Open access peer-reviewed chapter

Free Vibration of Axially Functionally Graded Beam

Written By

Dongxing Cao, Bin Wang, Wenhua Hu and Yanhui Gao

Submitted: November 6th, 2018 Reviewed: March 14th, 2019 Published: April 22nd, 2019

DOI: 10.5772/intechopen.85835

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Abstract

Axially functionally graded (AFG) beam is a special kind of nonhomogeneous functionally gradient material structure, whose material properties vary continuously along the axial direction of the beam by a given distribution form. There are several numerical methods that have been used to analyze the vibration characteristics of AFG beams, but it is difficult to obtain precise solutions for AFG beams because of the variable coefficients of the governing equation. In this topic, the free vibration of AFG beam using analytical method based on the perturbation theory and Meijer G-Function are studied, respectively. First, a detailed review of the existing literatures is summarized. Then, based on the governing equation of the AFG Euler-Bernoulli beam, the detailed analytic equations are derived on basis of the perturbation theory and Meijer G-function, where the nature frequencies are demonstrated. Subsequently, the numerical results are calculated and compared, meanwhile, the analytical results are also confirmed by finite element method and the published references. The results show that the proposed two analytical methods are simple and efficient and can be used to conveniently analyze free vibration of AFG beam.

Keywords

  • axially functionally graded beams
  • free vibration
  • natural frequency
  • asymptotic perturbation method
  • Meijer G-function
  • finite segment model

1. Introduction

Functionally gradient materials (FGMs) make a composite material by varying the microstructure from one material to another material with a specific gradient. It can be designed for specific function and applications. If it is for thermal or corrosive resistance or malleability and toughness, both strengths of the material may be used to avoid corrosion, fatigue, fracture, and stress corrosion cracking. FGMs are usually made into several structures, such as beams [1, 2, 3, 4], plates [5, 6, 7, 8], and shells [9, 10, 11, 12]. In this area, the variation of material properties in functionally graded beams may be oriented in transverse (thickness) direction or/and longitudinal/axial (length) direction.

For functionally graded beams with thickness-wise gradient variation, there have been many studies devoted to this topic. Lee et al. [13] establish an accurate transfer matrix method to analyze the free vibration characteristics of FGM beams whose Young’s modulus and density vary continuously with the height of the beam section through power law distribution. Su et al. [14] developed the dynamic stiffness method to investigate the free vibration behavior of FGM beams. Jing et al. [15] introduced a new approach by combining the cell-centered finite volume method and Timoshenko beam theory to analyze static and free vibration of FGM beams. Ait Atmane et al. [16] investigated the free vibration of a nonuniform FGM beams with exponentially varying width and material properties. Sina et al. [17] studied the free vibration of FGM beams by analytical method based on the traditional first-order shear deformation theory. Sharma [18] investigated the computational characteristics of harmonic differential quadrature method for free vibration of functionally graded piezoelectric material beam, which the material properties are assumed to have a power law or sigmoid law variation across the depth. Li et al. [19] proposed a high-order shear theory for free vibration of FGM beams with continuously varying material properties under different boundary conditions. Celebi et al. [20] employed the complementary function method to investigate the free vibration analysis of simply supported FGM beams, which the material properties change arbitrarily in the thickness direction. Chen et al. [21] studied the nonlinear free vibration behavior of shear deformable sandwich porous FGM beam based on the von kármán type geometric nonlinearity and Ritz method. Nazemnezhad and Hosseini-Hashemi [22] examined the nonlinear free vibration of FGM nanobeams with immovable ends using the multiple scale method.

As the FGMs are good for severe conditions, thermal-mechanical effect on FGM structures has attracted broad attention. In this field, Farzad Ebrahimi and Erfan Salari obtained outstanding achievements. Considering the thermal-mechanical effect and size-dependent thermo-electric effect, the buckling and vibration behavior of FGM nanobeams are studied [23, 24, 25, 26]. Considering the concept of neutral axis, they [27] studied the free buckling and vibration of FGM nanobeams using semi-analytical differential transformation method. To discuss the effect of the shear stress, Reddy’s higher-order shear deformation beam theory is introduced to study the vibration of the FGM structures [28, 29, 30]. Ebrahimi et al. [31, 32, 33] also studied vibration characteristics of FGM beams with porosities. Based on nonlocal elasticity theory, the nonlocal temperature-dependent vibration of FGM nanobeams were studied in thermal environment [34, 35, 36].

Another significant class of functionally graded beams is those with lengthwise varying material properties. It is difficult to obtain precise solutions for axially functionally graded (AFG) beams because of the variable coefficients of the governing equation. To solve this problem, a great deal of methods has been used to analyze the vibration characteristics of AFG beams. By assuming that the material constituents vary throughout the longitudinal directions according to a simple power law, Alshorbagy et al. [37] developed a two-node, six-degree-of-freedom finite element method (FEM) in conjunction with Euler-Bernoulli beam theory to detect the free vibration characteristics of a functionally graded beam. Shahba et al. [38, 39] used the FEM to study the free vibration of an AFG-tapered beam based on Euler-Bernoulli and Timoshenko beam theory. Shahba and Rajasekaran [40] studied the free vibration analysis of AFG-tapered Euler-Bernoulli beams employing the differential transform element method. Liu et al. [41] applied the spline finite point method to investigate the same problems. Rajasekaran [42] researched the free bending vibration of rotating AFG-tapered Euler-Bernoulli beams with different boundary conditions using the differential transformation method and differential quadrature element method. Rajasekaran and Tochaei [43] carried out the free vibration analysis of AFG Timoshenko beams using the same method. Huang and Li [44] studied the free vibration of variable cross-sectional AFG beams. The differential equation with variable coefficients is combined with the boundary conditions and transformed into Fredholm integral equation. By solving Fredholm integral equation, the natural frequencies of AFG beams can be obtained. Huang et al. [45] proposed a new approach for investigating the vibration behaviors of AFG Timoshenko beams with nonuniform cross section by introducing an auxiliary function. Huang and Rong [46] introduced a simple approach to deal with the free vibration of nonuniform AFG Euler-Bernoulli beams based on the polynomial expansion and integral technique. Hein and Feklistova [47] solved the vibration problems of AFG beams with various boundary conditions and varying cross sections via the Haar wavelet series. Xie et al. [48] presented a spectral collocation approach based on integrated polynomials combined with the domain decomposition technique for free vibration analyses of beams with axially variable cross sections, moduli of elasticity, and mass densities. Kukla and Rychlewska [49] proposed a new approach to study the free vibration analysis of an AFG beam; the approach relies on replacing functions characterizing functionally graded beams with piecewise exponential functions. Zhao et al. [50] introduced a new approach based on Chebyshev polynomial theory to investigate the free vibration of AFG Euler-Bernoulli and Timoshenko beams with nonuniform cross sections. Fang and Zhou [51, 52] researched the modal analysis of rotating AFG-tapered Euler-Bernoulli and Timoshenko beams with various boundary conditions employing the Chebyshev-Ritz method. Li et al. [53, 54] obtained the exact solutions for the free vibration of FGM beams with material profiles and cross-sectional parameters varying exponentially in the axial direction, where assumptions of Euler-Bernoulli and Timoshenko beam theories have been applied, respectively. Sarkar and Ganguli [55] studied the free vibration of AFG Timoshenko beams with different boundary conditions and uniform cross sections. Akgöz and Civalek [56] examined the free vibrations of AFG-tapered Euler-Bernoulli microbeams based on Bernoulli-Euler beam and modified couple stress theory. Yuan et al. [57] proposed a novel method to simplify the governing equations for the free vibration of Timoshenko beams with both geometrical nonuniformity and material inhomogeneity along the beam axis, and a series of exact analytical solutions are derived from the reduced equations for the first time. Yilmaz and Evran [58] investigated the free vibration of axially layered FGM short beams using experimental and FEM simulation, which the beams are manufactured by using the powder metallurgy technique using different weight fractions of aluminum and silicon carbide powders.

Till now, there also are plenty of literatures devoted to the free vibration for nonuniform beams. Boiangiu et al. [59] obtained the exact solutions for free bending vibrations of straight beams with variable cross section using Bessel’s functions and proposed a transfer matrix method to determine the natural frequencies of a complex structure of conical and cylindrical beams. Garijo [60] analyzed the free vibration of Euler-Bernoulli beams of variable cross section employing a collocation technique based on Bernstein polynomials. Arndt et al. [61] presented an adaptive generalized FEM to determine the natural frequencies of nonuniform Euler-Bernoulli beams. The spline-method of degree 5 defect 1 is proposed by Zhernakov et al. [62] to determine the natural frequencies of beam with variable cross section. Wang [63] studied the vibration of a cantilever beam with constant thickness and linearly tapered sides by means of a novel accurate, efficient initial value numerical method. Silva and Daqaq [64] solved the linear eigenvalue problem exactly of a slender cantilever beam of constant thickness and linearly varying width using the Meijer G-function approach. Rajasekaran and Khaniki [65] applied the FEM to research the vibration behavior of nonuniform small-scale beams in the framework of nonlocal strain gradient theory. Çalım [66] investigated the dynamic behavior of nonuniform composite beams employing an efficient method of analysis in the Laplace domain. Yang et al. [67] applied the power series method to investigate the natural frequencies and the corresponding complex mode functions of a rotating tapered cantilever Timoshenko beam. Clementi [68] analytically determined the frequency response curves of a nonuniform beam undergoing nonlinear oscillations by the multiple time scale method. Wang [69] proposed the differential quadrature element method for the natural frequencies of multiple-stepped beams with an aligned neutral axis. Abdelghany [70] utilized the differential transformation method to examine the free vibration of nonuniform circular beam.

The asymptotic development method, which is a kind of perturbation analysis method, is always used to solve nonlinear vibration equations. For example, Chen et al. [71, 72] studied the nonlinear dynamic behavior of axially accelerated viscoelastic beams and strings based on the asymptotic perturbation method. Ding et al. [73, 74] studied the influence of natural frequency of transverse vibration of axially moving viscoelastic beams and the steady-state periodic response of forced vibration of dynamic viscoelastic beams based on the multi-scale method. Chen [75] used the asymptotic perturbation method to analyze the finite deformation of prestressing hyperelastic compression plates. Hao et al. [76] employed the asymptotic perturbation method to obtain the nonlinear dynamic responses of a cantilever FGM rectangular plate subjected to the transversal excitation in thermal environment. Andrianov and Danishevs’kyy [77] proposed an asymptotic method for solving periodic solutions of nonlinear vibration problems of continuous structures. Based on the asymptotic expansion method of Poincaré-Lindstedt version [78], the longitudinal vibration of a bar and the transverse vibration of a beam under the action of a nonlinear restoring force are studied. The asymptotic development method is applied to obtain an approximate analytical expression of the natural frequencies of nonuniform cables and beams [79, 80]. Cao et al. [81, 82] applied the asymptotic development method to analyze the free vibration of nonuniform axially functionally graded (AFG) beams. Tarnopolskaya et al. [83] gave the first comprehensive study of the mode transition phenomenon in vibration of beams with arbitrarily varying curvature and cross section on the basis of asymptotic analysis.

The present topic focus on the free vibration of AFG beams with uniform or nonuniform cross sections using analytical method: the asymptotic perturbation method (APM) and Meijer G-function. First, the governing differential equation for free vibration of nonuniform AFG beam is summarized and rewritten in a form of a dimensionless equation based on Euler-Bernoulli beam theory. Second, the analytic equations are then derived in detail in Sections 3 and 4, respectively, where the nature frequencies are obtained and compared with the results of the finite element method and the published references. Finally, the conclusions are presented.

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2. Governing equation of the AFG beam

This studied free vibration of the axially functionally graded beam, which is a nonuniform and nonhomogeneous structure because of the variable width and height, as shown in Figure 1. Based on Euler-Bernoulli beam theory, the governing differential equation of the beam can be written as

Figure 1.

The geometry and coordinate system of an AFG beam.

2 x 2 E x I x 2 w x t x 2 + ρ x A x 2 w x t t 2 = 0 , 0 x L E1

where w x t is the transverse deflection at position x and time t; L is the length of the beams; E x I x is the bending stiffness, which is determined by Young’s modulus E x and the area moment of inertia I x ; and ρ x A x is the unit mass length of beam, which is determined by volume mass density ρ x and variable cross-sectional area A x .

Because of the particularity of AFG beam, bending stiffness E x I x and unit mass ρ x A x will change with the axis coordinates, which makes the original constant coefficient differential equation become variable coefficient differential equation and to some extent increases the difficulty of solving. In order to facilitate calculation, we simplify the calculation process by introducing dimensionless parameters. Reference flexural stiffness EI 0 and reference mass ρ A 0 are introduced, and the above two dimensionless parameters are invariant. Suppose E x I x = EI 0 + E x I x ¯ and ρ x A x = ρ A 0 + ρ x A x ¯ , where EI 0 and ρ A 0 are the invariant parts and E x I x ¯ and ρ x A x ¯ represent flexural stiffness and mass per unit length, respectively, and vary with the axial coordinates. Here, we introduce a dimensionless space variable ξ = x / L and a dimensionless time variable τ = t L 2 EI 0 ρ A 0 ; Eq. (1) can be rewritten in the dimensionless form:

2 ξ 2 1 + f 1 ξ 2 w ξ τ ξ 2 + 1 + f 2 ξ 2 w ξ τ τ 2 = 0 , 0 ξ 1 E2

where

f 1 ξ = E ξ I ξ ¯ EI 0 and f 2 ξ = ρ ξ A ξ ¯ ρ A 0 E3

are the nondimensional varying parts of the flexural stiffness and of the mass per unit length, respectively.

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3. Asymptotic perturbation method

3.1 Equation deriving

In this section, the APM is introduced to obtain a simple proximate formula for the nature frequency of the AFG beam. Firstly, we assume that

w ξ τ = W ξ sin ωτ E4

where W ξ is the amplitude of vibration and ω is the circular frequency of vibration. We obtain the following equation by substituting Eq. (4) with Eq. (2):

d 2 d ξ 2 1 + f 1 ξ d 2 W d ξ 2 ω 2 1 + f 2 ξ W = 0 , 0 ξ 1 E5

To use the APM, a small perturbation parameter ε is introduced:

f 1 ξ ε f 1 ξ , f 2 ξ ε f 2 ξ E6

According to the Poincaré-Lindstedt method [78, 79, 80, 81, 82], we assume the expansion for ω and W ξ as

ω = ω 0 + εω 1 + ε 2 ω 2 + , W ξ = W 0 ξ + ε W 1 ξ + ε 2 W 2 ξ + . E7

Substituting these expressions with governing Eq. (5) and then expanding the expressions into a ε -series, Eqs. (8) and (9) are obtained by equating the coefficients of ε 0 and ε 1 to zero, yielding a sequence of problems for the unknowns ω i and W i ξ :

d 4 W 0 d ξ 4 ω 0 2 W 0 = 0 E8
d 4 W 1 d ξ 4 ω 0 2 W 1 + h 1 ξ 2 ω 1 ω 0 W 0 = 0 E9

where

h 1 ξ = 2 df 1 ξ d 3 W 0 d ξ 3 + d 2 f 1 ξ d ξ 2 d 2 W 0 d ξ 2 + ω 0 2 f 1 ξ f 2 ξ W 0 E10

For Eq. (8), the following general solution can be obtained:

W 0 = A sin + B cos + C sinh + D cosh E11

where

k = ω 0 E12

For simplicity, we consider clamped-free (C-F) beams, and the corresponding boundary conditions are

W 0 = dW 0 = 0 , ξ = 0 E13
d 2 W 0 d ξ 2 = d 3 W 0 d ξ 3 = 0 , ξ = 1 E14

Then, the following equations from equation can be obtained:

A + C = 0 B + D = 0 C D = sin k sinh k cos k + cosh k E15

and the frequency equation is

cos k cosh k + 1 = 0 E16

The spatial mode shape can be obtained as

W 0 = cosh cos + C D sinh sin E17

Now, the solution of the first-order equation is analyzed. In Eq. (9), both h 1 ξ and W 1 are linearly correlated with W 0 . Based on the theory of ordinary differential equations [84], the solvability conditions of linear differential equations can be expressed by the orthogonality of solutions of homogeneous systems of equations. At the same time, according to the orthogonality of modal vibration theory, the solution of Eq. (9) exists under the condition of the solvability of differential equation:

0 1 h 1 ξ 2 ω 1 ω 0 W 0 W 0 = 0 E18

is satisfied. As a result,

ω 1 = 0 1 h 1 ξ W 0 2 ω 0 0 1 W 0 2 E19

Because h 1 ξ is linearly correlated with W 0 , the former equations indicate that the arbitrary amplitude of W 0 does not impact ω 1 . This finding yields the first-order correction of the natural frequency ω 0 corresponding to a nonuniform and homogeneous beam.

Integrating by parts, we obtain

0 1 h 1 ξ W 0 = df 1 d 2 W 0 d ξ 2 W 0 + f 1 d 3 W 0 d ξ 3 W 0 f 1 d 2 W 0 d ξ 2 dW 0 0 1 + 0 1 f 1 d 2 W 0 d ξ 2 2 ω 0 2 f 2 W 0 2 E20

By definition we have

f 1 ξ = E ξ I ξ ¯ E 0 I = E ξ I ξ EI 0 EI 0 E21

so that

df 1 d 2 W 0 d ξ 2 W 0 + f 1 d 3 W 0 d ξ 3 W 0 f 1 d 2 W 0 d ξ 2 dW 0 0 1 + 0 1 f 1 d 2 W 0 d ξ 2 2 = d E ξ I ξ EI 0 d 2 W 0 d ξ 2 W 0 + E ξ I ξ EI 0 d 3 W 0 d ξ 3 W 0 E ξ I ξ EI 0 d 2 W 0 d ξ 2 dW 0 + d 2 W 0 d ξ 2 dW 0 d 3 W 0 d ξ 3 W 0 0 1 + 1 EI 0 0 1 E ξ I ξ d 2 W 0 d ξ 2 2 0 1 d 2 W 0 d ξ 2 2 E22

Choosing the reference bending stiffness

EI 0 = d E ξ I ξ d 2 W 0 d ξ 2 W 0 + E ξ I ξ d 3 W 0 d ξ 3 W 0 E ξ I ξ d 2 W 0 d ξ 2 dW 0 0 1 + 0 1 E ξ I ξ d 2 W 0 d ξ 2 2 d 3 W 0 d ξ 3 W 0 d 2 W 0 d ξ 2 dW 0 0 1 + 0 1 d 2 W 0 d ξ 2 2 E23

we have df 1 d 2 W 0 d ξ 2 W 0 + f 1 d 3 W 0 d ξ 3 W 0 f 1 d 2 W 0 d ξ 2 dW 0 0 1 + 0 1 f 1 d 2 W 0 d ξ 2 2 = 0 .

Analogously, we choose

ρ A 0 = 0 1 ρ ξ A ξ W 0 2 0 1 W 0 2 E24

giving 0 1 f 2 W 0 2 = 0 . Then, we obtain ω 1 = 0 . These values are the properties of the equivalent homogeneous beam having the same frequency (at least up to the first order) as the given nonuniform AFG beam.

The nth natural circular frequency of the AFG beam can be derived as

λ n = 1 L 2 EI 0 ρ A 0 ω 0 E25

Each order of frequency of ω 0 can be determined by Eq. (16) (in turn, positive numbers from small to large). The required variables have been computed by the above expression. Eq. (25) is an approximate formula for the natural frequencies of variable cross-sectional AFG beams.

In order to show the applicability of this method, we study other supporting conditions, and we can easily get the corresponding boundary conditions of Eqs. (13) and (14). Due to the limited space, the detailed derivation process is omitted, and the final results are shown in Table 1.

Boundary conditions Frequency equation Mode shape
Simply supported (S-S) sin k = 0 W 0 = sin
Clamped-pinned (C-P) tan k tanh k = 0 W 0 = cosh cos cosh k cos k sinh k sin k sinh sin
Clamped-clamped (C-C) cos k cosh k 1 = 0 W 0 = cosh cos + sin k + sinh k cos k cosh k sinh sin

Table 1.

Frequency equations and mode shapes for various beams.

3.2 Numerical results and discussion

Based on the above analysis, four kinds of AFG beams with various taper ratios are considered, as shown in Figure 2. The numerical simulations are carried out, and the results are compared with the published literature results to verify the validity of the proposed method.

Figure 2.

Geometry and coordinate system of an AFG beam for different taper ratios: (a) case 1, c b = c h = 0 ; (b) case 2, c h = 0 , c b 0 ; (c) case 3, c b = 0 , c h 0 ; and (d) case 4, c b 0 , c h 0 .

In Figure 2, B L and B R are the width of the left and right ends of the beams, respectively, and H L and H R are the height of the left and right ends of the beams, respectively. Here, we assume that the geometric characteristics of AFG beams vary linearly along the longitudinal direction. Therefore, the variation of cross-sectional area A x and moment of inertia I x along the beam axis can be clearly expressed as follows:

A x = A L 1 c b x L 1 c h x L , I x = I L 1 c b x L 1 c h x L 3 E26

where c b = 1 B R B L and c h = 1 H R H L are the breadth and height taper ratios, respectively. A L and I L are cross-sectional area and area moment of inertia of the beam left sides, respectively. It is instructive to remember that if c b = c h = 0 , the beam would be uniform; if c h = 0 , c b 0 , the beam would be tapered with constant height; if c b = 0 , c h 0 , the beam would be tapered with constant width; and if c b 0 , c h 0 , the beam would be double tapered. These four cases are corresponding to Figure 2(a)–(d), respectively. Moreover, the material properties such as Young’s modulus E x and mass density ρ x along the beam axis are assumed as

E x = E L 1 + x L , ρ x = ρ L 1 + x L + x L 2 E27

where E L and ρ L are Young’s modulus and mass density of the beam left sides, respectively.

Based on the introduced analytical equation, the first third-order nondimensional natural frequencies ( Ω n = λ n L 2 ρ L A L / E L I L ) of the four cases of nonuniform AFG beams with different boundary configurations were obtained. The results were listed in Tables 27, respectively, and it also was compared with those of published work by Shahba et al. [38].

Boundary condition First mode Second mode Third mode
C-F Present 2.439 18.437 54.339
Ref. [38] 2.426 18.604 55.180
S-S Present 9.053 35.834 80.470
Ref. [38] 9.029 36.372 81.732
C-C Present 20.585 56.251 109.869
Ref. [38] 20.472 56.549 110.947

Table 2.

Nondimensional natural frequencies of the AFG uniform beam (case 1) with different boundary conditions.

c b C-F S-S C-C
First mode Second mode Third mode First mode Second mode Third mode First mode Second mode Third mode
0.2 Present 2.613 18.887 54.951 9.068 35.957 80.772 20.457 56.196 110.003
Ref. [38] 2.605 19.004 55.534 9.060 36.342 81.685 20.415 56.472 110.862
0.4 Present 2.854 19.483 55.753 9.088 36.117 81.165 20.294 56.124 110.177
Ref. [38] 2.851 19.530 56.023 9.087 36.315 81.645 20.288 56.298 110.671
0.6 Present 3.214 20.311 56.853 9.113 36.332 81.697 20.079 56.028 110.411
Ref. [38] 3.214 20.296 56.800 9.099 36.297 81.624 20.019 55.921 110.250
0.8 Present 3.832 21.542 58.453 9.147 36.638 82.456 19.783 55.892 110.743
Ref. [38] 3.831 21.676 58.435 9.069 36.277 81.639 19.385 54.971 109.142

Table 3.

Nondimensional natural frequencies of the AFG-tapered beam with constant height (case 2) and different boundary conditions.

c h C-F S-S C-C
First mode Second mode Third mode First mode Second mode Third mode First mode Second mode Third mode
0.2 Present 2.5054 17.2596 49.4982 8.1416 32.1888 72.2680 18.2420 50.1851 98.2992
Ref. [38] 2.5051 17.3802 50.0491 8.1341 32.5236 73.1138 18.2170 50.4801 99.1734
0.4 Present 2.6293 16.2995 45.3519 7.2793 28.9717 65.1203 16.3027 45.0600 88.4345
Ref. [38] 2.6155 16.0705 44.6181 7.1531 28.4747 63.9942 15.8282 44.0246 86.6272
0.6 Present 2.8535 15.6697 42.2358 6.4872 26.3694 59.4850 14.9152 41.2502 80.9747
Ref. [38] 2.7835 14.6508 38.7446 6.0357 24.1101 54.0921 13.2293 36.9653 72.8740
0.8 Present 3.2889 15.5662 40.6554 5.7966 24.6371 55.9734 14.2233 39.1823 76.7690
Ref. [38] 3.0871 13.1142 32.1309 4.6520 19.1314 42.6954 10.2235 28.7492 56.8109

Table 4.

Nondimensional natural frequencies of the AFG-tapered beam with constant width (case 3) and different boundary conditions.

c b 0.2 0.4 0.6 0.8
c h First mode
0.2 Present 2.6873 2.9380 3.3113 3.9455
Ref. [38] 2.6863 2.9336 3.2993 3.9219
0.4 Present 2.8226 3.0877 3.4796 4.1377
Ref. [38] 2.7987 3.0486 3.4181 4.0471
0.6 Present 3.0640 3.3506 3.7700 4.4625
Ref. [38] 2.9699 3.2237 3.5985 4.2355
0.8 Present 3.5271 3.8475 4.3081 5.0458
Ref. [38] 3.2794 3.5401 3.9232 4.5695
c h Second mode
0.2 Present 17.7225 18.3289 19.1598 20.3725
Ref. [38] 17.7501 18.2379 18.9501 20.2432
0.4 Present 16.7822 17.4061 18.2458 19.4418
Ref. [38] 16.4092 16.8571 17.5139 18.7164
0.6 Present 16.1771 16.8214 17.6687 18.8380
Ref. [38] 14.9567 15.3627 15.9616 17.0694
0.8 Present 16.0947 16.7493 17.5836 18.6877
Ref. [38] 13.3850 13.7466 14.2848 15.2955
c h Third mode
0.2 Present 50.2194 51.1534 52.4114 54.1995
Ref. [38] 50.3934 50.8645 51.6029 53.1332
0.4 Present 46.1970 47.2734 48.6925 50.6520
Ref. [38] 44.9504 45.4003 46.0957 47.5129
0.6 Present 43.2042 44.4117 45.9613 48.0269
Ref. [38] 39.0605 39.4844 40.1304 41.4236
0.8 Present 41.7065 42.9817 44.5636 46.5828
Ref. [38] 32.4229 32.8123 33.3986 34.5521

Table 5.

Nondimensional natural frequencies of the AFG double-tapered beam (case 4); boundary conditions: C-F.

c b 0.2 0.4 0.6 0.8
c h First mode
0.2 Present 8.1682 8.2018 8.2456 8.3051
Ref. [38] 8.1462 8.1498 8.1336 8.0646
0.4 Present 7.3172 7.3647 7.4262 7.5089
Ref. [38] 7.1455 7.1254 7.0794 6.9703
0.6 Present 6.5357 6.5960 6.6732 6.7754
Ref. [38] 6.0082 5.9638 5.8868 5.7351
0.8 Present 5.8537 5.9240 6.0128 6.1283
Ref. [38] 4.6046 4.5355 4.4264 4.2283
c h Second mode
0.2 Present 32.4133 32.7007 33.0819 33.6118
Ref. [38] 32.5123 32.5079 32.5164 32.5326
0.4 Present 29.2971 29.7076 30.2419 30.9665
Ref. [38] 28.4822 28.5003 28.5370 28.5928
0.6 Present 26.7834 27.2965 27.9493 28.8091
Ref. [38] 24.1371 24.1791 24.2469 24.3497
0.8 Present 25.1032 25.6683 26.3683 27.2590
Ref. [38] 19.1803 19.2509 19.3590 19.5300
c h Third mode
0.2 Present 72.8179 73.5237 74.4625 75.7732
Ref. [38] 73.0959 73.0903 73.1116 73.1855
0.4 Present 65.9158 66.9202 68.2291 70.0069
Ref. [38] 64.0054 64.0350 64.1007 64.2374
0.6 Present 60.4922 61.7392 63.3243 65.4089
Ref. [38] 54.1330 54.1992 54.3126 54.5207
0.8 Present 57.0969 58.4547 60.1303 62.2530
Ref. [38] 42.7677 42.8742 43.0436 43.3451

Table 6.

Nondimensional natural frequencies of the AFG double-tapered beam (case 4); boundary conditions: S-S.

c b 0.2 0.4 0.6 0.8
c h First mode
0.2 Present 18.2779 18.3231 18.3818 18.4612
Ref. [38] 18.1996 18.1286 17.9437 17.4566
0.4 Present 16.4975 16.7396 17.0484 17.4563
Ref. [38] 15.8498 15.8350 15.7367 15.4025
0.6 Present 15.2512 15.6622 16.1771 16.8423
Ref. [38] 13.2896 13.3319 13.3238 13.1529
0.8 Present 14.6662 15.2004 15.8587 16.6925
Ref. [38] 10.3229 10.4255 10.5168 10.5339
c h Second mode
0.2 Present 50.4430 50.7713 51.2035 51.7981
Ref. [38] 50.4565 50.3599 50.1017 49.3728
0.4 Present 45.6257 46.3346 47.2495 48.4763
Ref. [38] 44.0553 44.0370 43.9027 43.4066
0.6 Present 42.0890 43.1214 44.4245 46.1234
Ref. [38] 37.0509 37.1137 37.1104 36.8678
0.8 Present 40.2151 41.4614 42.9975 44.9420
Ref. [38] 28.8912 29.0409 29.1842 29.2402
c h Third mode
0.2 Present 98.2992 99.7466 100.8219 102.313
Ref. [38] 99.1474 99.0414 98.7543 97.9046
0.4 Present 88.4345 90.9806 92.8200 95.3023
Ref. [38] 86.6608 86.6414 86.4932 85.9176
0.6 Present 80.9747 84.4598 86.8967 90.0855
Ref. [38] 72.9681 73.0382 73.0375 72.7615
0.8 Present 76.7690 80.8426 83.5867 87.0576
Ref. [38] 56.9674 57.1341 57.2991 57.3787

Table 7.

Nondimensional natural frequencies of the AFG double-tapered beam (case 4); boundary conditions: C-C.

Table 2 shows the first third-order natural frequencies of the AFG beam, case of Figure 2(a), which is uniform but nonhomogeneous. It can be clearly seen that the analytical results obtained from the asymptotic development method are in good agreement with those given by Ref. [38].

As can be seen from Tables 3 and 4, the first third-order dimensionless natural frequencies of AFG conical beams with only varying width or height are studied, respectively. It is easy to find the following conclusions. This method has higher accuracy on the equal height AFG-tapered beam. When the height changes, there is a certain fractional error in the AFG-tapered beam.

According to Figure 2(d), when the height and width of AFG beams change simultaneously, we can see that AFG beams are not uniform. The natural frequencies of three boundary conditions (free clamping, simply supported, and clamping) are studied in Tables 5, 6, 7. From the data in the table, it can be clearly found that the natural frequencies of AFG beams at low order are in good agreement with Ref. [38], while at high order, there are some errors in the natural frequencies.

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4. The method of Meijer G-function

4.1 Equation deriving

In this section, the Meijer G-function is introduced to obtain the formula of the nature frequency of the AFG beam. Here, a special case of AFG beam is considered, where the cross section is uniform. Thus, in Eq. (1), Young’s modulus E x and material mass density ρ x are variable parameters, and the area moment of inertia I and the cross-sectional area A are invariant. To solve the governing equation, two parameters are firstly introduced to depict the functional gradient parameter equation:

E x = E L 1 f E x L , ρ x = ρ L 1 f ρ x L E28

where f E = 1 E R E L , ρ E = 1 ρ R ρ L . E L and E R are Young’s modulus at the left/right end of the beam, and ρ L and ρ R are the mass density at the left/right end of the beam. Eq. (2) is then rewritten as

1 f E x w + 1 f ρ x w ¨ = 0 E29

Based on the vibration theory, we assume w x t = ϕ x q t , where q n t = A n cos β n 2 t + B n sin β n 2 t and β n 2 is the modal frequency for dimensionless. The governing equation is then derived as

1 f E x ϕ n β n 4 1 f ρ x ϕ n = 0 E30

Next, Meijer G-function will be used to solve the linear partial differential equation. The general expression of Meijer G-function differential equation is written as

1 p m n z l = 1 p z d dz + 1 a l k = 1 q η d dz b k G z = 0 E31

where m , n , p and q are integers satisfying 0 m q , 0 n p , G is the dependent variable also known as the Meijer G-function, z is the independent variable, and a l and b k are real numbers.

A definition of the Meijer G-function is given by the following path integral in the complex plane, called the Mellin-Barnes type:

G p , q m , n a 1 a n , a n + 1 a p b 1 b m , b m + 1 b q | z = 1 2 πi τ k = 1 m Γ ξ b k k = 1 n Γ 1 a k + ξ k = 1 p Γ ξ a k k = m + 1 q Γ 1 b k + ξ z ξ E32

where an empty product is interpreted as 1, 0 m q , 0 n p , and the parameters are such that none of the poles of Γ b j ξ , j = 1 m coincides with the poles of Γ 1 a j + ξ , j = 1 n . Where i is a complex number such that i2 = −1.

A special case of Eq. (31) can be expanded by assuming n = p = 0 and q = 4 . We can get that

z 4 d 4 G dz 4 + 6 k = 1 4 b k z 3 d 3 G dz 4 + 7 3 n = 1 4 b k + k , l = 1 4 b k b l z 2 d 2 G dz 2 + 1 k = 1 4 b k + k , l = 1 4 b k b l b 1 b 2 + b 1 b 3 + b 2 b 3 b 4 z dG dz 1 m z k = 1 4 b k G = 0 E33

where k l . Although Eq. (30) is not similar to Eq. (33), the two equations can be similar by introducing some transformations:

ϕ n x = G z n x , z n x = β n 4 f E 4 1 f E x 4 E34

Eq. (30) is transformed into

η n 3 G + 5 η n 2 G + 69 16 η n G + 9 32 G 1 f ρ x 1 f E x G = 0 E35

Because of the difficulty of solving the differential equation with variable coefficients, we can simplify Eq. (35). Let 1 f E x = 1 f ρ x = 1 Fx ; it can be rewritten as

η n 3 G + 5 η n 2 G + 69 16 η n G + 9 32 G G = 0 E36

In order to solve the above equation, the coefficients of the ordinary differential Eqs. (33) and (36) are the same, so we can calculate the corresponding values, as shown in Table 8.

Case b 1 b 2 b 3 b 4
1 1/2 1/4 0 1/4
2 1/4 1/4 0 1/2
3 0 1/4 1/2 1/4
4 1/2 0 1/4 1/4
5 0 1/2 1/4 1/4
6 0 1/4 1/4 1/2
7 1/4 0 1/4 1/2
8 1/4 1/2 0 1/4
9 1/4 0 1/2 1/4

Table 8.

Possible case of unknown constant b k of Meijer G-function equation.

One set of data can be selected from Table 8 and expressed in the form of closed solutions of Meijer G-function:

φ 1 n x = G 0 , 4 2 , 0 1 4 1 4 0 1 2 β n 4 1 Fx 4 256 F 4 E37
φ 2 n x = G 0 , 4 1 , 0 1 4 0 1 2 1 4 β n 4 1 Fx 4 256 F 4 E38
φ 3 n x = G 0 , 4 1 , 0 0 1 2 1 4 1 4 β n 4 1 Fx 4 256 F 4 E39
φ 4 n x = G 0 , 4 1 , 0 1 2 1 4 1 4 0 β n 4 1 Fx 4 256 F 4 E40

Modal modes of beams:

ϕ n x = C 1 n φ 1 n x + C 2 n φ 2 n x + C 3 n φ 3 n x + C 4 n φ 4 n x , n = 1 , 2 , 3 , E41

In order to determine the undetermined coefficients C i and β n , the boundary conditions of beams need to be considered:

  1. C-F:

φ 1 n 0 φ 2 n 0 φ 3 n 0 φ 4 n 0 φ 1 n 0 φ 2 n 0 φ 3 n 0 φ 4 n 0 φ 1 n 1 φ 2 n 1 φ 3 n 1 φ 4 n 1 φ 1 n 1 φ 2 n 1 φ 3 n 1 φ 4 n 1 C 1 n C 2 n C 3 n C 4 n = 0 0 0 0 E42

  1. C-P:

φ 1 n 0 φ 2 n 0 φ 3 n 0 φ 4 n 0 φ 1 n 0 φ 2 n 0 φ 3 n 0 φ 4 n 0 φ 1 n 1 φ 2 n 1 φ 3 n 1 φ 4 n 1 φ 1 n 1 φ 2 n 1 φ 3 n 1 φ 4 n 1 C 1 n C 2 n C 3 n C 4 n = 0 0 0 0 E43

  1. S-S:

φ 1 n 0 φ 2 n 0 φ 3 n 0 φ 4 n 0 φ 1 n 0 φ 2 n 0 φ 3 n 0 φ 4 n 0 φ 1 n 1 φ 2 n 1 φ 3 n 1 φ 4 n 1 φ 1 n 1 φ 2 n 1 φ 3 n 1 φ 4 n 1 C 1 n C 2 n C 3 n C 4 n = 0 0 0 0 E44

  1. C-C:

φ 1 n 0 φ 2 n 0 φ 3 n 0 φ 4 n 0 φ 1 n 0 φ 2 n 0 φ 3 n 0 φ 4 n 0 φ 1 n 1 φ 2 n 1 φ 3 n 1 φ 4 n 1 φ 1 n 1 φ 2 n 1 φ 3 n 1 φ 4 n 1 C 1 n C 2 n C 3 n C 4 n = 0 0 0 0 E45

4.2 Numerical results and discussion

Based on the above analysis, the natural frequencies of beams under different boundary conditions can be solved. Meanwhile, the results of finite element method are also conducted to verify the accuracy of the analytical results. Here, we use the power law gradient of the existing AFG beams [44], and the material properties of AFG beams change continuously along the axial direction. Therefore, the expressions of Young’s modulus E x and mass density ρ x are listed in detail:

Y x = Y L 1 e αx / L 1 e α 1 + Y R e αx / L 1 e α 1 , α 0 , Y L 1 x L + Y R x L , α = 0 . E46

where Y L and Y R denote the corresponding material properties of the left and right sides of the beam, respectively. α is the gradient parameter describing the volume fraction change of both constituents involved. When gradient parameter α is equal and less than zero, Young’s modulus and mass density at the left end are less than those at the right end. When α equals zero, the beam is equivalent to a uniform Euler-Bernoulli beam, and Young’s modulus and mass density of the beam do not change with the length direction of the beam.

The variation of Y x along the axis direction of the beam can be shown in Figure 3 for Y R = 3 Y L . In order to show the practicability of this method, we choose the existing materials to study. The materials of AFG beams are composed of aluminum ( Al ) and zirconia ( ZrO 2 ).The left and right ends of the beam are pure aluminum and pure zirconia, respectively. The material properties of AFG beams are given in detail in Table 9. We choose the sizes of commonly used beams which are L = 0.2 m, B = 0.02 m, and H = 0.001 m.

Figure 3.

Variation of the material properties defined by Eq. (46) with Y R = 3 Y L .

Properties Unit Aluminum Zirconia
E GPa 70 200
ρ Kg/m3 2702 5700

Table 9.

Material properties of the AFG beam [44].

In order to verify the correctness of this method, some finite element simulation software is used to verify its correctness. In this paper, we analyze the natural frequencies of uniform AFG beams under different boundary conditions. In the process of finite element analysis, the AFG beam is transformed into a finite length model by using the delamination method [85]. At the same time, the AFG beam is delaminated along the axial direction. As shown in Figure 4, the material properties change along the axial direction, and the material properties of the adjacent layers are different. In order to analyze the performance of the beam, the uniform element is used to mesh each layer. In order to make the natural frequencies of AFG beams more precise, we can increase the number of layers and refine the finite element meshes.

Figure 4.

Finite segment model of the AFG beam.

In the Meijer G-function method, in order to solve the linear natural frequencies of beams under different boundary conditions, the determinant of the coefficient matrix of Eqs. (42)(45) is equal to zero. Finally, linear natural frequencies of beams with different boundary conditions of the first four orders are listed in Table 10.

F Order C-F C-P S-S C-C
Present FEM Present FEM Present FEM Present FEM
α = 0.9 1 2.4641 2.4651 4.0787 4.0789 3.0888 3.0891 4.5585 4.5579
2 5.2251 5.2265 7.1520 7.1516 6.2895 6.2893 7.6920 7.6908
3 8.2540 8.2560 10.2762 10.2778 9.4410 9.4420 10.8549 10.8560
4 11.3209 11.324 13.4075 13.4092 12.5854 12.5869 14.0136 14.0148
α = 0.5 1 2.0774 2.0772 4.0055 4.0056 3.1344 3.1344 4.7098 4.7096
2 4.8497 4.8491 7.1104 7.1100 6.2859 6.2861 7.8364 7.8360
3 7.9496 7.9501 10.2396 10.2409 9.4278 9.4294 10.9827 10.9836
4 11.0455 11.0652 13.3748 13.3760 12.5668 12.5707 14.1256 14.1273
α = 1.7 1 1.6098 1.6104 3.7738 3.7738 3.1279 3.1277 4.6896 4.6897
2 4.4786 4.4792 6.9816 6.9816 6.2887 6.2884 7.8189 7.8187
3 7.7326 7.7332 10.1477 10.1490 9.4315 9.4325 10.9679 10.9696
4 10.9067 10.9079 13.3045 13.3042 12.5726 12.5737 14.1139 14.1158
α = 2.7 1 1.5370 1.5377 3.7214 3.7217 3.1188 3.1212 4.6629 4.6630
2 4.4142 4.4142 6.9470 6.9471 6.2907 6.2904 7.7947 7.7948
3 7.6920 7.6931 10.1207 10.1222 9.4350 9.4360 10.9482 10.9497
4 10.8759 108,772 13.2807 13.2823 12.5762 12.5774 14.0972 14.0988

Table 10.

Comparisons between FEM and numerical calculation of linear dimensionless natural frequencies of AFG beams with different boundary conditions.

From Table 10, we can see that the results of finite element method are similar to those of Meijer G-function and the error is small. This can prove the accuracy of the method in frequency calculation on the one hand. In Figure 5, we can find that the first third-order dimensionless natural frequencies of C-F beams are in good agreement with FEM and numerical calculation. With the gradual increase of gradient parameter F, the dimensionless natural frequency of C-F beam increases gradually, and the change speed is accelerated. At the same time, the FEM and numerical simulation errors are very small, so the precise linear natural frequencies can be obtained.

Figure 5.

Dimensionless natural frequencies of C-F beams vary with parameter F: (a) fundamental frequencies, (b) second-order frequencies, and (c) third-order frequencies.

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5. Conclusions

FGMs are innovative materials and are very important in engineering and other applications. Despite the variety of methods and approaches for numerical and analytical investigation of nonuniform FG beams, no simple and fast analytical method applicable for such beams with different boundary conditions and varying cross-sectional area was proposed. In this topic, two analytical approaches, the asymptotic perturbation and the Meijer G-function method, were described to analyze the free vibration of the AFG beams.

Based on the Euler-Bernoulli beam theory, the governing differential equations and related boundary conditions are described, which is more complicated because of the partial differential equation with variable coefficients. For both the asymptotic perturbation and the Meijer G-function method, the variable flexural rigidity and mass density are divided into invariant parts and variable parts firstly. Different analytical processes are then carried out to deal with the variable parts applying perturbation theory and the Meijer G-function, respectively. Finally, the simple formulas are derived for solving the nature frequencies of the AFG beams with C-F boundary conditions followed with C-C, C-S, and C-P conditions, respectively. It is observed that natural frequency increases gradually with the increase of the gradient parameter.

Accuracy of the results is also examined using the available data in the published literature and the finite element method. In fact, it can be clearly found that result of the APM is more accurate in low-order mode, which is caused by the defect of the perturbation theory. However, the APM is simple and easily comprehensible, while the Meijer G-function method is more complex and unintelligible for engineers. In general, the results show that the proposed two analytical methods are efficient and can be used to analyze the free vibration of AFG beams.

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Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Grant Nos. 11672008, 11702188, and 11272016).

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Written By

Dongxing Cao, Bin Wang, Wenhua Hu and Yanhui Gao

Submitted: November 6th, 2018 Reviewed: March 14th, 2019 Published: April 22nd, 2019