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

Authors: Nguyen Van Dung; Nguyen Chi Tho; Nguyen Manh Ha; Vu Trong Hieu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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.


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## Body

## 1. 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 higher-order 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 platelet-reinforced 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 first-order 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 nanoplatelets-reinforced 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 [14–22]. Furthermore, the works [23–32] 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. Thus, 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, 35–37] 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 theory-type 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]). This 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.The 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 Section2. 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.


## 2. Finite Formulation of Rotating FG-GPL Beam

This work concentrates on the vibration response of the rotating FG-GPL beam with the rotational speedζ, in which the endpoint of the beam is distance r from a fixed axis Z. The beam contains an initial geometrical imperfection. The beam operates in a uniformly distributed thermal environment and is supported by a multiple-parameter elastic medium with kw and ks as shown in Figure 1. The dimensions of the beam are the length L, width b, and the thickness h.Figure 1
The model of the rotating FG-GPL beam with the initial geometrical imperfection. (a) General view. (b) xOz view. (c) Cross-sectional view.
(a)(b)(c)The beam is made fromNL GPLRC layers, with the assumption that the GPL reinforcements are randomly oriented and uniformly dispersed with the volume ratio VGPL. 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). The volume ratio GPL (VGPL) for the layer k is defined by the following formula [40]:(1)VGPLk=VGPL∗:U−GPLRC,VGPLk=2VGPL∗2k−NL−1NL:X−GPLRC,VGPLk=2VGPL∗1−2k−NL−1NL:O−GPLRC,where k = 1 − NL and VGPL∗ is the total GPL volume fraction, which is calculated as(2)VGPL∗=WGPLWGPL+ρGPL/ρm1−WGPL,where ρGPL and ρm are the mass densities of GPLs and matrix and WGPL is the total GPL weight fraction.This work employs the modified Halpin–Tsai model, and the GPLRC’s effective Young’s modulus is estimated by [40](3)E=381+ζLηLVGPL1−ηLVGPLEm+581+ζTηTVGPL1−ηTVGPLEm,where Em is Young’s modulus of matrix and ηL and ηT are calculated as(4)ηL=EGPL/Em−1EGPL/Em−ƛL,ηT=EGPL/Em−1EGPL/Em−ƛT,in which EGPL is Young’s modulus of GPLs and ƛL and ƛT are GPL geometry factors taking the following forms:(5)ƛL=2aGPLbGPLbGPLtGPL,ƛT=2bGPLtGPL,where aGPL, bGPL, and tGPL are the GPL length, width, and thickness, respectively. The mass density ρ, thermal expansion coefficient α, and Poisson’s ratio ν of GPLRCs are calculated according to the formula presented in [40].There 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. Therefore, the displacement field at any point M (x, z) in the beam has the following expression:(6)U1x,z=U1mx,y−z∂U3b∂x−fz∂U3s∂x,U3x,z=U3b+U3s+U3impx,in which U1m is the displacement in the x-direction at the neutral axis and fz=z−hsinhz/h+zcosh1/2.The expressions of longitudinal displacement℘x, shear displacement ξxz, and thermal displacement ℘T of the beam are calculated as(7)℘x=∂U1∂x=∂U1m∂x−z∂2U3b∂x2−gz∂2Uzs∂x2+dU3impdx∂U3b+U3s∂x=∂U1m∂x+z−∂2U3b∂x2+gz−∂2U3s∂x2+dU3impdx∂U3b+U3s∂x=℘1m+z℘1b+gz℘1s+℘1impξxz=∂gz∂z∂U3s∂x=∂gz∂zξ13s℘T=αT−T0=αΔT,in which ΔT is the temperature difference T compared to room temperature T0, and the strain components are expressed as(8)℘1m=∂U1m∂x,℘1b=−∂2U3b∂x2,℘1s=−∂2U3s∂x2,℘imp=dU3impdx∂U3b+U3s∂x.Assuming the material operates within an elastic limit, according to the Hook law, the normal stressΞx , thermal stress ΞT, and shear stress Θxz are expressed as follows:(9)Ξx=E℘x,ΞT=E℘T,Θxz=E21+νξxz.The elastic energy of the FGM beam is calculated as(10)ΠE=12∫VΞxT℘xx+ΘxzTξxzdV=12∫V℘1mTE℘1m+℘1mTzE℘1b+℘1mTgzE℘1s+℘1mTE℘1imp℘1bTEz℘1m+℘1bTEz2εxb+℘1bTEzgz℘1s+℘1bTEz℘1imp℘1sTEgz℘1m+℘1sTεxsTEzgzεxb+℘1sTEzg2z℘1s+℘1sTEgz℘1imp℘1xmpTE℘1m+℘1xmpTEzεxb+℘1impTEgz℘1s+℘1impTE℘1impdV+12∫V∂gz∂z2E21+νξ13sTξ13sdV.The energy of the elastic foundation is defined as(11)ΠF=12b∫LkwU3b+U3s2+ks∂U3b+U3s∂x2dx.When the FGM beam involves rotational movement around one fixed axis with the speedζ, the potential energy of the beam generated by this rotational movement is calculated as [43, 44](12)Πζ=12∫LPζx∂U3b+U3s∂x2dx,in which the centrifugal force Pζ is defined as [43](13)Pζ=12∫Sρzζ2rL−x+12L2−x2dS.The work done by temperature acting in the longitudinal direction of the GPLR beam is defined as(14)Πther=12∫VΞT∂U3b+U3s∂x2dV.The kinetic energy expression of the beam is calculated as(15)Πk=12∫Vρz∂U˙1∂tT∂U˙1∂t+∂U˙3∂tT∂U˙3∂tdV.To establish the equilibrium equation of the FGM beam, this paper employs Hamilton’s principle as follows:(16)δ∫t1t2Πk−ΠE−ΠF−ΠRodt=0.Herein, this paper employs a two-node beam element, in which each node has five degrees of freedom:(17)ue=∑i=12U1mU3biU3si∂U3b∂xi∂U3s∂xi,where the displacements at each point in the beam element are approximated through Lagrange and Hermite interpolation functions Ni and Hi.(18)U1m=∑i=12NiU1mi=Nuue,U3b=∑i=12HiU3bi+Hi+1∂U3b∂xi=Hbue,U3s=∑i=12HiU3si+Hi+1∂U3s∂xi=Hsue,∂U3b∂x=∑i=12∂Hi∂xU3bi+∂Hi+1∂x∂U3b∂xi=Hbxue,∂U3s∂x=∑i=12∂Hi∂xU3si+∂Hi+1∂x∂U3s∂xi=Hsxue,∂2U3b∂x2=∑i=12∂2Hi∂x2U3bi+∂2Hi+1∂x2∂U3b∂xi=Hb2xue,∂2U3s∂x2=∑i=12∂2Hi∂x2U3si+∂2Hi+1∂x2∂U3s∂xi=Hs2xue.Equation (18) can be shortened in the matrix form as follows:(19)u=U1mU3bU3s∂U3b∂x∂U3s∂x=NuHbHsHbxHsxue=Hue.The strains are defined according to the nodal displacement as follows:(20)℘1m=∂U1m∂x=∂Nu∂xue=Buue,℘1b=−∂2U3b∂x2=−Hb2xue=Bbue,℘1s=−∂2U3s∂x2=−Hs2xue=Bsue,℘1imp=∂U3b+U3s∂xdU3impdx=dU3impdxHb+Hsue=Bimpue,ξ13s=∂U3s∂x=Hsxue.Now, the expression of the elastic energy of the beam element is calculated as(21)ΠeE=12ueT∫VBuTEBu+BuTzEBb+BuTgzEBs+BuTEBimpBbTEzBu+BbTEz2Bb+BbTEzgzBs+BbTEzBimpBsTEgzBu+BsTEzgzBb+BsTEg2zBs+BsTEgzBimpBimpTEBu+BimpTEzBb+BimpTEgzBs+BimpTEBimpdVue+12ueT∫V∂gz∂z2E21+νHsxTHsxdVue=12ueT∫VBuTEBu+BuTzEBb+BuTgzEBs+BuTEBimpBbTEzBu+BbTEz2Bb+BbTEzgzBs+BbTEzBimpBsTEgzBu+BsTEzgzBb+BsTEg2zBs+BsTEgzBimpBimpTEBu+BimpTEzBb+BimpTEfzBs+BimpTEBimpdVue+12ueT∫V∂gz∂z2E21+νHsxTHsxdVue.Equation (21) can be shortened in the matrix form as(22)ΠeE=12ueTKeEue.The energies of the centrifugal inertia force and the elastic foundation are calculated, respectively, as follows:(23)ΠeF=12ueTb∫LkwHb+HsTHb+Hs+ksHbx+HsxTHbx+Hsxdxue=12ueTKeFue,ΠeR=12ueb∫LPΘxHbx+HsxTHbx+Hsxdxue=12ueTKeRue.The energy due to thermal strain is calculated as(24)Πther=12ueTb∫LEαΔTHbx+HsxTHbx+Hsxdxue=12ueTKetherue.The kinetic energy of the FGM beam element is defined as follows:(25)Πek=12∫Vu˙eTρzu˙edV=12u˙eT∫VHTHdVu˙e=12u˙eTMeu˙e,where Me=∫VρzHTHdV 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:(26)∑eKeE+KeF+KeR−Kether−ω2∑eMe∑eψe=0.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. The 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 (U3b) and shear (U3s) 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 asS):(27)u0=0,wb=0,ws=0.(ii)
Clamped (denoted asC):(28)u0=0,wb=0,ws=0,∂wb∂x=0,∂ws∂x=0.In this work, three boundary conditions are considered for calculations of FGM beams.(iii)
Fully clamped beam: C-C.(iv)
Fully simply supported beam: S-S.One side is clamped, the other side is free: C-F.


## 3. Verification Problems

Firstly, this section introduces five examples to evaluate the reality of the proposed approach and mechanical models, where the numerical results of these examples are compared with those of other exact works.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 lengthL, cross section b × h, L/h = 100, E = 380 GPa, ν = 0.3, and ρ = 3960 kg/m3. The structure is subjected to an axial compressive load N0, whereN¯=N0L2/π2EI0=0.6. The nondimensional fundamental frequency is normalized by the formula ω˜=ωρbhL4/EI0I0=bh3/12. The 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. The results in this work with the 8-element mesh size are different from the data of the same mesh size used in [47]. The 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.Table 1
First three nondimensional fundamental frequenciesω˜ of the beam subjected to the axial load N¯=0.6.
Boundary conditionVibration modeExact [45]Exact [46]FEM with 8 elements [47]This work4 elements6 elements8 elements10 elements12 elementsS-S16.24—6.246.2476.2436.2426.2426.241236.40—36.4136.50536.40236.38336.37736.375385.81—85.9361.89185.92485.76885.72185.703S-C1—13.0113.0113.01813.00613.00313.00313.0032—47.3547.3847.49347.35647.32347.31347.3093—101.54101.73101.053101.630101.450101.384101.358Example 2.
The 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. The geometry and material specifications are described in [3], with L/h = 10 and WGPL = 1% for a uniformly GPL-reinforced beam. The nondimensional fundamental frequencies ω^=ωL2ρmA/EmI are compared, where A = b.h, I = b.h3/12. 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). This shows that the approach employed in this study ensures the requisite level of dependability.Table 2
The dependence of the first nondimensional fundamental frequencyω^ of the FG-GPLRC beam on the mesh size.
MethodElement numbers23451012203035This work19.792819.755919.748819.746819.745419.745419.745419.745419.7454[3]20.861020.796120.784620.781420.779420.779320.779220.779220.7792Example 3.
Next, the fundamental frequency of the fully simply supported Al/Al2O3 beam resting on Winkler–Pasternak elastic foundation is considered. Geometrical and material characteristics are the length L, thickness h, L/h = 100, width b, Em = 70 GPa, ρm = 2702 kg/m3, Ec = 380 GPa, and ρm = 396 kg/m3. Two elastic foundation parameters are normalized as follows:(29)K¯w=K1L4EmI,K¯s=K2L2EmIπ2,with(30)I=bh312.
The nondimensional fundamental frequency of the beam is defined as follows:(31)ω⌢=ω1L2hρmEm.
Table3 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.Table 3
The first nondimensional fundamental frequencyω⌢ of the beam resting on Winkler–Pasternak elastic foundation, K¯w=10,K¯s=1, S-S.
a/h = 100nModeAnalytical method [48]This work6 elements8 elements10 elements12 elements14 elementsWithout elastic foundationK¯w=10K¯s=0015.4835.4835.4835.4835.4835.483221.93321.93121.92421.92221.92121.921349.35049.38949.31849.29649.28849.284114.2214.2204.2204.2204.2204.220216.88416.88416.87916.87716.87616.876337.98938.03037.97537.95837.95237.949213.8523.8513.8513.8513.8513.851215.40715.40815.40315.40215.40115.401334.66634.70734.65734.64134.63534.633With elastic foundationK¯w=10K¯s=1016.0156.0146.0146.0146.0146.014222.44522.44222.43622.43422.43322.433349.85849.89149.82449.80349.79649.792115.0085.0075.0075.0075.0075.007217.66617.66417.66017.65917.65817.658338.7738.80138.75338.73838.73238.729214.7524.7524.7524.7524.7524.752216.31616.31516.31116.31016.31016.309335.57735.60635.56435.55035.54535.543Example 4.
This example considers the fundamental frequency of the fully simply supported beam with an initial geometrical imperfection. The beam contains geometrical and material parametersL = 288.7 h, h = 0.02 m, b = 0.04 m, E = 971 GPa, and mass density ρ = 2300 kg/m3. An initial imperfection of the beam is expressed as wimx=J0sinπx, where J0 is the amplitude of the imperfection. The nondimensional fundamental frequency and the initial imperfection coefficient ς0 are normalized as follows:(32)ω∗∗=ω112ρL4Eh2,ς0=J0L.
The numerical results obtained from this example and the pseudo-arclength continuation technique [49] in the case of increasing gradually the value of ς0 are presented in Figure 2. An excellent match can be seen between the numerical findings and the outcome.Figure 2
The dependence of the first nondimensional fundamental frequency of the beam with the initial geometrical imperfection on the imperfection coefficientς0.Example 5.
Finally, this example presents a verification problem of the cantilever rotating beam with the rotational speedχ. 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=2700kg/m3. The nondimensional fundamental frequency ω˜i is defined as follows:(33)ω¯i=ωiL212ρEh2.
The first three nondimensional fundamental frequencies of the rotating beam with different values of the rotational speed ratioξ¯=L2Θ12ρ/Eh2 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.Table 4
The dependence of the first three nondimensional fundamental frequenciesω¯i of the cantilever rotational beam on rotational speed ξ¯.
Vibration modeMethodRotational speed ratioξ¯ξ¯=0ξ¯=3ξ¯=6ξ¯=12ω˜1This work3.51614.67907.073412.6098Exact [50]3.51604.79737.360413.1702DMM [51]3.51604.79737.3604—Isogeometric [43]3.51184.77337.3276—ω˜2This work22.024723.119726.111036.4500Exact [50]22.034523.320326.809137.6031DMM [51]————Isogeometric [43]————ω˜3This work61.625462.760266.024277.4512Exact [50]61.697262.985066.684079.6145DMM [51]————Isogeometric [43]————ω˜4This work120.647121.843125.348138.295Exact [50]120.902122.236126.140140.534DMM [51]————Isogeometric [43]————


## 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 lengthL, cross section b × h, thickness h, L/h = 10, aGPL=2.5μm, bGPL=1.5nm, tGPL=2.5μm, EGPL = 1.01 TPa, νGPL=0.186, αGPL=5.10−61/K, ρGPL=1062kg/m3, Em = 3.0 GPa, νm=0.34, αm=60.10−61/K, and ρm=1200kg/m3. The imperfection of beam is w0x=M0sinπx, in which M0 is the amplitude of imperfection and the imperfection ratio is γ0=M0/L. The nondimensional fundamental frequency and other parameters are calculated as follows:(34)ωi∗=ωiLρmEm,Kw∗=kwL4D¯m,Ks∗=ksL2D¯m,ϕ=L2ζ12ρmEmh02,D¯m=Emh0312.


### 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. The beam hasL/h = 10, imperfection ratio γ0=0.001, two elastic foundation parameters Kw∗=20, Ks∗=5, distance ratio a/L = 1, rotational speed ϕ=5, and ΔT=30K. We change the GPL weight fractions so that WGPL obtains the values in a range of 0–0.5%. The first five nondimensional frequencies of the rotating U-GPLRC beam depending on WGPL with two boundary conditions are presented in Figures 3, and 4 presents the dependence of the first five nondimensional frequencies on WGPL of the X- GPLRC, U-GPLRC, and O-GPLRC beams. One can see the following.Figure 3
The dependence of the first five natural frequencies of the rotating U-GPLRC beam on the GPL weight fractions,γ0=0.001, Kw∗=20, Ks∗=5. (a) S-S. (b) C-C.
(a)(b)Figure 4
The dependence of the first four natural frequencies of the rotating X-, U-, and O-GPLRC beams on the GPL weight fractions,γ0=0.001, Kw∗=20, Ks∗=5. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4.
(a)(b)(c)(d)When increasing the GPL weight fractionsWGPL, the frequencies of the structure rise as the GPL weight grows. WGPL, 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 WGPL coefficient.When increasing the coefficientWGPL, the X- GPLRC beam has the highest natural frequency, and the O-GPLRC beam has the lowest frequency. This 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.


### 4.2. Effect of Temperature

This section investigates the effect of the temperature on the free vibration response of the beam. Consider a beam withL/h = 10, γ0=0.001, and two elastic foundation parameters Kw∗=20, Ks∗=5. We change the acting temperature on the beam so that ΔT varies from 0 to 400 K. The 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. The first natural frequency of the beam depending on the WGPL ratio is presented in Figure 7. The numerical data point out the following.Figure 5
The dependence of the first five natural frequencies of the rotating U-GPLRC beam onΔT, γ0=0.001, Kw∗=20, Ks∗=5. (a) S-S. (b) C-C.
(a)(b)Figure 6
The dependence of the first four natural frequencies of the rotating X-, U-, and O-GPLRC beams onΔT, γ0=0.001, Kw∗=20, Ks∗=5. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4.
(a)(b)(c)(d)Figure 7
The dependence of the first natural frequency of the beam on temperature and theWGPL ratio (S-S), a/L = 1, ϕ=5, γ0=0.001, Kw∗=20, Ks∗=5. (a) X-GPLRC. (b) U-GPLRC. (c) O-GPLRC.
(a)(b)(c)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 Figure7, 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 ΔTcr.For the three material distribution cases of beams, the X-GPLRC beam has the largest value ofΔTcr and approximately 620 K, the O-GPLRC beam has the minimum value and approximately 480 K, and the value of ΔTcr for the U-GPLRC beam is approximately 550 K. This rule means that, before and after this temperature, the variation of the first natural frequency depends on the change in the WGPL ratio.


### 4.3. Effect of Rotational Speed

To understand the effect of the rotational speed on the free vibration response of the beam, changing the value ofϕ 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 WGPL ratio. Figure 10 presents the first natural frequency of the X-GPLR beam on temperature with different values of the rotational speed and the WGPL ratio. One can see the following.Figure 8
The dependence of the first five natural frequencies of the beam on the rotational speed (S-S),a/L = 1, γ0=0.001, Kw∗=20, Ks∗=5.
(a)(b)Figure 9
The dependence of the first four natural frequencies of the X-GPLRC beam on the rotational speed and theWGPL ratio (S-S), a/L = 1, γ0=0.001, Kw∗=20, Ks∗=5.
(a)(b)(c)(d)Figure 10
The dependence of the first natural frequency of the X-GPLRC beam on the rotational speed and temperature (S-S),a/L = 1, γ0=0.001, Kw∗=20, Ks∗=5.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. The inertia impact on the rotation speed is only noticeable when the rotation speed coefficient is less than 2, but whenϕ > 2, the natural frequency of the beam changes strongly.As the speed of rotation increases, the value ofΔTcr also increases, i.e., the temperature value, so that the first natural frequency of the beam independent of the graphene content also increases.


### 4.4. 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. Letkw∗ increase gradually from 0 (no elastic foundation) to 100, and ks∗ gets the values from 0 to 10. The dependence of the first natural frequency of the beam on the elastic foundation parameters and the WGPL ratio is plotted in Figure 11. This 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 ΔTcr; when the stiffness coefficient of the elastic foundation increases, the value of ΔTcr also increases, but the increase of ΔTcr is not much.Figure 11
The dependence of the first natural frequency of the X-GPLRC beam on the temperature and the elastic foundation parameters (S-S),a/L = 1, ϕ=5, γ0=0.001.


## 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. The beam hasL/h = 10, imperfection ratio γ0=0.001, two elastic foundation parameters Kw∗=20, Ks∗=5, distance ratio a/L = 1, rotational speed ϕ=5, and ΔT=30K. We change the GPL weight fractions so that WGPL obtains the values in a range of 0–0.5%. The first five nondimensional frequencies of the rotating U-GPLRC beam depending on WGPL with two boundary conditions are presented in Figures 3, and 4 presents the dependence of the first five nondimensional frequencies on WGPL of the X- GPLRC, U-GPLRC, and O-GPLRC beams. One can see the following.Figure 3
The dependence of the first five natural frequencies of the rotating U-GPLRC beam on the GPL weight fractions,γ0=0.001, Kw∗=20, Ks∗=5. (a) S-S. (b) C-C.
(a)(b)Figure 4
The dependence of the first four natural frequencies of the rotating X-, U-, and O-GPLRC beams on the GPL weight fractions,γ0=0.001, Kw∗=20, Ks∗=5. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4.
(a)(b)(c)(d)When increasing the GPL weight fractionsWGPL, the frequencies of the structure rise as the GPL weight grows. WGPL, 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 WGPL coefficient.When increasing the coefficientWGPL, the X- GPLRC beam has the highest natural frequency, and the O-GPLRC beam has the lowest frequency. This 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.


## 4.2. Effect of Temperature

This section investigates the effect of the temperature on the free vibration response of the beam. Consider a beam withL/h = 10, γ0=0.001, and two elastic foundation parameters Kw∗=20, Ks∗=5. We change the acting temperature on the beam so that ΔT varies from 0 to 400 K. The 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. The first natural frequency of the beam depending on the WGPL ratio is presented in Figure 7. The numerical data point out the following.Figure 5
The dependence of the first five natural frequencies of the rotating U-GPLRC beam onΔT, γ0=0.001, Kw∗=20, Ks∗=5. (a) S-S. (b) C-C.
(a)(b)Figure 6
The dependence of the first four natural frequencies of the rotating X-, U-, and O-GPLRC beams onΔT, γ0=0.001, Kw∗=20, Ks∗=5. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4.
(a)(b)(c)(d)Figure 7
The dependence of the first natural frequency of the beam on temperature and theWGPL ratio (S-S), a/L = 1, ϕ=5, γ0=0.001, Kw∗=20, Ks∗=5. (a) X-GPLRC. (b) U-GPLRC. (c) O-GPLRC.
(a)(b)(c)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 Figure7, 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 ΔTcr.For the three material distribution cases of beams, the X-GPLRC beam has the largest value ofΔTcr and approximately 620 K, the O-GPLRC beam has the minimum value and approximately 480 K, and the value of ΔTcr for the U-GPLRC beam is approximately 550 K. This rule means that, before and after this temperature, the variation of the first natural frequency depends on the change in the WGPL ratio.


## 4.3. Effect of Rotational Speed

To understand the effect of the rotational speed on the free vibration response of the beam, changing the value ofϕ 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 WGPL ratio. Figure 10 presents the first natural frequency of the X-GPLR beam on temperature with different values of the rotational speed and the WGPL ratio. One can see the following.Figure 8
The dependence of the first five natural frequencies of the beam on the rotational speed (S-S),a/L = 1, γ0=0.001, Kw∗=20, Ks∗=5.
(a)(b)Figure 9
The dependence of the first four natural frequencies of the X-GPLRC beam on the rotational speed and theWGPL ratio (S-S), a/L = 1, γ0=0.001, Kw∗=20, Ks∗=5.
(a)(b)(c)(d)Figure 10
The dependence of the first natural frequency of the X-GPLRC beam on the rotational speed and temperature (S-S),a/L = 1, γ0=0.001, Kw∗=20, Ks∗=5.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. The inertia impact on the rotation speed is only noticeable when the rotation speed coefficient is less than 2, but whenϕ > 2, the natural frequency of the beam changes strongly.As the speed of rotation increases, the value ofΔTcr also increases, i.e., the temperature value, so that the first natural frequency of the beam independent of the graphene content also increases.


## 4.4. 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. Letkw∗ increase gradually from 0 (no elastic foundation) to 100, and ks∗ gets the values from 0 to 10. The dependence of the first natural frequency of the beam on the elastic foundation parameters and the WGPL ratio is plotted in Figure 11. This 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 ΔTcr; when the stiffness coefficient of the elastic foundation increases, the value of ΔTcr also increases, but the increase of ΔTcr is not much.Figure 11
The dependence of the first natural frequency of the X-GPLRC beam on the temperature and the elastic foundation parameters (S-S),a/L = 1, ϕ=5, γ0=0.001.


## 5. Conclusions

This 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. The numerical results point out that(i)
With the increase of the weight fractionsWGPL, the natural frequency of the beam increases, and the first five natural frequencies of the beam depend nonlinearly on the WGPL 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.



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*Source: 1586388-2021-07-27.xml*