A simplified-nonlocal model for transverse vibration of nanotubes acted upon by a moving nanoparticle

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T E C H N I C A L P A P E R

A simplified-nonlocal model for transverse vibration of nanotubes acted upon by a moving nanoparticle

Ali Nikkhoo¹^•Saber Zolfaghari¹^•Keivan Kiani²

Received: 17 March 2017 / Accepted: 12 August 2017 / Published online: 30 August 2017 ÓThe Brazilian Society of Mechanical Sciences and Engineering 2017

Abstract This study provides a simplified solution for estimating the dynamic response of a single-walled carbon nanotube when excited by a moving nanoparticle. At first, the strong form of the equation of motion for a nonlocal Rayleigh nanotube is deduced, and the inertia effect of a moving nanoparticle along a nanobeam is then considered.

For obtaining a weak form of the above nonlocal model, we use the Galerkin method, where the test functions are a set of orthogonal polynomials generated from a polynomial satisfying given boundary conditions. This process leads to a second-order differential equation which for a moving load the matrix coefficients are time dependent. In the state-space formulation, the forced response depends upon a transition matrix that can be locally approximated by the matrix exponential by assuming that the coefficients are locally constant. The normalized frequencies for a moving force are calculated and compared to those obtained in previous studies, and good agreement between them was observed. After acquiring the dynamic responses of a nanotube for a wide range of velocities and weights of moving nanoparticles, as well as for the nonlocal effects on

a nanobeam, a nonlinear regression analysis is adapted to estimate the response of a nanobeam according to an analogous classical Rayleigh beam. These equivalent results in three multipliers (a, b, and c) are functions of kinetic parameters and nonlocal effects. Due to the nor- malization of the variables, these multipliers can be used for various types of beam-like structures in both the nano- and macro-domains. The accuracy of these coefficients is evaluated using the results gained by the analytical solution. This paper offers a remedy for a time-consuming process by means of some simple substitutions.

Keywords Single-walled carbon nanotubeNonlocal Rayleigh beam theoryMoving load Moving mass Conversion coefficient

1 Introduction

In recent years, much attention has been given to nanoscale technology. This interest was triggered by the discovery of single-walled carbon nanotubes (SWCNTs) [1]. These became very popular because of their unique mechanical, physical, and chemical properties. Since their discovery, applications of SWCNTs in fields such as drug and particle delivery systems [2], energy harvesting [3], nanosensors [4], and nano-electromechanical systems [5] were demonstrated experimentally. The dynamic analysis of an SWCNT when excited by a moving nanoparticle became important, especially in drug and particle delivery systems.

At the nanoscale, many aspects of carbon nanotubes have so far been studied. For instance, Wang and Varadan [6] by modeling a nonlocal continuum model, the vibration of SWCNTs and double-walled carbon nanotubes (DWCNTs) was compared to the results of published experimental Technical Editor: Ka´tia Lucchesi Cavalca Dedini.

& Keivan Kiani

[email protected]; [email protected] Ali Nikkhoo

[email protected] Saber Zolfaghari [email protected]

1 Department of Civil Engineering, University of Science and Culture (USC), P.O. Box 13145-871, Tehran, Iran

2 Department of Civil Engineering, K.N.Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran

https://doi.org/10.1007/s40430-017-0892-8

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reports. Numerical simulation was conducted to show small- scale effects on free dynamic analysis of SWCNTs and DWCNTs with different lengths and diameters. Reddy and Pang [7] investigated the effect of nonlocality on the deflections, buckling loads, and natural frequencies of carbon nanotubes. Eltaher et al. [8] employed finite-element method to study nonlocal vibrations of nanobeams via Euler–Ber- noulli beam theory. The influences of the nonlocal parameter, slenderness ratio, rotary inertia, and boundary conditions on the transverse dynamic behavior of the nanostructure were explained. In a more comprehensive work, using finite-element method, Eltaher et al. [9] examined free vibrations of nanobeams accounting for both surface energy effect and nonlocality. The capability of the suggested model in ana- lyzing complex geometry was also illustrated. By employing differential quadrature methodology, static bending of microtubules was studied by Civalek and Demir [10] using a nonlocal Euler–Bernoulli beam model. The role of the small- scale parameter on the resulted deflection and nonlocal bending moment of the nanostructure for various boundary conditions and load patterns was addressed. Kiani and Mehri [11] studied mechanical vibrations of SWCNTs under excitation of a moving nanoparticle. The obtained results revealed the necessity of using nonlocal shear deformation beam theories for very stocky nanotubes excited by moving nanoparticles with a low velocity. The free vibration of an SWCNT was studied by Behera and Chakraverty [12] exploiting simple and orthogonal polynomials based on Euler–Bernoulli nonlocal beam theory. The results showed the efficiency of the exploited polynomials in describing the vibration behavior of the short nanotubes, where the effects of small-scale, transverse shear deformation and rotary inertia are significant.

S¸ims¸ek [13] explored the forced vibration of an SWCNT under a harmonic load. The SWCNT was modeled as an Euler–Bernoulli nanobeam. The results show that the dynamic response increases with the increase in the nonlocal parameter; thus, the local theories are insufficient in the nanodomain. S¸ims¸ek [14] studied the effect of nonlocality on the forced vibration of an elastically connected DWCNT system under a moving nanoparticle. The undertaken study explained that the classical beam models are unsuitable for modeling carbon nanotubes and the velocity of a nanoparticle, and the stiffness of elastic layer has an extensive effect on the dynamic behavior of DWCNTs. Eltaher et al. [15] also investigated static, buckling, and dynamic behavior of nano- scaled beams via a higher order gradient model in the context of nonlocal continuum-field theory of Eringen. The effec- tiveness of the proposed model was then presented for various boundary conditions.

The inertia effects of moving nanoparticles were modeled in the nanoscale domain by Kiani [16], in which longitudinal and transverse vibrations of an SWCNT under excitation of a moving nanoparticle were examined, taking into account

both inertia and nonlocal effects. It was displayed that by increasing the mass of the moving nanoparticle, the dynamic responses increase, as well as the contact force. In addition, the possibility of separation between the nanoparticle and the nanotube becomes greater as the size of the moving mass increases. Kiani [17] analyzed an SWCNT by taking into account both longitudinal and transverse inertial effects of the moving nanoparticle and the friction force. The given study showed that SWCNT’s nonlinear analysis is required when the nanotube is traversed by nanoparticles with large masses and high velocities.

Most of the above-mentioned works have been developed via a simple version of the nonlocal elasticity theory of Eringen [18]. Recently, some of researchers have been interested in working with the original integral form of the nonlocal governing equations [19–24]. The main feature of such a complete version is that the paradoxial results of can- tilevered nanobeams via previous simple version would be resolved [19,20]. Herein, we develop a nonlocal-regression- based model using existing common nonlocal models for vibration of nanotubes under a moving nanoparticle. Devel- opment of more advanced nonlocal-integro-based models as well as corresponding regression models could be considered for future works.

At the macro-scale, there have been numerous studies on the moving load case [25–29]. However, several published articles have considered the inertia effects [30–39].

Unfortunately, the proposed methods in all of the afore- mentioned research required time-dependent complex mathematical calculations. In contrast, Nikkhoo et al. [40]

proposed a simplified solution in which a simple formula obtained from nonlinear regression analysis converts the moving load cases to the moving mass ones for classical Euler–Bernoulli beams.

In this paper, dynamic analysis of an SWCNT traversed by a moving nanoparticle was conducted based on nonlocal Rayleigh beam theory. The inertial effect is considered in the theoretical formulation of the problem. A formula comprising all parameters was achieved. This formula can easily and very quickly convert the moving load cases to a moving mass case. The resultant values can then be converted to the nonlocal moving nanoparticle by taking into account the small-scale effect. In addition, the accuracy of the regression analysis has been discussed.

2 Problem modeling

2.1 Mathematical model description

Let us consider an SWCNT subjected to a moving nanoparticle of mass weightMgand constant velocity ofv (Fig.1). The SWCNT is restrained by two springs, one

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torsional and one transversal, at each end. The springs’

constants areK_zandK_y, respectively, and the values ofK_z and Ky are demonstrated in Table 1. The SWCNT is a

cylindrical tube of diameter d and thickness t_b. The following assumptions are made in this study: (1) the SWCNT is both geometrically and materially linear with Young’s modulus ofEb. (2) The vibration of nanotube is modeled as a nonlocal Rayleigh nanotube. (3) The initial conditions of the nanotube are zero (the nanotube is at rest condition), and the moving nanoparticle enters from the left end of the nanotube. (4) The nanoparticle would be in contact with nanotube during excitation.

2.2 Equation of motion by considering nonlocal continuum theory

Based on the nonlocal continuum theory of Eringen [18], at an arbitrary point x of a continuum, the nonlocal stress tensor rij is related to the local stress tensor tij by

½1 ðe0aÞ²r²rijðxÞ ¼tijðxÞ, wherea is the internal characteristic length of the nanotube, r² is the Laplacian operator, ande0is a constant appropriate to each material.

The values ofe0are estimated for deriving dispersion curve obtained by atomic models from nonlocal continuum theory. Much research was conducted to see the effect on these parameters of the dynamic behavior of nanotubes. In general, the value ofe₀ais considered between 0 and 2 nm for dynamic analysis of nanotubes [16].

For an elastic homogeneous nanotube, the local stress field is t_xx¼E_bexx. Therefore, by substituting the local stress in equations mentioned before, one could obtain the nonlocal stress asrxx ðe0aÞ²rxx;xx ¼Ebexx. consequently, the relation between local and nonlocal stresses could be achieved using the nonlocal bending moment R

A_bzrxxdA and local bending moment R

Abz txxdA as

M_b^nl ðe0aÞ²M_b;xx^nl ¼ EbI_bo²wðx;tÞ

ox² : ð1Þ

The equation of motion by considering nonlocal effects and Rayleigh beam theory could be expressed as

q_b Ab

o²wðx;tÞ ot² Ib

o⁴wðx;tÞ ot²ox²

M_b;xx^nl ¼fðx;tÞ; ð2Þ whereq_bis the mass density of nanotube material, andI_bis the relevant moment of inertia andAbis the cross-sectional area. Material damping has been disregarded to simplify the analysis. After substituting Eq. (2) into Eq. (1) and Fig. 1 Schematic representation of an SWCNT under a moving

nanoparticle excitation

Table 1 K_zandK_y values for various boundary conditions Boundary conditions Simply supported Clamped Free

Kz 1 1 0

Ky 0 1 0

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taking the inertia effects into account, the external excitation can be written as

fðx;tÞ ¼M gd²wðxM;tÞ dt²

dðxxMÞ

¼M go²wðx;tÞ

ot² 2vo²wðx;tÞ

otox v²o²wðx;tÞ ox²

x¼vt

dðxxMÞ;

ð3Þ wherexM denotes the location of the moving nanoparticle (i.e., x_M¼vt); by combining Eqs. (1), (2), (3), the governing equation is provided by

q_b Ab

o²wðx;tÞ ot² Ib

o⁴wðx;tÞ ot²ox²

ðe0aÞ²q_b Ab

o⁴wðx;tÞ ot²ox² Ib

o⁶wðx;tÞ ot²ox⁴

þEbIb

o⁴wðx;tÞ ox⁴

¼M

"

go²wðx;tÞ

ot² 2vo²wðx;tÞ

x¼vt

dðxxMÞ ðe0aÞ² o²

ox² go²wðx;tÞ

ot² 2vo²wðx;tÞ

x¼vt

dðxxMÞÞ

# :

ð4Þ Equation (4) represents the strong form of motion equation for an SWCNT under a moving nanoparticle with the inertia effect consideration. If the value of the small-scale parameter (e0a) is set to zero, the motion equations of a classical beam under a moving mass excitation will be achieved.

3 Solving the governing equations 3.1 General solution

For solving the strong form of the equations of motion, the Galekin approximation method is utilized herein; thus, the unknown variablew(x,t) can be written as

wðx;tÞ ¼Xⁿ

j¼1

u_jðxÞAjðtÞ; ð5Þ

whereu_jðxÞis thejth assumed mode function of the beam andAjðxÞis the corresponding jth time-dependent ampli- tude. Now by substituting Eq. (5) into Eq. (4) the motion equation yields as:

q_b Ab

Xⁿ

j¼1

u_jðxÞAj;ttðtÞ Ib

Xⁿ

j¼1

d²u_jðxÞ dx² Aj;ttðtÞ

!

ðe0aÞ²q_b Ab

Xⁿ

j¼1

d²u_jðxÞ

dx² ðxÞAj;ttðtÞ Ib

Xⁿ

j¼1

d⁴u_jðxÞ

dx⁴ ðxÞAj;ttðtÞ

!

þEbIb

Xⁿ

j¼1

d⁴u_jðxÞ dx⁴ ðxÞAjðtÞ

¼M

"

gXⁿ

j¼1

u_jðxÞAj;ttðtÞ 2vXⁿ

j¼1

du_jðxÞ dx Aj;tðtÞ v²Xⁿ

j¼1

d²u_jðxÞ dx² ðxÞAjðtÞ

!

x¼vt

dðxxMÞ

ðe0aÞ²o²

ox² gXⁿ

j¼1

u_jðxÞAj;ttðtÞ 2vXⁿ

j¼1

du_jðxÞ dx Aj;tðtÞ v²Xⁿ

j¼1

d²u_jðxÞ dx² AjðtÞ

!

x¼vt

dðxxMÞ

!#

:

ð6Þ Multiplying both sides of Eq. (6) byu_iðxÞ and then inte- grating the resultant equation over the beam’s length and doing some simplifications:

q_b Ab

Xⁿ

j¼1

Z lb

0

u_iðxÞujðxÞdx

Aj;ttðtÞ

þIb

Xⁿ

j¼1

Z l_b 0

u_i;xðxÞuj;xðxÞdx

Aj;ttðtÞ

!

ðe0aÞ²q_b A_bXⁿ

j¼1

Z l_b 0

u_i;xxðxÞu_jðxÞdx

Aj;ttðtÞ

Ib

Xⁿ

j¼1

Z lb

0

u_i;xxðxÞu_j;xxðxÞdx

Aj;ttðtÞ

!

þEbIb

Xⁿ

j¼1

Z l_b 0

u_i;xxðxÞu_j;xxðxÞdx

AjðtÞ

¼M ( Xⁿ

i¼1

gu_iðvtÞ Xⁿ

j¼1

u_iðvtÞu_jðvtÞAj;ttðtÞ

2vXⁿ

j¼1

u_iðvtÞuj;xðvtÞAj;tðtÞ v²Xⁿ

j¼1

u_iðvtÞuj;xxðvtÞAjðtÞ

!

ðe0aÞ² gu_i;xxðvtÞ Xⁿ

j¼1

u_i;xxðvtÞujðvtÞAj;ttðtÞ

2vXⁿ

j¼1

u_i;xxðvtÞu_j;xðvtÞAj;tðtÞ

v²Xⁿ

j¼1

u_i;xxðvtÞu_j;xxðvtÞAjðtÞ

!)

; ð7Þ

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by rearranging the above equation in a matrix form:

½MðtÞAðtÞ þ ½CðtÞ€ AðtÞ þ ½KðtÞAðtÞ ¼_ fðtÞ; ð8Þ where

½M_ij¼ Z lb

0

q_bAbu_iðxÞu_jðxÞ þIbui;xðxÞuj;xðxÞ ðe0aÞ²q_bu_i;xxðxÞAbu_jðxÞ Ibu_j;xxðxÞ

dx

þMu_iðvtÞ ðe0aÞ²u_i;xxðvtÞ u_jðvtÞ;

ð9Þ

½K_ij¼ Z l_b

0

E_bI_b u_i;xxðxÞu_j;xxðxÞdx þMv²u_iðvtÞ ðe0aÞ²u_i;xxðvtÞ

u_j;xxðvtÞ;

ð10Þ

like velocity of

½C_ij¼2Mvu_iðvtÞ ðe0aÞ²u_i;xxðvtÞ

u_j;xðvtÞ; ð11Þ

½f_i¼Mgu_iðvtÞ ðe0aÞ²u_i;xxðvtÞ

: ð12Þ

It is worth mentioning that for the moving load case, the values of stiffness, mass, and damping matrices depend on the mechanical characteristic of the SWCNT only; therefore, they are independent of other parameters such as velocity of moving force, its location, and magnitude. In contrary, this is not the case in the moving mass loading; wherein, the inertial effects alter the system matrices components during the course of vibration.

There are several numerical methods to solve Eq. (8) in the time domain (e.g., Newmark-Beta and Crank–Nichol- son approaches); however, we use the matrix exponential approach presented in [41] and already used by the authors in the works [33,39,40,42]. To this end, Eq. (8) should be transferred to the state space as

XðtÞ ¼_ AðtÞXðtÞ þ EðtÞfðtÞ; ð13Þ in which

X¼ A

A_ _2p1;A¼ 0 I M¹K M¹C

2p2p

;E¼ 0

M¹

2pp

:

IfA¹ exists, Eq. (13) can be solved easily as follows:

Xðtkþ1Þ ¼A₁ðtkÞXðtkÞ þE₁ðtkÞfðtkÞ: ð14Þ where

A1ðtkÞ ﬃe^Aðt ^k^ÞDt^k;

E1ðtkÞ ﬃ ½A1ðtkÞ IA¹ðtkÞEðt kÞ;

in whichDtk¼tkþ1tk is an assumed time interval.

3.2 Generating assumed shape functions using characteristic orthogonal polynomials (COPs)

There are different ways to determine mode shape functions, Behera and Chakraverty [12] proposed COPs that can both satisfy natural and geometrical boundary conditions, and in combination with Gram–Schmidt procedure, it could produce higher mode shape functions. General first mode shape functions could be shown as

u₁ðxÞ ¼a0þa1xþa2x²þa3x³þa4x⁴ ð15Þ Z l_b

0

u₁ðxÞ²dx¼1: ð16Þ

After satisfying all natural and geometrical boundary conditions in Eq. (15), the assumed first mode shape function for all of the considered boundary conditions is shown in Table2.

Higher mode shape functions could increase the accuracy of the solution and can be obtained utilizing the Gram–Schmidt procedure as

u₂ðxÞ ¼ ðxP2Þu1ðxÞ: ð17Þ u_nðxÞ ¼ ðxPnÞu_n1ðxÞ Qnu_n2ðxÞ; n¼3;4;5;. . .

ð18Þ

Table 2 First mode shape function for various boundary conditions

u₁ðxÞ Boundary conditions

3 ffiffiffiffiffiffiffi

2170 p 31 ffiffiffi_l

b

p x

lb

2 _l^x

b

3

þ _l^x

b

4

S-S (simply supported)

3^pffiffiffiffi₇₀ ffiffiffil_b

p x

lb

2

2 _l^x

b

3

þ _l^x

b

4

C-C (clamped–clamped)

3^pffiffiffiffiffiffiffi₁₃₃₀

19 ffiffiffi_l

b

p x

lb

3 _l^x

b

3

þ2 _l^x

b

4

S-C (simple-clamped)

3 ffiffiffiffiffiffi

p130 52 ffiffiffi_l

b p 6 _l^x

b

2

4 _l^x

b

3

þ _l^x

b

4

C-F (clamped-free)

Table 3 Assumed static deflection and location of the reference point for various boundary conditions

Boundary conditions

Location of the reference point

Static deflection

S-S 0:5l_b P₀l³_b

48EbIb

C-C 0:5l_b P₀l³_b

192EbIb

S-C 0:5l_b 7P0l³_b

768EbIb

C-F lb P0l³_b

3E_bI_b

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where

P_n¼ Rlb

0 x½u_n1ðxÞ²WðxÞdx Rl_b

0 ½u_n1ðxÞ²WðxÞdx : ð19Þ Qn¼

Rlb

0 xu_n1ðxÞun2ðxÞWðxÞdx Rl_b

0 u_n1ðxÞu_n2ðxÞWðxÞdx : ð20Þ

W(x) represents the weight function which is assumed to be unit in this study. The obtained polynomials are always orthogonal, that is

Z l_b 0

u_aðxÞu_bðxÞWðxÞdx ¼0 ifa6¼b 6¼0 ifa¼b:

ð21Þ Table 4 Comparison of the first three dimensionless frequencies of the present study with those of another study

l_b ffiffiffiffiffiffiffiffiffiffiffi I_b=A_b

p l_b

d

l= 0 l= 0.1

Present study (3 Modes)

Ref.

[11]

Ref.

[11]

10 3.35 3.0688 3.0685 3.0685 3.0685 2.9975 2.9972 2.9972 2.9972

6.0918 5.7841 5.7817 5.7817 5.5743 5.3222 5.3202 5.3202

9.0835 8.0665 8.0400 8.0400 7.4807 6.8756 6.8587 6.8587

30 10.05 3.1334 3.1330 3.1330 3.1330 3.0605 3.0602 3.0602 3.0602

6.5605 6.2188 6.2161 6.2161 6.0217 5.7223 5.7199 5.7199

10.4925 9.2385 9.2056 9.2056 8.8093 7.8777 7.8530 7.8530

50 16.76 3.1388 3.1385 3.1385 3.1385 3.0658 3.0655 3.0655 3.0655

6.6066 6.2613 6.2586 6.2586 6.0661 5.7615 5.7591 5.7591

10.6630 9.3767 9.3429 9.3429 8.9799 7.9959 7.9701 7.9701

70 23.46 3.1403 3.1400 3.1400 3.1400 3.0673 3.0670 3.0670 3.0670

6.6196 6.2733 6.2706 6.2706 6.0786 5.7725 5.7701 5.7701

10.7125 9.4166 9.3825 9.3825 9.0299 8.0301 8.0040 8.0040

Table 5 Comparison of the first five normalized frequencies for a nanotube whenl_b=d¼10 for three mode numbers and various boundary conditions

Boundary conditions l¼0 l¼0:1

5 Modes 10 Modes 15 Modes Ref. [44] 5 Modes 10 Modes 15 Modes Ref. [44]

S-S 3.1400 3.1400 3.1400 3.1416 3.0670 3.0670 3.0670 3.0685

6.2733 6.2706 6.2706 6.2832 5.7725 5.7701 5.7701 5.7817

9.4166 9.3825 9.3825 9.4248 8.0301 8.0040 8.0040 8.0400

15.4899 12.4671 12.4671 12.5660 11.8307 9.8378 9.8378 9.9161

20.9772 15.5348 15.5162 15.7080 14.4202 11.3817 11.3706 11.5111

C-C 4.7271 4.7271 4.7271 4.7300 4.5908 4.5908 4.5908 4.5945

7.8362 7.8349 7.8349 7.8532 7.1209 7.1193 7.1193 7.1402

10.9512 10.9408 10.9408 10.9956 9.2129 9.2011 9.2011 9.2583

14.7753 14.0160 14.0160 14.1372 11.6019 10.9008 10.9008 11.0160

18.5650 17.0586 17.0535 17.2787 13.5050 12.3302 12.3245 12.5200

S-C 3.9243 3.9243 3.9243 3.9266 3.8185 3.8185 3.8185 3.8209

7.0549 7.0532 7.0532 7.0686 6.4510 6.4493 6.4493 6.4649

10.1892 10.1618 10.1618 10.2102 8.6281 8.6062 8.6062 8.6517

14.2446 13.2418 13.2417 13.3518 11.2324 10.3737 10.3737 10.4690

21.2152 16.2930 16.2851 16.4934 14.5568 11.8589 11.8519 12.0180

C-F 1.8747 1.8747 1.8747 1.8751 1.8787 1.8787 1.8866 1.8792

4.6876 4.6864 4.6864 4.6941 4.5410 4.5399 4.5397 4.5475

7.9028 7.8241 7.8241 7.8548 7.2010 7.1175 7.1175 7.1459

11.3501 10.9169 10.9168 10.9955 9.6288 9.1893 9.1893 9.2569

20.8631 13.9801 13.9772 14.1372 19.6028 10.8927 10.8893 11.0160

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4 Numerical results

In the numerical results, the maximum response of an SWCNT at a reference point subjected to a moving mass by consideration of the inertia effect is presented. To normalize the results and make them independent of material and geometrical properties, the maximum dynamic deflection of the beam was divided by the static deflection, as presented in Table3. This normalized value is generally called the dynamic amplification factor (DAF).

To calculate natural frequencies of the nanotube, we need free vibration of the undamped nanotube; thus, Eq. (8) could be written as

½MðtÞAðtÞ þ ½KðtÞAðtÞ ¼€ 0: ð22Þ Harmonic motion is assumed to represent the free-vibration solutions to the structure , so that½A€ is taken asx²½U, wherex² is an eigenvalue (with units of reciprocal time squared, e.g., s²), and the equation reduces to

hx²½M þ ½Ki

½U ¼ ½0; ð23Þ

which has a solution only when

x²½M þ ½K¼ ½0: ð24Þ Eigenvalues (natural frequencies) and eigenvectors (modes of free vibration) of the nanotubes are obtained by solving this set of eigenvalue equations. A dimensionless fre- quency associated with the nth mode of vibration of a nonlocal nanotube is defined as

kn¼ q_bAb

EbIb

1=4

ffiffiffiffiffiffi xn

p : ð25Þ

There are other normalized parameters, defined as:

MN¼ M

q_bA_bl_b; VN¼ v

v_cr; l¼e0a

l_b ; ð26Þ

where v_cr¼x1l_b=p, and M_N, V_N, and l are normalized mass, normalized velocity, and the small-scale effect,

Fig. 2 Effect of normalized velocity and normalized small-scale parameter on the normalized deflection of a nanobeam with different small- scale factors for different normalized masses

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respectively. Recently, it was proven that this normaliza- tion describes the beam responses independent of the geometrical and material properties of the beam [11,40,43].

For the purpose of verifying the results, the first three dimensionless frequencies were compared with the results of [11] in Table4for the different values of the normalized small-scale parameters.

Close agreement between the present frequencies and those of Kiani and Mehri [11] can be seen in Table4. Table5 illustrates the comparison of the first five dimensionless frequencies obtained by the current study with those presented in Ref. [44] for various boundary conditions. The small discrepancies are caused by the difference in nonlocal theories, namely, the nonlocal Rayleigh theory for the present study and the nonlocal Euler theory used in Ref. [44].

Considering an SWCNT with a thickness, outer diameter, and length equals 0.34, 2.34, and 30 nm, respectively [16]. The density and Young modulus are equal to 2500 kg=m³ and 1 TPa, respectively. In this study, all the normalized spectra of the beam deflections are calculated for the moving nanoparticle with the values of dimensionless masses as MN¼0:001, 0.05, 0.1, 0.15, and 0.20, the dimensionless velocities as 0 to 1, and the normalized small-scale parameters between 0 and 0.10 (see Fig.2).

For verification of the obtained results when l¼0 (classical continuum structure), the results are compared with those presented in [40] (Fig.3). For a nanotube under a moving nanoparticle of MN¼0:2, the results are compared with the results of Ref. [16] (Fig.4).

In Fig.5, the ratio between the normalized deflection of a nanobeam with different small-scale factors and the normalized deflection of a classical beam (l¼0) for various boundary conditions are depicted.

As shown in Fig.5, increase in values oflleads to grow in M_N except for the CF boundary condition and causes higher results as well.

5 Regression model

In this section, a simplified formula to convert the dimensionless response of a classical elasticity-based beam excited by a moving nanoparticle to the dimensionless response of a nanobeam with different small-scale parameters is obtained. By this strategy, the nanobeam response can be obtained by simply substituting dimensionless parameters of the formula proposed herein.

At this stage, a nonlinear regression analysis based on the information depicted in Fig. 5 could be performed.

Primary variables of the regression analysis to the best fit of the shown spectra are based on polynomial functions of

order 3 inVN, order 2 inland the first order ofMN. Three coefficients are obtained from the regression analysis: (1) the parametera, which converts the response spectra of a classical beam under excitation of a moving load to the response spectra of a classical beam under a moving mass excitation; (2) the factorb, which converts response spectra of a classical beam under a moving mass excitation to the response spectra of a nanobeam under moving inertial nanoparticles with various values of l; and (3) the parameter c, which converts the response spectra of a classical beam under a moving load directly to the response spectra of a nanobeam under a moving inertial nanoparticle with various values ofl.

Fig. 3 Comparison between the results gained by Ref. [40] (continuous lines) and present study (dashed lines) using three modes shape (l¼0)

Fig. 4 Comparison between the results presented in [16] (continuous lines) and present study (dashed lines) whenMN¼0:2 for different small-scale parameters

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The general form of the fitting function can be written as

a¼DAFMoving mass

DAFMoving load

¼P₀₀þP₁₀V_NþP₀₁M_N þP20VN2þP11VNMN

þP₀₂M_N²þP₃₀V_N³þP₂₁V_N²M_N þP12VNMN2;

ð27Þ

b¼ DAF_l

DAFl¼0¼P000þP100VNþP010

MNþP001lþP110VNMNþP101VNl þP₀₁₁M_NlþP₁₁₁V_NM_NlþP₂₀₀ VN2þP210VN2MN

þP201VN2lþP211VN2MNlþP300

V_N³þP₃₁₀V_N³M_N

þP002l²þP102VNl²þP012MNl² þP112VNMNl²;

ð28Þ

c¼ab; ð29Þ

where the constants P_ij and P_ijk are the regression coefficients.

The values of dynamic amplification factors presented in Fig. 5 gained after hours of calculations using computer model for all the considered boundary conditions. How- ever, by the proposed regression-based formulas, it takes only 0.0122, 0.0115, 0.0118, and 0.0111 of the time spent for obtaining the results used to illustrate Fig.5 based on the Galerkin-based assumed mode method for SS, CC, SC, and CF boundary conditions, respectively.

Table 6 presents the regression coefficients for the considered boundary conditions, which are calculated by the least-square procedure along with the goodness of fit parameters including summed square of residuals (SSE), R² (R square), R² (adjusted R²), and root-mean-squared error (RMSE) tests. These parameters confirm good agreement between the regression model and the results obtained by the numerical procedure, always givingR²,R² values of close to 0.9 and SSE and RMSE values of less than 0.02.

Fig. 5 Effect of the normalized velocityV_Nand normalized small-scale parameter lon the dynamic response of the nanotube for different normalized masses

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Table 7 displays the regression coefficients for the considered boundary conditions, calculated first utilizing the least-square procedure for the first two normalized parameters, namely, the normalized velocity and the small-effect parameter. After this, the resultant regression model was fitted again to be consistent with the various normalized mass parameters. Therefore, the final form of the regression model composed of three independent coefficients (Table 7). These formulas could be obtained for other dynamic fields, such as stress and strain, for different boundary conditions.

6 Accuracy of of the proposed regression-based approach

To determine the accuracy of the regression analysis in Sect. 5, the normalized deflection spectra of a classical beam under a moving load are converted to the normalized deflection spectra of a nanobeam (l¼0:10) ) under a moving nanoparticle featuringMN¼0:2 via the proposed formulas. The obtained results are illustrated in Fig.6 for different boundary conditions. At first, the results pertinent to the closed-form solution for the moving load case of a Table 6 Regression

coefficients of the parametera for considered boundary conditions

Boundary conditions SS CC SC CF

Regression coefficients

P00 0.99940 1.02400 0.99380 0.97750

P10 -0.01017 -0.25190 0.03395 0.28030 P01 -0.05470 -0.19270 -0.10080 0.05747

P20 0.11910 0.75990 0.11560 -0.81900

P11 0.33360 1.40600 0.80900 -0.27440

P02 -0.15020 0.39070 0.10560 0.05838 P30 -0.06933 -0.49570 -0.10930 0.45430 P21 -0.04271 -0.87850 -0.42010 -0.64020

P12 2.20100 1.42800 1.46700 -1.70800

Goodness of fit

SSE 0.03149 0.06936 0.05295 0.02301

R-square 0.92660 0.90650 0.89080 0.98660

Adjusted R-square 0.92360 0.90270 0.88630 0.98610

RMSE 0.01271 0.01886 0.01648 0.01086

Table 7 Regression

coefficients of the parameterb for considered boundary conditions

Regression coefficients SS CC SC CF

P000 0.99720 0.97450 0.97540 1.01500

P100 0.03199 0.23210 0.18350 -0.13750

P010 -0.03815 -0.15030 0.09164 -0.02814

P001 0.03638 0.50290 0.96830 -0.26590

P110 0.22080 1.47500 -0.73430 0.19840

P101 -0.35950 -3.67600 -5.34600 1.57800

P011 1.70500 -1.10000 -3.87700 0.45040

P111 -8.34700 -0.98760 22.58000 -1.31400 P200 -0.07317 -0.48630 -0.29080 0.29040 P210 -0.24490 -3.07700 1.22100 -0.29480

P201 0.39040 4.38300 4.29800 -2.38300

P211 7.19000 -4.59200 -13.54000 -0.67370

P300 0.04531 0.26710 0.12330 -0.15620

P310 0.04617 1.90300 -0.66010 0.12980

P002 11.12000 47.44000 8.82200 0.56260

P102 -1.64600 -29.61000 24.04000 4.80600 P012 -10.59000 19.16000 31.36000 -7.55700

P112 23.15000 -85.79000 -147.30000 21.74000

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classical beam were compared to the result obtained by the semi-analytical process. Eventually, the value gained byb andccoefficients was compared to the results of the semi- analytical procedure.

Table8presents the exact values of normalized deflection (obtained by a semi-analytical procedure) at some selected normalized velocity against the values obtained by the proposed conversion coefficientsa,b, andc. The high perfor- mance of the simplified procedure could be clearly seen.

7 Conclusions and remarks

In this paper, a single-walled carbon nanotube under excitation by a moving nanoparticle was modeled. The mathematical model was based on the nonlocal continuum

theory of Eringen. The nanobeam was modeled as a nonlocal Rayleigh beam under four boundary conditions, i.e., simply supported, clamped–clamped, simply clamped, and clamped-free. Utilizing the separation of variables tech- nique, the strong form of equation of motion is transformed into a number of ODEs. Assumed shape functions were generated utilizing COPs. After substituting the COPs into the ODEs and then using the matrix exponential approach, the equations were numerically solved. Normalized deflection spectra for the classical and nonlocal beams under various boundary conditions and excitations were computed. All variables (e.g., mass, velocity, etc.) were normalized, so that the obtained results became independent of the problem at hand and can thus be applied to most beam-like structures.

Fig. 6 Effect of normalized velocityVNon the normalized deflection spectra for different boundary conditions (results by Fryba [29]

(continuous lines), moving load case generated by analytical process (dashed lines), values obtained by semi-analytical method (dash with

single dot lines), usingbcoefficients (dotted lines) or generated viac coefficients (dash with double dot lines) when MN¼0:2 and l¼0:10)

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Regression analysis is utilized to obtain three coefficients (i.e.,a, b, andc), which convert different states of the problem to another state. The mentioned coefficients can easily convert the responses of a classical beam under a moving load case to the complicated responses of a nonlocal beam under excitation by a moving nanoparticle, taking into account the inertia effects. Responses of the moving load case can be readily calculated by commercial software or using other researchers studies.

Finally, it is worth mentioning that the proposed method can be used to reduce the needed time for dynamic analysis of nanoscale beams and to obtain the desired results with sat- isfactory accuracy. By following the given procedure in this work, nonlocal regression-based models could be readily developed for vibrations of nonlocal shear deformable beams and plates under an inertial moving nanoparticle.

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