On the Finite Element Model of Rotating Functionally Graded Graphene Beams Resting on Elastic Foundation

Rotating structures can be easily encountered in engineering practice such as turbines, helicopter propellers, railroad tracks in turning positions, and so on. In such cases, it can be seen as a moving beam that rotates around a ﬁxed axis. These structures commonly operate in hot weather; as a result, the arising temperature signiﬁcantly changes their mechanical response, so studying the mechanical behavior of these structures in a temperature environment has great implications for design and use in practice. This work is the ﬁrst exploration using the new shear deformation theory-type hyperbolic sine functions to carry out the free vibration analysis of the rotating functionally graded graphene beam resting on the elastic foundation taking into account the eﬀects of both temperature and the initial geometrical imperfection. Equations for determining the fundamental frequencies as well as the vibration mode shapes of the beam are established, as mentioned, by the ﬁnite element method. The beam material is reinforced with graphene platelets (GPLs) with three types of GPL distribution ratios. The numerical results show numerous new points that have not been published before, especially the inﬂuence of the rotational speed, temperature, and material distribution on the free vibration response of the structure.


Introduction
Recently, due to the development of science and technology, new materials have been invented and widely applied in engineering practice, in which, materials reinforced by graphene platelets (GPLs) are one of the next-generation structural forms. Graphene-reinforced materials have remarkable characteristics such as a very high Young's modulus, great strength, and excellent thermal conductivity. In addition to the low production cost, the material reinforced by GPL is a relatively good choice to fabricate details and structures in a high load-bearing environment. As a result, researchers worldwide are interested in the mechanical behavior of these structures. Yas and Rahimi [1] carried out the thermal vibration analysis of functionally graded porous nanocomposite beams reinforced by graphene platelets using the Timoshenko beam theory and the generalized differential quadrature method. Also, based on the first-order shear deformation beam theory combined with the differential quadrature method, Song et al. [2] investigated the nonlinear free vibration of edge-cracked graphene nanoplatelet-(GPL-) reinforced composite laminated beams resting on a two-parameter elastic foundation in thermal environments. Barati and Shahverdi [3] studied the forced vibration problem of a nanocomposite beam reinforced with different distributions of graphene platelets in thermal environments using the development of a higherorder refined beam element. Wang and his co-workers [4] used a new Ritz-solution shape function and an improved third-order shear deformation theory to capture the solution for free and forced vibrations of a functionally graded polymer nanocomposite beam reinforced with a low content of graphene oxide and excited by a moving load with a constant velocity. Mojiri and Salami [5] combined both Timoshenko beam theory and generalized differential quadrature method to examine the free vibration and dynamic transient response of a multilayer polymer nanocomposite beam resting on an elastic foundation reinforced by graphene platelets nonuniformly distributed through the thickness direction in a thermal environment. Mohammadimehr et al. [6] also used Timoshenko beam theory to conduct a vibration analysis of single-/three-layered micro sandwich beams with porous core and graphene plateletreinforced composite face sheets under magnetic field and elastic foundation. Liu [7] introduced an exact solution of the vibrational characteristics of multilayer magnetic nanocomposite beams reinforced by graphene nanoplatelets. Nematollahi et al. [8] employed a higher-order laminated beam model to find an analytical solution for the nonlinear vibration behavior of thick sandwich nanocomposite beams reinforced by functionally graded graphene nanoplatelet sheets. Tabatabaei-Nejhada and colleagues [9] investigated the out-of-plane vibration characteristics of laminated functionally graded graphene platelets-reinforced composite curved beams bonded by piezoelectric layers using the firstorder shear deformation theory. Manickam et al. [10] used the trigonometric shear flexible beam theory and direct iterative procedure to explore the nonlinear flexural free vibration behavior of size-dependent curved nano/ microbeams with reinforcement of graphene platelets. Based on a refined higher-order shear deformation theory, Arshid and Amir [11] used an analytical method to carry out a vibration analysis of three-layered fluid-infiltrated porous curved microbeams, which were integrated with nanocomposite face sheets reinforced by graphene platelets as lightweight and high-stiffness reinforcements. Zhang et al. [12] presented the vibration analysis of functionally graded graphene platelet-reinforced porous beams resting on a Winkler-Pasternak elastic foundation under a moving load based on the Timoshenko beam theory. Arefi and Najafitabar [13] investigated the buckling and free vibration behavior of sandwich beam, which is composed of a soft core integrated with functionally graded graphene nanoplateletsreinforced composite face sheets. Recently, a variety of studies and presentations of the mechanical response of FG graphene plates with and without elastic foundations have been published [14][15][16][17][18][19][20][21][22]. Furthermore, the works [23][24][25][26][27][28][29][30][31][32] including further intriguing information on the mechanical reaction of beam and plate structures can also be considered.
In practice, some structures can be involved in the rotational movements such as turbines, helicopter propellers, railroad tracks in turning positions, and so on. us, it can be seen as a moving beam that rotates around a fixed axis. Due to the presence of additional centrifugal force when participating in the rotation, these beam structures have much nonidentical mechanical behavior compared to conventional beams. Khosravi et al. [33] employed the Timoshenko beam theory and von Kármán type of kinematic assumptions to explore the effect of uniform temperature elevation on the vibration response of the rotating composite beam reinforced with carbon nanotubes. Timoshenko beam theory and von Kármán type of kinematic assumptions were also employed by Kiani et al. [34] to study the buckling behavior of an anisotropic rotating annular plate under a uniformly compressive load on both inner and outer edges. Arvin et al. [35] investigated the free vibration treatment of pre-and post-buckled rotating functionally graded beams by using the Euler-Bernoulli beam theory. Furthermore, the works [33,[35][36][37] including further intriguing information on the mechanical reaction of beam and plate structures with rotating motion can also be considered. A study of the free vibration of a beam rotating around a fixed axis was recently published [38]; however, it only considered the porous FGM material.
From the short review above, one can see that there are no works dealing with the mechanical behavior of a functionally graded GPL-reinforced composite (FG-GPLRC) beam resting on an elastic foundation, in which the whole structure is embled in a temperature environment, and rotating around one fixed axis. In addition, the initial geometrical imperfection is taken into calculations. So, this work aims to solve this problem using the finite element method combined with the new shear deformation theorytype hyperbolic sine functions, which do not require any shear correction factor but still simulate accurately the mechanical responses of structures. By the numerical results, this paper aims to clearly show the free vibration behavior of the structure taking into account the effect of temperature; thus, there will be special features that have not been mentioned by any work. In addition, the data play an important role in the design and use of GPL-reinforced structures. For example, functionally graded materials with varying microstructures from one material to another one and microstructure sizes ranging over several orders of magnitude may not be adequately modeled using classical continuum mechanics alone but are likely more accurately analyzed using nonclassical continuum mechanics as well as spatial variation (readers can see more in [39]). is study is only concerned with calculating large-scale structures without taking into account the macro level for structures based on continuous mechanics theory, in order to meet the required accuracy. e rest of this paper is organized as follows. Finite element formulations for free vibration analysis of the rotating FG-GPL beam are clearly presented in Section 2. Verification problems are introduced in Section 3. Numerical data and discussions are given in Section 4. Section 5 draws out some main important novel investigations of this paper.

Finite Formulation of Rotating FG-GPL Beam
is work concentrates on the vibration response of the rotating FG-GPL beam with the rotational speed ζ, in which the endpoint of the beam is distance r from a fixed axis Z. e beam contains an initial geometrical imperfection. e beam operates in a uniformly distributed thermal environment and is supported by a multiple-parameter elastic medium with k w and k s as shown in Figure 1.
e dimensions of the beam are the length L, width b, and the thickness h. e beam is made from N L GPLRC layers, with the assumption that the GPL reinforcements are randomly oriented and uniformly dispersed with the volume ratio 2 Mathematical Problems in Engineering V GPL . In general, three kinds of volume fraction distribution are considered as follows: uniformly distributed GPL type (designated as U), functionally graded type (designated as X and O), and layer-wise graded type (designated as X and O). e volume ratio GPL (V GPL ) for the layer k is defined by the following formula [40]: where k � 1 − N L and V * GPL is the total GPL volume fraction, which is calculated as where ρ GPL and ρ m are the mass densities of GPLs and matrix and W GPL is the total GPL weight fraction. is work employs the modified Halpin-Tsai model, and the GPLRC's effective Young's modulus is estimated by [40] E � 3 8 where E m is Young's modulus of matrix and η L and η T are calculated as in which E GPL is Young's modulus of GPLs and ƛ L and ƛ T are GPL geometry factors taking the following forms: where a GPL , b GPL, and t GPL are the GPL length, width, and thickness, respectively. e mass density ρ, thermal expansion coefficient α, and Poisson's ratio ] of GPLRCs are calculated according to the formula presented in [40].
ere have been numerous shear deformation theories to explore the mechanical behavior of beam structures, in which each one includes its advantages. Recently, there is a new type of shear deformation theory that has been developed and employed widely called hyperbolic sine functions [41,42]. So, this paper also aims to use this new theory to establish the finite element formulations of free vibration problems of rotating FGM beams. erefore, the displacement field at any point M (x, z) in the beam has the following expression: in which U 1m is the displacement in the x-direction at the neutral axis and f(z) � z − h sinh(z/h) + z cosh(1/2). e expressions of longitudinal displacement ℘ x , shear displacement ξ xz , and thermal displacement ℘ T of the beam are calculated as in which ΔT is the temperature difference T compared to room temperature T 0 , and the strain components are expressed as Assuming the material operates within an elastic limit, according to the Hook law, the normal stress Ξ x , thermal stress Ξ T , and shear stress Θ xz are expressed as follows: e elastic energy of the FGM beam is calculated as e energy of the elastic foundation is defined as When the FGM beam involves rotational movement around one fixed axis with the speed ζ, the potential energy of the beam generated by this rotational movement is calculated as [43,44] in which the centrifugal force P ζ is defined as [43] e work done by temperature acting in the longitudinal direction of the GPLR beam is defined as e kinetic energy expression of the beam is calculated as To establish the equilibrium equation of the FGM beam, this paper employs Hamilton's principle as follows: Herein, this paper employs a two-node beam element, in which each node has five degrees of freedom: where the displacements at each point in the beam element are approximated through Lagrange and Hermite interpolation functions N i and H i .
Equation (18) can be shortened in the matrix form as follows: e strains are defined according to the nodal displacement as follows: Mathematical Problems in Engineering Now, the expression of the elastic energy of the beam element is calculated as Equation (21) can be shortened in the matrix form as e energies of the centrifugal inertia force and the elastic foundation are calculated, respectively, as follows: 6 Mathematical Problems in Engineering e energy due to thermal strain is calculated as e kinetic energy of the FGM beam element is defined as follows: where M e � V ρ(z)(H T H)dV is the element stiffness matrix. By substituting equations (21)-(23) into (16), the equation to obtain the fundamental frequency, as well as the vibration mode shapes of the rotating FGM beam, can be expressed as follows: Equation (26) shows that all constituents relating to rotational movement, elastic foundation, and initial geometrical imperfection are presented in the equation for finding the fundamental vibration behavior of the FGM beam, and this is completely different from conventional beam structures; therefore, this makes the computation more complicated in comparison with previous works. By looking at the established calculation formulas, the reader may see that the approach employed in this work is based on the theory of shear strain hyperbolic sine functions, which does not require shear correction. e reason is that the shear correction factor is dependent on the structure's material, and it is difficult to predict its value precisely; therefore, the theory utilized in this study will more accurately explain the structure's reaction than the first-order shear strain theory. Furthermore, the separation of z-axis displacement into two components related to bending (U 3b ) and shear (U 3s ) allows the mechanical response of the beam to be precisely described. Because the shear strain theory used in this study has fewer components than higher-order shear strain theories, it takes less time to compute.
Boundary conditions are defined by the following expressions: (i) Simply supported (denoted as S): (ii) Clamped (denoted as C): In this work, three boundary conditions are considered for calculations of FGM beams. Example 1. At first, this example collates the natural frequency of the homogeneous beam subjected to the axial compressive load. Geometrical and material characteristics are the length L, cross section b × h, L/h � 100, E � 380 GPa, ] � 0.3, and ρ � 3960 kg/m 3 . e structure is subjected to an axial compressive load N 0 , whereN � (N 0 L 2 /π 2 EI 0 ) � 0.6. e nondimensional fundamental frequency is normalized by the formula ω � ω (ρbhL 4 /EI 0 ) (I 0 � bh 3 /12). e data obtained from this example, exact expressions [45,46], and FEM modeling [47] are presented in Table 1, when this example calculates using a variety of different meshes. It can be observed that when the mesh size increases, the numerical results ensure the necessary convergence. e results in this work with the 8-element mesh size are different from the data of the same mesh size used in [47]. e cognition is that reference [47] employed Timoshenko's first-order shear deformation theory. So, for the 8-element mesh, the accuracy is fine; thus, this mesh size is going to be used for all the following related explorations.

Example 2.
e natural frequency of the FG-GPLRC beam generated from this case is then compared to the finite element method (FEM) [3], which utilized a refined beam theory. e geometry and material specifications are described in [3], with L/h � 10 and W GPL � 1% for a uniformly GPL-reinforced beam. e nondimensional fundamental frequencies ω � ωL 2 ρ m A/E m I are compared, where A � b.h, I � b.h 3 /12. Table 2 shows the calculation and comparison results, which demonstrate that as the mesh size increases, the results converge to a frequency value that is near to the frequency calculated using the finite element approach [3] (based on a refined beam theory which is different from the beam theory used in this work). is shows that the approach employed in this study ensures the requisite level of dependability.
Example 3. Next, the fundamental frequency of the fully simply supported Al/Al 2 O 3 beam resting on Winkler-Pasternak elastic foundation is considered.
Geometrical and material characteristics are the length L, thickness h, L/h � 100, width b, E m � 70 GPa, ρ m � 2702 kg/ m 3 , E c � 380 GPa, and ρ m � 396 kg/m 3 . Two elastic foundation parameters are normalized as follows: with I � bh 3 12 .
e nondimensional fundamental frequency of the beam is defined as follows: Table 3 shows the comparative first fundamental frequency of the beam obtained from this work and the analytical method [48]. It can be seen that with 8 elements, the reliability is acceptable; thus, this work will use this mesh for all the following related investigations.

Example 4.
is example considers the fundamental frequency of the fully simply supported beam with an initial geometrical imperfection. e beam contains geometrical and material parameters L � 288.7 h, h � 0.02 m, b � 0.04 m, E � 971 GPa, and mass density ρ � 2300 kg/m 3 . An initial imperfection of the beam is expressed as w im (x) � J 0 sin(πx), where J 0 is the amplitude of the imperfection. e nondimensional fundamental frequency and the initial imperfection coefficient ς 0 are normalized as follows: e numerical results obtained from this example and the pseudo-arclength continuation technique [49] in the case of increasing gradually the value of ς 0 are presented in      Example 5. Finally, this example presents a verification problem of the cantilever rotating beam with the rotational speed χ. Let us consider a beam with the following geometrical and material properties: the length L, thickness h � b � L/100, r � 0, E � 70 GPa, and mass density ρ � 2700 kg/m 3 . e nondimensional fundamental frequency ω i is defined as follows: e first three nondimensional fundamental frequencies of the rotating beam with different values of the rotational speed ratio ξ � L 2 Θ 12ρ/Eh 2 obtained from this example, an exact solution [50], an isogeometric analysis [43], and a new dynamic modeling method (DMM) [51] are presented in Table 4.

Numerical Results
Now, this section presents the vibration analysis of the rotating porosity FGM beam resting on Winkler-Pasternak elastic foundation under the pre-axial compressive load, in which the initial geometrical imperfection is taken into account. Let us consider a beam with the following geometrical and material characteristics: the length L, cross section b × h, thickness h, L/ h � 10, a GPL � 2.5 μm, b GPL � 1.5 nm, t GPL � 2.5 μm, E GPL � 1.01 TPa, ] GPL � 0.186, α GPL � 5.10 − 6 1/K, ρ GPL � 1062 kg/m 3 , E m � 3.0 GPa, ] m � 0.34, α m � 60.10 − 6 1/K, and ρ m � 1200 kg/m 3 . e imperfection of beam is w 0 (x) � M 0 sin(πx), in which M 0 is the amplitude of imperfection and the imperfection ratio is c 0 � M 0 /L. e nondimensional fundamental frequency and other parameters are calculated as follows: 4.1. Effect of the GPL Weight Fractions. Firstly, this section examines the effect of the GPL weight fractions on the vibration behavior of the GPLR beam. e beam has L/ h � 10, imperfection ratio c 0 � 0.001, two elastic foundation parameters K * w � 20, K * s � 5, distance ratio a/L � 1, rotational speed ϕ � 5, and ΔT � 30 K. We change the GPL weight fractions so that W GPL obtains the values in a range of 0-0.5%. e first five nondimensional frequencies of the rotating U-GPLRC beam depending on W GPL with two boundary conditions are presented in Figures 3, and 4 presents the dependence of the first five nondimensional frequencies on W GPL of the X-GPLRC, U-GPLRC, and O-GPLRC beams. One can see the following.
When increasing the GPL weight fractions W GPL , the frequencies of the structure rise as the GPL weight grows. W GPL , in turn, determines the first five frequencies of the beam in a nonlinear manner. However, it is much more obvious when looking at the higher frequencies, since there is a nonlinear relationship between these dependence on the W GPL coefficient.
When increasing the coefficient W GPL , the X-GPLRC beam has the highest natural frequency, and the O-GPLRC beam has the lowest frequency. is proves that the law of material distribution affects the mass and stiffness of the GPLR beam at the same time; as a result, the frequencies of this beam are different.

Effect of Temperature.
is section investigates the effect of the temperature on the free vibration response of the beam. Consider a beam with L/h � 10, c 0 � 0.001, and two elastic foundation parameters K * w � 20, K * s � 5. We change the acting temperature on the beam so that ΔT varies from 0 to 400 K. e dependencies of the first five natural frequencies of the beam on the temperature with two cases of boundary conditions S-S and C-C are shown in Figure 5. Figure 6 presents the dependence of the first four natural frequencies of the beam on the temperature with three cases of material distribution for S-S boundary. e first natural frequency of the beam depending on the W GPL ratio is presented in Figure 7. e numerical data point out the following.
As the temperature increases, the natural frequencies for all three cases of material distribution decrease. At different temperatures, the natural frequency of the X-GPLRC beam has the largest natural frequency, while the natural frequency of the O-GPLRC beam has the smallest value.
Interestingly, drawn from Figure 7, there exists a value of temperature for any value of the WGPL coefficient the first natural frequency of the beam does not change; in other words, there exists a value of temperature whose first natural frequency of the beam does not depend on the ratio of the graphene material, called ΔT cr . For the three material distribution cases of beams, the X-GPLRC beam has the largest value of ΔT cr and approximately 620 K, the O-GPLRC beam has the minimum value and approximately 480 K, and the value of ΔT cr for the U-GPLRC beam is approximately 550 K. is rule means that, before and after this temperature, the variation of the first natural frequency depends on the change in the W GPL ratio.

Effect of Rotational Speed.
To understand the effect of the rotational speed on the free vibration response of the beam, changing the value of ϕ in a range of 0 to 15, the dependencies of the first five natural frequencies of the O-GPLRC and U-GPLRC beams are shown in Figure 8. Figure 9 shows the dependence of the first four natural frequencies of the X-GPLR beam on the rotational speed and the W GPL ratio. Figure 10 presents the first natural frequency of the X-GPLR beam on temperature with different values of the rotational speed and the W GPL ratio. One can see the following.
As the speed of rotation of the beam increases, the stiffness of the beam increases due to the influence of the inertial force component, so the natural frequency of the beam increases. e inertia impact on the rotation speed is  only noticeable when the rotation speed coefficient is less than 2, but when ϕ > 2, the natural frequency of the beam changes strongly. As the speed of rotation increases, the value of ΔT cr also increases, i.e., the temperature value, so that the first natural frequency of the beam independent of the graphene content also increases.

Effect of Elastic Foundation
Parameters. Finally, this work considers the effect of the elastic foundation parameters of the GPLRC beam on the free vibration response of the structure. Let k * w increase gradually from 0 (no elastic foundation) to 100, and k * s gets the values from 0 to 10. e dependence of the first natural frequency of the beam on the elastic foundation parameters and the W GPL ratio is plotted in Figure 11. is numerical result shows that when increasing the stiffness coefficient of the elastic foundation, the natural frequency of the beam increases. In addition, the elastic foundation also affects the value of ΔT cr ; when the stiffness coefficient of the elastic foundation increases, the value of ΔT cr also increases, but the increase of ΔT cr is not much.

Conclusions
is work firstly uses the finite element method combined with the new shear deformation theory-type hyperbolic sine functions to investigate the free vibration analysis of rotating FG-GPLRC beams resting on two-parameter elastic foundations in the thermal environment, in which the initial geometrical imperfection is taken into account. e numerical results point out that (i) With the increase of the weight fractions W GPL , the natural frequency of the beam increases, and the first five natural frequencies of the beam depend nonlinearly on the W GPL ratio. (ii) As the temperature increases, the natural frequency of the GPLRC beam decreases, and there exists a value of temperature so that the natural frequency of the beam does not depend on the proportions of the graphene material.   (iii) When the speed of rotation of the GPLRC beam increases, the natural frequency of the beam increases; however, when the rotational speed is small, the natural frequency increases slightly, and the natural frequency increases significantly when the rotational speed is high.
Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.