Document Type : Research Paper
Authors
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Abstract
Keywords
Application of BiDirectional Functionally Graded Material Model for Free Vibration Analysis of Rotating EulerBernoulli Nanobeams
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
KEYWORDS 

ABSTRACT 
Rotating nanobeam Generalized differential quadrature method Nonlocal elasticity theory

In this work, mechanical vibration analysis of rotating bidirectional functionally graded EulerBernoulli nanobeams is investigated, which has not already been studied deeply based on the latest authors’ knowledge. Material properties vary along the thickness and axis directions based on powerlaw distribution. The nonlocal elasticity theory of Eringen (NET) is utilized for modeling smallscale effects. Different boundary conditions are considered as clampedclamped (CC), clampedsimply (CS), and clampedfree (CF). Governing equations and associated boundary conditions are derived based on minimum total potential energy, and the generalized differential quadrature (GDQ) method is employed for the solution process. Convergence and verification studies are accomplished to affirm this work, and in the continuation, the effects of various parameters, namely hub ratio, rotation speed, and power indexes along x and z directions on the dimensionless natural frequencies, are investigated. It is revealed that the decrement made by the different values of in the natural frequency, parameter is more effective than the reduction caused by the , especially for the higher rotation speed.

Beams and columns made of functionally graded materials have been found in many different crafts such as civil, mechanical, and aerospace engineering and so on for the reason of desirable performance, namely eliminating delamination unfavorable events due to loading conditions, high strength to weight ratio, high corrosion and heat resistance, high machining capability. It is expected to expire the bending [1, 2] vibration [37] and buckling [8] analyses of beams and columns due to their simplicity and vast application for many years in the endurance of compressive and tensile forces and also as more than three decades go by introducing the concept of functionally graded materials [9]. However, their studies are the best topics of researchers even until now, and research is tracked to fill the gaps. Sherafatnia et al. [10] extended an analytical solution for the free vibration and buckling analysis of functionally graded material beams with edge cracks. They used different theories such as EulerBernoulli, Rayleigh, shear, and Timoshenko. Due to the hard operating situations, namely temperature and stress distribution in two or three directions, it is necessary to analyze the structures made of multidirectional functionally graded materials. Moreover, in practical cases, material properties vary in multidirection [1113]. Li Li et al.[14] developed the nonlinear bending behavior of bidirectional functionally graded EulerBernoulli beam throughout the GDQ method. They [15] also extended the free torsional vibration behavior of nanotubes made of a bidirectional functionally graded material model. Properties were supposed to change along the length and radius directions based on exponential and powerlaw distribution functions, respectively. It is found that the torsional frequencies are increased by decreasing the nonlocal parameter.
The rotation effect on the natural frequency is the most important parameter with various applications in engineering science, such as gas turbines, helicopter rotors, and wind turbines. Rotary structures have more complicated behavior than static ones, which is because of the rotation parameter. Therefore, vast efforts have been made to analyze these structures in recent years [1619]. The static and dynamic analyses of micronano structures have attracted the consideration of engineering scientists with the increase of the usage of nanoelectromechanical systems in the automotive industry. However, it is not accurate to apply classical theories for the analysis of micronano structures. On the other hand, dynamic simulations implementation on the nanostructures is computationally expensive. Thus the usage of sizedependent theories is suggested. Up to now, various sizedependent theories such as the modified couple stress theory (MCST) [20, 21], the modified strain gradient theory (MSGT) [22, 23], the NET [24, 25] and the nonlocal strain gradient theory (NSGT) [2628] are proposed.
In 2016, Ebrahimi and Shafiei [29] performed the free vibration of rotating functionally graded nano cantilever beam with the aid of the GDQ method. After that, researchers have performed much research on the vibration of micronano rotating beams made of functionally graded materials based on MCST, MSGT, and NET. Karamanli and Aydogdu [30] exploited a bidirectional functionally graded porous sandwich materials model on the rotating microbeams. In their research, a fourvariable refined and normal deformation beam theory was used based on MCST via the finite element method.
By considering the surveyed literature review, it can be concluded that there is no comprehensive research on the simultaneous effect investigation of axial functionally graded (AFG) and conventional functionally graded (CFG) power indexes and angular velocity on the free vibration of EulerBernoulli nanobeam via NET. The first accomplished works on the free [35] and forced [36] vibrations of isotropic rotary nanobeams and free vibration of rotary FG [29] nanobeam has been performed with cantilever boundary conditions. Therefore, to fill the detected gap, the free vibration of twodimensional functionally graded material rotary nanobeam is presented here under CC, CS, and CF boundary conditions employing the GDQ method. Material properties vary in both axial and thickness directions according to the powerlaw distribution. Numerical results indicate that increasing the AFG and CFG indexes decreases the first natural frequency. In CC and CS boundary conditions, it is observed that the stiffnesssoftening effect switches to the stiffnesshardening effect. But increasing the rotary speed in the CF boundary condition vanishes the effect of the AFG index on the natural frequency. Moreover, with increasing the angular velocity, the effect of the functionally gradient index along xdirection on the natural frequency is more noticeable than that of the zdirection.
A rotating EulerBernoulli nanobeam made of bidirectional functionally graded is considered as shown in Fig. 1. The Cartesian coordinate system is set at the end of the beam connected to a rigid hub such that its xaxis is taken perpendicular to the crosssectional area of the beam. L, Ω, and R are total length, constant angular velocity, and hub radius.
The material properties are varied in both axial and thickness directions according to the powerlaw distribution [32, 33, 37] as follows:

(1) 
where , , and are Young’s modulus and density of metal and ceramic, respectively and and denote the AFG and CFG indexes. It is also mentioned that Poisson's ratio is assumed to be constant. If is set to zero, the nanobeam is made of conventional FGMs, and if and are set to zero simultaneously, the nanobeam is made of pure ceramic.
Fig. 1. The geometry of rotary EulerBernoulli nanobeam
According to EulerBernoulli nanobeam, and are the displacement fields of an arbitrary point of the beam along x and z directions which are described by:

(2) 
in which u(x,t) and w(x,t) show the displacements of the central plane along with x and z directions, respectively. Based on the EulerBernoulli displacement field with two unknown, the linear straindisplacement relation are obtained by the following equation:

(3) 
Hamilton’s principle is used to derive the equations and related boundary conditions as follows:

(4) 
where U, T, and V are strain, kinetic and potential energies due to constant angular speed around the zaxis. The variation of strain energy is:

(5) 
where N and M are the axial compressive force and bending moment, respectively as follows:

(6) 
According to linear elastic solids, the constitutive equation for FG material can be expressed as:

(7) 
and the variation of kinetic energy is

(8) 
where , , are the mass moments of inertia, defined as:

(9) 
The variation of potential energy corresponding to the angular velocity can be written as:

(10) 
is the rotational force and the following equation determines it:

(11) 
By inserting relations 8, 9, and 11 into hamilton’s principle, performing some mathematical manipulation, and setting the coefficients of and to zero, the governing equations and related boundary conditions can be obtained by following equations:

(12) 
with the related boundary conditions (at x=0, L)

(13) 
NET is an appropriate theory for the analysis of nanostructures propounded by Eringen [38]. In this theory, the empty spaces between atoms are considered by smallscale parameters. Based on this theory, the stress of each arbitrary point x is supposed to be a function of the strain of all the points in the vicinity of x. Hence, the nonlocal stress tensor at any point x is defined as:

(14) 
where, is local stress tensor. denotes nonlocal kernel function and shows a material constant for applying the small scale factor, where is a material constant obtained experimentally, and represent internal and external characteristic lengths of the nanobeam, respectively. The local stress is defined based on Hook’s law as:

(15) 
According to the differential form of nonlocal elasticity theory proposed by Eringen and due to the complexity of integral form, the only stress relation based on EulerBernoulli nonlocal functionally graded material is obtained as :

(16) 
Multiplying and on relation. (16) yield to stress resultants. The governing equations of the nanobeam made of bidirectional functionally graded materials in terms of the displacements are obtained by substituting the stress resultants in equations 12 and 13, as follows:

(17) 
Moreover, the following boundary conditions are obtained as follows:

(18) 
In order to investigate the harmonic vibrations of the present nanobeam, the displacement fields are considered as follows:

(19) 
where ω is the natural frequency, nondimensional relations are used to obtain the dimensionless equations in the following form:

(20) 
By inserting Eqs. 19 and 20 into Eqs. 17 and 18 and applying some mathematical manipulations, the dimensionless forms of the governing equations and related boundary conditions can be rewritten as follows:

(21) 
Related boundary conditions are:

(22) 
It is difficult to obtain an analytical solution for the free vibration of bidirectional functionally graded nanobeam considering various boundary conditions due to the extreme coupling of the governing equations. Hence, the GDQ method is used to obtain the natural frequency response. Recently, many researchers have used this method due to low computational cost and remarkable convergence in low degrees of freedom[31, 32, 37]. In this method, the derivative of a function f(x) can be written as:

(23) 
where N is several grid points in the xdirection, and the position of each point is determined by the ChebyshevGaussLobatto technique as:

(24) 
Also, are the weighting coefficients along the xdirection, and the first weighting coefficient is calculated as:

(25) 
where is determined as:

(26) 
In addition, the recurrence relationship is used to calculate the higherorder weighting coefficients as follows:

(27) 
where m is the order derivative of a function; finally, by implementation the GDQ method on boundary condition equations and assembling the governing equations and related boundary conditions equations, the eigenvalues problem are obtained as:

(28) 
The values of dimensionless natural frequencies are calculated by solving the eigenvalue problem.
This section presents the vibrational behavior of the EulerBernoulli nanobeam made of bidirectional functionally graded materials for general boundary conditions with developed Matlab code. The material properties of the bidirectional FG nanobeam are given in Table 1. Convergence studies and validation are carried out to attain sufficient numbers of grid points. Finally, it graphically shows the effects of nonlocal parameters, powerlaw indexes, nx, nz, and hub ratio.
Table 1. Material properties
Material 
E(Pa) 
) 
SUS304 

8166 
Al_{2}O_{3} 

3800 
Considering the accuracy dependence of the numerical results on the number of grid points obtained from the GDQ method, convergence studies on the first nondimensional frequency are performed to determine the required grid numbers for different boundary conditions in Table 2. In the convergence test, various dimensionless parameters such as τ,δ, ϕ and are considered, represented nonlocal parameter, nondimensional hub ratio, nondimensional angular velocity, and thickness to length, respectively. The dimensionless format of mentioned parameters is shown in Eq. 20. It is observed from Table 2 that all numerical results may be converged in N=16. Obtained numerical results are compared with those available in the literature for the first nondimensional frequency, which demonstrates acceptable accuracy for all different numerical results. Moreover, Fig. 2 shows the results for evaluating the nondimensional natural frequency and its comparison with Ref [39] under the SS boundary condition. The nondimensional parameters are considered as:
Furthermore, good agreement between the obtained results can be seen.
In this section, the free vibration analysis of rotating bidirectional functionally graded EulerBernoulli nanobeam is presented. The effect of various parameters, namely AFG and CFG power indexes and nonlocal and rotary parameters, is studied for CS, CC, and CF boundary conditions. Figs. 36 depict the effect of nondimensional angular velocity on the first nondimensional frequency versus the AFG index considering different nonlocal parameters under CC, CS, and CF boundary conditions, respectively. In these figures, the invariable values of and are selected as 1, 0, and 0.05, respectively. As can be observed, by increasing the angular velocity, the natural frequency of the nanobeam increases. In other words, the stiffness of the 2DFG nanobeam increases for all the mentioned boundary conditions. Increasing the powerlaw index decreases the natural frequency due to the variation of the material properties from ceramic to metal and Young's modulus. For n_{x} >2, increasing n_{x} has a much slower effect on the fundamental frequency and converges to a constant value. The values of frequency for the CC beam are higher than those for the remaining boundary conditions. The reason is that the C and F boundary conditions have the fewest and most degree of freedom.
Table 2. Convergence study and verification




Number of grid points 

ref 

BC 



10 
14 
16 



CF 
0.4 
1 
1 
4.81 
4.70 
4.70 
4.70 
[36] 

CC 
0 
0 
5 
24.55 
24.55 
24.55 
24.54 
[24] 

CS 
0 
0 
5 
17.62 
17.62 
17.62 
17.61 
[24] 

Fig. 2. Comparison of natural frequency parameter results with Ref [39]
Increasing the angular velocity has different trends for the first natural frequency depending on the nonlocal parameter. By observing Figs. 3 and 4, it can be understood that increasing the angular velocity leads to switching the stiffnesssoftening effect of the nonlocal parameter to the stiffnesshardening effect related to CC and CS boundary conditions, respectively. This trend has also been observed for the isotropic nanobeam by Talebitooti et al. [24]. Below a specific value (here Φ≈6), the natural frequency decreases by increasing the nonlocal parameter, whereas, for the larger Φ, it increases. Increasing the angular velocity with fundamental frequency is not monotonous and depends upon the boundary condition. Interchanging behavior of nonlocal parameter happens faster for CC beam.
Figure 5 shows the variation of the first nondimensional natural frequency under the CF boundary condition. It is seen that increasing the nonlocal parameter results in the stiffnesshardening effect in contrast with the other boundary conditions. By increasing the angular velocity and the stiffnesshardening effect of the nanobeam, the effect of the nonlocal parameter becomes more noticeable and does not switch. Similar to this result, it was also reported for the isotropic nanobeam by Murmu et al. [35]. It can be concluded from Figs. 35 that increasing the angular velocity has a stiffnesshardening effect on the nanocal parameter. In other words, it eliminates the stiffnesssoftening effect for CC and CS boundary conditions and increases the stiffnesshardening effect for the CF. The comparison of the results in Fig. 5 for different Φ shows that for Φ >6, the effect of axial FG index becomes to be inconsiderable on the fundamental frequency. By making a comparison
between Figs. 3 to 5, it can be shown that for the CF beam, the effect of velocity dominates on the effect of n_{x} that is much more noticeable than those for other boundary conditions. The main reason that can interpret this phenomenon is that increasing the angular velocity has a more significant effect on increasing the natural frequency of CF nanobeams than those of other boundary conditions. As , the natural frequency increases by approximately four times in comparison to . While in other boundary conditions, it increases by approximately two times. As a result, the stiffnesssoftening effect of the n_{x} index decreases more rapidly in the CF boundary condition than those in other boundary conditions.
Fig. 3. The effect of nondimensional angular velocity on the first nondimensional frequency of CC nanobeam concerning the AFG index with consideration different nonlocal parameters
Figure 6 shows the effect of the hub ratio on the first nondimensional natural frequency for the CC boundary condition. It is mentioned that the CFG index is constant, and the only axial FG index varies. The linear relations are indicated for δ and ψ_{1}. The investigation of this figure shows that by increasing the hub ratio, the first nondimensional frequency increases that is caused by the rising of the beam’s stiffness. By increasing the angular velocity and hub ratio simultaneously, the increasing effect of the hub ratio on the first nondimensional natural frequency happens more rapidly. Again, the effect of this parameter on the first nondimensional natural frequency can be observed in Fig. 6.
On the other hand, the CFG index can vary, and the AFG index is kept constant, Fig. 7. In this figure, the hub ratio acts as Fig. 6. By comparing Figs. 6 and 7, it is revealed that the effect of the AFG index is more significant than that of the CFG index in higher angular velocities. In other words, the effect of disappears in higher angular velocities, and the results related to the thickness FG index become closer to each other than those related to the axial FG index.
In this study, a nonlocal elasticity theory is developed to study the vibrational behavior of rotary bidirectional functionally graded EulerBernoulli nanobeam, including CC, CS, and CF conditions. The prime objective of the current study is the utilization of a bidirectional functionally graded material model in the analysis of rotating nanobeam. The governing equations are solved to obtain the frequency response by using the GDQ method. The effects of material variation, thickness and length direction, nonlocal parameters, angular velocity, and hub ratio are reported. The most important results are shown as follows:
Fig. 4. The effect of nondimensional angular velocity on the first nondimensional frequency of CS nanobeam concerning the AFG index with consideration different nonlocal parameters
Fig. 5. The effect of nondimensional angular velocity on the first nondimensional frequency of CF nanobeam concerning the hub ratio with consideration different conventional FG index
Fig. 6. The effect of nondimensional angular velocity on the first nondimensional frequency of CC nanobeam concerning the hub ratio with consideration different axial FG index
Fig. 7. The effect of nondimensional angular velocity on the first nondimensional frequency of CC nanobeam concerning the hub ratio with consideration different conventional FG index
References
[1] Nguyen DK, Nguyen KV, Gan BS, Alexandrov S, 2018. Nonlinear bending of elastoplastic functionally graded ceramicmetal beams subjected to nonuniformly distributed loads. Applied Mathematics and Computation, 333, pp. 44359.
[2] Vinyas M, Kattimani SC, 2017. Static studies of a stepped functionally graded magnetoelectroelastic beam subjected to different thermal loads. Composite Structures, 163, pp 21637.
[3] Mirzabeigy A, 2014. Semianalytical approach for free vibration analysis of variable crosssection beams resting on elastic foundation and under axial force. International Journal of Engineering, 27(3), pp.385394.
[4] Jafari A, 2015. Size effect on free transverse vibration of cracked nanobeams using couple stress theory. International Journal of Engineering, 28(2), pp.296304.
[5] Keshavarzian M, Najafizadeh MM, Khorshidi K, Yousefi P, Alavi M, 2020. Highorder Analysis of Linear Vibrations of a ModeratleyThick Sandwich Panel With an Electrorheological Core. Mechanics of Advanced Composite Structures, 7(2), pp.177188.
[6] Mohammadrezazadeh S, Jafari AA, 2020. Active control of free and forced vibration of rotating laminated composite cylindrical shells embedded with magnetostrictive layers based on classical shell theory. Mechanics of Advanced Composite Structures, 7(2), pp.355369.
[7] Nejati M, GhasemiGhalebahman A, Soltanimaleki A, Dimitri R, Tornabene F, 2019. Thermal vibration analysis of SMA hybrid composite double curved sandwich panels. Composite Structures, 224, pp.111035.
[8] Babaei H, Kiani Y, Eslami MR, 2019. Thermal buckling and postbuckling analysis of geometrically imperfect FGM clamped tubes on nonlinear elastic foundation. Applied Mathematical Modelling, 71, pp.1230.
[9] Koizumi M, 1997. FGM activities in Japan. Composites Part B: Engineering, 28(12), pp.14.
[10] Sherafatnia K, Farrahi G, Faghidian SA, 2014. Analytic approach to free vibration and buckling analysis of functionally graded beams with edge cracks using four engineering beam theories. International Journal of Engineering, 27(6), pp.97990.
[11] Zafarmand H, Kadkhodayan M, 2015. Threedimensional elasticity solution for static and dynamic analysis of multidirectional functionally graded thick sector plates with general boundary conditions. Composites Part B: Engineering, 69, pp.592602.
[12] Adineh M, Kadkhodayan M, 2017. Threedimensional thermoelastic analysis of multidirectional functionally graded rectangular plates on elastic foundation. Acta Mechanica, 228(3), pp.88199.
[13] Adineh M, Kadkhodayan M, 2017. Threedimensional thermoelastic analysis and dynamic response of a multidirectional functionally graded skew plate on elastic foundation. Composites Part B: Engineering, 125, pp.22740.
[14] Li L, Li X, Hu Y, 2018. Nonlinear bending of a twodimensionally functionally graded beam. Composite Structures, 184, pp.104961.
[15] Li L, Hu Y, 2017. Torsional vibration of bidirectional functionally graded nanotubes based on nonlocal elasticity theory. Composite Structures, 172, pp.24250.
[16] Zarrinzadeh H, Attarnejad R, Shahba A, 2012. Free vibration of rotating axially functionally graded tapered beams. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 226(4), pp.36379.
[17] Chen Y, Zhang J, Zhang H, 2016. Flapwise bending vibration of rotating tapered beams using variational iteration method. Journal of Vibration and Control, 22(15), pp.338495.
[18] Ghadiri M, Shafiei N, Safarpour H, 2017. Influence of surface effects on the vibration behavior of a rotary functionally graded nanobeam based on Eringen’s nonlocal elasticity. Microsystem Technologies, 23(4), pp.104565.
[19] Ghadiri M, Shafiei N, 2016. Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method. Microsystem Technologies, 22(12), pp.285367.
[20.] Fang J, Gu J, Wang H, 2018. Sizedependent threedimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory. International Journal of Mechanical Sciences, 136, pp.18899.
[21] Fang J, Gu J, Wang H, Zhang X, 2019. Thermal effect on vibrational behaviors of rotating functionally graded microbeams. European Journal of MechanicsA/Solids, 75, pp.497515.
[22] Shenas AG, Ziaee S, Malekzadeh P, 2016. Vibrational behavior of rotating pretwisted functionally graded microbeams in thermal environment. Composite Structures, 157, pp.22235.
[23] Shenas AG, Ziaee S, Malekzadeh P, 2019. Postbuckling and vibration of postbuckled rotating pretwisted FG microbeams in thermal environment. ThinWalled Structures, 138, pp.33560.
[24] Talebitooti R, Rezazadeh SO, Amiri A, 2019. Comprehensive semianalytical vibration analysis of rotating tapered AFG nanobeams based on nonlocal elasticity theory considering various boundary conditions via differential transformation method. Composites Part B: Engineering, 160, pp.41235.
[25] Azimi M, Mirjavadi SS, Shafiei N, Hamouda A, 2017. Thermomechanical vibration of rotating axially functionally graded nonlocal Timoshenko beam. Applied Physics A, 123(1), pp.104.
[26] Amiri A, Talebitooti R, Li L, 2018. Wave propagation in viscousfluidconveying piezoelectric nanotubes considering surface stress effects and Knudsen number based on nonlocal strain gradient theory. The European Physical Journal Plus, 133(7), pp.117.
[27] Amiri A, Vesal R, Talebitooti R, 2019. Flexoelectric and surface effects on sizedependent flowinduced vibration and instability analysis of fluidconveying nanotubes based on flexoelectricity beam model. International Journal of Mechanical Sciences, 156, pp.47485.
[28] Li L, Hu Y, 2016. Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material. International Journal of Engineering Science, 107, pp.7797.
[29] Ebrahimi F, Shafiei N, 2016. Application of Eringen's nonlocal elasticity theory for vibration analysis of rotating functionally graded nanobeams. Smart structures and systems, 17(5), pp.83757.
[30] Karamanli A, Aydogdu M, 2019. Sizedependent flapwise vibration analysis of rotating twodirectional functionally graded sandwich porous microbeams based on a transverse shear and normal deformation theory. International Journal of Mechanical Sciences, 159, pp.16581.
[31] Lal R, Dangi C, 2019. Thermomechanical vibration of bidirectional functionally graded nonuniform Timoshenko nanobeam using nonlocal elasticity theory. Composites Part B: Engineering, 172, pp.72442.
[32] Lei J, He Y, Li Z, Guo S, Liu D, 2019. Postbuckling analysis of bidirectional functionally graded imperfect beams based on a novel thirdorder shear deformation theory. Composite Structures, 209, pp.81129.
[33.] Karami B, Janghorban M, Rabczuk T, 2020. Dynamics of twodimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory. Composites Part B: Engineering, 182, pp.107622.
[34] Golmakani M, Rezatalab J, 2015. Nonuniform biaxial buckling of orthotropic nanoplates embedded in an elastic medium based on nonlocal Mindlin plate theory. Composite Structures, 119, pp.23850.
[35.] Pradhan S, Murmu T, 2010. Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever. Physica E: LowDimensional Systems and Nanostructures, 42(7), pp.19449.
[36] Atanasov MS, Stojanović V, 2020. Nonlocal forced vibrations of rotating cantilever nanobeams. European Journal of MechanicsA/Solids, 79, pp.103850.
[37] Shafiei N, Mirjavadi SS, MohaselAfshari B, Rabby S, Kazemi M, 2017. Vibration of twodimensional imperfect functionally graded (2DFG) porous nano/microbeams. Computer Methods in Applied Mechanics and Engineering, 322, pp.61532.
[38] Eringen AC. Nonlocal polar elastic continua, 1972. International journal of engineering science, 10(1), pp.116.
[39] Ehyaei J, Akbarshahi A, Shafiei N, 2017. Influence of porosity and axial preload on vibration behavior of rotating FG nanobeam. Advances in nano research, 5(2), pp.141.