Comprehensive semi-analytical vibration analysis of rotating tapered AFG nanobeams based on nonlocal elasticity theory considering various boundary conditions via differential transformation method

Accepted Manuscript Comprehensive semi-analytical vibration analysis of rotating tapered AFG nanobeams based on nonlocal elasticity theory considering various boundary conditions via differential transformation method Roohollah Talebitooti, Seyed Omid Rezazadeh, Ahad Amiri PII:

S1359-8368(18)32229-7

DOI:

https://doi.org/10.1016/j.compositesb.2018.12.085

Reference:

JCOMB 6414

To appear in:

Composites Part B

Received Date: 18 July 2018 Revised Date:

15 December 2018

Accepted Date: 17 December 2018

Please cite this article as: Talebitooti R, Rezazadeh SO, Amiri A, Comprehensive semi-analytical vibration analysis of rotating tapered AFG nanobeams based on nonlocal elasticity theory considering various boundary conditions via differential transformation method, Composites Part B (2019), doi: https://doi.org/10.1016/j.compositesb.2018.12.085. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Comprehensive semi-analytical vibration analysis of rotating tapered AFG nanobeams based on nonlocal elasticity theory considering various boundary conditions via differential transformation method

1

RI PT

Roohollah Talebitooti1*, Seyed Omid Rezazadeh1, Ahad Amiri1 Noise and Vibration Control Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran *

Corresponding Author: Roohollah Talebitooti,

SC

Email address: [email protected]

M AN U

Abstract

This paper is concerned with the vibration analysis of a rotating tapered axially functionally graded (AFG) nanobeam. The material properties of the nanobeam are assumed to vary along its length. Accordingly, Newtonian method is employed to derive the governing equation of the system considering Euler-Bernoulli beam assumptions. Also, the nonlocal elasticity theory (NET) is used in order to take the small scale effects into

TE D

account in the modeling. In addition, the differential transformation method (DTM) is hired to solve the attained differential equation semi-analytically. Therefore, the obtained equations as well as boundary conditions are transformed into algebraic equations by the aid of DTM. Then, the characteristics equation can be solved to gain the non-dimensional frequencies of the system. After presenting the convergence and verification illustrations,

EP

some numerical results are discussed in detail to investigate the influences of various parameters namely nonlocal parameter, tapered ratio, angular velocity and FG index on the first three non-dimensional frequencies.

AC C

As a principal result, it is disclosed that the nonlocal parameter shows both stiffness-hardening and stiffnesssoftening behavior in different bounds of the angular velocity. Totally, the numerical analysis reveals some important findings which are hoped to be used in efficient design of nano-structures benefited from the rotating nanobeams.

Keywords: Rotating tapered nanobeam, AFG, DTM, Non-dimensional frequency, Stiffness-hardening, Stiffness-softening.

ACCEPTED MANUSCRIPT Nomenclature e0 a

δ ψ2

nonlocal parameter Young’s modulus moment of inertia of the cross-section natural frequency dimensionless x coordinate dimensionless nonlocal parameter dimensionless hub radius dimensionless natural frequency parameter

z

z coordinate

λ2

dimensionless angular velocity parameter

x

x coordinate

A0

cross-sectional area at the root

ρ ( x)

I0

A( x )

variable density variable cross-sectional area

c

moment of inertia at the root tapered ratio

w( x, t )

transverse displacement

Tc

material properties of ZrO2

σ xx

axial stress time axial strain

Ta

material properties of Aluminum material non-homogeneity parameter

R S N

t

ε xx

I ( x)

ω ζ µ

n

SC

Ω

M AN U

b( x )

E ( x)

RI PT

M

length variable thickness variable width angular speed hub radius shear force axial force bending moment

L t ( x)

1. Introduction

Nanotechnology is a field of engineering science which has attracted the attentions of researchers during recent years [1]. The fast growth of nanotechnology brings us more

TE D

interest in design and examination of devices related to micro/ nano electromechanical systems (MEMS/ NEMS). As a result, the mechanical problems of nano-structures namely

EP

nanobeams, nanoplates, nanotubes, nanorods and nanoshells which are tremendously used in various MEMS/NEMS devices have been a hot topic for researchers. Sensors, actuators,

AC C

transistors, probes and resonators are some potential applications of nano-structures in nanotechnology [2-6].

Among these nano-structures are rotating nanobeams playing a crucial role in design

of various systems such as nano-turbines, nano-shafts, nano-gears and nano-motors which contain rotating parts [7, 8]. Totally, motivated by the fact that having enough knowledge about rotating nanobeams could be helpful in manufacturing and designing accurate nanosensors, biosensors, atomic microscopes and other NEMS devices benefiting from

ACCEPTED MANUSCRIPT rotating nanobeams , a great deal of research has been devoted to study the mechanical properties of rotary nanobeams. This research could be detected not only for cantilever rotating nanobeams but also for other boundary conditions such as simply supported-simply supported and clamped-clamped [9-11]. For instance, rotary nanomotors play a vital role in

RI PT

designing some advanced nano-devices used for converting electric energy to nanoscale mechanical motions for nanomachines and nanofactories [12]. Additionally, improvements in nanotechnology lead us to model and use the rotating nanobeams as nano-blades in nano-

SC

turbines used for investigating the nano-scale water flow and drag force on such rotating nanobeams. It should be mentioned that, this application is of significant importance in

M AN U

rotating nano-devices and molecular motors [13].

It is valuable to note that after conducting many research engaged with the mechanical behavior of nano-structures, it is now proven that for exact predicting the characteristics of such systems, the small-scale effects (size effects) should be taken into

TE D

account. As a consequence, so far, various non-classical theories including couple stress theory, modified couple stress theory, nonlocal elasticity theory (NET) and nonlocal strain

EP

gradient theory (NSGT) have been introduced and hired by scholars to investigate the exact response of the nano-structures from the size-dependency viewpoint [14-18]. Also, it should

AC C

be noted that recently some works have considered the thickness effect (strain gradient in zdirection) in their non-classical modeling of nanobeams and nanoplates to obtain more precise results [19-21]. Additionally, some papers could be detected in the literature in which the integral models of the non-classical theories are used instead of differential models [2225]. In size-dependent theories, small-scale parameters play an important role in mechanical behavior of the nano-structures. Therefore, it is of great importance to choose the correct value for small-scale parameter. The size-dependent static and vibrational behavior has been experimentally observed in some materials. These parameters are often difficult to predict

ACCEPTED MANUSCRIPT and determine experimentally and theoretically specially for advanced materials. As a result, other methods including atomic lattice dynamics and molecular dynamics are exploited to choose a proper value of small-scale parameters. In other words, in some cases it is seen that these parameters are chosen by matching and comparing the wave dispersion curves obtained

RI PT

by various methods [20, 26].

Nowadays, due to the impressive advances in material science, the advanced materials such as smart materials and functionally graded materials (FGMs) have been discovered and

SC

introduced to the scientific society. In other words, these materials are of great importance

M AN U

and interest. Additionally, the modified, extraordinary and adaptable mechanical properties give advanced materials distinct advantages over other materials which results in great applications of these materials in the mechanical systems especially for ones with nano dimensions. Therefore, the mechanical problems associated with the structures constructed from the advanced materials for example FGM have gained a huge deal of attention during

TE D

the recent years. Particularly, FGMs are multiphase composite based materials which their properties vary along one or more directions. Some of FGMs known as axially FGMs are

[27-31].

EP

such that their volume fraction as well as properties differs along the length of the structure

AC C

Up to now, researchers have used various approaches to solve the differential

equations in engineering problems. Some of these implemented methods are differential quadrature method (DQM), generalized deferential quadrature method (GDQM), finite element method, homotopy perturbation method (HPM), homotopy analysis method (HAM), Adomian decomposition method (ADM), optimal homotopy asymptotic method (OHAM) and differential transformation method (DTM) which have been used by various scholars to solve the governing equations on mechanical problems. Each of mentioned methods has its own advantages over the other methods [32-36]. DTM is an authentic, reliable and accurate

ACCEPTED MANUSCRIPT method which has the ability of simple formulation and low computational cost. DTM utilizes the polynomial form to approximate the exact solution based on Tailor series expansion. Throughout this method, the obtained governing differential equation as well as boundary conditions is transformed into algebraic equations. Therefore, the precise and

RI PT

reliable results can be achieved throughout solving the resulted algebraic equations needing less computational operations. In addition, it is worth mentioning that DTM is a semianalytical approach which has the ability of solving the nonlinear differential equation as well

SC

as the differential equations with variable coefficients. By and large, the simplicity and high accuracy are the most outstanding aspects of DTM [37-39]. There are some works in the

M AN U

literature that have used DTM as a solution procedure to solve the governing differential equation in mechanical systems. For instance, Ebrahimi et al. [40] utilized the DTM in order to investigate the thermo-mechanical vibration behavior of nonlocal FG nanobeams for different boundary conditions. They analyzed the influences of various parameters on the

TE D

vibration properties of the system and also showed that the DTM can result in the more accurate results in comparison with the analytical methods. In other work, Ebrahimi and salari [41] explored the vibration analysis of FG Euler-Bernoulli nanobeams using the NET.

EP

They considered the semi-analytical DTM to solve the governing differential equation which

AC C

yields the numerical results with high precision and computational efficiency. Additionally, Ebrahimi et al. [42] analyzed the applicability of the DTM for solving a problem concentrated on the vibration characteristics of FG Euler-Bernoulli nonlocal nanobeams. Herewith, they also used an analytical method known as Navier method where the analytically obtained results were compared with those concluded from the DTM to show the accuracy and applicability of the semi-analytical solution. Rahmani et al. [43] employed the DTM to solve a thermo-mechanical vibration problem for FG mass sensor considering the NET incorporated with the Euler-Bernoulli beam theory under two various thermal loadings.

ACCEPTED MANUSCRIPT They reported some numerical results to demonstrate the influences of the tip mass and its position, thermal loading, material distribution and non-locality on the frequencies of the nano mass sensor. In addition, a DTM based solution was implemented by Zarepour et al. [37] to investigate the transverse free vibration of a nano mass sensor modeled using the NET

RI PT

and Euler-Bernoulli beam model. They used DTM to obtain accurate results with minimum mathematical calculations.

There are also some papers in the literature in which the researchers have focused on

SC

the mechanical problems related to rotating nanobeams. Among these studies, Pradhan and

M AN U

Murmu [44] investigated the flapwise bending vibration of rotating nano-cantilevers on the basis of nonlocal Euler-Bernoulli beam model. The obtained differential equation was solved by DQM to investigate the effects of small scale, hub radius and angular velocity on the frequencies. The flapwise bending vibration characteristics of non-uniform rotating nanocantilevers was examined by Aranda-Ruiz et al. [45] considering Euler-Bernoulli beam

TE D

theory and NET. They used pseudo-spectral collocation method which is based upon the Chebyshev polynomials to solve the governing differential equation containing variable

EP

coefficients. They investigated the influences of various involved parameters including the nonlocal parameter, angular velocity, nonuniformity of the section and hub radius on the first

AC C

three non-dimensional frequencies. Pourasghar et al. [46] analyzed the bending vibration of Euler-Bernoulli nonlocal rotating nanobeam applied to follower force in order to discover the different parameters effects on the frequencies of the system. They utilized DQM to obtain some numerical results for clamped-free boundary condition. Safarabadi et al. [11] employed DQM in order to study the role of surface energy on the vibrational characteristics of rotating nanobeams. The effects of various parameters including boundary conditions and angular velocity were investigated throughout analysis. Considering thermal effects, Ghadiri et al. [47] hired power series method to solve the mechanical vibration problem for Euler-Bernoulli

ACCEPTED MANUSCRIPT rotating nano-cantilever based on the NET. Comparing the obtained results with those of exact DQM solutions, the effects of different parameters namely temperature and small scale on the first three frequencies were investigated in the analysis. Employing DQM and direct iterative method, Ghadiri and Shafiei [7] solved a nonlinear problem for rotating cantilever

RI PT

and propped cantilever nanobeams. They used Euler-Bernoulli beam theory in conjunction with the NET by taking into account the von Karman geometric nonlinearity assumptions. Considering surface elasticity and the NET, Baghani et al. [10] investigated the dynamic and

SC

stability of rotating Euler-Bernoulli nanobeams subjected to nonuniform magnetic field and compressive axial loading. By making use of GDQM, Ghadiri et al. [2] tried to solve the

M AN U

vibration analysis problem for a nonlocal Timoshenko rotating nanobeam. Thermal and surface effects were taken into account for clamped-simply and clamped-free nanobeams and the influences of various effective parameters were revealed in the analysis. Vibration analysis of FG rotating nanobeams was studied by Ghadiri et al. [48] according to the

TE D

nonlocal Euler-Bernoulli beam model and considering the surface effects. They solved the problem for different boundary conditions with the aid of GDQM. Taking attention to thermal stress effects and geometrical nonlinearity, Shafiei et al. [8] investigated the flapwise

EP

bending vibration of a nonlocal Euler-Bernoulli nanobeam for cantilever and propped

AC C

cantilever boundary conditions. Azimi et al. [49] discussed the vibration of rotating AFG Timoshenko nanobeams on the basis of the NET when an in-plane nonlinear thermal loading is applied. They considered clamped-free boundary condition and utilized DQM as a solution method in their analysis. Shafiei et al. [50] took a rotary tapered FG nanobeam into consider in a thermal environment to discover its flapwise bending vibrational characteristics at low temperature based on nonlocal Euler-Bernoulli beam model. DQM was exploited to solve the governing differential equation for cantilever and propped cantilever boundary conditions. Preethi et al. [51] presented a finite element approach for investigating the bending and free

ACCEPTED MANUSCRIPT vibration of nonuniform rotating Timoshenko nanobeams. Size effects, surface elasticity effects and geometrical nonlinearity were taken into account in their modeling. Bakhshi Khaniki [13] utilized Eringen’s two-phase local/ nonlocal model for investigating the vibration response of rotating nanobems. Modified generalized differential quadrature

RI PT

method was exploited in their investigation. Furthermore, some other works focused on the vibration analysis of rotating microbeams and nanobeams can be detected in the literature in which various theories have been used [9, 52-54]. It is worth mentioning that, however

SC

various papers have been done for investigating the mechanical properties of rotating nanobeams, most of them have not presented a detailed and comprehensive review on the

M AN U

exact effect of the nonlocal parameter. The researchers have only concentrated on the overall effect of the nonlocal parameter. However, sometimes an intersection point might be appeared leading to the vice versa effect of nonlocal parameter on the frequencies. Therefore, in order to tackle this deficiency, this paper is allocated to study the mechanics of rotating

TE D

nanobeams from this viewpoint. For instance, it is investigated that how the intersection point’ location changes when the angular velocity or tapered ratio changes. For this aim, different boundary conditions are taken into account to present a detailed investigation on

EP

tapered AFG rotating nanobeams as complementary study.

AC C

As it was discussed in the literature review, so far some research related to the vibration characteristics of the rotating nanobeams made from different materials has been done by the researchers using various theories. However, according to the best knowledge of authors, there is no comprehensive investigation on the vibration characteristics of rotating tapered AFG nonlocal nanobeams considering various boundary conditions. Therefore, in this paper, this issue is investigated to fill the detected gap in the open literature related to the mechanical vibration of rotating nanobeams. It is worth pointing out that AFG and variable section property of the considered nanobeam leads to a partial differential equation with

ACCEPTED MANUSCRIPT variable coefficients. For solving such equation, the DTM is employed as a semi-analytical approach which is less used by the other researchers. It should be also noted that, DTM outcomes precise results throughout less computational calculations.

RI PT

Herewith, the governing differential equation is derived throughout the Newtonian method based on nonlocal Euler-Bernoulli beam model. Subsequently, the separation of variables is applied to the equation that yields an equation containing the frequency of the system. Thereafter, non-dimensional parameters are defined to obtain the dimensionless form

SC

of the governing equation as well as the related boundary conditions. The algebraic equations

M AN U

obtained by means of the transformation rules lead to a characteristic equation which should be solved numerically to gain the non-dimensional frequencies. For this aim, all possible boundary conditions are taken into account for presenting a comprehensive study. To evaluate the performance of the numerical method, verification and convergence study is adapted. The results show that the presented methodology proves to be accurate. Finally,

TE D

some detailed numerical results are represented in which the first three non-dimensional frequencies are investigated under various boundary conditions. Consequently, the effects of

EP

various involved parameters including nonlocal parameter, tapered ratio, angular velocity and FG index are explored comprehensively. Some important results are revealed which may be

AC C

useful for efficient designing of the nano-structures benefiting from rotating nanobeams. It is fascinatingly seen that the nonlocal parameter does not exhibit only stiffness-softening effect. In fact, this behavior depends on the value of tapered ratio and angular velocity. In some cases (specific angular velocities and tapered ratios), it is shown that the stiffness-softening effect switches to the stiffness-hardening effect. 2. Mathematical modeling

ACCEPTED MANUSCRIPT The studied model is a rotating tapered Euler-Bernoulli nanobeam with total length L , variable thickness t(x) and variable width b( x) . The schematic and side view of the model is demonstrated in Figs. (1a) and (1b). The nanobeam is fixed to a rigid hub at at constant angular speed

Ω

x=0

and rotates

around rotation axis. It should be noted that the considered

RI PT

nanobeam can be pinned or clamped at location x = 0 . Also, the distance between where the nanobeam is connected to hub ( x = 0 ) and rotation axis is R . In our model, the tip of the

TE D

M AN U

SC

nanobeam takes different boundary conditions (i.e. free, pinned or clamped).

AC C

EP

(a)

(b)

RI PT

ACCEPTED MANUSCRIPT

(c)

Fig. 1. Rotating tapered nanobeam made of AFG material: (a) Schematic view, (b) Side view, (c) Differential slice of the

SC

nanobeam.

M AN U

Figure (1c) demonstrates a differential nanobeam’s slice of length dx . In this figure, S , N and M are respectively shear force on the cross section, axial force and resultant

bending moment. Therefore, by taking the Newtonian’s method into consideration, the

TE D

equation of vertical motion for the differential slice can be written as: ∂S ( x , t ) ∂ 2 w( x , t ) = ρ ( x ) A( x ) ∂x ∂t 2

(1)

EP

where ρ(x) and A(x) are variable density and variable cross-sectional area, respectively.

AC C

Also, w(x, t) denotes transverse displacement. In the analysis, it is supposed that the rotational inertia could be ignorable. Additionally, higher order differential terms leading to nonlinear equation are neglected. Therefore, the momentum balance of the slice gives rise to following relation:

S ( x, t ) = N ( x)

∂w( x, t ) ∂M ( x, t ) − ∂x ∂x

where the classical resultant bending moment M(x,t) is defined as:

(2)

ACCEPTED MANUSCRIPT M = − ∫ σ xx zdA

(3)

A

Substituting Eq. (1) to Eq. (2), the following relation is obtained: ∂ 2 M ( x, t ) ∂ ∂w( x , t ) ∂ 2 w( x , t ) ( N ( x ) ) ρ ( x ) A ( x ) = − ∂x 2 ∂x ∂x ∂t 2

RI PT

(4)

According to the Euler-Bernoulli beam theory, the non-zero axial strain is expressed as:

SC

∂ 2 w( x , t ) ∂x 2

ε xx = − z

(5)

σ xx − (e0 a ) 2

∂ 2σ xx = Eε xx ∂x 2

in which e0 a is nonlocal parameter.

M AN U

On the basis of nonlocal elasticity theory, the constitutive relation is written as [55]: (6)

Now, using Eqs. (3) and (6) incorporated with Eq. (5) results as follows: ∂ 2 M ( x, t ) ∂ 2 w( x , t ) = − E ( x ) I ( x ) ∂x 2 ∂x 2

TE D

− M + (e0 a ) 2

(7)

Finally, when Eq. (7) is applied to Eq. (4), the governing equation of motion is derived as:

EP

3 ∂2 ∂ 2 w( x, t ) ∂ ∂w( x, t ) ∂w( x, t ) 2 ∂ − + ( E ( x ) I ( x ) ) ( N ( x ) ) ( e a ) ( N ( x) ) 0 2 2 3 ∂x ∂x ∂x ∂x ∂x ∂x 2 ∂ 2 w( x , t ) ∂ 2 w( x , t ) 2 ∂ + ρ ( x ) A( x ) − ( e a ) ( ρ ( x ) A ( x ) )=0 0 ∂t 2 ∂x 2 ∂t 2

(8)

AC C

where N(x) is produced axial force attributed to the rotation. This force is defined as follows [28, 56]:

L

N ( x ) = ∫ ρ ( x ) A( x )Ω 2 ( R + x ) dx

(9)

x

According to the separation of variables method, the solution of Eq. (8) may be expressed as following:

ACCEPTED MANUSCRIPT w( x , t ) = W ( x ) e iω t

(10)

where W(x) is a function satisfying the related boundary conditions; and ω is the natural frequency of the system.

RI PT

Therefore, by substituting Eq. (10) into Eq. (8), the following equation is deduced:

(11)

SC

3 d2 d 2W ( x ) d dW ( x ) dW ( x ) 2 d ( E ( x ) I ( x ) ) − ( N ( x ) ) + ( e a ) ( N ( x) ) 0 2 2 3 dx dx dx dx dx dx d2 − ρ ( x ) A( x )ω 2W ( x ) + (e0 a ) 2 2 ( ρ ( x ) A( x )ω 2W ( x )) = 0 dx

In order to streamline the calculations in the analysis, the non-dimensional parameters are

M AN U

defined as follows:

e0 a ρ c A0 L4 2 4 ρ c A0 L4 2 x W R 4 ζ = ;W = ; µ= ω ;λ = ; δ = ;ψ = Ω . L L L L Ec I 0 Ec I 0

(12)

where the index c in ρc and Ec refers to the ceramic part of the axially FG nanobeam.

TE D

In addition, A0 and I0 are respectively the cross-sectional area and moment of inertia at the root of the nanobeam (where the nanobeam is fixed to the hub).

EP

For a tapered nanbeam with linear variation in width and height, the cross-sectional area and the moment of inertia are given as follows:

AC C

x x A = A0 (1 − c )m ; I = I 0 (1 − c ) m+ 2 L L

(13)

It is notable that when the nanobeam is tapered only in height direction, m is taken to be 1. Additionally, for the nanobeam which is linearly variable in both height and width directions, m is equal to 2. Thus, the resulted cross-sectional area and the moment of inertia for these

two cases are represented by Eqs. (14) and (15) respectively as [45]:

ACCEPTED MANUSCRIPT x x A = A0 (1 − c ); I = I 0 (1 − c ) 3 L L

(14)

x x A = A0 (1 − c )2 ; I = I 0 (1 − c ) 4 L L

(15)

RI PT

in which the positive constant value c ( 0 ≤ c ≤ 1 ) is known as nanobeam tapered ratio. By taking Eqs. (14) and (15) into consideration, it is clear that when the tapered ratio c is set to be 0, the considered nanobeam is uniform.

SC

In addition to the geometrical properties (i.e. cross-sectional area and the momentum inertia) which are variable along the nanobeam, the material properties are also x -dependent

M AN U

and this is due to the fact that nanobeam is constructed from axially FG material. Therefore, the material properties are considered to be as follows [57]:

AC C

EP

TE D

x T = (Ta − Tc )( )n + Tc L

(16)

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

where

T

TE D

Fig. 2. Material property of AFG material [57].

is a function defining the material property; Ta and Tc are respectively the material

properties of Aluminum and ZrO2 which the nanobeam is made of; and n is material non-

EP

homogeneity parameter. The Young’s modulus of considered AFG material for various FG

AC C

indexes is illustrated in Fig. (2).

The flexural rigidity is defined as:

d ( x) = E( x) I ( x)

(17)

Therefore, according to Eqs. (13) and (16), Eq. (17) yields as: x x    d ( x ) =  ( Ea − Ec )( ) n + E z   I 0 (1 − c ) m + 2  L L   

(18)

ACCEPTED MANUSCRIPT Now, utilizing the non-dimensional parameters defined in Eq. (12), the flexural rigidity can be re-written as following:

 ( E − Ec ) n  d (ζ ) = Ec I 0  a ζ + 1 (1 − cζ ) m + 2 = Ec I 0 d (ζ )  Ec 

RI PT

(19)

Also by inserting the non-dimensional parameters into Eq. (9), we have: 1 Ec I 0  ( ρ a − ρ c ) n  EI ζ + 1 (1 − cζ ) m (δ + ζ )λ 2 d ζ = c 2 0 N (ζ ) 2 ∫ L ζ ρc L 

(20)

SC

N (ζ ) =

According to Eq. (11), the other effective property on the governing equation is the

M AN U

mass per unit length of the nanotube. In order to write this property in non-dimensional form, the following function ma(x) is allotted to this property:

ma( x) = ρ ( x) A( x)

(21)

Afterwards, by using the non-dimensional parameters and also considering Eqs. (13) and (16)

TE D

, the non-dimensional form of nanobeam’s mass per unit length can be written as:

(22)

EP

 ( ρ − ρc ) n  ma( x) = ρc A0  a ζ + 1 (1 − cζ )m = ρc A0 ma (ζ )  ρc 

Finally, substituting Eqs. (19), (20) and (22) results in the non-dimensional governing motion

AC C

equation as follows:

Ez I 0 d 2 d 2W (ζ ) Ez I 0 d dW (ζ ) ζ ( d ( ) )− 4 ( N (ζ ) ) 4 2 2 L dζ dζ L dζ dζ +

Ez I 0 e0 a 2 d 3 dW (ζ ) ( ) ( N (ζ ) ) − ρ z A0ω 2 ma (ζ )W (ζ ) 4 3 L L dζ dζ

+ ρ z A0ω 2 (

e0 a 2 d 2 ) ( ma (ζ )W (ζ )) = 0 L dζ 2

(23)

ACCEPTED MANUSCRIPT Thereafter, by applying some other non-dimensional parameters defined in Eq. (12) and after simple manipulation, the final non-dimensional equation is obtained: 3 d2 d 2W (ζ ) d dW (ζ ) dW (ζ ) 2 d ( d ( ζ ) ) − ( N ( ζ ) ) + µ ( N (ζ ) ) 2 2 3 dζ dζ dζ dζ dζ dζ

d2 −ψ ma (ζ )W (ζ ) + ψ µ ( ma (ζ )W (ζ )) = 0 dζ 2 2

(24)

2

RI PT

2

In order to solve Eq. (24), four boundary conditions for each type of nanobeam is

SC

requisite. Totally, the considered nannobeam can take simply supported and clamped boundary conditions at the root (where the nanobeam is fixed to the hub). In addition, the

M AN U

nanobeam at the tip can be free, simply supported or clamped. The non-classical boundary conditions for various types of boundary conditions are as follows: Clamped: dW (ζ ) =0 dζ

Free:

d dW (ζ ) d 2W (ζ ) ( N (ζ ) ) −ψ 4 ma (ζ )W (ζ )) + d (ζ ) =0 dζ dζ dζ 2

EP

M = µ2(

(25)

TE D

W (ζ ) = 0;

d d dW (ζ ) d d 2W (ζ ) V =µ ( ( N (ζ ) ) −ψ 4 ma (ζ )W (ζ )) + ( d (ζ ) )=0 dζ dζ dζ dζ dζ 2

AC C

2

(26)

Simply supported: W (ζ ) = 0 M = µ2(

d dW (ζ ) d 2W (ζ ) ( N (ζ ) ) − ψ 4 ma (ζ )W (ζ )) + d (ζ ) =0 dζ dζ dζ 2

(27)

However, there are some papers in literature in which the non-classical boundary conditions are replaced by the classical ones. In other words, it is shown that the difference between the obtained results by classical and non-classical boundary conditions is negligible.

ACCEPTED MANUSCRIPT The mentioned classical corresponding boundary conditions for various nanobeams are defined as follows [40, 41]: Clamped-Free: dW (0) =0 dζ

RI PT

W (0) = 0;

d 2W (1) d 3W (1) 0 = ; =0 dζ 2 dζ 3

dW (0) =0 dζ

dW (1) W (1) = 0; =0 dζ Clamped-Simply supported: dW (0) =0 dζ

d 2W (1) W (1) = 0; =0 dζ 2

(29)

(30)

TE D

W (0) = 0;

M AN U

W (0) = 0;

SC

Clamped-Clamped:

(28)

Simply supported-Simply supported: d 2W (0) =0 dζ 2

EP

W (0) = 0;

(31)

AC C

d 2W (1) W (1) = 0; =0 dζ 2

3. Solution procedure

In this section, DTM is applied to solve the obtained equations. It should be

mentioned that two transformations are required for applying DTM in order to solve the differential equations. These two transformations known as differential transformation (DT) and inverse differential transformation (IDT) are defined as following [58]:

ACCEPTED MANUSCRIPT DT:

F [k ] =

1  d k f ( x)  k !  dx k  x = x

(32)

0

RI PT

IDT: ∞

f ( x ) = ∑ F [ k ] ( x − x0 ) k k =0

SC

Generally, Eq. (33) can be considered as finite series: nt

f ( x ) = ∑ F [ k ] ( x − x0 ) k

where nt is number of terms. Merging Eqs. (32) and (33) results in: ( x − x0 ) k  d k f ( x)  k !  dx k  x = x k =0 0 ∞

TE D

f ( x) = ∑

M AN U

k =0

(33)

(34)

(35)

Some essential transformations used for applying DTM to transfer the governing equation and boundary conditions are included in Tables 1 & 2.

EP

Table 1. Fundamental Transformation rules for one-dimensional DTM [59] Original functions

Transformed functions F [k ] = G[k ] ± H [ k ]

f ( x ) = ag ( x )

F [ k ] = aG [ k ]

f ( x) = g ( x)h( x)

F [k ] =

∑ G [ k − i ] H [i ]

F [k ] =

(k + p )! G [k + p] k!

AC C

f ( x) = g ( x) ± h( x)

f ( x) =

d p g ( x) dx p

f ( x) = x p

k

i=0

0 k ≠ p F [ k ] = δ (k − p ) =  1 k = p

ACCEPTED MANUSCRIPT Table 2. Transformed classical boundary conditions based on DTM [59] Transformed B.C.

Original B.C.

x=L

Transformed B.C.

Original B.C.

f (0) = 0

F [ 0] = 0

df (0) =0 dx d 2 f (0) =0 dx 2 d 3 f (0) =0 dx 3

F [1] = 0

∞

∑ F [k ] = 0

f ( L) = 0

k =0

df ( L) =0 dx d 2 f ( L) =0 dx 2 d 3 f (L) =0 dx 3

F [ 2] = 0

∞

∑ kF [ k ] = 0 k =0

∞

∑ k ( k − 1) F [ k ] = 0 k =0

∞

∑ k ( k − 1)( k − 2) F [ k ] = 0 k =0

SC

F [3] = 0

RI PT

x=0

relation: k (k + 1)(k + 2)(k + 3)d [1]W [ k + 4] + SS1 k +1

− k ∑ (k − i + 2) N [ i ]W [ k − i + 3] i =1

M AN U

Applying the presented transformation rules to Eq. (24) gives rise to the following algebraic

−ψ

k

4

∑ ma [i ]W [ k − i + 1] i =1

k +2

(36)

TE D

+ µ 2 k (k + 1)(k + 2)(k + 3) N [1]W [ k + 4] + SS 2

+ψ 4 µ 2 k (k + 1)∑ ma [ i ]W [ k − i + 3] = 0 where k +2

EP

i =1

AC C

SS1 = k ( k + 1)∑ ( k − i + 3)(k − i + 4)d [i ]W [ k − i + 5]

(37)

i=2

k +3

SS 2 = µ 2 k ( k + 1)( k + 2)∑ ( k − i + 4) N [i ]W [ k − i + 5]

(38)

i=2

where

W [ k ] , d [i ] , N [ i ]

and ma [i ] are transformed functions of W (ζ ), d (ζ ), N(ζ ) and ma(ζ )

respectively. The transformed non-classical boundary conditions could be written as:

ACCEPTED MANUSCRIPT Clamped-Free:

W [ 0] = 0; W [1] = 0 nt

(k − 1)t1 + µ 2ψ 4t2 + t3 = 0

2

k =1 nt

∑µ

(39)

(k − 1)(k − 2)t1 + µ 2ψ 4 (k − 1)t2 + (k − 1)t3 = 0

2

k =1

RI PT

∑µ

where k

t1 = ∑ (k − i + 1) W [ k − i + 2]N [ i ]

SC

i =1 k

t2 = ∑ W [ k − i + 1]ma [ i ] i =1

M AN U

k

(40)

t3 = ∑ (k − i + 1)(k − i + 2)W [ k − i + 3]d [i ] i =1

Clamped-Clamped:

W [ 0] = 0; W [1] = 0

∑W [ k ] = 0; k =1

nt

∑ (k − 1)W [ k ] = 0 k =1

(41)

TE D

nt

Clamped-Simply supported:

nt

∑W [ k ] = 0;

∑µ k =1

2

(k − 1)t 1 + µ 2ψ 4t 2 + t 3 = 0

(42)

AC C

k =1

nt

EP

W [1] = 0; W [ 2] = 0

Simply supported-Simply supported:

W [1] = 0; W [3] = − nt

nt

∑W [ k ] = 0; ∑ µ k =1

k =1

2

µ 2 N [2]

W [2] (2µ 2 N [1] + d [1])

(k − 1)t 1 + µ ψ t 2 + t 3 = 0 2

4

In addition, the transformed classical boundary conditions are as follows: Clamped-Free:

(43)

ACCEPTED MANUSCRIPT W [ 0] = 0; W [1] = 0 nt

nt

k =1

k =1

(44)

∑ (k −1)(k − 2)W [ k ] = 0; ∑ (k − 1)(k − 2)(k − 3)W [ k ] = 0 Clamped-Clamped:

nt

∑W [ k ] = 0; k =1

RI PT

W [ 0] = 0; W [1] = 0 nt

∑ (k − 1)W [ k ] = 0 k =1

Clamped-Simply supported:

nt

nt

k =1

k =1

SC

W [1] = 0; W [ 2] = 0

Simply supported-Simply supported:

W [1] = 0; W [3] = 0 nt

∑W [ k ] = 0; k =1

nt

∑ (k − 1)(k − 2)W [ k ] = 0 k =1

M AN U

∑W [ k ] = 0; ∑ (k − 1)(k − 2)W [ k ] = 0

(45)

(46)

(47)

TE D

By making use of Eq. (36) in conjunction with the converted boundary conditions for each beam, the following characteristic equation is extracted:

EP

 A11 (Ψ ) A12 (Ψ )   A (Ψ ) A (Ψ )  [C ] = 0  21 22 

(48)

AC C

For calculating the nontrivial solution of Eq. (48), the determinant of the coefficient matrix should be equal to zero. Hence, considering this fact leads to the following characteristic equation:

A11 (Ψ ) A12 (Ψ ) A21 (Ψ ) A22 (Ψ )

=0

(49)

ACCEPTED MANUSCRIPT th It should be accentuated that throughout solving Eq. (49) the i eigenvalue is estimated for

nth iteration of calculations. The final and desirable value of iterations depending on the accuracy of calculations is determined through the following relation: Ψ in − Ψ in −1 ≤ ε

RI PT

(50)

In this paper, ε is equal to 0.0001 .

SC

4. Results and discussions

M AN U

This section presents comprehensive numerical results for vibration analysis of the considered nanobeam. The nanobeam is axially FG constructed from Aluminum and ceramic ZrO2 of material properties included in Table 3. First of all, in order to examine the accuracy

and convergence of the numerical model and verify it, a validation study is exploited. For this aim, some numerical data related to the first non-dimensional frequency of a P-P nanobeam

TE D

are listed in Table 4. In addition, the mentioned table encompasses some results obtained by Reddy et al. [60]. As it is seen, by increasing k (i.e. the number of terms in the

EP

transformation method) a good convergence is detectible in the results. Also, by comparing the results obtained here with those reported by Reddy et al. [60], a good degree of accuracy

AC C

is found showing the reliability of the used numerical method. Figure (3) indicates the diagrams of the first three non-dimensional frequencies of a cantilever uniform isotropic nanobeam as a function of the angular velocity considering various nonlocal parameters. The plotted diagrams employing the numerical solution method used in the present paper are compared with the results reported in Ref. [44]. It can be found that the obtained results agree well with those described by Pradhan and Murmu [44]. Additionally, in order to guarantee the reliability and accuracy of the numerical method, Fig. (4) is presented as a further verification. In the mentioned figure, the variation of the first non-dimensional frequency of a

ACCEPTED MANUSCRIPT cantilever isotropic non-rotating nanobeam with tapered ratio is plotted when m =1.

λ = 0, δ = 0

and

It is indicated that the obtained results by DTM are in a good agreement with those of

Ref [61]. Finally, Table 5 is presented to show that the numerical solution is accurate enough for various boundary conditions. As it is shown, various boundary conditions are taken into λ 2 = 5, δ = 0, c = 0

. Consequently, the obtained results using the numerical

RI PT

account when

solution are tabulated and compared with those presented by Rajasekaran [28]. It is seen that,

SC

the obtained results are in good agreement with reference. Table 3. Material properties [28] Properties

Value

2702 ( Kg / m 3 )

Ceramic density ( ρc )

5700 ( Kg / m 3 )

M AN U

Aluminum density ( ρa )

Aluminum Young’s modulus ( E a )

70 ( Gpa )

Ceramic Young’s modulus ( Ec )

200 ( Gpa )

Poisson ratio

0.3

0 0.01 0.02 0.03 0.04 0.05

k = 12 9.8901 9.4357 9.0385 8.6877 8.3748 8.0934

k = 13

9.8901 9.4357 9.0385 8.6877 8.3748 8.0934

EP

1

µ2

AC C

Ψ

TE D

Table 4. Convergence study and comparison between the obtained results and the results obtained by Reddy et al. [60] Number of terms ( nt ) k = 14 k = 15 9.8683 9.4147 9.0182 8.6681 8.3557 8.0750

9.8683 9.4147 9.0182 8.6681 8.3557 8.0750

k = 16 9.8697 9.4159 9.0195 8.6693 8.3569 8.0762

k = 17 9.8697 9.4159 9.0195 8.6693 8.3569 8.0762

Reddy et al. [60] 9.8696 9.4159 9.0195 8.6693 8.3569 8.0761

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 3. Variation of first, second and third non-dimensional frequencies of a cantilever uniform isotropic nanobeam versus

AC C

EP

TE D

angular velocity for various nonlocal parameters: filled marker: present study; unfilled marker: Pradhan & Murmu [44].

Fig. 4. Variation of first non-dimensional frequency ( ψ12 ) versus the tapered ratio for a cantilever isotropic non-rotating nanobeam, when δ = 0, m = 1 : filled marker: present study; unfilled marker: Murmu and Pradhan [61]

ACCEPTED MANUSCRIPT Table 5. Verification study for different boundary conditions according to present study and results reported by [28] considering λ 2 = 5, δ = 0, c = 0

1 2

C-F

Present study reference Present study reference Present study reference

ψ2

ψ1

ψ2

6.4496 6.4495 6.9716 6.9717 6.9289 6.9289

25.4461 25.4461 25.6519 25.6522 26.0436 26.0440

5 5 5 5 5 5

19.9197 19.9197 19.5320 19.5321 20.0838 20.0839

n 0 1 2

C-C Present study reference Present sudy reference Present study reference

ψ2

24.5410 24.5442 22.7496 22.7497 22.3545 22.3542

64.7998 64.8012 60.6390 60.6397 60.6607 60.6611

2

C-P Present sudy reference Present study reference Present study reference

ψ1

ψ2

17.6125 17.6130 17.2463 17.2465 17.1413 17.1414

53.0701 53.0707 50.6605 50.6611 51.1017 51.1021

M AN U

1

P-P

ψ1

n 0

P-F

ψ1

RI PT

0

from

ψ1

ψ2

13.0954 13.0953 12.1032 12.1033 12.2333 12.2333

43.3513 43.3513 40.6536 40.6541 41.3792 41.3794

SC

n

P-C

ψ1

ψ2

18.5679 18.5698 16.2857 16.2858 16.2650 16.2652

53.7787 53.7785 49.4333 49.4338 49.7853 49.7864

TE D

Figures (5)-(8) illustrate the variation of the first non-dimensional frequency ψ 1 with non-dimensional angular velocity λ considering different boundary conditions (C-C, C-F, CP and P-P). It should be mentioned that, n , m and δ are respectively set to 0, 1 and 1 in

EP

these figures. Also, various non-dimensional nonlocal parameters µ and various tapered ratios c are taken into account. The figures highlight the increasing effect of the non-

AC C

dimensional angular velocity on the fundamental frequency. By considering the results related to C-C, C-P and P-P boundary conditions, one can understand that for each c there is an intersection point. The intersection point divides the diagram to two zones where the response of the non-dimensional frequency to nonlocal parameter is vice versa. In other words as a prominent result it is seen that for low angular velocities, the nonlocal parameter exhibits a decreasing impact on the fundamental frequency. However when the angular velocity increases and surpasses specific value, the nonlocal parameter acts inversely and

ACCEPTED MANUSCRIPT increases the fundamental frequency of the nanobeam. Furthermore, by following the trend of different diagrams plotted for various tapered ratios, it is totally revealed that the intersection point shifts to left side when the tapered ratio c ranges from 0 to 0.8. Compared to the other boundary conditions, the response of fundamental frequency of cantilever nanobeam to the

RI PT

nonlocal parameter is somehow different. Figure (6) discloses the fact that for lower values of tapered ratio, the nonlocal parameter has an increasing effect on the fundamental frequency. However, as it is clear, an intersection point appears for higher values of c in the diagram.

SC

Therefore, it is evident that for the angular velocities before the intersection point, an increasing in nonlocal parameter value gives rise to reduction in the fundamental frequency.

M AN U

While by further increment in angular velocity, the hardening effect of the nonlocal parameter on the fundamental frequency is emerged again. As it is observed, the diagrams reveal a region (bound for angular velocity) where the response of the frequencies to the nonlocal parameter is fascinatingly against our expectations. It is obviously shown that there

TE D

is an intersection point which specifies a particular transition angular velocity in which the stiffness-softening behavior switches to stiffness-softening behavior. This region of the

AC C

EP

diagrams is less discussed in the available literature.

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

EP

Fig. 5. Variation of first non-dimensional frequency of C-C nanobeam versus the angular velocity for various nonlocal parameters when

AC C

n = 0, m = 1 and δ = 1 : (a) c = 0 , (b) c = 0.2 , (c) c = 0.4 , (d) c = 0.8 .

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 6. Variation of first non-dimensional frequency of C-F nanobeam versus the angular velocity for various nonlocal parameters when

AC C

n = 0, m = 1 and δ = 1 : (a) c = 0 , (b) c = 0.2 , (c) c = 0.4 , (d) c = 0.8 .

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 7. Variation of first non-dimensional frequency of C-P nanobeam versus the angular velocity for various nonlocal parameters when

AC C

n = 0, m = 1 and δ = 1 : (a) c = 0 , (b) c = 0.2 , (c) c = 0.4 , (d) c = 0.8 .

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

Fig. 8. Variation of first non-dimensional frequency of P-P nanobeam versus the angular velocity for various nonlocal parameters when n = 0, m = 1 and δ = 1 : (a) c = 0 , (b) c = 0.2 , (c) c = 0.4 , (d) c = 0.8 .

After discussing about the first non-dimensional frequency, it is of importance to

investigate the behavior of second frequency. Therefore, the variation of the second nondimensional frequency with respect to the angular velocity is plotted in Figs. (9) and (10). It is notable that the diagrams are indicated for different values of nonlocal parameters as well as various tapered ratios. Moreover, due to the fact that the behavior of C-C, C-P and P-P

ACCEPTED MANUSCRIPT nanobeams are relatively similar, only the diagrams of C-C and C-F nanobeams are presented. In the case of C-C nanobeams, it is obviously seen that the stiffness softening is the predominant effect of the nonlocal parameter on the second non-dimensional frequency. In comparison with the first frequency, it is predicted that the intersection point for the

RI PT

second non-dimensional frequency will appear in higher angular velocities. Additionally, for C-F nanobeams it is totally seen that the nonlocal parameter has a decreasing influence on the second frequency. Consequently, by increasing the nonlocal parameter, the second frequency

SC

decreases. However, for some nonlocal parameters, stiffness hardening effect is observed at higher angular velocities. For nanobeams having higher tapered ratio c , all the curves

M AN U

intersect each other in a particular angular velocity. In other words, the intersection point

AC C

EP

TE D

becomes more visible.

SC

RI PT

ACCEPTED MANUSCRIPT

M AN U

Fig. 9. Variation of second non-dimensional frequency of C-C nanobeam versus the angular velocity for various nonlocal parameters when

AC C

EP

TE D

n = 0, m = 1 and δ = 1 : (a) c = 0 , (b) c = 0.2 , (c) c = 0.4 , (d) c = 0.8 .

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 10. Variation of second non-dimensional frequency of C-F nanobeam versus the angular velocity for various nonlocal parameters when n = 0, m = 1 and δ = 1 : (a) c = 0 , (b) c = 0.2 , (c) c = 0.4 , (d) c = 0.8 .

The calculated results for first three non-dimensional frequencies (ψ 2 ) of C-F, C-P and

TE D

P-P rotating tapered AFG nanobeams are respectively tabulated in Tables 6, 7 and 8. It should be noted that various nonlocal parameters µ , various tapered ratios c , various angular

and

δ

λ

as well as different FG indexes

n

are taken into account. Also the parameters

m

EP

velocities

are chosen 1 and 0.5 respectively. Thus, the non-dimensional frequencies for each case

AC C

has been determined and inserted to the tables. It should be emphasized that one can understand and realize the decreasing and increasing effect of each effective parameter by pursuing the represented results which have been also shown symbolically throughout some figures.

ACCEPTED MANUSCRIPT Table 6. First three non-dimensional frequencies (ψ ) of C-F AFG nanobeam considering different λ , different 2

n , different c and different µ when m = 1 and δ = 0.5 .

4

0.2

0

2

4

0

2

4

Ψ 32

n =1

n=3

n=0

n =1

n=3

n=0

n =1

n=3

4.3098 4.2569 4.1096 3.8957 3.6467 5.1715 5.1868 5.2325 5.3078 5.4102 7.1386 7.2765 7.6618 8.2206 8.8686 4.3942 4.3296 4.1518 3.8992 3.6129 5.2601 5.2671 5.2902 5.3328 5.3957 7.2376 7.3704 7.7379 8.2569 8.8332 4.5132 4.4291 4.2024 3.8914 3.5529 5.3839 5.3780 5.3667 5.3614 5.3691 7.3741 7.5003 7.8411 8.2990 8.7745

4.1816 4.0245 3.6413 3.1900 2.7701 5.0557 4.9833 4.8191 4.6518 4.5263 7.0377 7.1077 7.2934 7.5397 7.7934 4.2823 4.1165 3.7141 3.2437 2.8096 5.1552 5.0762 4.8985 4.7193 4.5860 7.1405 7.2062 7.3855 7.6261 7.8711 4.4209 4.2419 3.8110 3.3132 2.8590 5.2918 5.2039 5.0077 4.8119 4.6664 7.2807 7.3422 7.5129 7.7403 7.9646

18.9335 17.7693 15.1025 12.1903 09.6176 20.0416 19.1951 17.2614 15.0507 12.7733 23.0417 22.9560 22.6319 21.6659 19.4278 17.7190 16.5652 13.9944 11.2964 09.0116 18.8494 18.0127 16.1789 14.2233 12.3757 21.8885 21.8063 21.5928 20.9768 19.4898 16.4239 15.2527 12.7468 10.2557 08.2504 17.5800 16.7327 14.9793 13.2756 11.8404 20.6612 20.5724 20.4376 20.1024 19.2587

22.7404 20.9927 17.4110 14.0383 11.4346 23.5579 22.0348 18.9692 16.1656 14.0615 25.8549 24.9115 23.0963 21.4981 20.2258 21.2469 19.5418 16.1126 12.9522 10.5481 22.0921 20.6190 17.7251 15.1632 13.3000 24.4527 23.5732 21.9605 20.6423 19.6573 19.6553 17.9641 14.6627 11.7179 09.5262 20.5343 19.0916 16.3622 14.0602 12.4580 22.9706 22.1496 20.7441 19.6895 18.9538

23.4128 20.9842 16.6792 13.1694 10.6785 24.2130 21.9891 18.1270 15.0961 13.0499 26.4664 24.7662 21.9469 19.8833 18.5658 21.9008 19.5505 15.4510 12.1608 09.8462 22.7232 20.5812 16.9366 14.1450 12.2998 25.0277 23.4157 20.8461 19.0689 17.9885 20.2888 17.9970 14.0955 11.0365 08.9133 21.1387 19.0669 15.6498 13.1275 11.5119 23.5047 21.9859 19.6986 18.2323 17.3900

53.0145 43.8775 31.6323 24.1256 20.1043 54.1452 45.5826 34.4542 28.3798 57.3867 50.3089 41.6691 38.7675 48.2842 39.8141 28.5728 21.6687 17.7730 49.4297 41.5209 31.3772 25.8429 52.7017 46.2482 38.6061 36.1104 43.2673 35.4328 25.2214 18.9684 15.3034 44.4324 37.1636 28.0628 23.1378 21.6395 47.7449 41.9425 35.4053 33.3175 -

60.3781 49.2825 34.9431 26.0540 20.6514 61.1948 50.5003 36.9515 28.9638 24.5628 63.5765 53.9837 42.4196 36.4343 34.2769 54.9910 44.7017 31.5590 23.4691 18.5585 55.8321 45.9507 33.6152 26.4404 22.5269 58.2777 49.5139 39.2145 34.0773 32.3235 49.2706 39.7580 27.8568 20.6275 16.2561 50.1438 41.0637 30.0205 23.7480 20.3821 52.6724 44.7702 35.8717 31.6593 30.2408

62.6924 49.8892 34.5514 25.4154 19.9039 63.4959 51.0732 36.4558 28.0994 23.3639 65.8409 54.4521 41.5900 34.8563 31.5444 57.0787 45.2219 31.2085 22.9291 17.9482 57.9004 46.4206 33.1274 25.6311 21.4321 60.2922 49.8383 38.3250 32.4955 29.7453 51.1235 40.2002 27.5759 20.2182 15.8131 51.9697 41.4359 29.5612 23.0171 19.4187 54.4246 44.9487 34.9370 30.1168 27.9531

RI PT

n=0 3.0212 3.0343 3.0757 3.1523 3.2817 4.1228 4.2047 4.4560 4.9015 5.6237 6.3402 6.5380 7.1370 8.1828 9.8878 3.1005 3.1020 3.1073 3.1195 3.1449 4.2000 4.2733 4.4937 4.8664 5.4174 6.4216 6.6115 7.1770 8.1124 9.4635 3.2112 3.1931 3.1416 3.0636 2.9681 4.3072 4.3672 4.5421 4.8182 5.1818 6.5334 6.7130 7.2312 8.0223 9.0118

AC C

0.4

Ψ 22

SC

2

0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4

Ψ12

M AN U

0

µ

TE D

0.0

λ

EP

c

ACCEPTED MANUSCRIPT Table 7. First three non-dimensional frequencies (ψ ) of C-P AFG nanobeam considering different λ , different 2

n , different c and different µ when m = 1 and δ = 0.5 .

4

0.2

0

2

4

0

2

4

Ψ 32

n =1

n=3

n=0

n =1

n=3

n=0

n =1

n=3

15.3936 14.5844 12.7498 10.7958 09.1324 15.9418 15.2760 13.7894 12.2565 11.0177 17.4696 17.1725 16.5257 15.9108 15.4848 14.2349 13.4759 11.7609 09.9424 08.4004 14.7975 14.1828 12.8188 11.4280 10.3207 16.3551 16.1093 15.5971 15.1495 14.8785 12.9799 12.2753 10.6903 09.0193 07.6094 13.5595 13.0036 11.7809 10.5540 09.5977 15.1502 14.9716 14.6282 14.3734 14.2547

15.3391 14.5354 12.7128 10.7697 09.1142 15.8686 15.1998 13.7002 12.1422 10.8696 17.3460 17.0206 16.2860 15.5462 14.9967 14.1499 13.3994 11.7027 09.9012 08.3714 14.6918 14.0733 12.6973 11.2739 10.1294 16.1944 15.9115 15.2963 14.7182 14.3280 12.8564 12.1648 10.6076 08.9626 07.5712 13.4135 12.8537 11.6138 10.3540 09.3572 14.9455 14.7204 14.2592 13.8741 13.6553

42.9332 35.9129 26.0676 19.5242 15.3836 43.9474 37.3492 28.3074 22.5754 19.1617 46.8335 41.2904 33.9328 29.5160 27.0139 38.8771 32.4305 23.4666 17.5535 13.8250 39.8979 33.8538 25.6658 20.5512 17.5494 42.7902 37.7522 31.2306 27.4723 25.4313 34.5513 28.6956 20.6561 15.4132 12.1266 35.5802 30.1165 22.8449 18.4067 15.8609 38.4782 33.9938 28.4093 25.3870 23.8474

48.0941 40.1871 29.1010 21.7410 17.0949 48.8215 41.2071 30.6998 23.9660 19.9246 50.9316 44.1052 35.0061 29.5582 26.5357 43.6055 36.3333 26.2134 19.5412 15.3464 44.3490 37.3660 27.8240 21.7861 18.2094 46.4983 40.2914 32.1676 27.4538 24.9309 38.8158 32.1956 23.0869 17.1456 13.4353 39.5796 33.2542 24.7431 19.4673 16.4067 41.7766 36.2388 29.1997 25.3079 23.3269

48.8992 40.8888 29.6311 22.1410 17.4082 49.6201 41.8986 31.2035 24.3143 20.1566 51.7123 44.7619 35.4022 29.7035 26.4849 44.2848 36.9366 26.6751 19.8869 15.6135 45.0176 37.9469 28.2327 22.0426 18.3521 47.1375 40.8072 32.4160 27.4545 24.7510 39.3691 32.7074 23.4841 17.4370 13.6531 40.1172 33.7260 25.0542 19.6238 16.4504 42.2718 36.6091 29.2879 25.1540 23.0168

89.5767 64.3178 41.1523 29.2766 22.5359 90.6983 66.1393 44.1633 33.4469 27.7297 93.9685 71.2300 51.7766 42.9047 38.4484 80.7314 57.7801 36.9164 26.2664 20.2250 81.8540 59.5611 39.8359 30.3117 25.2781 85.1202 64.5412 47.2801 39.6236 35.8944 71.2965 50.7222 32.3082 22.9832 17.7027 72.4220 52.4791 35.1776 26.9650 22.6872 75.6861 57.3862 42.5341 36.2235 33.2895

99.5372 71.3740 45.6238 32.4598 24.9945 100.329 72.6435 47.7402 35.4504 28.8066 102.664 76.2984 53.4762 42.9594 37.6684 89.7309 64.0881 40.8754 29.0764 22.3951 90.5375 65.3618 42.9883 32.0609 26.2008 92.9105 69.0239 48.7246 39.5887 35.1044 79.2683 56.2186 35.7098 25.3904 19.5631 80.0943 57.5164 37.8668 28.4387 23.4472 82.5181 61.2371 43.7104 36.1101 32.5111

101.906 73.1612 46.8117 33.3234 25.6695 102.694 74.4276 48.9170 36.2887 29.4379 105.016 78.0646 54.5729 43.6389 38.0760 91.8302 65.6913 41.9430 29.8527 23.0035 92.6265 66.9431 44.0064 32.7605 26.7067 94.9701 70.5386 49.5885 40.0620 35.3387 81.0881 57.6309 36.6449 26.0689 20.0961 81.8969 58.8844 38.7085 28.9826 23.8130 84.2718 62.4817 44.3161 36.3605 32.5648

RI PT

n=0 13.2484 12.5445 10.9520 09.2602 07.8246 13.9906 13.4918 12.3861 11.2646 10.3787 15.9807 15.9639 15.8941 15.8228 15.8118 12.2511 11.5870 10.0906 08.5170 07.1773 13.0016 12.5383 11.5231 10.5143 09.7390 14.9978 15.0075 15.0327 15.1001 15.2201 11.1622 10.5416 09.1506 07.6935 06.4726 11.9209 11.5006 10.5936 09.7174 09.0691 13.9189 13.9702 14.1196 14.3416 14.5853

AC C

0.4

Ψ 22

SC

2

0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4

Ψ12

M AN U

0

µ

TE D

0.0

λ

EP

c

ACCEPTED MANUSCRIPT Table 8. First three non-dimensional frequencies (ψ ) of P-P AFG nanobeam considering different λ , different 2

n , different c and different µ when m = 1 and δ = 0.5 .

4

0.2

0

2

4

0.4

0

4

Ψ 32

n =1

n=3

n=0

n =1

n=3

n=0

n =1

n=3

9.3371 8.9124 7.9191 6.8143 5.8364 10.2560 9.8856 9.0354 8.1247 7.3639 12.5655 12.3097 11.7548 11.2234 10.8418 8.2906 7.9172 7.0426 6.0671 5.2014 9.2449 8.9253 8.1915 7.4070 6.7552 11.5968 11.3906 10.9485 10.5330 10.2387 7.1548 6.8377 6.0927 5.2584 4.5147 8.1554 7.8917 7.2862 6.6400 6.1058 10.5553 10.4057 10.0879 9.7884 9.5681

9.6904 9.2479 8.2139 7.0648 6.0487 10.5709 10.1761 9.2685 8.2932 7.4741 12.7968 12.4976 11.8431 11.2084 10.7483 8.6272 8.2364 7.3219 6.3036 5.4011 9.5359 9.1907 8.3973 7.5468 6.8373 11.7936 11.5408 10.9967 10.4858 10.1307 7.4658 7.1323 6.3500 5.4756 4.6980 8.4124 8.1228 7.4569 6.7455 6.1572 10.7085 10.5101 10.0928 9.7159 9.4616

33.9226 28.7233 21.1229 15.8979 12.5410 35.2364 30.2964 23.2575 18.6500 15.8725 38.8724 34.5055 28.5476 24.9098 22.8775 30.4570 25.7824 18.9524 14.2602 11.2472 31.7610 27.3362 21.0523 16.9709 14.5411 35.3518 31.4824 26.2808 23.2106 21.5743 26.8037 22.6658 16.6335 12.5001 9.8513 28.0965 24.2062 18.7206 15.2089 13.1607 31.6328 28.2976 23.9290 21.4841 20.2656

37.5189 31.7858 23.3833 17.5942 13.8728 38.4475 32.8925 24.8896 19.5698 16.3208 41.0956 35.9866 28.9060 24.5315 22.0932 33.6992 28.5477 20.9905 15.7811 12.4336 34.6361 29.6629 22.5071 17.7742 14.9116 37.2960 32.7684 26.5488 22.7952 20.7756 29.6741 25.1174 18.4351 13.8338 10.8824 30.6219 26.2507 19.9871 15.8875 13.4491 33.2965 29.3879 24.1086 21.0388 19.4629

38.6260 32.7284 24.0862 18.1318 14.3026 39.5629 33.8373 25.5797 20.0728 16.6894 42.2324 36.9286 29.5294 24.8840 22.2318 34.6748 29.3824 21.6186 16.2650 12.8224 35.6146 30.4884 23.0988 18.1886 15.1962 38.2818 33.5639 27.0281 23.0099 20.7907 30.5144 25.8432 18.9893 14.2641 11.2292 31.4589 26.9535 20.4753 16.2048 13.6398 34.1254 30.0289 24.4308 21.1068 19.3563

76.3257 55.5443 35.7701 25.4498 19.5691 77.6636 57.3943 38.5801 29.2221 24.1994 81.5273 62.5235 45.6637 37.7871 33.7831 68.5097 49.8239 32.0705 22.8143 17.5418 69.8328 51.6306 34.7953 26.4754 22.0496 73.6438 56.6391 41.7175 34.9061 31.5418 60.2250 43.6781 28.0567 19.9461 15.3332 61.5335 45.4549 30.7340 23.5527 19.7870 65.2893 50.3724 37.5686 31.9385 29.2753

84.3094 61.3617 39.4996 28.0905 21.5935 85.2408 62.6420 41.4605 30.7789 24.9748 87.9703 66.3088 46.7633 37.5352 32.8594 75.6683 55.0059 35.3607 25.1299 19.3112 76.6062 56.2875 37.3180 27.8156 22.6941 79.3486 59.9518 42.6177 34.5924 30.6252 66.5100 48.1783 30.8711 21.9096 16.8269 67.4587 49.4788 32.8680 24.6585 20.2935 70.2238 53.1842 38.2613 31.5731 28.3826

86.8323 63.2265 40.7198 28.9645 22.2670 87.7752 64.5168 42.6850 31.6457 25.6257 90.5377 68.2018 47.9528 38.2973 33.3476 77.9279 56.6929 36.4710 25.9260 19.9250 78.8697 57.9662 38.3969 28.5565 23.2303 81.6238 61.6019 43.5912 35.1617 30.9475 68.4956 49.6799 31.8614 22.6174 17.3719 69.4395 50.9502 33.7850 25.2564 20.6999 72.1921 54.5723 38.9935 31.9312 28.5290

RI PT

n=0 8.4807 8.0908 7.1809 6.1716 5.2807 9.6783 9.3771 8.6969 7.9902 7.4224 12.4974 12.3628 12.1011 11.9022 11.8093 7.6013 7.2521 6.4372 5.5330 4.7347 8.8169 8.5547 7.9655 7.3597 6.8804 11.6270 11.5319 11.3619 11.2566 11.2302 6.6417 6.3378 5.6280 4.8397 4.1429 7.8788 7.6607 7.1737 6.6794 6.2952 10.6712 10.6235 10.5560 10.5399 10.5615

AC C

2

Ψ 22

SC

2

0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4

Ψ12

M AN U

0

µ

TE D

0.0

λ

EP

c

The other parameter that plays a key role in this study is the tapered ratio of the nanobeam. Therefore, it is of interest to evaluate the influences of this parameter exactly. By this way, Figs. (11)- (16) are allocated to explore the exact effect of the tapered ratio c on the

ACCEPTED MANUSCRIPT non-dimensional frequencies. Among the mentioned figures, Figs. (10)- (14) are dedicated to variation of the first non-dimensional frequency versus the tapered ratio respectively for C-C, C-F, C-P and P-P boundary conditions. It should be mentioned that the other effective parameters n , m and δ are all chosen 1. Particularly, the results related to C-C, C-P and P-P

RI PT

nanobeams specify that the increment in the tapered ratio makes the non-dimensional frequency decrease. This behavior may be attributed to the reduction in the overall structural stiffness. Furthermore, comparing the allotted graphs to various angular velocities highlights

SC

the fact that by increasing the angular velocity gradually, the stiffness-softening effect of the nonlocal parameter alters to the stiffness-hardening effect. This paramount behavior (i.e.

M AN U

phase transition in the nonlocal parameter effect) can be revealed by following the diagrams carefully plotted for a particular boundary condition when different nonlocal parameters and angular velocities are employed. It is clear that similar to the previous diagrams, there exists intersection point in the non-dimensional frequency-tapered ratio diagrams. The results for C-

λ2 = 0 .

TE D

F nanobeams (Fig. 12) are somehow different where the transition tapered ratio appears at Hence, both stiffness-softening and stiffness-hardening trend can be seen in the

diagram. At higher velocities, the transition tapered ratio point disappears and the stiffness-

EP

hardening behavior is only seen in the figures. Additionally, it is indicated that for lower

AC C

values of µ , the slope of the diagrams is positive and this trend is contrary to the higher nonlocal parameters. The similar tendency for decreasing effect of the tapered ratio can be seen in the results obtained for non-dimensional second frequency. In order to show this trend, the graphs associated with C-P and P-P nanobeams are presented as an instance in Figs. (15) and (16). In these figures, the non-dimensional second frequency as a function of the tapered ratio is displayed. As it is evident, compared to the results for the first nondimensional frequency, the phase transition in nonlocal parameter’s behavior will appear in higher angular velocities. Clearly discussing, by following the results associated with various

ACCEPTED MANUSCRIPT angular velocities, it can be understood that by increasing and tending the velocities to higher values, the related diagrams for different nonlocal parameters tend (get closer) to each other and as a result an intersection point and also the phase transition behavior will be revealed in

AC C

EP

TE D

M AN U

SC

RI PT

the plots.

Fig. 11. Variation of first non-dimensional frequency of C-C nanobeam against variation of the tapered ratio c for various nonlocal parameters when n = 1, m = 1 and δ = 1 : (a) λ 2 = 2 , (b) λ 2 = 4 , (c) λ 2 = 6 , (d) λ 2 = 8 .

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 12. Variation of first non-dimensional frequency of C-F nanobeam against variation of the tapered ratio c for various nonlocal parameters when n = 1, m = 1 and δ = 1 : (a) λ 2 = 2 , (b) λ 2 = 4 , (c) λ 2 = 6 , (d) λ 2 = 8 .

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 13. Variation of first non-dimensional frequency of C-P nanobeam against variation of the tapered ratio c for various nonlocal parameters when n = 1, m = 1 and δ = 1 : (a) λ 2 = 2 , (b) λ 2 = 4 , (c) λ 2 = 6 , (d) λ 2 = 8 .

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 14. Variation of first non-dimensional frequency of P-P nanobeam against variation of the tapered ratio c for various nonlocal parameters when n = 1, m = 1 and δ = 1 : (a) λ 2 = 2 , (b) λ 2 = 4 , (c) λ 2 = 6 , (d) λ 2 = 8 .

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

Fig. 15. Variation of second non-dimensional frequency of C-P nanobeam against variation of the tapered ratio c for various nonlocal parameters when n = 1, m = 1 and δ = 1 : (a) λ 2 = 2 , (b) λ 2 = 4 , (c) λ 2 = 6 , (d) λ 2 = 8 .

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 16. Variation of second non-dimensional frequency of P-P nanobeam against variation of the tapered ratio c for various nonlocal parameters when n = 1, m = 1 and δ = 1 : (a) λ 2 = 2 , (b) λ 2 = 4 , (c) λ 2 = 6 , (d) λ 2 = 8 .

Finally, it is of substantial importance to investigate the effects of the FG index as another effective parameter in the analysis. Therefore, Figs. (17) and (18) are represented as an example to demonstrate the evolution of the non-dimensional frequencies with the FG index regarding C-F boundary condition. It should be noted that different nonlocal

ACCEPTED MANUSCRIPT parameters as well as various angular velocities are taken into account in the analysis. Also, the parameters c , m and δ are set to 0.2, 1 and 1 respectively. It is totally revealed that when the FG index increases, the first and second non-dimensional frequencies tend to decrease. Comparing the diagrams of first and second dimensionless frequencies reveals the fact that

RI PT

the variation rates of the lower modes are higher than those of the other modes. In other words, the first frequency is more sensitive to the FG index variations. Totally, the dominant trend observed for the variation of the non-dimensional frequency with the FG index is

SC

decreasing trend. To put it differently, as it was previously mentioned, the nanobeam is made from Ceramic and Aluminum which result the axially graded properties. Setting

n=0

means

M AN U

that the nanobeam is of pure Aluminum. Also, when the FG index n increases and tends to higher values, the volume fraction of the Ceramic increases. Therefore, by increasing the FG index, the effective Young’s modulus (consequently the overall bending stiffness) and the total mass of the nanobeam increase. As anticipated, the total mass and overall bending

TE D

stiffness act in an opposite manner in the way of affecting the frequencies. In other words, increasing the total mass/the overall bending stiffness can result in a decrease/an increase in the frequencies. As it is seen, in our case, the total mass effect overcomes the overall stiffness

EP

effect and consequently the frequencies decrease. However, it is valuable to note that the

AC C

observed behavior (decreasing trend of the frequencies with n ) depends strongly on the distribution of the materials constructing the AFG nanobeam. Clearly discussing, by choosing other materials in the AFGM and also other distribution, an increasing trend for the frequencies may be seen. (This behavior can be seen in the results obtained by Mirjavadi et al. [57]). Furthermore, it is concluded that the variation rate is more considerable at lower ranges of the FG index and enhancement of this parameter leads to the moderate variations.

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 17. Variation of first non-dimensional frequency of C-F nanobeam with the FG index n for various angular velocities when c = 0.2, m = 1 and

δ = 1 : (a) µ = 0 , (b) µ = 0.15 , (c) µ = 0.3 , (d) µ = 0.45 .

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 18. Variation of second non-dimensional frequency of C-F nanobeam with the FG index n for various angular velocities when c = 0.2, m = 1 and

δ = 1 : (a) µ = 0 , (b) µ = 0.15 , (c) µ = 0.3 , (d) µ = 0.45 .

ACCEPTED MANUSCRIPT 5. Conclusions Utilizing DTM, a semi-analytical approach was presented in this paper in order to investigate the vibration characteristics of rotating tapered AFG nanobeams considering various

RI PT

boundary conditions. The governing partial equation was derived using the Newtonian method in conjunction with the Euler-Bernoulli beam theory as well as the nonlocal effects. Afterwards, the separation of variables solution was applied to obtain an equation containing the frequency. Then, DTM and related rules were employed to convert the obtained equation

SC

as well as the boundary conditions to algebraic equations, from which, a characteristic

M AN U

equation was extracted. Solving numerically the resulted characteristic equation yielded the non-dimensional frequencies. After investigating the convergence and accuracy of the numerical method by means of the verification study, some important features were drawn. To this end, comprehensive illustrations were presented to investigate the influences of different effective parameters on the first three non-dimensional frequencies of various

TE D

nanobeams in detail. Results showed that the frequencies of the system increase as the angular velocity increases. Besides, as an important result it was indicated that the nonlocal

EP

parameter exhibits both stiffness-softening and stiffness-hardening effect. In other words, an intersection point corresponds to the transition angular velocity is observed for diagrams in

AC C

which the non-dimensional frequency behavior against the angular velocity variation is examined. The mentioned point divides the region to two territories where both softening and hardening effect of the nonlocal parameter can be separately seen. To put it another way, after the transition point, the stiffness-softening behavior of the nonlocal parameter switches to the stiffness-hardening behavior. The position of this point depends on the type of boundary condition and the value of tapered ratio. In addition, the results revealed the decreasing effect of the tapered ratio on the non-dimensional frequencies. However in the case of C-F boundary condition, the increasing effect is also discovered. Moreover, the

ACCEPTED MANUSCRIPT variation of the non-dimensional frequencies with the FG index variation was analyzed by some illustrations. The results given in this paper present some complementary information about the exact size-dependent vibration behavior of the tapered AFG rotating nanobeams.

structures in which the rotating nanobeams play a pivotal role.

References

RI PT

Therefore, the paper and its results can be applicable in an efficient design of the nano-

SC

[1]Lu L, Guo X, Zhao J. Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. International Journal of Engineering Science. 2017;116:12-24.

M AN U

[2]Ghadiri M, Shafiei N, Akbarshahi A. Influence of thermal and surface effects on vibration behavior of nonlocal rotating Timoshenko nanobeam. Applied Physics A. 2016;122(7):673. [3]Amiri A, Pournaki I, Jafarzadeh E, Shabani R, Rezazadeh G. Vibration and instability of fluid-conveyed smart micro-tubes based on magneto-electro-elasticity beam model. Microfluidics and Nanofluidics. 2016;20(2):38.

[4]Amiri A, Shabani R, Rezazadeh G. Coupled vibrations of a magneto-electro-elastic micro-diaphragm in

TE D

micro-pumps. Microfluidics and Nanofluidics. 2016;20(1):18. [5]Taghavi N, Nahvi H. Pull-in instability of cantilever and fixed–fixed nano-switches. European Journal of Mechanics-A/Solids. 2013;41:123-33.

EP

[6]Malekzadeh P, Shojaee M. Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams. Composites Part B: Engineering. 2013;52:84-92.

AC C

[7]Ghadiri M, Shafiei N. Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method. Microsystem Technologies. 2016;22(12):2853-67. [8]Shafiei N, Kazemi M ,Ghadiri M. Nonlinear vibration behavior of a rotating nanobeam under thermal stress using Eringen’s nonlocal elasticity and DQM. Applied Physics A. 2016;122(8):728. [9]Mohammadi M, Safarabadi M, Rastgoo A, Farajpour A. Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment. Acta Mechanica. 2016;227(8):2207-32.

ACCEPTED MANUSCRIPT [10]Baghani M, Mohammadi M, Farajpour A. Dynamic and stability analysis of the rotating nanobeam in a nonuniform magnetic field considering the surface energy. International Journal of Applied Mechanics. 2016;8(04):1650048. [11]Safarabadi M, Mohammadi M, Farajpour A, Goodarzi M. Effect of surface energy on the vibration analysis of rotating nanobeam. Journal of Solid Mechanics. 2015;7(3):299-311.

from nanoscale building blocks. Nature communications. 2014;5:3632.

RI PT

[12]Kim K, Xu X, Guo J, Fan D. Ultrahigh-speed rotating nanoelectromechanical system devices assembled

[13]Khaniki HB. Vibration analysis of rotating nanobeam systems using Eringen's two-phase local/nonlocal

SC

model. Physica E: Low-dimensional Systems and Nanostructures. 2018;99:310-9.

[14]Aydogdu M. A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E: Low-dimensional Systems and Nanostructures. 2009;41(9):1651-5.

M AN U

[15]Zeighampour H, Beni YT. Free vibration analysis of axially functionally graded nanobeam with radius varies along the length based on strain gradient theory. Applied Mathematical Modelling. 2015;39(18):5354-69. [16]Şimşek M. Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory. Composites Part B: Engineering. 2014;56:621-8.

[17]Amiri A, Talebitooti R, Li L. Wave propagation in viscous-fluid-conveying piezoelectric nanotubes

TE D

considering surface stress effects and Knudsen number based on nonlocal strain gradient theory. The European Physical Journal Plus. 2018;133(7):252.

[18]Ghayesh MH, Farajpour A. Nonlinear mechanics of nanoscale tubes via nonlocal strain gradient theory.

EP

International Journal of Engineering Science. 2018;129:84-95. [19]Deng W, Li L, Hu Y, Wang X, Li X. Thermoelastic damping of graphene nanobeams by considering the

AC C

size effects of nanostructure and heat conduction. Journal of Thermal Stresses. 2018;41(9):1182-200. [20]Li L, Tang H, Hu Y. The effect of thickness on the mechanics of nanobeams. International Journal of Engineering Science. 2018;123:81-91. [21]Tang H, Li L, Hu Y. Coupling effect of thickness and shear deformation on size-dependent bending of micro/nano-scale porous beams. Applied Mathematical Modelling. 2019;66:527-47. [22]Zhu X, Li L. Twisting statics of functionally graded nanotubes using Eringen’s nonlocal integral model . Composite Structures. 2017;178:87-96. [23]Zhu X, Li L. Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity. International Journal of Mechanical Sciences. 2017;133:639-50.

ACCEPTED MANUSCRIPT [24]Zhu X, Li L. Closed form solution for a nonlocal strain gradient rod in tension. International Journal of Engineering Science. 2017;119:16-28. [25]Zhu X, Li L. On longitudinal dynamics of nanorods. International Journal of Engineering Science. 2017;120:129-45. [26]Li L, Hu Y. Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain

RI PT

gradient theory. Computational materials science. 2016;112:282-8.

[27]Najibi A, Talebitooti R. Nonlinear transient thermo-elastic analysis of a 2D-FGM thick hollow finite length cylinder. Composites Part B: Engineering. 2017;111:211-27.

SC

[28]Rajasekaran S. Differential transformation and differential quadrature methods for centrifugally stiffened axially functionally graded tapered beams. International Journal of Mechanical Sciences. 2013;74:15-31. [29]Kiani K. Thermo-elasto-dynamic analysis of axially functionally graded non-uniform nanobeams with

M AN U

surface energy. International Journal of Engineering Science. 2016;106:57-76.

[30]Daneshjou K, Bakhtiari M, Tarkashvand A. Wave propagation and transient response of a fluid-filled FGM cylinder with rigid core using the inverse Laplace transform. European Journal of Mechanics-A/Solids. 2017;61:420-32.

[31]Daneshjou K, Talebitooti R, Tarkashvand A. An exact solution of three-dimensional elasticity for sound

TE D

transmission loss through FG cylinder in presence of subsonic external flow. International Journal of Mechanical Sciences. 2017;120:105-19.

[32]Sheikholeslami M, Ganji D. Nanofluid hydrothermal behavior in existence of Lorentz forces considering

EP

Joule heating effect. Journal of Molecular Liquids. 2016;224:526-37. [33]Sheikholeslami M, Ganji DD. Nanofluid flow and heat transfer between parallel plates considering

AC C

Brownian motion using DTM. Computer Methods in Applied Mechanics and Engineering. 2015;283:651-63. [34]Sheikholeslami M, Ganji D. Heat transfer of Cu-water nanofluid flow between parallel plates. Powder Technology. 2013;235:873-9.

[35]Sheikholeslami M, Ellahi R, Ashorynejad H, Domairry G, Hayat T. Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium. Journal of Computational and Theoretical Nanoscience. 2014;11(2):486-96. [36]Sheikholeslami M, Ganji D, Ashorynejad H. Investigation of squeezing unsteady nanofluid flow using ADM. Powder Technology. 2013;239:259-65.

ACCEPTED MANUSCRIPT [37]Zarepour M, Hosseini SA, Ghadiri M. Free vibration investigation of nano mass sensor using differential transformation method. Applied Physics A. 2017;123(3):181. [38]Ebrahimi F, Salari E. A semi-analytical method for vibrational and buckling analysis of functionally graded nanobeams considering the physical neutral axis position. Comput Model Eng Sci(CMES). 2015;105(2):151-81. [39]Balazadeh N, Sheikholeslami M, Ganji DD, Li Z. Semi analytical analysis for transient Eyring-Powell

RI PT

squeezing flow in a stretching channel due to magnetic field using DTM. Journal of Molecular Liquids. 2018;260:30-6.

[40]Ebrahimi F, Salari E. Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG

SC

nanobeams with various boundary conditions. Composites Part B: Engineering. 2015;78:272-90.

[41]Ebrahimi F, Salari E. Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method. Composites Part B: Engineering. 2015;79:156-69.

M AN U

[42]Ebrahimi F, Ghadiri M, Salari E, Hoseini SAH, Shaghaghi GR. Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams. Journal of Mechanical Science and Technology. 2015;29(3):1207-15.

[43]Rahmani O, Niaei AM, Hosseini S, Shojaei M. In-plane vibration of FG micro/nano-mass sensor based on nonlocal theory under various thermal loading via differential transformation method. Superlattices and

TE D

Microstructures. 2017;101:23-39.

[44]Pradhan S, Murmu T. Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever. Physica E: Low-dimensional Systems and Nanostructures. 2010;42(7):1944-9.

EP

[45]Aranda-Ruiz J, Loya J, Fernández-Sáez J. Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory. Composite Structures. 2012;94(9):2990-3001.

AC C

[46]Pourasghar A, Homauni M, Kamarian S. Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanobeam using the eringen nonlocal elasticity theory under axial load. Polymer Composites. 2016;37(11):3175-80. [47]Ghadiri M, Hosseini S, Shafiei N. A power series for vibration of a rotating nanobeam with considering thermal effect. Mechanics of Advanced Materials and Structures. 2016;23(12):1414-20. [48]Ghadiri M, Shafiei N, Safarpour H. Influence of surface effects on vibration behavior of a rotary functionally graded nanobeam based on Eringen’s nonlocal elasticity. Microsystem Technologies. 2017;23(4):1045-65.

ACCEPTED MANUSCRIPT [49]Azimi M, Mirjavadi SS, Shafiei N, Hamouda A. Thermo-mechanical vibration of rotating axially functionally graded nonlocal Timoshenko beam. Applied Physics A. 2017;123(1):104. [50]Shafiei N, Ghadiri M, Mahinzare M. Flapwise bending vibration analysis of rotary tapered functionally graded nanobeam in thermal environment. Mechanics of Advanced Materials and Structures. 2017:1-17. [51]Preethi K, Raghu P, Rajagopal A, Reddy J. Nonlocal nonlinear bending and free vibration analysis of a

RI PT

rotating laminated nano cantilever beam. Mechanics of Advanced Materials and Structures. 2018;25(5):439-50. [52]Ghafarian M, Ariaei A. Free vibration analysis of a multiple rotating nano-beams system based on the Eringen nonlocal elasticity theory. Journal of Applied Physics. 2016;120(5):054301.

SC

[53]Ebrahimi F, Shafiei N. Application of Eringens nonlocal elasticity theory for vibration analysis of rotating functionally graded nanobeams. Smart Structures and Systems. 2016;17(5):837-57.

[54]Ebrahimi F, Barati MR. Free vibration analysis of couple stress rotating nanobeams with surface effect

M AN U

under in-plane axial magnetic field. Journal of Vibration and Control. 2017:1077546317744719. [55]Eltaher M, Emam SA, Mahmoud F. Static and stability analysis of nonlocal functionally graded nanobeams. Composite Structures. 2013;96:82-8.

[56]Zarrinzadeh H, Attarnejad R, Shahba A. Free vibration of rotating axially functionally graded tapered

2012;226(4):363-79.

TE D

beams. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering.

[57]Mirjavadi SS, Rabby S, Shafiei N, Afshari BM, Kazemi M. On size-dependent free vibration and thermal buckling of axially functionally graded nanobeams in thermal environment. Applied Physics A.

EP

2017;123(5):315.

[58]Sheikholeslami M, Ganji D, Rashidi M. Magnetic field effect on unsteady nanofluid flow and heat transfer

AC C

using Buongiorno model. Journal of Magnetism and Magnetic Materials. 2016;416:164-73. [59]Ju S-P. Application of differential transformation to transient advective–dispersive transport equation. Applied Mathematics and Computation. 2004;155(1):25-38. [60]Reddy J. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science. 2007;45(2-8):288-307. [61]Murmu T, Pradhan S. Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory. Physica E: Low-dimensional Systems and Nanostructures. 2009;41(8):1451-6.