beta - \cos \beta }} \right)\left( {\sinh \left( {\frac{\beta x}{L}} \right) - sin\left( {\frac{\beta x}{L}} \right)} \right)} \right\} \\ \end{aligned}$$
(40)

and for clamped–simply support, one obtains

$$\begin{aligned} w\left( {x,t} \right) & = \left[ {Xcos\tau + \frac{{f_{2} X^{3} }}{{32f_{1} + 24f_{2} X^{2} }}\left( {cos3\tau - \cos \tau } \right)} \right] \\ & \quad \sqrt {\frac{I}{A}} \left\{ {\left( {\cosh \left( {\frac{\beta x}{L}} \right) - \cos \left( {\frac{\beta x}{L}} \right)} \right)} \right. \\ & \quad \left. { - \left( {\frac{\cosh \beta - \cos \beta }{\sinh \beta - \sin \beta }} \right)\left( {\sinh \left( {\frac{\beta x}{L}} \right) - sin\left( {\frac{\beta x}{L}} \right)} \right)} \right\} \\ \end{aligned}$$
(41)
Table 1 The basic functions corresponding to the above boundary conditions [37]
Full size table

3.1.2 Analysis of double-walled carbon nanotube

For a DWCNT, the governing equation is given by

$$\begin{aligned} & \frac{{d^{2} W_{1} }}{{dt^{2} }} + \left( {\frac{{\alpha_{1} EI_{1} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA_{1} \alpha_{x} T_{t} + \eta A_{1} H_{x}^{2} } \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{1} }}} \right)W_{1} \\ & \quad - \frac{{c_{1} }}{{\rho A_{1} }}(W_{2} - W_{1} ) + \left( {\frac{{(\alpha_{8} \mu - \alpha_{4} )\frac{{EA_{1} }}{2L}}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{1} }}} \right)W_{1}^{3} = 0 \\ \end{aligned}$$
(42)
$$\begin{aligned} & \frac{{d^{2} W_{2} }}{{dt^{2} }} + \left( {\frac{{\alpha_{1} EI_{2} + \alpha_{2} k_{1} - \alpha_{5} \mu k_{1} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA_{2} \alpha_{x} T_{t} + \eta A_{2} H_{x}^{2} } \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{2} }}} \right)W_{2} \\ & \quad + \frac{{c_{1} }}{{\rho A_{2} }}(W_{2} - W_{1} ) + \left( {\frac{{\alpha_{3} k_{3} - \alpha_{4} \frac{{EA_{2} }}{2L} - 6\alpha_{6} \mu k_{3} - 3\alpha_{7} \mu k_{3} + \alpha_{8} \mu \frac{{EA_{2} }}{2L}}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{2} }}} \right)W_{2}^{3} = 0 \\ \end{aligned}$$
(43)

Using the following dimensionless parameters,

$$r = \sqrt {\frac{{I_{1} }}{{A_{1} }}} ,\quad a_{1} = \frac{{W_{1} }}{r},\quad a_{2} = \frac{{W_{2} }}{r}\quad {\text{and}}\quad \tau = \omega_{0} t$$
(44)

On substituting the dimensionless parameters in Eq. (44) into Eqs. (42) and (43), we have the following dimensionless nonlinear system of equations

$$\omega_{0}^{2} \frac{{d^{2} a_{1} }}{{d\tau^{2} }} + f_{1} a_{1} + f_{2} a_{1}^{3} - f_{3} a_{2} = 0,$$
(45)
$$\omega_{0}^{2} \frac{{d^{2} a_{2} }}{{d\tau^{2} }} + g_{1} a_{2} + g_{2} a_{2}^{3} - g_{3} a_{1} = 0.$$
(46)

where

$$\begin{aligned} f_{1} & = \frac{{\alpha_{1} EI_{1} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA_{1} \alpha_{x} T_{t} + \eta A_{1} H_{x}^{2} } \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{1} }} + \frac{{c_{1} }}{{\rho A_{1} }}, \\ f_{2} & = \frac{{(\alpha_{8} \mu - \alpha_{4} )EI_{1} }}{{2L\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{1} }},\quad f_{3} = \frac{{c_{1} }}{{\rho A_{1} }} \\ g_{1} & = \left( {\frac{{\alpha_{1} EI_{2} + \alpha_{2} k_{1} - \alpha_{5} \mu k_{1} + (\alpha_{1} \mu - \alpha_{5} )\left( {EA_{1} \alpha_{x} T_{t} + \eta A_{1} H_{x}^{2} } \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{2} }}} \right) + \frac{{c_{1} }}{{\rho A_{2} }} \\ g_{2} & = \frac{{\left( {\alpha_{3} k_{3} - \alpha_{4} \frac{{EA_{2} }}{2L} - 6\alpha_{6} \mu k_{3} - 3\alpha_{7} \mu k_{3} + \alpha_{8} \mu \frac{{EA_{2} }}{2L}} \right)}}{{\left( {\alpha_{2} - \alpha_{5} \mu } \right)\rho A_{2} }}\left( {\frac{{I_{1} }}{{A_{1} }}} \right), \\ g_{3} & = \frac{{c_{1} }}{{\rho A_{2} }} \\ \end{aligned}$$
(47)

In a similar manner to SWCNT, we construct a homotopy on Eqs. (45) and (46) as follows

$$\left( {1 - p} \right)\left\{ {\omega_{0}^{2} \left( {\frac{{d^{2} a_{1} }}{{d\tau^{2} }} + a_{1} } \right)} \right\} + p\left\{ {\omega_{0}^{2} \frac{{d^{2} a_{1} }}{{d\tau^{2} }} + f_{1} a_{1} + f_{2} a_{1}^{3} - f_{3} a_{2} } \right\} = 0,$$
(48a)
$$\left( {1 - p} \right)\left\{ {\omega_{0}^{2} \left( {\frac{{d^{2} a_{2} }}{{d\tau^{2} }} + a_{2} } \right)} \right\} + p\left\{ {\omega_{0}^{2} \frac{{d^{2} a_{2} }}{{d\tau^{2} }} + g_{1} a_{2} + g_{2} a_{2}^{3} - g_{3} a_{1} } \right\} = 0.$$
(48b)

The solutions of \(a_{1} = a_{1} (\tau ,p)\), \(a_{2} = a_{2} (\tau ,p)\) and \(\upomega =\upomega(p)\) of the homotopy change from their initial approximations \(a_{10} = a_{1} (\tau )\),\(a_{20} = a_{2} (\tau )\) and \(\omega_{0}\) to the required solutions \(a_{1} \left( \tau \right),a_{2} \left( \tau \right)\) and \(\omega_{o}\) of Eqs. (49) and (50) as the embedding parameter p travels from 0 to 1.

Assuming the solution of Eqs. (45) and (46) to be in the following form

$$a_{1} \left( \tau \right) = a_{10} \left( \tau \right) + pa_{11} \left( \tau \right) + p^{2} a_{12} \left( \tau \right) + p^{3} a_{13} \left( \tau \right) + \cdots ,$$
(49a)
$$a_{2} \left( \tau \right) = a_{20} \left( \tau \right) + pa_{21} \left( \tau \right) + p^{2} a_{22} \left( \tau \right) + p^{3} a_{23} \left( \tau \right) + \cdots ,$$
(49b)
$$\upomega =\upomega_{0} + p\upomega_{1} + p^{2}\upomega_{2} + p^{3}\upomega_{3} + \cdots$$
(49c)

Substituting Eqs. (45a–c) into the homotopy in Eqs. (44a) and (44b), collecting and rearranging the coefficients of the terms with identical powers of p, we have a series of linear differential equations

$$p^{0} :\left\{ {\begin{array}{*{20}c} {\frac{{d^{2} a_{10} }}{{d\tau^{2} }} + a_{10} = 0,} & {a_{10} \left( 0 \right) = X_{1} ,} & {\frac{{da_{10} \left( 0 \right)}}{d\tau } = 0,} \\ {\frac{{d^{2} a_{20} }}{{d\tau^{2} }} + a_{20} = 0,} & {a_{20} \left( 0 \right) = X_{1} ,} & {\frac{{da_{20} \left( 0 \right)}}{d\tau } = 0} \\ \end{array} } \right.$$
(50a)
$$p^{1} :\left\{ {\begin{array}{*{20}l} {\omega_{0}^{2} \left\{ {\frac{{d^{2} a_{11} }}{{d\tau^{2} }} + a_{11} } \right\} - \omega_{0}^{2} a_{10} + f_{1} a_{10} + f_{2} a_{10}^{3} - f_{3} a_{20} = 0,} \hfill & {a_{11} \left( 0 \right) = 0} \hfill & {\frac{{da_{11} \left( 0 \right)}}{d\tau } = 0,} \hfill \\ {\omega_{0}^{2} \left\{ {\frac{{d^{2} a_{21} }}{{d\tau^{2} }} + a_{21} } \right\} - \omega_{0}^{2} a_{20} + g_{1} a_{20} + g_{2} a_{20}^{3} - g_{3} a_{10} = 0,} \hfill & {a_{21} \left( 0 \right) = 0,} \hfill & {\frac{{da_{21} \left( 0 \right)}}{d\tau } = 0} \hfill \\ \end{array} } \right.$$
(50b)
$$p^{2} :\left\{ {\begin{array}{*{20}l} {\omega_{0}^{2} \left\{ {\frac{{d^{2} a_{12} }}{{d\tau^{2} }} + a_{12} } \right\} - \omega_{0}^{2} a_{11} + 2\omega_{0} \omega_{1} \frac{{d^{2} a_{10} }}{{d\tau^{2} }} + f_{1} a_{11} + 3f_{2} a_{10}^{2} a_{11} - f_{3} a_{21} = 0,} \hfill & {a_{12} \left( 0 \right) = 0} \hfill & {\frac{{da_{12} \left( 0 \right)}}{d\tau } = 0,} \hfill \\ {\omega_{0}^{2} \left\{ {\frac{{d^{2} a_{22} }}{{d\tau^{2} }} + a_{22} } \right\} - \omega_{0}^{2} a_{21} + 2\omega_{0} \omega_{1} \frac{{d^{2} a_{20} }}{{d\tau^{2} }} + g_{1} a_{21} + 3g_{2} a_{20}^{2} a_{21} - g_{3} a_{11} = 0,} \hfill & {a_{22} \left( 0 \right) = 0,} \hfill & {\frac{{da_{22} \left( 0 \right)}}{d\tau } = 0} \hfill \\ \end{array} } \right.$$
(50c)

The solution of the initial zeroth approximation in Eq. (50a) is simply given by

$$a_{10} (0) = X_{1} cos\tau ,$$
(51)
$$a_{20} (0) = X_{2} cos\tau ,$$
(52)

Substituting Eqs. (51) and (52) into the first approximation in Eq. (50b), eliminating the coefficient of cos \(\tau\) in the above system to avoid the secular terms, we have the following nonlinear system of equations:

$$- X_{1} \omega^{2}_{0} + f_{1} X_{1} + \frac{3}{4}f_{2} X_{1}^{3} - f_{3} X_{2} = 0$$
(53)
$$- X_{2} \omega^{2}_{0} + g_{1} X_{2} + \frac{3}{4}g_{2} X_{2}^{3} - g_{3} X_{1} = 0$$
(54)

From Eq. (53),

$$X_{2} = \frac{{ - X_{1} \omega^{2}_{0} + f_{1} X_{1} + \frac{3}{4}f_{2} X_{1}^{3} }}{{f_{3} }}.$$
(55)

After the substitution of Eq. (55) into Eq. (53), we have

$$\begin{aligned} & - \omega^{2}_{0} \left( {\frac{{ - X_{1} \omega^{2}_{0} + f_{1} X_{1} + \frac{3}{4}f_{2} X_{1}^{3} }}{{f_{3} }}} \right) + g_{1} \left( {\frac{{ - X_{1} \omega^{2}_{0} + f_{1} X_{1} + \frac{3}{4}f_{2} X_{1}^{3} }}{{f_{3} }}} \right) \\ & \quad + \frac{3}{4}g_{2} \left( {\frac{{ - X_{1} \omega^{2}_{0} + f_{1} X_{1} + \frac{3}{4}f_{2} X_{1}^{3} }}{{f_{3} }}} \right)^{3} - g_{3} X_{1} = 0 \\ \end{aligned}$$
(56)

After expansion of Eq. (56) and collecting like terms, we arrived at

$$\begin{aligned} & \frac{3}{4}\frac{{g_{2} }}{{f_{3}^{3} }}X_{1}^{3} \omega_{0}^{6} + \left( {\frac{{X_{1} }}{{f_{3} }} - \frac{3}{4}\frac{{g_{2} }}{{f_{3}^{3} }}X_{1}^{2} \left( {f_{1} X_{1} + \frac{3}{2}f_{2} X_{1}^{3} } \right)} \right. \\ & \quad \left. { - \frac{9}{8}\frac{{g_{2} }}{{f_{3}^{3} }}f_{1} f_{2} X_{1}^{8} } \right)\omega_{0}^{4} + \left( {\frac{{f_{1} X_{1} + \frac{3}{4}f_{2} X_{1}^{3} + g_{1} X_{1} }}{{f_{3} }}} \right. \\ & \quad + \frac{3}{4}\frac{{g_{2} }}{{f_{3}^{3} }}f_{1}^{2} X_{1}^{3} + \frac{9}{16}f_{2}^{2} X_{1}^{7} + 2X_{1} \left( {f_{1}^{2} X_{1}^{2} + \frac{9}{16}f_{2}^{2} X_{1}^{6} } \right) \\ & \quad \left. { + \frac{9}{2}f_{1} f_{2} X_{1}^{5} } \right)\omega_{0}^{2} + g_{3} X_{1} - g_{1} \left( {\frac{{f_{1} X_{1} + \frac{3}{4}f_{2} X_{1}^{3} }}{{f_{3} }}} \right) \\ & \quad - \frac{3}{4}\frac{{g_{2} }}{{f_{3}^{3} }}\left\{ {f_{1}^{3} X_{1}^{3} + \frac{27}{64}f_{2}^{3} X_{1}^{6} + \frac{3}{4}f_{1}^{2} f_{2} X_{1}^{5} } \right. \\ & \quad \left. { + \frac{9}{16}f_{1} f_{2}^{2} X_{1}^{7} + 2\left[ {\frac{3}{4}f_{1}^{2} f_{2} X_{1}^{5} + \frac{9}{16}f_{1} f_{2}^{2} X_{1}^{7} } \right]} \right\} = 0 \\ \end{aligned}$$
(57)

Equation (57) can be written as

$$\lambda_{1} \omega_{0}^{6} + \lambda_{2} \omega_{0}^{4} + \lambda_{3} \omega_{0}^{2} + \lambda_{4} = 0$$
(58)

where

$$\begin{aligned} \lambda_{1} & = \frac{3}{4}\frac{{g_{2} }}{{f_{3}^{3} }}X_{1}^{3} \\ \lambda_{2} & = \left( {\frac{{X_{1} }}{{f_{3} }} - \frac{3}{4}\frac{{g_{2} }}{{f_{3}^{3} }}X_{1}^{2} \left( {f_{1} X_{1} + \frac{3}{2}f_{2} X_{1}^{3} } \right) - \frac{9}{8}\frac{{g_{2} }}{{f_{3}^{3} }}f_{1} f_{2} X_{1}^{8} } \right) \\ \lambda_{3} & = \left( {\frac{{f_{1} X_{1} + \frac{3}{4}f_{2} X_{1}^{3} + g_{1} X_{1} }}{{f_{3} }} + \frac{3}{4}\frac{{g_{2} }}{{f_{3}^{3} }}f_{1}^{2} X_{1}^{3} + \frac{9}{16}f_{2}^{2} X_{1}^{7} } \right. \\ & \quad \left. { + 2X_{1} \left( {f_{1}^{2} X_{1}^{2} + \frac{9}{16}f_{2}^{2} X_{1}^{6} } \right) + \frac{9}{2}f_{1} f_{2} X_{1}^{5} } \right) \\ \lambda_{4} & = g_{3} X_{1} - g_{1} \left( {\frac{{f_{1} X_{1} + \frac{3}{4}f_{2} X_{1}^{3} }}{{f_{3} }}} \right) - \frac{3}{4}\frac{{g_{2} }}{{f_{3}^{3} }}\left\{ {f_{1}^{3} X_{1}^{3} } \right. \\ & \quad + \frac{27}{64}f_{2}^{3} X_{1}^{6} + \frac{3}{4}f_{1}^{2} f_{2} X_{1}^{5} + \frac{9}{16}f_{1} f_{2}^{2} X_{1}^{7} \\ & \quad \left. { + 2\left[ {\frac{3}{4}f_{1}^{2} f_{2} X_{1}^{5} + \frac{9}{16}f_{1} f_{2}^{2} X_{1}^{7} } \right]} \right\} \\ \end{aligned}$$
(59)

The roots of the sexic equations are

$$\left( {\omega_{0} } \right)_{1} = \sqrt {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} }$$
(60a)
$$\left( {\omega_{0} } \right)_{2} = - \sqrt {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} }$$
(60b)
$$\left( {\omega_{0} } \right)_{3} = \sqrt {\begin{array}{*{20}l} {\frac{ - 1}{{2\lambda_{1} }}\left[ \begin{aligned} \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ \end{aligned} \right]} \hfill \\ { + \frac{{\sqrt { - 3} }}{{2\lambda_{1} }}\left[ \begin{aligned} \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ - \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ \end{aligned} \right]} \hfill \\ { - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} }$$
(60c)
$$\left( {\omega_{0} } \right)_{4} = - \sqrt {\begin{array}{*{20}l} {\frac{ - 1}{{2\lambda_{1} }}\left[ \begin{aligned} \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ \end{aligned} \right]} \hfill \\ { + \frac{{\sqrt { - 3} }}{{2\lambda_{1} }}\left[ \begin{aligned} \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ - \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ \end{aligned} \right]} \hfill \\ { - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} }$$
(60d)
$$\left( {\omega_{0} } \right)_{5} = \sqrt {\begin{array}{*{20}l} {\frac{ - 1}{{2\lambda_{1} }}\left[ \begin{aligned} \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ \end{aligned} \right]} \hfill \\ {\frac{{ - \sqrt { - 3} }}{{2\lambda_{1} }}\left[ \begin{aligned} \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ - \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ \end{aligned} \right]} \hfill \\ { - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} }$$
(60e)
$$\left( {\omega_{0} } \right)_{6} = - \sqrt {\begin{array}{*{20}l} {\frac{ - 1}{{2\lambda_{1} }}\left[ \begin{aligned} \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ \end{aligned} \right]} \hfill \\ {\frac{{ - \sqrt { - 3} }}{{2\lambda_{1} }}\left[ \begin{aligned} \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ - \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ \end{aligned} \right]} \hfill \\ { - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} }$$
(60f)

The nonlinear natural frequency (\(\omega_{0}\)) for embedded DWCNTs is obtained from the above solution. The smallest real value of \(\omega_{0}\) is the nonlinear natural frequency for DWCNTs. From Eq. (60),

$$\omega_{0} = \sqrt {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}}{ - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \end{array} }$$
(61)

To calculate the linear natural frequencies for DWNT, substitute

$$a_{10} \left( \tau \right) = X_{1} cos\omega t$$
(62a)
$$a_{20} \left( \tau \right) = X_{2} cos\omega t$$
(62b)

into Eq. (50b) and neglecting the nonlinear terms give

$$- X_{1} \omega^{2} + f_{1} X_{1} - f_{3} X_{2} = 0$$
(63)
$$- X_{2} \omega^{2} + g_{1} X_{2} - g_{3} X_{1} = 0$$
(64)

which can be written in matrix form as

$$\left[ {\begin{array}{*{20}c} { - \omega^{2} + f_{1} } & { - f_{1} } \\ { - g_{3} } & { - \omega^{2} + g_{1} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {X_{1} } \\ {X_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right]$$
(65)

Since \(\left[ {\begin{array}{*{20}c} {X_{1} } \\ {X_{2} } \\ \end{array} } \right]\) cannot be equal to zero, for nontrivial case to occur, then

$$\left[ {\begin{array}{*{20}c} { - \omega^{2} + f_{1} } & { - f_{1} } \\ { - g_{3} } & { - \omega^{2} + g_{1} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right] .$$
(66)

By equating the determinant of the matrix in Eq. (66) to zero, the frequency characteristic equation is obtained as

$$\omega^{4} - \left( {f_{1} + g_{1} } \right)\omega^{2} + f_{1} g_{1} - f_{3} g_{3} = 0$$
(67)

where the roots of the quartic equation are

$$\left( \omega \right)_{1} = \sqrt {\frac{{\left( {f_{1} + g_{1} } \right) + \sqrt {\left( {f_{1} + g_{1} } \right)^{2} - 4\left( {f_{1} g_{1} - f_{3} g_{3} } \right)} }}{2}}$$
(68a)
$$\left( \omega \right)_{2} = - \sqrt {\frac{{\left( {f_{1} + g_{1} } \right) + \sqrt {\left( {f_{1} + g_{1} } \right)^{2} - 4\left( {f_{1} g_{1} - f_{3} g_{3} } \right)} }}{2}}$$
(68b)
$$\left( \omega \right)_{3} = \sqrt {\frac{{\left( {f_{1} + g_{1} } \right) - \sqrt {\left( {f_{1} + g_{1} } \right)^{2} - 4\left( {f_{1} g_{1} - f_{3} g_{3} } \right)} }}{2}}$$
(68c)
$$\left( \omega \right)_{4} = - \sqrt {\frac{{\left( {f_{1} + g_{1} } \right) - \sqrt {\left( {f_{1} + g_{1} } \right)^{2} - 4\left( {f_{1} g_{1} - f_{3} g_{3} } \right)} }}{2}}$$
(68d)

The linear natural frequency of DWNTs is the lowest root of the Eq. (67). From Eq. (68), it is

$$\omega = \sqrt {\frac{{\left( {f_{1} + g_{1} } \right) - \sqrt {\left( {f_{1} + g_{1} } \right)^{2} - 4\left( {f_{1} g_{1} - f_{3} g_{3} } \right)} }}{2}}$$
(69)

We should recall that frequency ratio is given by \(\psi = \frac{{\omega_{o} }}{\omega }\). Therefore

$$\psi = \sqrt {\frac{{\left( {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} } \right)}}{{\left( {\frac{{\left( {f_{1} + g_{1} } \right) - \sqrt {\left( {f_{1} + g_{1} } \right)^{2} - 4\left( {f_{1} g_{1} - f_{3} g_{3} } \right)} }}{2}} \right)}}}$$
(70)

On substituting Eqs. (51) and (52) into Eq. (50b), we have

$$\left\{ {\begin{array}{*{20}l} {\omega_{0}^{2} \left\{ {\frac{{d^{2} a_{11} }}{{d\tau^{2} }} + a_{11} } \right\} - \omega_{0}^{2} X_{1} cos\tau + f_{1} X_{1} cos\tau + f_{2} \left( {X_{1} cos\tau } \right)^{3} - f_{3} X_{2} cos\tau = 0,} \hfill & {a_{11} \left( 0 \right) = 0} \hfill & {\frac{{da_{11} \left( 0 \right)}}{d\tau } = 0,} \hfill \\ {\omega_{0}^{2} \left\{ {\frac{{d^{2} a_{21} }}{{d\tau^{2} }} + a_{21} } \right\} - \omega_{0}^{2} X_{2} cos\tau + g_{1} X_{2} cos\tau + g_{2} \left( {X_{2} cos\tau } \right)^{3} - g_{3} X_{1} cos\tau = 0,} \hfill & {a_{21} \left( 0 \right) = 0,} \hfill & {\frac{{da_{21} \left( 0 \right)}}{d\tau } = 0} \hfill \\ \end{array} } \right.$$
(71)

After the application of trigonometry identities to the fourth-term in the LHS

$$\left\{ \begin{aligned} \omega_{0}^{2} \left\{ {\frac{{d^{2} a_{11} }}{{d\tau^{2} }} + a_{11} } \right\} - \omega_{0}^{2} X_{1} cos\tau + f_{1} X_{1} cos\tau + \frac{3}{4}f_{2} X_{1}^{3} cos\tau + \frac{1}{4}f_{2} X_{1}^{3} cos3\tau - f_{3} X_{2} cos\tau = 0,\,\,\,\,\,\,\,\,a_{11} \left( 0 \right)\, = 0\,\,\,\,\,\,\,\frac{{da_{11} \left( 0 \right)}}{d\tau } = 0,\, \hfill \\ \,\,\,\,\, \hfill \\ \omega_{0}^{2} \left\{ {\frac{{d^{2} a_{21} }}{{d\tau^{2} }} + a_{21} } \right\} - \omega_{0}^{2} X_{2} cos\tau + g_{1} X_{2} cos\tau + \frac{3}{4}g_{2} X_{2}^{3} cos\tau + \frac{1}{4}g_{2} X_{2}^{3} cos3\tau - g_{3} X_{1} cos\tau = \,0,\,\,\,\,\,\,\,\,\,a_{21} \left( 0 \right)\, = 0,\,\,\,\,\,\,\,\frac{{da_{21} \left( 0 \right)}}{d\tau } = 0 \hfill \\ \end{aligned} \right.$$
(72)

The solutions of Eqs. (72) are

$$a_{11} (\tau ) = \frac{{f_{2} X^{3} \left( {cos3\tau - \cos \tau } \right)}}{{32\left( {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} } \right)}}$$
(73)
$$a_{21} (\tau ) = \frac{{g_{2} X^{3} \left( {cos3\tau - \cos \tau } \right)}}{{32\left( {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} } \right)}}$$
(74)

Therefore, the first approximate solution of Eqs. (45) and (46) can be written as follows:

$$a_{1} \left( {x,t} \right) = X_{1} cos\tau + \frac{{f_{2} X^{3} \left( {cos3\tau - \cos \tau } \right)}}{{32\left( {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} } \right)}}$$
(75)
$$a_{2} \left( {x,t} \right) = X_{2} cos\tau + \frac{{g_{2} X^{3} \left( {cos3\tau - \cos \tau } \right)}}{{32\left( {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} } \right)}}$$
(76)

The displacements of the nanotubes to be expressed as for simple simply support as

$$w_{1} \left( {x,t} \right) = \left( \begin{aligned} X_{1} cos\tau + \frac{{f_{2} X^{3} \left( {cos3\tau - \cos \tau } \right)}}{{32\left( {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} } \right)}} \hfill \\ \hfill \\ \end{aligned} \right)\sqrt {\frac{I}{A}} sin\left( {\frac{n\pi x}{l}} \right)$$
(77)
$$w_{2} \left( {x,t} \right) = \left( \begin{aligned} X_{2} cos\tau + \frac{{g_{2} X^{3} \left( {cos3\tau - \cos \tau } \right)}}{{32\left( {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} } \right)}} \hfill \\ \hfill \\ \end{aligned} \right)\sqrt {\frac{I}{A}} sin\left( {\frac{n\pi x}{l}} \right)$$
(78)

while for clamped–clamped are given as

$$w_{1} \left( {x,t} \right) = \left( \begin{aligned} X_{1} cos\tau + \frac{{f_{2} X^{3} \left( {cos3\tau - \cos \tau } \right)}}{{32\left( \begin{aligned} \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} \hfill \\ \, + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }} \hfill \\ \end{aligned} \right)}} \hfill \\ \hfill \\ \end{aligned} \right)\sqrt {\frac{I}{A}} \left\{ {\begin{array}{*{20}l} {\left( {\cosh \left( {\frac{\beta x}{L}} \right) - \cos \left( {\frac{\beta x}{L}} \right)} \right)} \hfill \\ { - \left( {\frac{\sinh \beta + \sin \beta }{\cosh \beta - \cos \beta }} \right)\left( {\sinh \left( {\frac{\beta x}{L}} \right) - sin\left( {\frac{\beta x}{L}} \right)} \right)} \hfill \\ \end{array} } \right\}$$
(79)
$$w_{2} \left( {x,t} \right) = \left( \begin{aligned} X_{2} cos\tau + \frac{{g_{2} X^{3} \left( {cos3\tau - \cos \tau } \right)}}{{32\left( {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} } \right)}} \hfill \\ \hfill \\ \end{aligned} \right)\sqrt {\frac{I}{A}} \left\{ {\begin{array}{*{20}l} {\left( {\cosh \left( {\frac{\beta x}{L}} \right) - \cos \left( {\frac{\beta x}{L}} \right)} \right)} \hfill \\ { - \left( {\frac{\sinh \beta + \sin \beta }{\cosh \beta - \cos \beta }} \right)\left( {\sinh \left( {\frac{\beta x}{L}} \right) - sin\left( {\frac{\beta x}{L}} \right)} \right)} \hfill \\ \end{array} } \right\}$$
(80)

And for clamped–simply supports, we have

$$w_{1} \left( {x,t} \right) = \left( \begin{aligned} X_{1} cos\tau + \frac{{f_{2} X^{3} \left( {cos3\tau - \cos \tau } \right)}}{{32\left( {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} } \right)}} \hfill \\ \hfill \\ \end{aligned} \right)\sqrt {\frac{I}{A}} \left\{ {\begin{array}{*{20}l} {\left( {\cosh \left( {\frac{\beta x}{L}} \right) - \cos \left( {\frac{\beta x}{L}} \right)} \right)} \hfill \\ { - \left( {\frac{\cosh \beta - \cos \beta }{\sinh \beta - \sin \beta }} \right)\left( {\sinh \left( {\frac{\beta x}{L}} \right) - sin\left( {\frac{\beta x}{L}} \right)} \right)} \hfill \\ \end{array} } \right\}$$
(81)
$$w_{2} \left( {x,t} \right) = \left( \begin{aligned} X_{2} cos\tau + \frac{{g_{2} X^{3} \left( {cos3\tau - \cos \tau } \right)}}{{32\left( {\begin{array}{*{20}l} {\sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) + \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }}} \hfill \\ { + \sqrt[3]{{\left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right) - \sqrt {\left( {\frac{{\lambda_{3} }}{{3\lambda_{1} }} - \frac{{\lambda_{2}^{2} }}{{9\lambda_{1}^{2} }}} \right)^{3} + \left( {\frac{{ - \lambda_{2}^{3} }}{{27\lambda_{1}^{3} }} + \frac{{\lambda_{2} \lambda_{3} }}{{6\lambda_{1}^{2} }} - \frac{{\lambda_{4} }}{{2\lambda_{1} }}} \right)^{2} } }} - \frac{{\lambda_{2} }}{{3\lambda_{1} }}} \hfill \\ \end{array} } \right)}} \hfill \\ \hfill \\ \end{aligned} \right)\sqrt {\frac{I}{A}} \left\{ {\begin{array}{*{20}l} {\left( {\cosh \left( {\frac{\beta x}{L}} \right) - \cos \left( {\frac{\beta x}{L}} \right)} \right)} \hfill \\ { - \left( {\frac{\cosh \beta - \cos \beta }{\sinh \beta - \sin \beta }} \right)\left( {\sinh \left( {\frac{\beta x}{L}} \right) - sin\left( {\frac{\beta x}{L}} \right)} \right)} \hfill \\ \end{array} } \right\}$$
(82)

4 Results and discussion

Using the material and geometric parameters of the carbon nanotubes, E = 1.1 TPa, ρ = 1300 kg/m3, l = 45 nm, the outer diameter do = 3 nm, and the thickness of each layer, h = 0.68 nm, the frequency ratio against non-dimensional maximum amplitude for the nonlinear vibrations of SWCNTs and DWCNTs in a thermal and magnetic environment are given in Figs. 3, 4, 5, 6, 7, 8, 9 and 10. The results of the simulation and the effects of various parameters on the frequency ratio of nonlinear vibrations of embedded single- and double-walled carbon nanotubes in a thermal and magnetic environment are presented and discussed.

Fig. 3
figure 3

Frequency ratio versus non-dimensional amplitude for SWCNT under various boundary conditions

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Fig. 4
figure 4

Frequency ratio versus non-dimensional amplitude for DWCNT under various boundary conditions

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Fig. 5
figure 5

Effect of Winkler constant (k1) on amplitude–frequency ratio curve for SWCNT

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Fig. 6
figure 6

Effect of Winkler constant (k1) on amplitude–frequency ratio curve for DWCNTs

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Fig. 7
figure 7

Effect of nonlinear spring constant constant (k3) on amplitude–frequency ratio curve for DWCNTs

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Fig. 8
figure 8

Effect of Van der waals force on amplitude–frequency ratio curve for DWCNTs

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Fig. 9
figure 9

Effect of temperature on amplitude–frequency ratio curve on outerwall of DWCNTs

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Fig. 10
figure 10

Effect of magnetic force strength on amplitude–frequency ratio curve on outerwall of DWCNTs

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4.1 Effects of boundary conditions on the frequency ratio of the carbon nanotubes

Figures 3 and 4 show the effects of boundary conditions on the frequency ratio for the nonlinear vibrations of SWCNTs and DWCNTs, respectively in thermal and magnetic environment (k1 = 107 N/m2, k3 = 108 N/m2, T = 40 K, Hx = 107A/m, eoa = 1.5 × 10−9 and c1 = c2 = c3 = 0.3 × 1012 N/m2). As it is depicted in the figures, the frequency ratio for all boundary conditions decreases as the number of wall increases. This is due to the fact that carbon nanotubes generally have weak shear interactions between adjacent tubes and become more predominant as the number of walls increases. It could therefore be inferred that in an application where linear vibration is preferred for system stability, DWCNTs will perform better than SWCNTs of the same geometry and size. Also, the figures show that for both the SWCNTs and DWCNTs, the frequency ratio is highest for clamped simple supported and least for clamped–clamped supported. This establishes that the clamped–clamped supported system provided the best grip (support) for the nanotubes and this can be used to control nonlinear vibration of the system.

4.2 Effects of spring stiffness (k1) on the frequency ratio of the carbon nanotubes

The impacts of the spring stiffness (k1) on the dimensionless frequency ratio of the single- and double-walled carbon nanotubes in thermal and magnetic environment are shown in Fig. 5 and 6. It is depicted that he frequency ratio decreases with increases in the value of spring constant (k1) for CNTs. This is because, the linear frequency increases as the value spring constant increases. At large value of k1 (k1 = 1010 N/m2), the vibration of the system becomes stable and this can be used as good measure in controlling nonlinear vibration of the system.

4.3 Effects of nonlinear spring stiffness (k3) on the frequency ratio of the carbon nanotubes

Figure 7 shows the effect of nonlinear spring stiffness (k3) on the frequency ratio of outer walled of embedded DWCNTs in a thermal and magnetic environment. It could be seen that the frequency ratio increases with increases in the value of the nonlinear spring constant. This is because the nonlinear frequency increases as the value of the nonlinear spring constant increases without producing any effect on the linear frequency. The value of nonlinear spring constant should be kept as low as possible since it makes the vibration of the system to be nonlinear and this can lead to the instability of the system.

4.4 Effects of Van der Waal force on the frequency ratio of the carbon nanotubes

Figure 8 presents the effects of Van der Waal force on the frequency ratio of the SWCNTs and DWCNTs, respectively, in a thermal and magnetic environment. It can be seen that when the coefficient of the van der Waals forces is zero i.e. c = 0 N/m2, it means a single-walled carbon nanotube with the same dimension and geometry with double-walled carbon nanotubes. The results reveals that the frequency ratio decreases as the number of walls increases. Increasing the number of walls can be used as a good parameter to control the nonlinear vibration of the system.

4.5 Effects of temperature on the frequency ratio of outer wall of DWCNTs

Figure 9 illustrates the influence of temperature on the frequency ratio on the outer wall of DWCNTs in a thermal and magnetic environment. The result presents that as the temperature increases, the frequency ratio decreases. This shows that increase in temperature leads to increase in the value of linear natural frequency of the system.

4.6 Effects of magnetic force strength on the frequency ratio of outer wall of DWCNTs

Figure 10 presents the impact of magnetic force strength on the dimensionless frequency ratio. From the figure, it is established that as the magnetic field strength increases, the vibration of the system approaches linear vibration and become linear at higher value of magnetic force strength, H = 109 A/m.

4.7 The linear and nonlinear vibration deflection-time curve of outer wall of DWCNTs

Figure 11 shows the comparison of the linear vibration with nonlinear vibration of the DWCNT. It could be seen in the figure that the discrepancy between the linear and nonlinear amplitudes increases and the vibration time progresses.

Fig. 11
figure 11

The linear and nonlinear vibration deflection-time curve of outer wall of DWCNTs

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5 Conclusion

In this work, nonlocal elasticity theory has been used to analyze the nonlinear vibrations of single and double-walled carbon nanotubes resting on two-parameter foundation in a thermal and magnetic environment. The effects of the various controlling parameters on the nonlinear to linear frequency ratio have been analyzed and discussed. The results established that the frequency ratio for all boundary conditions decreases as the number of walls increases from single to double. Also, it is established that the frequency ratio is highest for clamped–simple supported and least for clamped–clamped supported. Additionally, the results revealed that the frequency ratio decreases with increase in the value of spring constant (k1) temperature and magnetic field strength. This work will enhance the applications of carbon nanotubes in structural, electrical, mechanical and biological applications especially in a thermal and magnetic environment.