# Free Vibration Exploration of Rotating FGM Porosity Beams under Axial Load considering the Initial Geometrical Imperfection

Authors: Nguyen Thi Giang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In practice, some components in large structures such as the connecting rods between the rotating parts in the engines, turbines, and so on, can model as beam structures rotating around the fixed axis and subject to the axial compression load; therefore, the study of mechanical behavior to these structures has a significant meaning in practice. This paper analyzes the vibration responses of rotating FGM beams subjected to axial compressive loads, in which the beam is resting on the two-parameter elastic foundation, taking into account the initial geometrical imperfection. Finite element formulations are established by using the new shear deformation theory type of hyperbolic sine functions and the finite element method. The materials are assumed to be varied smoothly in the thickness direction of the beam based on the power-law function with the porosity. Verification problems are conducted to evaluate the accuracy of the theory, proposed mechanical structures, and the calculation programs coded in the MATLAB environment. Then, a parameter study is carried to explore the effects of geometrical and material properties on the vibration behavior of FGM beams, especially the influences of the rotational speed and axial compressive load.


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## Body

## 1. Introduction

Functionally graded material (FGM) is commonly made of ceramic and metal in a certain proportion of ingredients; this is a new material that fully carries the characteristics of the material components such as good heat resistance, corrosive resistance, and withstanding thermal stress. Therefore, studying the mechanical behavior of structures made of FG materials has great practical value, especially in designing, manufacturing, and using them in the most effective way. Also, large mechanical systems such as helicopter blades, aircraft, and ship propellers and even nanosize devices such as microturbines may include blade-like structures that occur in rotational movements, where they can be considered as rotating beam structures. Hence, investigations of rotating integrated beam structures play a very important role in science. There have been a number of research projects and exploration results on these issues. Saito and Otomi [1] researched the stability and vibration of a beam under axial and tangential loads, where the analysis is based on the Timoshenko beam theory, and the effects of the attached mass are expressed with Dirac delta functions. Esmailzadeh and Ohadi [2] introduced a solution for stability and vibration response analysis of a beam subjected to both tangential and axial loads using the Frobenius method and Timoshenko beam theory, in which the axial force is assumed to be tangential to the axis of the beam-column. Sloss et al. [3] used Green’s function to find the solution of the vibration control problem of piezoelectric beams subjected to axial load. Zhang et al. [4] studied the vibration problem of carbon nanotubes subjected to axial load using Bernoulli–Euler theory. Alkharabsheh and Younis [5] investigated the effect of axial forces on the static behavior and the fundamental natural frequency of electrostatically actuated MEMS arches using Euler–Bernoulli theory. Babilio [6] employed the Galerkin procedure to study the dynamics of an FGM beam under axial load, in which Taylor expansion was used to establish the partial differential equation. Barry et al. [7] gave out equations of motion of a beam under axial loads; then, the solution of the vibration response of this beam was introduced, in which this beam was attached to a mass-spring-mass system. Farokhi and Ghayesh [8] researched the nonlinear bending and resonant behavior of a microarch based on Euler–Bernoulli theory and the Galerkin method. Ghannadiasl and Mofid [9] used Green function to investigate the free vibration of a beam subjected to axial loads using Timoshenko beam theory and an analytical method. Nejati et al. [10] employed the generalized differential quadrature approach to explore the buckling and vibration behavior of the FGM beam reinforced with carbon nanotubes. A two-node C1 beam element was used by Thuc et al. [11] to analyze the free vibration of composite beams under axial load using a four-unknown shear and normal deformation theory. Waddar et al. [12] showed an experimental study of buckling and the dynamic response of the cenosphere reinforced epoxy composite core sandwich beam with sisal fabric/epoxy composite facings subjected to axial load. Zhao et al. [13] presented the exact solutions of the forced transverse vibration of a coupled Timoshenko double-beam system under compressive axial load based on Timoshenko beam theory and Green’s functions; the solution could be suitable for any boundary conditions. Szymczak and Kujawa [14] presented an exact formulation of the frequency for free torsional vibration analysis of laminated beams subjected to axial load. Xu et al. [15] used the energy principle in conjunction with the Rayleigh–Ritz procedure to investigate the flexural vibration and its power flow of an axially loaded beam with arbitrary boundary and nonuniform elastic foundation. Brahimi-Mamaghani et al. [16] employed an analytical approach to finding the solution of dynamic and vibration stability of a viscoelastic FGM beam under axial load, where the Laplace transform and a Galerkin discretization scheme were combined to capture the solution. Khosravi et al. [17] used the first-order shear deformation plate theory to investigate the nonaxisymmetric buckling behaviour of isotropic homogeneous annular plates subjected to simultaneous effects of uniform temperature rise and constant angular speed.Recently, the explorations of mechanical responses of rotating beam structures have also been in interest and researched by scientists. Khosravi et al. [18] employed Timoshenko beam theory and von Kármán type of kinematic assumptions to examine the influence of uniform temperature elevation on the vibration of rotating composite beams reinforced with carbon nanotubes which may lead to instability. Timoshenko beam theory and von Kármán type of kinematic assumptions were also used by Khosravi and his coworkers [19] to study the thermal buckling of rotating carbon nanotube-reinforced composite beams. Kiani et al. [20] investigated the buckling behaviour of an isotropic homogeneous rotating annular plate subjected to uniform compression on both inner and outer edges. Bagheri et al. [21] researched the free vibration treatment of pre- and postbuckled rotating functionally graded beams using the Euler–Bernoulli beam theory. Hosseini et al. [22] analyzed the buckling and postbuckling of rotating clamped-clamped functionally graded beams due to the interaction between the thermal environment and the rotation based on the Euler–Bernoulli beam theory. From these works, one can see that the boundary condition has a significant influence on the mechanical responses of the rotating beam, especially to the natural frequencies as well as the vibration mode shapes of this structure. In addition, boundary details and material degradation also have a significant effect on these responses, which can be better understood in [23, 24].In the fabrication process, there can be many types of defects, in which the porosities are easy to appear, dramatically altering the mechanical response of these structures. There have been a number of scientific studies on the free and forced vibration responses and static bending of beams and plates made of FG materials with pore defects [25–32].From the published research studies, readers can find that previous studies only mentioned the mechanical behavior of porosity beam structures made of FG materials, whether or not resting on elastic foundations, while some works investigated the effect of initial compressive force with rotational movement around one fixed axis, where these studies have not fully presented beams with both pore defects and imperfections subjected to both axial compression and elastic foundation effect. Besides, this proposed mathematical model is close to the technical reality models, which can help to guide in designing, manufacturing, and using them effectively.Therefore, this work is about to focus on these proposed problems by using the finite element method and a new type of shear deformation theory in the form of hyperbolic sine functions. This is a simple theory, which takes into account the effect of shear strain without any shear correction factors, and it is an advantage of computational theory in this work. This is the first work using this beam theory to solve the problem of free vibration of rotating FGM beams considering both shape and material defects, thereby giving the influence of different parameters on the free vibration response of this structure, especially the influence of the rotational movement. This is the new point of this work, in which the calculation results are of great value for the designers and users as mentioned above.The rest of this work is structured as follows. Finite element formulations for free vibration analysis of rotating FGM beams resting on two-parameter elastic foundations are fully presented in Section2, in which the initial geometrical imperfection and the porosity are considered. Verification examples are introduced in Section 3. Numerical results and discussion are abundantly presented in Section 4. Section 5 draws out some main important highlight explorations of this paper.


## 2. Finite Element Model of Rotating FGM Beams with Geometrical Imperfection and Porosity

This work is about to focus on the vibration analysis of the FGM beam rotating with the rotational speedΘ, in which one side of the beam is at distance r from a fixed axis Φ. The beam has an initial geometrical imperfection and the porosity. The structure is subjected to an axial compressive load N0 and resting on the two-parameter elastic foundation with two parameters kw and ks as shown in Figure 1. The geometrical dimensions are the length L, width b, and thickness h.Figure 1
A rotating FGM beam subjected to axial compressive loads: (a) 3D view, (b)xOz view, and (c) cross-sectional view.
(a)(b)(c)The beam is made from two main components including ceramic (denoted asc) and metal (denoted as m) with the corresponding proportions Vc and Vm, which are calculated by the following function [33–38]:(1)Vc=zh+12n;Vm=1−Vcwithn≥0,in which z is the thickness coordinate variable with −h/2≤z≤h/2 and n is the volume fraction gradient index and its variation.Then, the mechanical properties are changed in thez-direction based on the power-law distribution as(2)Ez=Em+Ec−Em12+zhn−α2Ec+Emρz=ρm+ρc−ρm12+zhn−α2ρc+ρmνz=νm+νc−νm12+zhn−α2νc+νm,where E is Young’s modulus, ρ is the density, ν is Poisson’s ratio, and α is a porosity volume fraction (α < 1).There have been many shear deformation theories to investigate the mechanical responses of beam structures, where each one has its advantages. Recently, a new type of shear deformation theory has been developed and used widely, which is hyperbolic sine functions [39, 40]. This work also employs this new theory to establish the finite element equations of free vibration problems of rotating FGM beams. So, the displacement field at any point (x, z) within the beam has the following form:(3)uxx,z=ux0x,y−z∂uzb∂x−fz∂uzs∂x,uzx,z=uzb+uzs+uzimx,where fz=z−hsinhz/h+zcosh1/2.The expressions of strain components of the beam are calculated as(4)εx=∂ux∂x=∂ux0∂x−z∂2uzb∂x2−gz∂2uzs∂x2+duzimdx∂uzb+uzs∂x=∂ux0∂x+z−∂2uzb∂x2+gz−∂2uzs∂x2+duzimdx∂uzb+uzs∂x=εx0+zεxb+gzεxs+εximγxz=∂gz∂z∂uzs∂x=∂gz∂zγxz0,in which(5)εx0=∂ux0∂x,εxb=−∂2uzb∂x2,εxs=−∂2uzs∂x2,εim=duzimdx∂uzb+uzs∂x.For the linear problem, the stress-strain relation according to Hooke’s law is expressed as follows:(6)Tx=Ezεx,℘xz=Ez21+νzγxz,where Tx is the normal stress and ℘xz is the shear stress.The elastic strain generates the elastic energy of the FGM beam which is calculated as(7)UE=12∫VTxxTεxx+℘xzTγxzdV=12∫Vεx0zεxbgzεxsεimTEzεx0zεxbgzεxsεimdV+12∫V∂gz∂zγxz0TEz21+νz∂gz∂zγxz0dV=12∫Vεx0TEzεx0+εx0TzEzεzb+εx0TgzEzεxs+εx0TEzεximεxbTEzzεx0+εxbTEzz2εxb+εxbTEzzgzεxs+εxbTEzzεximεxsTEzgzεx0+εxsTEzzgzεxb+εxsTEzg2zεxs+εxsTEzgzεximεximTEzεx0+εximTEzzεxb+εximTEzgzεxs+εximTEzεximdV+12∫V∂gz∂z2Ez21+νzγxz0Tγxz0dV.The work done by the external compressive load is expressed as follows [41]:(8)UAx=12∫VN0∂wb+ws∂x2dx.The energy of the elastic foundation is calculated as(9)UF=12b∫Lkwwb+ws2+ks∂wb+ws∂x2dx.When the FGM beam is rotating around one fixed axis with the rotational speedΘ, the potential energy of this beam generated by the rotational movement is calculated as [41, 42](10)UΘ=12∫LPΘx∂wb+ws∂x2dx,with the centrifugal force PΘ [42].(11)PΘ=12∫SρzΘ2rL−x+12L2−x2dS,and ρz is the density of the material.The kinetic energy of the beam is expressed as(12)K=12∫Vρz∂u˙x∂tT∂u˙x∂t+∂u˙z∂tT∂u˙z∂tdV.To determine the equilibrium equation of the FGM beam, this work uses Hamilton’s principle as follows:(13)δ∫t1t2K−UE−UF−URo−UAxdt=0.Herein, this paper uses a two-node beam element, where each node contains five degrees of freedom:(14)ψe=∑i=12ux0uzbiuzsi∂uzb∂xi∂uzs∂xi,in which the displacement components at each point in the beam element are calculated through Lagrange and Hermite interpolation functions Ni and Hi:(15)ux0=∑i=12Niu0i=Nxuψe,uzb=∑i=12Hiuzbi+Hi+1∂uzb∂xi=Hbψe,uzs=∑i=12Hiuzsi+Hi+1∂uzs∂xi=Hsψe,∂uzb∂x=∑i=12∂Hi∂xuzbi+∂Hi+1∂x∂uzb∂xi=Hbxψe,∂uzs∂x=∑i=12∂Hi∂xuzsi+∂Hi+1∂x∂uzs∂xi=Hsxψe,∂2uzb∂x2=∑i=12∂2Hi∂x2uzbi+∂2Hi+1∂x2∂uzb∂xi=Hb2xψe,∂2uzs∂x2=∑i=12∂2Hi∂x2uzsi+∂2Hi+1∂x2∂uzs∂xi=Hs2xψe.Equation (15) can be rewritten in the matrix form as follows:(16)ψ=uz0uzbuzs∂uzb∂x∂uzs∂x=NxuHbHsHbxHsxψe=Hψe.Strain components are expressed according to the nodal displacement as(17)ε0x=∂ux0∂x=∂Nxu∂xψe=Buψe,εxb=−∂2uzb∂x2=−Hb2xψe=Bbψe,εxs=−∂2uzs∂x2=−Hs2xψe=Bsψe,εxim=∂uzb+uzs∂xduzimdx=duzimdxHb+Hsψe=Bimψe,γxz0=∂uzs∂x=Hsxψe.And now, the elastic energy of the FGM beam element has the following expression:(18)UeE=12ψeT∫VBuTEBu+BuTzEBb+BuTgzEBs+BuTEBimBbTEzBu+BbTEz2Bb+BbTEzgzBs+BbTEzBimBsTEgzBu+BsTEzgzBb+BsTEg2zBs+BsTEgzBimBimTEBu+BimpTEzBb+BimpTEgzBs+BimTEBimdVψe+12ψeT∫V∂gz∂z2E21+νHsxTHsxdVψe=12ψeT∫VBuTEBu+BuTzEBb+BuTgzEBs+BuTEBimBbTEzBu+BbTEz2Bb+BbTEzgzBs+BbTEzBimBsTEgzBu+BsTEzgzBb+BsTEg2zBs+BsTEgzBimBimTEBu+BimpTEzBb+BimpTEfzBs+BimTEBimdVψe+12ψeT∫V∂gz∂z2E21+νHsxTHsxdVψe.Equation (18) can be rewritten in the matrix form as(19)UeE=12ψeTKeEψe.The energies of the centrifugal inertia force and the elastic foundation are defined as follows:(20)UeF=12ψeTb∫LkwHb+HsTHb+Hs+ksHbx+HsxTHbx+Hsxdxψe=12ψeTKeFψeUeR=12ψeb∫LPΘxHbx+HsxTHbx+Hsxdxψe=12ψeTKeRψe.The energy expression due to the axial compressive load has the following form:(21)UAx=12ψeTb∫LN0Hbx+HsxTHbx+Hsxdxψe=12ψeTKeAxψe.The kinetic energy expression of the beam element is calculated as follows:(22)Ke=12∫Vψ˙eTρzψ˙edV=12ψ˙eT∫VHTHdVψ˙e=12ψ˙eTMeψ˙e,in which Me=∫VρzHTHdV is the element stiffness matrix.Substituting equations (18)–(20) into equation (13), the equation to get the fundamental frequency, as well as the vibration mode shape of the rotating FGM beam, is obtained as follows:(23)∑eKeE+KeF+KeR−KeAx−ω2∑eMe∑eψe=0.Equation (23) demonstrates that all components relating to rotational speed, elastic foundation parameter, initial geometrical imperfection, and axial compressive loads are presented in the equation for finding the fundamental vibration response of the FGM beam, which is completely different from traditional problems; this makes the computation more difficult in comparison with previous publications.Each side of the beam is under the following boundary conditions:Simply supported (denoted as S):(24)u0=0,wb=0,ws=0.Clamped (denoted as C):(25)u0=0,wb=0,ws=0,∂wb∂x=0,∂ws∂x=0.This paper employs three boundary conditions for calculations of FGM beams.One side is clamped, and the other side is free: C-F.Fully simply supported beam: S-S.Fully clamped beam: C-C.


## 3. Verification Examples

This section conducts five examples to evaluate the accuracy of the proposed theory and mathematical models, in which the numerical data of these examples are compared with those of other trustful publications.Example 1.
This example compares the frequency of the homogeneous beam under the axial compressive load. Geometrical and material properties are the lengthL, cross section bxh, L/h = 100, E = 380 GPa, ν = 0.3, and ρ = 3960 kg/m3. The beam is subjected to an axial compressive load N0 with N¯=N0L2/π2EI0 = 0.6. The nondimensional frequency is defined by the formula ω˜=ωρbhL4/EI0 (with I0=bh3/12). The results obtained from this work, exact expressions [43, 44], and FEM modeling [45] are listed in Table 1, in which they are calculated with different meshes. It can be seen that when the mesh size increases, the calculation results ensure the necessary convergence. The computational results in this work with the 8-element mesh are different from the results of the same mesh in reference [45]. The reason is that document [45] used the first-order shear deformation theory of Timoshenko. And for the 8-element mesh, the accuracy is acceptable; therefore, this mesh will be used for all the following related investigations.Table 1
First three nondimensional frequenciesω˜ of the beam under the axial load N¯ = 0.6.
BCModeExact [43]Exact [44]FEM with 8 elements [45]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—85.9361.89185.92485.76885.72185.703C-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.
Next, this problem compares the results of porosity FGM beams from this work and Chebyshev collocation solution (CCS) [46]. The ratio L/h obtains the values of 5, 10, and 20; the porosity parameter ϑ is 0 and 0.2. Geometrical and material characteristics of the fully clamped FGM beam are Ec = 380 GPa, νc = 0.3, ρc = 3960 kg/m3, Em = 70 GPa, νm = 0.3, and ρm = 2702 kg/m3. The comparative nondimensional frequencies ω^=ωLρm/Em are presented in Table 2. It can be observed that the data have a very good similarity.Table 2
The first nondimensional frequencyω^ of the FGM beam depends on the porosity parameter ϑ.
ϑL/hn = 0.2n = 0.5n = 1.0CCS [46]This workCCS [46]This workCCS [46]This work051.88231.88921.73921.74161.58441.5820101.08841.08960.99921.00220.90560.9109200.56910.56930.52130.52170.47160.47240.251.92051.92721.74021.73951.52101.5129101.10921.11050.99560.99890.86060.8661200.57970.57990.51860.51920.44650.4477Example 3.
Now, the frequency of the fully simply supported FGM (Al/Al2O3) beam resting on the two-parameter elastic foundation is considered. Geometrical and material properties 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 nondimensional elastic foundation parameters are defined as follows:(26)K¯w=K1L4EmI,K¯s=K2L2EmIπ2,with(27)I=bh312.
The nondimensional frequency of the beam is calculated as follows:(28)ω⌢=ω1L2hρmEm.
Table3 presents the comparative first frequencies of the beam between this work and the analytical method [47]. One can see that, with the 10 elements, the accuracy is acceptable; therefore, this work will use this mesh for all the following related explorations.Table 3
The first nondimensional frequenciesω⌢ of the beam resting on the two-parameter elastic foundation, K¯w=10,K¯s=1, S-S.
a/h = 100nModeExact [47]This work6 elements8 elements10 elements12 elements14 elementsWithout the elastic foundationK¯w=0K¯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 the 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.
Next, this example compares the frequency of the S-S beam with an initial geometrical imperfection. The beam has the following geometrical and material parameters:L = 288.7h, h = 0.02 m, b = 0.04 m, E = 971 GPa, and mass density ρ = 2300 kg/m3. The initial imperfection of the beam is wimx=J0sinπx, in which J0 is the amplitude of the imperfection. The nondimensional frequency and the initial imperfection coefficient ς0 are defined as follows:(29)ω∗∗=ω112ρL4Eh2,ς0=J0L.
The numerical data obtained from this work and the pseudo-arc-length continuation technique [48] in the case of increasing the value of ς0 are shown in Figure 2. It can be seen that the data meet a good agreement.Figure 2
The dependence of the first nondimensional frequency of the beam with the initial geometrical imperfection onς0.Example 5.
Finally, this last example carries the verification of the nondimensional frequency of the cantilever rotating beam with the speedχ. Consider a beam with the following geometrical and material parameters: length L, thickness h = b = L/100, r = 0, E = 70 GPa, and mass density ρ = 2700 kg/m3. The nondimensional frequency ω˜i is normalized as follows:(30)ω¯i=ωiL212ρEh2.
The first three nondimensional frequencies of the rotating beam with different values of the speed ratioξ¯=L2Θ12ρ/Eh2 obtained from this work, an exact solution [49], a new dynamic modeling method (DMM) [50], and an isogeometric analysis [42] are presented in Table 4.Table 4
The dependence of the first three nondimensional frequenciesω¯i of the cantilever rotating beam on ϑ.
ModeMethodRotational speed ratioξ¯ξ¯ = 0ξ¯ = 3ξ¯ = 6ξ¯ = 12ω˜1This work3.51614.67907.073412.6098Exact [49]3.51604.79737.360413.1702DMM [50]3.51604.79737.3604–Isogeometric [42]3.51184.77337.3276–ω˜2This work22.024723.119726.111036.4500Exact [49]22.034523.320326.809137.6031DMM [50]––––Isogeometric [42]––––ω˜3This work61.625462.760266.024277.4512Exact [49]61.697262.985066.684079.6145DMM [50]––––Isogeometric [42]––––ω˜4This work120.647121.843125.348138.295Exact [49]120.902122.236126.140140.534DMM [50]––––Isogeometric [42]––––


## 4. Numerical Results

This section carries out the vibration analysis of the rotating FGM porosity beam resting on the two-parameter elastic foundation subjected to the preaxial compressive load, where the initial geometrical imperfection is taken into calculations. Consider a beam with the following geometrical and material properties: lengthL, cross section bxh, Ec = 380 GPa, νc = 0.3, ρc = 3960 kg/m3, Em = 70 GPa, νm = 0.3, andρm = 2702 kg/m3. The imperfection of the beam is w0x=Y0sinπx, in which G0 is the amplitude of the imperfection, and the imperfection ratio μ0=Y0/L. The nondimensional frequency and other parameters are normalized as follows:(31)ωi∗=ωiLρmEm,Kw∗=kwL4D¯m,Ks∗=ksL2D¯m,δ=L2Θ12ρmEmh02,D¯m=Emh0312,N∗=N0L2π2EmI0,I0=bh0312,h0=L20.


### 4.1. Effect of the Rotational Speed

Firstly, the influence of the rotational speed on the free vibration response of the rotating FGM beam is investigated. The rotational speedΘ is increased gradually so that the coefficient δ changes in a range of 0 to 10; the volume fraction index n obtains the values from 0 to 10; other parameters are L/h = 20, imperfection ratio λ0 = 0.001, axial compressive load N∗ = 0.5, Kw∗ = 5, Ks∗ = 1, distance ratio r/L = 1, and ϑ = 0.2. Figures 3–5 present the dependence of the first frequency of the beam on the rotational speed ratio δ and the volume fraction index n. The first four vibration mode shapes of the S-S and C-C rotating FGM beams with the speed ratio δ = 0 and 8 are presented in Figures 6 and 7. These figures point out the following.Figure 3
The dependence of the first nondimensional frequency of the rotating FGM beam on the volume fraction indexn, λ0 = 0.001, Kw∗ = 5, and Ks∗ = 1. (a) C-C. (b) S-S.
(a)(b)Figure 4
The dependence of the first nondimensional frequency of the rotating FGM beam on the volume fraction indexn and the rotational speed, λ0 = 0.001, Kw∗ = 5, and Ks∗ = 1. (a) S-S. (b) C-C.
(a)(b)Figure 5
The dependence of the first nondimensional frequency of the rotating FGM beam on the volume fraction indexn and the rotational speed, λ0 = 0.001, Kw∗ = 5, and Ks∗ = 1.
(a)(b)Figure 6
The dependence of the first four vibration mode shapes of the rotating FGM beam on the rotational speed, S-S,n = 1, and N∗ = 0.5. (a) S-S, δ = 0, and r = 0. (b) S-S, η = 8, and r/L = 1.
(a)(b)Figure 7
The dependence of the first four vibration mode shapes of the rotating FGM beam on the rotational speed, C-C,n = 1, and N∗ = 0.5. (a) C-c, δ = 0, and r = 0. (b) C-C, δ = 8, and r/L = 1.
(a)(b)When the rotational speed increases, nondimensional frequencies of the beam in the case of both boundary conditions (S-S and C-C) increase. This is because as the rotational speed of the beam increases, the inertial force component increases; therefore, the stiffness matrix of the FGM beam is increased by the addition of this inertial component. The change of the first nondimensional frequency of the structure depends strongly on the boundary conditions and the volume fraction indexn. For the S-S boundary, when n increases, the first frequency of the beam decreases. In contrast, for the C-C beam, when n increases, the first frequency of the beam decreases sharply at first, then its value reaches the minimum point, and it increases gradually.Besides, the rotational speed has a strong effect on the vibration mode shape of the beam, especially the first vibration mode shape. Also, due to the appearance of the inertial force, the first vibration mode shape is no longer symmetrical through the positionx = L/2. The influence of the rotational speed on the vibration mode shapes depends strongly on the boundary conditions; for the S-S boundary condition, the influence level is more pronounced than the beam under the C-C boundary condition.


### 4.2. Influence of the Axial Load

This section is about to capture the effect of the axial compressive load on the vibration behavior of the FGM beam. The axial compressive load is changed so that the nondimensional loadN∗ obtains the values from 0 to 0.8. The dependence of the first frequency of the FGM beam on the axial load N∗ and the volume fraction index n is presented in Table 5, and the dependence of the first frequency on the axial load N∗ and the rotational speed is presented in Table 6. As the axial compressive load increases, the frequency of the FGM beam decreases; this is because as the compressive load increases, the stiffness matrix of this beam decreases. Table 7 shows that when the value of the compression load N∗ is increased, the first nondimensional frequency changes the most, while the higher-order frequencies change slightly.Table 5
The dependence of the first nondimensional frequency of the FGM beam on the axial compressive loadN∗ and the volume fraction index n, L/h = 20, δ = 5, α = 0.2, and r/L = 1.
BCN∗N00.512510S-S00.34260.29510.26340.24590.25400.26260.20.33770.28850.25540.23640.24370.25220.40.33280.28180.24710.22640.23300.24140.60.32770.27490.23850.21600.22170.22990.80.32260.26790.22960.20500.20980.2179C-C00.66720.56970.50860.45040.42040.41600.20.66410.56560.50360.44440.41330.40860.40.66100.56150.49860.43820.40610.40110.60.65790.55730.49360.43200.39880.39340.80.65480.55310.48850.42560.39130.3855Table 6
The dependence of the first nondimensional frequency of the FGM beam on the axial compressive loadN∗ and δ, n = 0.5, α = 0.2, and r/L = 1.
BCN∗δ025101520S-S00.21720.23200.29510.43970.59070.73890.20.20830.22360.28850.43510.58700.73570.40.19890.21490.28180.43050.58340.73260.60.18910.20590.27490.42580.57970.72940.80.17880.19640.26790.42100.57590.7262C-C00.52140.52960.56970.68830.83750.99630.20.51690.52510.56560.68500.83490.99410.40.51230.52060.56150.68170.83220.99180.60.50770.51610.55730.67830.82950.98950.80.50300.51150.55310.67490.82670.9873Table 7
The dependence of the first six nondimensional frequencies of the FGM beam on the axial compressive loadN∗, L/h = 20, n = 2, δ = 5, α = 0.2, and r/L = 1.
BCN∗ωi∗ω1∗ω2∗ω3∗ω4∗ω5∗ω6∗S-S00.24590.78521.60372.69854.02165.47630.40.22640.76231.57902.67333.99655.45260.80.20500.73871.55402.64773.97125.42871.00.19340.72671.54132.63493.95855.4167C-C00.45041.13452.08463.29474.65966.03660.40.43821.11652.06413.27314.63826.01690.80.42561.09822.04343.25154.61675.99721.00.41921.08892.03303.24064.60605.9873


### 4.3. Influence of the Thickness of the FGM Beam

Consider an FGM beam withL/h varying from 5 to 80, imperfect parameter λ0 = 0.001, N∗ = 0.5, Kw∗ = 5, Ks∗ = 1, distance ratio r/L = 1, and volume fraction index n = 0.2. The dependence of the first nondimensional frequency of the FGM beam on L/h and α is presented in Table 8, and the dependence of the first nondimensional frequency on L/h and the rotational speed parameter δ is presented in Table 9. Table 10 shows the dependence of the first six nondimensional frequencies of the FGM beam on L/h. These numerical results point out the following:When the thickness of the FGM beam is reduced, the first nondimensional frequency of the beam decreases; however, the strongest decrease of the first frequency is when the thickness decreases fromL/h = 5 to L/h = 10. The frequency of the C-C beam is always greater than that of the S-S beam in the same other conditions.When increasing parameterα, the first nondimensional frequency of the beam increases; this is because when increasing the porosity ratio of the material, both mass and stiffness of the structure decrease; however, the stiffness of the beam decreases sharply than the decrease of the mass. This also means that the porosity ratio is a parameter to control the vibration of the FGM beam; in other words, changing the porosity ratio will change the working performance of the structure.Table 8
The dependence of the first nondimensional frequency of the FGM beam onL/h and α, n = 0.2, N∗ = 0.5, δ = 5, λ0 = 10−3, and r/L = 1.
BCαL/h51020406080S-S00.91980.50990.30790.22740.20890.20360.10.92240.51130.30870.22850.21050.20570.20.92550.51300.30970.22970.21240.20830.30.92920.51500.31090.23120.21460.21140.40.93370.51740.31240.23310.21750.21520.50.93930.52050.31420.23550.22100.2201C-C01.86631.09580.60520.36530.29900.27220.11.88051.10390.60940.36800.30160.27510.21.89751.11350.61450.37120.30470.27850.31.91811.12530.62070.37510.30840.28260.41.94371.13990.62830.37990.31320.28780.51.97641.15850.63820.38620.31920.2945Table 9
The dependence of the first nondimensional frequency of the FGM beam onL/h and δ, n = 0.2, α = 0.2, N∗ = 0.5, λ0 = 10−3, and r/L = 1.
BCδL/h51020406080S-S00.90480.47400.24130.13280.11070.110120.90820.48050.25380.15410.13500.134150.92550.51300.30970.22970.21240.2083100.98440.61190.44520.37330.35020.3397151.07330.74110.59110.51330.48280.4664C-C01.88571.09160.57300.29930.21610.183321.88761.09520.57990.31240.23360.203351.89751.11350.61450.37120.30470.2785101.93241.17580.72030.51540.45610.4279151.98841.27000.85880.67160.60880.5745Table 10
The dependence of the first nondimensional frequency of the FGM beam onL/h and α, n = 0.2, N∗ = 0.5, α = 0.2, δ = 5, λ0 = 10−3, and r/L = 1.
BCωi∗L/h51020406080S-Sω1∗0.92550.51300.30970.22970.21240.2083ω2∗3.33531.96431.08470.65640.53950.4935ω3∗6.15534.13982.32471.30950.99850.8648ω4∗8.44066.78143.98912.20471.61381.3447ω5∗9.89429.60336.01513.30322.23501.6755ω6∗10.781410.28236.78293.41872.40601.9483C-Cω1∗1.89751.11350.61450.37120.30470.2785ω2∗4.46662.90411.62290.91890.70540.6140ω3∗6.78455.16513.04931.69451.24691.0439ω4∗8.79097.83114.87392.70611.94361.5828ω5∗10.045310.38116.96023.93042.79002.2308ω6∗10.884010.80699.24025.32713.76022.9688


## 4.1. Effect of the Rotational Speed

Firstly, the influence of the rotational speed on the free vibration response of the rotating FGM beam is investigated. The rotational speedΘ is increased gradually so that the coefficient δ changes in a range of 0 to 10; the volume fraction index n obtains the values from 0 to 10; other parameters are L/h = 20, imperfection ratio λ0 = 0.001, axial compressive load N∗ = 0.5, Kw∗ = 5, Ks∗ = 1, distance ratio r/L = 1, and ϑ = 0.2. Figures 3–5 present the dependence of the first frequency of the beam on the rotational speed ratio δ and the volume fraction index n. The first four vibration mode shapes of the S-S and C-C rotating FGM beams with the speed ratio δ = 0 and 8 are presented in Figures 6 and 7. These figures point out the following.Figure 3
The dependence of the first nondimensional frequency of the rotating FGM beam on the volume fraction indexn, λ0 = 0.001, Kw∗ = 5, and Ks∗ = 1. (a) C-C. (b) S-S.
(a)(b)Figure 4
The dependence of the first nondimensional frequency of the rotating FGM beam on the volume fraction indexn and the rotational speed, λ0 = 0.001, Kw∗ = 5, and Ks∗ = 1. (a) S-S. (b) C-C.
(a)(b)Figure 5
The dependence of the first nondimensional frequency of the rotating FGM beam on the volume fraction indexn and the rotational speed, λ0 = 0.001, Kw∗ = 5, and Ks∗ = 1.
(a)(b)Figure 6
The dependence of the first four vibration mode shapes of the rotating FGM beam on the rotational speed, S-S,n = 1, and N∗ = 0.5. (a) S-S, δ = 0, and r = 0. (b) S-S, η = 8, and r/L = 1.
(a)(b)Figure 7
The dependence of the first four vibration mode shapes of the rotating FGM beam on the rotational speed, C-C,n = 1, and N∗ = 0.5. (a) C-c, δ = 0, and r = 0. (b) C-C, δ = 8, and r/L = 1.
(a)(b)When the rotational speed increases, nondimensional frequencies of the beam in the case of both boundary conditions (S-S and C-C) increase. This is because as the rotational speed of the beam increases, the inertial force component increases; therefore, the stiffness matrix of the FGM beam is increased by the addition of this inertial component. The change of the first nondimensional frequency of the structure depends strongly on the boundary conditions and the volume fraction indexn. For the S-S boundary, when n increases, the first frequency of the beam decreases. In contrast, for the C-C beam, when n increases, the first frequency of the beam decreases sharply at first, then its value reaches the minimum point, and it increases gradually.Besides, the rotational speed has a strong effect on the vibration mode shape of the beam, especially the first vibration mode shape. Also, due to the appearance of the inertial force, the first vibration mode shape is no longer symmetrical through the positionx = L/2. The influence of the rotational speed on the vibration mode shapes depends strongly on the boundary conditions; for the S-S boundary condition, the influence level is more pronounced than the beam under the C-C boundary condition.


## 4.2. Influence of the Axial Load

This section is about to capture the effect of the axial compressive load on the vibration behavior of the FGM beam. The axial compressive load is changed so that the nondimensional loadN∗ obtains the values from 0 to 0.8. The dependence of the first frequency of the FGM beam on the axial load N∗ and the volume fraction index n is presented in Table 5, and the dependence of the first frequency on the axial load N∗ and the rotational speed is presented in Table 6. As the axial compressive load increases, the frequency of the FGM beam decreases; this is because as the compressive load increases, the stiffness matrix of this beam decreases. Table 7 shows that when the value of the compression load N∗ is increased, the first nondimensional frequency changes the most, while the higher-order frequencies change slightly.Table 5
The dependence of the first nondimensional frequency of the FGM beam on the axial compressive loadN∗ and the volume fraction index n, L/h = 20, δ = 5, α = 0.2, and r/L = 1.
BCN∗N00.512510S-S00.34260.29510.26340.24590.25400.26260.20.33770.28850.25540.23640.24370.25220.40.33280.28180.24710.22640.23300.24140.60.32770.27490.23850.21600.22170.22990.80.32260.26790.22960.20500.20980.2179C-C00.66720.56970.50860.45040.42040.41600.20.66410.56560.50360.44440.41330.40860.40.66100.56150.49860.43820.40610.40110.60.65790.55730.49360.43200.39880.39340.80.65480.55310.48850.42560.39130.3855Table 6
The dependence of the first nondimensional frequency of the FGM beam on the axial compressive loadN∗ and δ, n = 0.5, α = 0.2, and r/L = 1.
BCN∗δ025101520S-S00.21720.23200.29510.43970.59070.73890.20.20830.22360.28850.43510.58700.73570.40.19890.21490.28180.43050.58340.73260.60.18910.20590.27490.42580.57970.72940.80.17880.19640.26790.42100.57590.7262C-C00.52140.52960.56970.68830.83750.99630.20.51690.52510.56560.68500.83490.99410.40.51230.52060.56150.68170.83220.99180.60.50770.51610.55730.67830.82950.98950.80.50300.51150.55310.67490.82670.9873Table 7
The dependence of the first six nondimensional frequencies of the FGM beam on the axial compressive loadN∗, L/h = 20, n = 2, δ = 5, α = 0.2, and r/L = 1.
BCN∗ωi∗ω1∗ω2∗ω3∗ω4∗ω5∗ω6∗S-S00.24590.78521.60372.69854.02165.47630.40.22640.76231.57902.67333.99655.45260.80.20500.73871.55402.64773.97125.42871.00.19340.72671.54132.63493.95855.4167C-C00.45041.13452.08463.29474.65966.03660.40.43821.11652.06413.27314.63826.01690.80.42561.09822.04343.25154.61675.99721.00.41921.08892.03303.24064.60605.9873


## 4.3. Influence of the Thickness of the FGM Beam

Consider an FGM beam withL/h varying from 5 to 80, imperfect parameter λ0 = 0.001, N∗ = 0.5, Kw∗ = 5, Ks∗ = 1, distance ratio r/L = 1, and volume fraction index n = 0.2. The dependence of the first nondimensional frequency of the FGM beam on L/h and α is presented in Table 8, and the dependence of the first nondimensional frequency on L/h and the rotational speed parameter δ is presented in Table 9. Table 10 shows the dependence of the first six nondimensional frequencies of the FGM beam on L/h. These numerical results point out the following:When the thickness of the FGM beam is reduced, the first nondimensional frequency of the beam decreases; however, the strongest decrease of the first frequency is when the thickness decreases fromL/h = 5 to L/h = 10. The frequency of the C-C beam is always greater than that of the S-S beam in the same other conditions.When increasing parameterα, the first nondimensional frequency of the beam increases; this is because when increasing the porosity ratio of the material, both mass and stiffness of the structure decrease; however, the stiffness of the beam decreases sharply than the decrease of the mass. This also means that the porosity ratio is a parameter to control the vibration of the FGM beam; in other words, changing the porosity ratio will change the working performance of the structure.Table 8
The dependence of the first nondimensional frequency of the FGM beam onL/h and α, n = 0.2, N∗ = 0.5, δ = 5, λ0 = 10−3, and r/L = 1.
BCαL/h51020406080S-S00.91980.50990.30790.22740.20890.20360.10.92240.51130.30870.22850.21050.20570.20.92550.51300.30970.22970.21240.20830.30.92920.51500.31090.23120.21460.21140.40.93370.51740.31240.23310.21750.21520.50.93930.52050.31420.23550.22100.2201C-C01.86631.09580.60520.36530.29900.27220.11.88051.10390.60940.36800.30160.27510.21.89751.11350.61450.37120.30470.27850.31.91811.12530.62070.37510.30840.28260.41.94371.13990.62830.37990.31320.28780.51.97641.15850.63820.38620.31920.2945Table 9
The dependence of the first nondimensional frequency of the FGM beam onL/h and δ, n = 0.2, α = 0.2, N∗ = 0.5, λ0 = 10−3, and r/L = 1.
BCδL/h51020406080S-S00.90480.47400.24130.13280.11070.110120.90820.48050.25380.15410.13500.134150.92550.51300.30970.22970.21240.2083100.98440.61190.44520.37330.35020.3397151.07330.74110.59110.51330.48280.4664C-C01.88571.09160.57300.29930.21610.183321.88761.09520.57990.31240.23360.203351.89751.11350.61450.37120.30470.2785101.93241.17580.72030.51540.45610.4279151.98841.27000.85880.67160.60880.5745Table 10
The dependence of the first nondimensional frequency of the FGM beam onL/h and α, n = 0.2, N∗ = 0.5, α = 0.2, δ = 5, λ0 = 10−3, and r/L = 1.
BCωi∗L/h51020406080S-Sω1∗0.92550.51300.30970.22970.21240.2083ω2∗3.33531.96431.08470.65640.53950.4935ω3∗6.15534.13982.32471.30950.99850.8648ω4∗8.44066.78143.98912.20471.61381.3447ω5∗9.89429.60336.01513.30322.23501.6755ω6∗10.781410.28236.78293.41872.40601.9483C-Cω1∗1.89751.11350.61450.37120.30470.2785ω2∗4.46662.90411.62290.91890.70540.6140ω3∗6.78455.16513.04931.69451.24691.0439ω4∗8.79097.83114.87392.70611.94361.5828ω5∗10.045310.38116.96023.93042.79002.2308ω6∗10.884010.80699.24025.32713.76022.9688


## 5. Conclusions

By using the new shear deformation theory type of hyperbolic sine functions, finite element formulations for free vibration analysis of rotating FGM beams resting on two-parameter elastic foundations are established in this paper, in which the initial geometrical imperfection and the porosity of the material are considered. Based on the numerical results, some important novel explorations are drawn as follows:The rotational speed has a significant effect on the free vibration response as well as the vibration mode shape of the structures. When the rotational speed increases, the frequencies of the beam increase, especially the first frequency, and the vibration mode shape is no longer symmetrical through positionx = L/2.As the axial compressive load value increases, the frequency decreases, and the strongest change manifests itself at the first frequency as the axial load increases.When reducing the beam thickness and the porosity ratio, the frequencies of the FGM beam are reduced.



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*Source: 5519946-2021-03-18.xml*