A.A.Yinusa,M.G.Sobamowo

Department of Mechanical Engineering,University of Lagos,Akoka,Lagos State,Nigeria

Keywords:

ABSTRACT Single-walled carbon nanotubes(SWCNTs)are receiving immense research attention due to their tremendous thermal,electrical,structural and mechanical properties.In this paper,an exact solution of the dynamic response of SWCNT with a moving uniformly distributed load is presented.The SWCNT is modelled via the theories of Bernoulli-Euler-thermal elasticity mechanics and solved using Integral transforms.The developed closed-form solution in the present work is compared with existing results and excellent agreements are established.The parametric studies show that as the magnitude of the pressure distribution at the surface increases,the deflection associated with the single walled nanotube increases at any mode whilst a corresponding increase in temperature and foundation parameter have an attenuating effect on deflection.Moreover,an increase in the Winkler parameter,as well as a decrease in the SWCNT mass increases its frequency of vibration.Furthermore,an increase in the speed of the external agent decreases the total external pressure as a result of the removal of dead loads.The present work is envisaged to improve the application of SWCNT as nanodevices for structural,electrical and mechanical systems.

1.Introduction

Following the discovery of carbon nanotube(CNT)by Iijima,considerable amount of studies on CNTs with multi branched arrangements have been studied[1-4].With the functionality of CNTs in transistors and diodes,several investigations on different aspects of carbon nanotube including vibrations of micro-resonator excited by electrostatic and piezoelectric actuations have been carried out.Other interesting research areas on CNT include beams,nano-wires,nano-rods and nano-beam so as to specifically understand and achieve their area of best fit.To achieve the best fit,different established beam models have been employed with effective dynamic ranges obtained within the scope of studied structures[5-22].To this end,various authors including Abdel-Rahman,Hawwa,Hajnayeb,and Belhadj performed a vibration and instability studies of a DWCNT using a nonlinear model and considering an electrostatic actuation as an external excitation agent[23-26].In these works,a dual-walled CNT(DWCNT)is situated and conditioned to a direct and alternating voltage and different behaviours of the nanotube is recorded as the exciting agent is varied.These studies further investigate the bifurcation point of the DWCNT and conclude that both walls of the nanotube possess the same vibration frequency under the two considered resonant conditions.

Nevertheless in Ref.[26],Belhadj et al.present a pinned-pinned supported SWCNT employing non-local theory of elasticity and obtained natural frequency up to the third mode.Their work establishes how the high frequency obtained in their work may be harnessed for optical applications.Moreover,Lei et al.[27]studied the dynamic behaviour of DWCNT.Lei et al.employed the popular Timoshenko theory of beam to analyse the vibration of a moderately thick beam Asgharifard et al.derived different relationship in application to nanobeams that are graded and possess surface roughness[28].Wang developed a close form model for the aforementioned surface roughness effect for an unforced fluid conveying nanotubes and beams based on nonlocal theory of elasticity[29].The study further ascertains the significance of surface integrity for reasonably small thickness of the tubes.Sheikholeslami et al.[30],did an experimental investigation on the use of nano-refrigerants for enhancement of boiling heat transfer processes.Additionally,the consideration of nanofluids in compound tubular simulation[31,32],the use of innovative schemes for nanoparticles heat transfer analysis[33],application of nanoparticles to heat storage[34]as well as incorporation of nanoparticles in solidification processes[35]have been explicitly considered and presented.In an attempt to study the dynamics of these nanoparticles,several interesting works considering the modelling of CNTs as structures resting on or embedded in elastic foundations such as Winkler,Pasternak and Visco-Pasternak medium have been carried out.These led to the presentation of various dimensionalized nanotubes using different experimental or modelling approaches to justify the widespread application of SWCNTs[36-41].The reason for the selection of this nanoparticle in the current scope of study is because the nanoparticles under investigation are currently receiving immense research attention due to their tremendous thermal,electrical,structural and mechanical properties.The high strength to weight ratio is also a factor of importance.The excellent electrical property possessed by this nanotubes has also made them gain applications in the current design of Nano-diodes,Nano-transistors as well as Nano-switches.However,the vibration of this nanoparticle when excited is in the order of GHz and THz.In view of the abovementioned,the present work is aimed at understanding the dynamic response of a SWCNT under uniform external pressure.The developed models are solved using exact method via Integral transform.The results of effect of external uniform pressure as well as other parameters are presented,interpreted and discussed.

2.Problem formulation

Consider a SWCNT with a uniformly distributed surface pressure as shown in Fig.1(a)and(b).

To simplify the problem formulation,the following assumptions are made as:

1.The SWCNT is modelled according to Euler-Bernoulli beam

2.The CNT is homogenous and possesses constant properties.

3.The CNT has a constant cross-sectional area,and therefore constant moment of inertia.

4.The exciting agent is a uniform pressure distribution at the SWCNT surface governed by:

5.The SWCNT is supported at both ends.

Case 1.SWCNT with static or non-moving uniformly distributed pressure at the surface

By incorporating the above assumptions in the classic Euler-Bernoulli beam model,the vibration of Fig.(1a)can be described by the model:

on substituting Eq.(1)in Eq.(3)and simplifying the resulting expression,the model becomes:

3.Methods of solution using Integral transforms

In this present study,the linear transient model in Eq.(4)will be solved exactly using Laplace and Fourier transform since the resulting model obtained in Eq.(4)contains both spatial and temporal part.The temporal part is solved using the Laplace transform,whilst the spatial terms are solved using the Fourier transform.

3.1.Basic principle of the Integral transforms

Fourier transforms are generally used to transform second and higherorder spatial derivatives.Although,the inversion of Fourier transform is usually easier than the inverse of Laplace transform,there are some functions which do not have Fourier transform as they have in Laplace transform.

Fourier transform could be used to solve problems in.

i.Finite Domains(0≤x≤L).

ii.Semi-Infinite Domain,and（0≤x≤∞.-∞≤x≤0）.

iii.Infinite Domains unlike Laplace transform（-∞≤x≤∞）.

The transform of a cosine finite Fourier function is:

Operational properties

Generally,

The Fourier finite cosine inverse transform is

Fig.1.Schematic of the SWCNT with Static and Moving uniformly distributed pressure at the surface.

The Fourier finite sine transform of a function is defined as

using the initial condition and applying Fourier transform to the spatial terms,we have

Generally,we have

if the boundary value problem is a 2nd-order derivative and extends over a finite domain 0≤x≤L and has 1st-type boundary condition at both ends［f（0=f0at x=0，f（L）=fLat x=L）］,then the Fourier finite sine transform can be used to transform the 2nd-order derivatives.

Therefore,the Fourier finite cosine inverse transform is

3.2.Method of solution:Laplace and Fourier transform

Recall that the linear transient governing equation as shown in Eq.(4)may be expressed as,

Applying the boundary conditions and grouping like terms,

3.3.Determination of the natural frequency

The natural frequencies can be determined from the poles of Eq.(14)

The displacement of the nanotube will be realized by finding inverse Laplace Transform to the already established Eq.(13)which is

Subject to the pinned-pinned conditions:Applying Laplace transform on the temporal term,gives and the Fourier inverse of Eq.(14)becomes:

Therefore,Eq.(17)is the desired exact solution that represents deflection of the SWCNT.

Case 2.SWCNT with a moving uniformly distributed pressure at the surface

The vibration of Fig.(1b)can be described by the model:

3.4.Determination of the natural frequency

If the mass of the moving external agent is also considered,the external agent becomes The second term of Eq.(19)after neglecting the term with negligible impact from the application point of view may be expressed as;

Where the first term on the right hand side is the acceleration in the

The natural frequencies can be determined from the poles of Eq.(14)

The displacement of the nanotube will be realized by finding inverse Laplace Transform to the already established Eq.(22)which is

deflection direction and the second in the part curvature due to centripetal acceleration.Substituting Eqs.(19-20)into(18)and collecting like terms yields;and the Fourier inverse of Eq.(24)becomes:

Equation(25)is the desired exact solution that represents deflection of the SWCNT under moving external load.

3.5.Determination of the SWCNT bending moment and shear force

The Bending moment is related to the SWCNT deflection by

Similarly,the Shear force is related to the SWCNT deflection by

Substituting Eq.(25)into Eqs.(26-27),we have

It is important to know that the Bending moment and Shear force analysis are best performed at the mid-point of the SWCNT span.

4.Results and discussion

4.1.Convergence criteria

Fig.(2)illustrates the convergence criteria based on the number of iterations(i)in the closed-form solution for the deflection of the simply supported SWCNT.The computational time associated with each iteration is shown in Table 1.It is obvious that at i=3,the solution has already converged,hence,extending the iteration above three will only increase the computational time and cost with negligible effect on the improvement of the established solution.

4.2.Effect of the modal number on the deflection of the SWCNT

Fig.3 displays the simply supported SWCNT deflection along its length for the first five mode shapes.Critical visualization shows that as the modal number associated with the kernel that defines the boundary condition increases,the stability of the SWCNT under study decreases as a result of an increase in the cycles covered by the SWCNT for the same length.These occur because the kernel depends on the modal number.

4.3.Influence of surface uniform pressure on the SWCNT deflection

In Figs.4-7 the influence of magnitude of uniformly distributed external pressure on the deflection of the SWCNT for first four modes is illustrated.Increasing external pressure results in a corresponding increase in the deflection of the SWCNT.The distributed surface pressure when converted into a force will act at the center of the SWCNT span.At that point,the shearing force will be zero while bending moment will be maximum.This maximum moment induces large deflection in the SWCNT that continues to increase as the external uniform pressure increases.

Table 1Convergence criteria based on the number of iteration(i)in the close form solution for the deflection of the SWCNT.

4.4.Effect of change in temperature on the deflection of the SWCNT

Figs.8-11 depict the influence of change in temperature on the deflection of the SWCNT for the first four modes.The SWCNT deflection decreases as the temperature change at room condition increase as a result of thermal expansion coefficient being negative when the SWCNT is considered at room temperature and positive when temperature is very high.Furthermore,at low value of temperature,the flexural rigidity of the SWCNT decreases.This leads to an increase in the flexibility of the SWCNT and consequently increases its deflection.

4.5.Effect of pre-tension on the deflection of the SWCNT

Figs.12-15 depict the influence of pre-tension on deflection of the SWCNT for the first four modes.A careful study of dynamic analysis of the SWCNT shows that as the magnitude of the pre-tension is increased,the deflection decreases.This is because the pre-tension as modelled in the governing equation tends to annul the initial static deflection induced in the SWCNT as a result of the reaction from the support.

4.6.Effect of foundation parameter on the deflection of the SWCNT

Fig.2.Convergence criteria.

Fig.3.SWCNT modes with pinned-pinned condition.

Fig.4.Influence of uniformly distributed pressure on the deflection.

Fig.5.Influence of uniformly distributed pressure on the deflection of the SWCNT for mode 1 of the SWCNT for mode 2.

Fig.6.Influence of uniformly distributed pressure on the deflection.

Fig.7.Influence of uniformly distributed pressure on the deflection of the SWCNT for mode 3 of the SWCNT for mode 4.

Fig.8.Influence of change in temperature on the deflection.

Fig.9.Influence of change in temperature on the deflection of the for mode 1 of the SWCNT for mode 2.

Fig.10.Influence of change in temperature on the deflection.

Fig.11.Influence of change in temperature on the deflection of the SWCNT for mode 3 of the SWCNT for mode 4.

Fig.12.Influence of pre-tension on the deflection.

Fig.13.Influence of pre-tension on the deflection of the SWCNT for mode 1 of the SWCNT for mode 2.

Fig.14.Influence of pre-tension on the deflection.

Fig.15.Influence of pre-tension on the deflection of the SWCNT for mode 1 of the SWCNT for mode 2.

Fig.16.Effect of foundation parameter on the deflection.

Fig.17.Effect of foundation parameter on the deflection of the SWCNT for mode 1 of the SWCNT for mode 2.

Fig.18.Effect of foundation parameter on the deflection.

Fig.19.Effect of foundation parameter on the deflection of the SWCNT for mode 3 of the SWCNT for mode 4.

Figs.16-19 depict the influence of Winkler foundation parameter on the deflection of the single walled carbon nanotube for the first four modes.An increase in the Winkler parameter makes the foundation of the SWCNT stiffer and consequently attenuates its deflection.

4.7.Dynamic response of the SWCNT

Figs.20-23 depict the three dimensional dynamic response associated with the SWCNT for the first four modes.A critical assessment shows that it is possible to track the behaviour of the CNT in any instance.Fig.23 shows that the system at forth mode is on its route to chaos and as such requires adequate damping if the mode and exciting agent are to be maintained.The dynamic analysis is important as it helps in the quick monitory and adjustment of the CNT during application.

The present study agrees excellently with the established result of Cos?kun et al.beam model as shown in Table 2:

4.8.Effect of modal number and length on the frequency of the SWCNT

Fig.20.Dynamic response of the SWCNT.

Fig.21.Dynamic response of the SWCNT for mode 1 for mode 2.

Fig.22.Dynamic response of the SWCNT.

Fig.23.Dynamic response of the SWCNT for mode 3 for mode 4.

Table 2Comparison of present study with Cos?kun et al.,2011[40]Exact method for pinned-pinned condition.

Fig.24.Effect of modal number and length on the frequency of the SWCNT.

Fig.24 depicts the influence of modal number and length on the frequency of the SWCNT.A careful study helps visualizes the effect of these two important parameters on stability of the SWCNT.The frequency of SWCNT which is a vital parameter in the study of the SWCNT stability reaches some THz and continues to increase as the modal number increases.This astonishing property enables SWCNT to offer exceptional optical and mechanical properties although there is always a need to dampen the frequency to an application limit.However,an antonymous effect is realized as the length of the SWCNT increases.These two parameters as a result of their tremendous effects on frequency may be used to annul the effect on each other when one of them is desired based on the requirement of the engineering design and applications.

Fig.25.Effect of foundation parameter on the frequency.

Fig.26.Effect of SWCNT mass on the frequency of the SWCNT of the SWCNT.

4.9.Influence of foundation variable and mass on SWCNT frequency

Figs.25-26 depict the impact of foundation variable and mass on the excitation frequency of the SWCNT.From the plots,it is obvious that an increase in foundation variable or parameter as well as a reduction in the mass of the single walled carbon nanotube increases its frequency.For both parameters,extreme values should be avoided to prevent instability for very low values of the foundation parameter and over excitation for very high values.Moderate and intelligent choice of the mass of the SWCNT may also be used to annul the above mentioned effects.

4.10.Effect of pre-tension and temperature on the frequency of the SWCNT

Fig.27.Effect of pre-tension on the frequency.

Fig.28.Influence of temperature change on the frequency of the SWCNT of the SWCNT.

Figs.27-28 depict the influence of pre-tension and temperature on the dimensional frequency of operation of the present study SWCNT.As the value of pre-tension increases,the operating frequency of vibration of the SWCNT also increases as a result of the decrease in the initial deflection due to the mass of the SWCNT which consequently increases the operational and dimensional frequency.It is worth noting that the convention of the pre-tension is very important.If the convention of the pre-tension in the governing equation of the SWCNT in question is reversed,the tension becomes compressive.This increases the initial deflection and consequently increases the frequency and instability of the SWCNT.However,change in temperature has negligible impact on the SWCNT frequency for the range used.One of the reason is that the carbon nanotube is not flow induced.

4.11.Effect of moving mass velocity on the frequency of the SWCNT

Fig.29 depicts the influence of external mass velocity on the dimensional frequency of operation of the present study SWCNT.As the value of velocity increases,the operating frequency of vibration of the SWCNT decreases.This is because an increase in the velocity of the moving mass causes the centripetal force to increase which consequently decreases the total external pressure.Since an increase speed reduces the external pressure on the SWCNT,static deflection will be annulled due to the absence of dead load and hence the reason for the reduction in frequency.

Fig.29.Effect of external mass velocity on the frequency of the SWCNT.

Fig.30.Shear force Diagram of the SWCNT.

Fig.31.Shear force Diagram of the SWCNT for mode 1 for mode 2.

Fig.32.Bending moment Diagram of the SWCNT.

Fig.33.Bending moment Diagram of the SWCNT for mode 1 for mode 2.

Fig.34.Stability validation of the SWCNT.

The dimensional frequency model obtained in the present study is also reduced to the level of Belhadj et al.SWCNT model and a superb agreement is obtained as shown in Fig.34:

4.12.The shear force and bending moment of the SWCNT

Figs.30-33 depict the three dimensional Shear force and bending moment diagram of the SWCNT for the first two modes.A critical assessment shows that it is possible to track the positions of maximum shear and maximum moment for proper design of the CNT device.The dynamic analysis is important as it helps in the quick monitory and adjustment of the CNT during application.

4.13.Validation

Fig.34 depicts the validation of this work.In order to adequately validate the stability solution obtained in the present study,the dimensional frequency model is also reduced to the level of Belhadj et al.[26],SWCNT model and a superb agreement is obtained as shown.

5.Conclusion

The present paper presents an analytical investigation of the dynamic response of a SWCNT exposed to an external uniform pressure.The developed models are solved using Integral Transform.The exact solution of the present study is reduced to the level of Cos?kun et al.[40],beam model with excellent agreement established.The natural frequency obtained in the present study is also reduced to the level of Belhadj et al.

[26],with excellent agreement equally established.The present study also establishes that the Integral transform produces reliable result and is efficient for the dynamic response problem.From the study,it is established that as the magnitude of the pressure distribution at the surface of the SWCNT increases,the deflection of the single walled carbon nanotube increases at any mode.Moreover,a corresponding increase in temperature and foundation parameter have an attenuating effect on the deflection of the SWCNT.Furthermore,an increase in the foundation parameter as well as a decrease in the SWCNT mass increases the frequency of vibration of the nanotube.The present work is envisaged to improve the applicability of SWCNT for structural,electrical and mechanical systems.

Declaration of competing interest

None.

Nomenclature

P（x） Pressure distribution

ACNTArea of the SWCNT

ICNTSWCNT Inertia

EICNTFlexural rigidity

t Time coordinate

T Axial pre-tension

Fc,FsFourier cosine and sine functions

n Modal number

PoMagnitude of pressure distribution

LCNTSWCNT length

E Young modulus

x Space coordinate

M Mass of SWCNT

K Foundation parameter

i Number of iterations

V Velocity

Greek letters

μ Pressure coefficient

v Poisson ratio

χ Deflection of the SWCNT

ω Natural frequency