Frequency equations and mode shapes for various beams.
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 GFunction are studied, respectively. First, a detailed review of the existing literatures is summarized. Then, based on the governing equation of the AFG EulerBernoulli beam, the detailed analytic equations are derived on basis of the perturbation theory and Meijer Gfunction, 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 Gfunction
 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 thicknesswise 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 cellcentered 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 firstorder 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 highorder 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 HosseiniHashemi [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, thermalmechanical effect on FGM structures has attracted broad attention. In this field, Farzad Ebrahimi and Erfan Salari obtained outstanding achievements. Considering the thermalmechanical effect and sizedependent thermoelectric 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 semianalytical differential transformation method. To discuss the effect of the shear stress, Reddy’s higherorder 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 temperaturedependent 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 twonode, sixdegreeoffreedom finite element method (FEM) in conjunction with EulerBernoulli 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 AFGtapered beam based on EulerBernoulli and Timoshenko beam theory. Shahba and Rajasekaran [40] studied the free vibration analysis of AFGtapered EulerBernoulli 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 AFGtapered EulerBernoulli 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 crosssectional 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 EulerBernoulli 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 EulerBernoulli and Timoshenko beams with nonuniform cross sections. Fang and Zhou [51, 52] researched the modal analysis of rotating AFGtapered EulerBernoulli and Timoshenko beams with various boundary conditions employing the ChebyshevRitz method. Li et al. [53, 54] obtained the exact solutions for the free vibration of FGM beams with material profiles and crosssectional parameters varying exponentially in the axial direction, where assumptions of EulerBernoulli 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 AFGtapered EulerBernoulli microbeams based on BernoulliEuler 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 EulerBernoulli 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 EulerBernoulli beams. The splinemethod 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 Gfunction approach. Rajasekaran and Khaniki [65] applied the FEM to research the vibration behavior of nonuniform smallscale 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 multiplestepped 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 steadystate periodic response of forced vibration of dynamic viscoelastic beams based on the multiscale 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 PoincaréLindstedt 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 Gfunction. 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 EulerBernoulli 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.
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 EulerBernoulli beam theory, the governing differential equation of the beam can be written as
where
Because of the particularity of AFG beam, bending stiffness
where
are the nondimensional varying parts of the flexural stiffness and of the mass per unit length, respectively.
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
where
To use the APM, a small perturbation parameter
According to the PoincaréLindstedt method [78, 79, 80, 81, 82], we assume the expansion for
Substituting these expressions with governing Eq. (5) and then expanding the expressions into a
where
For Eq. (8), the following general solution can be obtained:
where
For simplicity, we consider clampedfree (CF) beams, and the corresponding boundary conditions are
Then, the following equations from equation can be obtained:
and the frequency equation is
The spatial mode shape can be obtained as
Now, the solution of the firstorder equation is analyzed. In Eq. (9), both
is satisfied. As a result,
Because
Integrating by parts, we obtain
By definition we have
so that
Choosing the reference bending stiffness
we have
Analogously, we choose
giving
The nth natural circular frequency of the AFG beam can be derived as
Each order of frequency of
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 (SS) 


Clampedpinned (CP) 


Clampedclamped (CC) 


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.
In Figure 2,
where
where
Based on the introduced analytical equation, the first thirdorder nondimensional natural frequencies (

CF  SS  CC  

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 

CF  SS  CC  

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 

0.2  0.4  0.6  0.8  


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  

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  

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 

0.2  0.4  0.6  0.8  


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  

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  

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 

0.2  0.4  0.6  0.8  


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  

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  

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 2 shows the first thirdorder 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 thirdorder 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 AFGtapered beam. When the height changes, there is a certain fractional error in the AFGtapered 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.
4. The method of Meijer Gfunction
4.1 Equation deriving
In this section, the Meijer Gfunction 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
where
Based on the vibration theory, we assume
Next, Meijer Gfunction will be used to solve the linear partial differential equation. The general expression of Meijer Gfunction differential equation is written as
where
A definition of the Meijer Gfunction is given by the following path integral in the complex plane, called the MellinBarnes type:
where an empty product is interpreted as 1,
A special case of Eq. (31) can be expanded by assuming
where
Eq. (30) is transformed into
Because of the difficulty of solving the differential equation with variable coefficients, we can simplify Eq. (35). Let
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 





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 
One set of data can be selected from Table 8 and expressed in the form of closed solutions of Meijer Gfunction:
Modal modes of beams:
In order to determine the undetermined coefficients
CF:
CP:
SS:
CC:
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
where
The variation of
Properties  Unit  Aluminum  Zirconia 

E  GPa  70  200 

Kg/m^{3}  2702  5700 
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.
In the Meijer Gfunction 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  CF  CP  SS  CC  

Present  FEM  Present  FEM  Present  FEM  Present  FEM  

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  

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  

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  

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 
From Table 10, we can see that the results of finite element method are similar to those of Meijer Gfunction 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 thirdorder dimensionless natural frequencies of CF beams are in good agreement with FEM and numerical calculation. With the gradual increase of gradient parameter F, the dimensionless natural frequency of CF 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.
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 crosssectional area was proposed. In this topic, two analytical approaches, the asymptotic perturbation and the Meijer Gfunction method, were described to analyze the free vibration of the AFG beams.
Based on the EulerBernoulli 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 Gfunction 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 Gfunction, respectively. Finally, the simple formulas are derived for solving the nature frequencies of the AFG beams with CF boundary conditions followed with CC, CS, and CP 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 loworder mode, which is caused by the defect of the perturbation theory. However, the APM is simple and easily comprehensible, while the Meijer Gfunction 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.
Acknowledgments
The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Grant Nos. 11672008, 11702188, and 11272016).
References
 1.
Nie GJ, Zhong Z, Chen S. Analytical solution for a functionally graded beam with arbitrary graded material properties. Composites Part B: Engineering. 2013; 44 (1):274282. DOI: 10.1016/j.compositesb.2012.05.029  2.
Nguyen DK. Large displacement response of tapered cantilever beams made of axially functionally graded material. Composites Part B: Engineering. 2013; 55 (9):298305. DOI: 10.1016/j.compositesb.2013.06.024  3.
Calim FF. Transient analysis of axially functionally graded Timoshenko beams with variable crosssection. Composites Part B: Engineering. 2016; 98 (2015):472483. DOI: 10.1016/j.compositesb.2016.05.040  4.
Navvab S, Mohammad K, Majid G. Nonlinear vibration of axially functionally graded tapered microbeams. International Journal of Engineering Science. 2016; 102 (2016):1226. DOI: 10.1016/j.ijengsci.2016.02.007  5.
Hao YX, Chen LH, Zhang W, Lei JG. Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate. Journal of Sound and Vibration. 2008; 312 (4–5):862892. DOI: 10.1016/j.jsv.2007.11.033  6.
Hao YX, Zhang W, Yang J, Li S. Nonlinear dynamics of a functionally graded thin simplysupported plate under a hypersonic flow. Mechanics of Advanced Materials and Structures. 2015; 22 (8):619632. DOI: 10.1080/15376494.2013.828817  7.
Niu Y, Hao Y, Yao M, Zhang W, Yang S. Nonlinear dynamics of imperfect FGM conical panel. Shock and Vibration. 2018; 2018 :120. DOI: 10.1155/2018/4187386  8.
Zhang W, Yang J, Hao Y. Chaotic vibrations of an orthotropic FGM rectangular plate based on thirdorder shear deformation theory. Nonlinear Dynamics. 2010; 59 (4):619660. DOI: 10.1007/s110710099568y  9.
Hao YX, Li ZN, Zhang W, Li SB, Yao MH. Vibration of functionally graded sandwich doubly curved shells using improved shear deformation theory. Science China Technological Sciences. 2018; 61 (6):791808. DOI: 10.1007/s1143101690977  10.
Hao YX, Zhang W, Yang J. Nonlinear dynamics of cantilever FGM cylindrical shell under 1:2 internal resonance relations. Mechanics of Advanced Materials and Structures. 2012; 20 (10):819833. DOI: 10.1080/15376494.2012.676717  11.
Zhang W, Hao YX, Yang J. Nonlinear dynamics of FGM circular cylindrical shell with clampedclamped edges. Composite Structures. 2012; 94 (3):10751086. DOI: 10.1016/j.compstruct.2011.11.004  12.
Hao YX, Cao Z, Zhang W, Chen J, Yao MH. Stability analysis for geometric nonlinear functionally graded sandwich shallow shell using a new developed displacement field. Composite Structures. 2019; 210 :202216. DOI: 10.1016/j.compstruct.2018.11.027  13.
Lee JW, Lee JY. Free vibration analysis of functionally graded BernoulliEuler beams using an exact transfer matrix expression. International Journal of Mechanical Sciences. 2017; 122 :117. DOI: 10.1016/j.ijmecsci.2017.01.011  14.
Su H, Banerjee JR, Cheung CW. Dynamic stiffness formulation and free vibration analysis of functionally graded beams. Composite Structures. 2013; 106 (12):854862. DOI: 10.1016/j.compstruct.2013.06.029  15.
Ll J, Pj M, Wp Z, Lr F, Yp C. Static and free vibration analysis of functionally graded beams by combination Timoshenko theory and finite volume method. Composite Structures. 2016; 138 :192213. DOI: 10.1016/j.compstruct.2015.11.027  16.
Ait Atmane H, Tounsi A, Meftah SA, Belhadj HA. Free vibration behavior of exponential functionally graded beams with varying crosssection. Journal of Vibration and Control. 2010; 17 (2):311318. DOI: 10.1177/1077546310370691  17.
Sina SA, Navazi HM, Haddadpour H. An analytical method for free vibration analysis of functionally graded beams. Materials and Design. 2009; 30 (3):741747. DOI: 10.1016/j.matdes.2008.05.015  18.
Sharma P. Efficacy of harmonic differential quadrature method to vibration analysis of FGPM beam. Composite Structures. 2018; 189 (2018):107116. DOI: 10.1016/j.compstruct.2018.01.059  19.
Li XF, Wang BL, Han JC. A higherorder theory for static and dynamic analyses of functionally graded beams. Archive of Applied Mechanics. 2010; 80 (10):11971212. DOI: 10.1007/s0041901004356  20.
Celebi K, Yarimpabuc D, Tutuncu N. Free vibration analysis of functionally graded beams using complementary functions method. Archive of Applied Mechanics. 2018; 88 (5):729739. DOI: 10.1007/s0041901713386  21.
Chen D, Kitipornchai S, Yang J. Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core. ThinWalled Structures. 2016; 107 (2016):3948. DOI: 10.1016/j.tws.2016.05.025  22.
Nazemnezhad R, HosseiniHashemi S. Nonlocal nonlinear free vibration of functionally graded nanobeams. Composite Structures. 2014; 110 :192199. DOI: 10.1016/j.compstruct.2013.12.006  23.
Ebrahimi F, Salari E, Hosseini SAH. Thermomechanical vibration behavior of FG nanobeams subjected to linear and nonlinear temperature distributions. Journal of Thermal Stresses. 2015; 38 (12):13601386. DOI: 10.1080/01495739.2015.1073980  24.
Ebrahimi F, Salari E. Sizedependent thermoelectrical buckling analysis of functionally graded piezoelectric nanobeams. Smart Materials and Structures. 2015; 24 (12):117. DOI:10.1088/09641726/24/12/125007  25.
Ebrahimi F, Salari E. Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperaturedependent functionally graded nanobeams. Mechanics of Advanced Materials and Structures. 2016; 23 (12):13791397. DOI: 10.1080/15376494.2015.1091524  26.
Ebrahimi F, Salari E, Hosseini SAH. Inplane thermal loading effects on vibrational characteristics of functionally graded nanobeams. Meccanica. 2015; 51 (4):951977. DOI: 10.1007/s1101201502483  27.
Salari FEaE. A semianalytical method for vibrational and buckling analysis of functionally graded nanobeams considering the physical neutral Axis position. Computer Modeling in Engineering and Sciences. 2015; 105 (2):151181. DOI: 10.3970/cmes.2015.105.151  28.
Ebrahimi F, Shafiei N. Influence of initial shear stress on the vibration behavior of singlelayered graphene sheets embedded in an elastic medium based on Reddy’s higherorder shear deformation plate theory. Mechanics of Advanced Materials and Structures. 2016; 24 (9):761772. DOI: 10.1080/15376494.2016.1196781  29.
Ebrahimi F, Farazmandnia N. Thermomechanical vibration analysis of sandwich beams with functionally graded carbon nanotubereinforced composite face sheets based on a higherorder shear deformation beam theory. Mechanics of Advanced Materials and Structures. 2017; 24 (10):820829. DOI: 10.1080/15376494.2016.1196786  30.
Ebrahimi F, Salari E. Thermomechanical vibration analysis of a singlewalled carbon nanotube embedded in an elastic medium based on higherorder shear deformation beam theory. Journal of Mechanical Science and Technology. 2015; 29 (9):37973803. DOI: 10.1007/s1220601508262  31.
Ebrahimi F, Ghasemi F, Salari E. Investigating thermal effects on vibration behavior of temperaturedependent compositionally graded Euler beams with porosities. Meccanica. 2015; 51 (1):223249. DOI: 10.1007/s110120150208y  32.
Ebrahimi F, Mokhtari M. Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method. Journal of The Brazilian Society of Mechanical Sciences and Engineering. 2015; 37 (4):14351444  33.
Ebrahimi F, Zia M. Large amplitude nonlinear vibration analysis of functionally graded Timoshenko beams with porosities. Acta Astronautica. 2015; 116 :117125. DOI: 10.1016/j.actaastro.2015.06.014  34.
Ebrahimi F, Salari E. Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments. Composite Structures. 2015; 128 :363380. DOI: 10.1016/j.compstruct.2015.03.023  35.
Ebrahimi F, Salari E. Thermomechanical vibration analysis of nonlocal temperaturedependent FG nanobeams with various boundary conditions. Composites Part B: Engineering. 2015; 78 :272290. DOI: 10.1016/j.compositesb.2015.03.068  36.
Ebrahimi F, Salari E. Nonlocal thermomechanical vibration analysis of functionally graded nanobeams in thermal environment. Acta Astronautica. 2015; 113 :2950  37.
Alshorbagy AE, Eltaher MA, Mahmoud FF. Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling. 2011; 35 (1):412425. DOI: 10.1016/j.apm.2010.07.006  38.
Shahba A, Attarnejad R, Hajilar S. Free vibration and stability of axially functionally graded tapered EulerBernoulli beams. Shock and Vibration. 2011; 18 (5):683696. DOI: 10.3233/sav20100589  39.
Shahba A, Attarnejad R, Marvi MT, Hajilar S. Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and nonclassical boundary conditions. Composites Part B: Engineering. 2011; 42 (4):801808. DOI: 10.1016/j.compositesb.2011.01.017  40.
Shahba A, Rajasekaran S. Free vibration and stability of tapered EulerBernoulli beams made of axially functionally graded materials. Applied Mathematical Modelling. 2012; 36 (7):30943111. DOI: 10.1016/j.apm.2011.09.073  41.
Liu P, Lin K, Liu H, Qin R. Free transverse vibration analysis of axially functionally graded tapered EulerBernoulli beams through spline finite point method. Shock and Vibration. 2016; 2016 (5891030):123. DOI: 10.1155/2016/5891030  42.
Rajasekaran S. Differential transformation and differential quadrature methods for centrifugally stiffened axially functionally graded tapered beams. International Journal of Mechanical Sciences. 2013; 74 :1531. DOI: 10.1016/j.ijmecsci.2013.04.004  43.
Rajasekaran S, Tochaei EN. Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowestorder. Meccanica. 2014; 49 (4):9951009. DOI: 10.1007/s110120139847z  44.
Huang Y, Li XF. A new approach for free vibration of axially functionally graded beams with nonuniform crosssection. Journal of Sound and Vibration. 2010; 329 (11):22912303. DOI: 10.1016/j.jsv.2009.12.029  45.
Huang Y, Yang LE, Luo QZ. Free vibration of axially functionally graded Timoshenko beams with nonuniform crosssection. Composites Part B: Engineering. 2013; 45 (1):14931498. DOI: 10.1016/j.compositesb.2012.09.015  46.
Huang Y, Rong HW. Free vibration of axially inhomogeneous beams that are made of functionally graded materials. The International Journal of Acoustics and Vibration. 2017; 22 (1):6873. DOI: 10.20855/ijav.2017.22.1452  47.
Hein H, Feklistova L. Free vibrations of nonuniform and axially functionally graded beams using Haar wavelets. Engineering Structures. 2011; 33 (12):36963701. DOI: 10.1016/j.engstruct.2011.08.006  48.
Xie X, Zheng H, Zou X. An integrated spectral collocation approach for the static and free vibration analyses of axially functionally graded nonuniform beams. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2016; 231 (13):24592471. DOI: 10.1177/0954406216634393  49.
Kukla S, Rychlewska J. An approach for free vibration analysis of axially graded beams. Journal of Theoretical and Applied Mechanics. 2016; 54 (3):859870. DOI: 10.15632/jtampl.54.3.859  50.
Zhao Y, Huang Y, Guo M. A novel approach for free vibration of axially functionally graded beams with nonuniform crosssection based on Chebyshev polynomials theory. Composite Structures. 2017; 168 :277284. DOI: 10.1016/j.compstruct.2017.02.012  51.
Fang J, Zhou D. Free vibration analysis of rotating axially functionally gradedtapered beams using ChebyshevRitz method. Materials Research Innovations. 2015; 19 (5):S51255S5S562. DOI: 10.1179/1432891714Z.0000000001289  52.
Fang JS, Zhou D. Free vibration analysis of rotating axially functionally graded tapered Timoshenko beams. International Journal of Structural Stability and Dynamics. 2016; 16 (05):119. DOI: 10.1142/S0219455415500078  53.
Li XF, Kang YA, Wu JX. Exact frequency equations of free vibration of exponentially functionally graded beams. Applied Acoustics. 2013; 74 (3):413420. DOI: 10.1016/j.apacoust.2012.08.003  54.
Tang AY, Wu JX, Li XF, Lee KY. Exact frequency equations of free vibration of exponentially nonuniform functionally graded Timoshenko beams. International Journal of Mechanical Sciences. 2014; 89 :111. DOI: 10.1016/j.ijmecsci.2014.08.017  55.
Sarkar K, Ganguli R. Closedform solutions for axially functionally graded Timoshenko beams having uniform crosssection and fixed–fixed boundary condition. Composites Part B: Engineering. 2014; 58 :361370. DOI: 10.1016/j.compositesb.2013.10.077  56.
Akgöz B, Civalek Ö. Free vibration analysis of axially functionally graded tapered BernoulliEuler microbeams based on the modified couple stress theory. Composite Structures. 2013; 98 (3):314322. DOI: 10.1016/j.compstruct.2012.11.020  57.
Yuan J, Pao YH, Chen W. Exact solutions for free vibrations of axially inhomogeneous Timoshenko beams with variable cross section. Acta Mech. 2016; 227 (9):26252643. DOI: 10.1007/s0070701616586  58.
Yilmaz Y, Evran S. Free vibration analysis of axially layered functionally graded short beams using experimental and finite element methods. Science and Engineering of Composite Materials. 2016; 23 (4):453460. DOI: 10.1515/secm20140161  59.
Boiangiu M, Ceausu V, Untaroiu CD. A transfer matrix method for free vibration analysis of EulerBernoulli beams with variable cross section. Journal of Vibration and Control. 2016; 22 (11):25912602. DOI: 10.1177/1077546314550699  60.
Garijo D. Free vibration analysis of nonuniform EulerBernoulli beams by means of Bernstein pseudospectral collocation. Engineering with Computers. 2015; 31 (4):813823. DOI: 10.1007/s0036601504016  61.
Arndt M, Machado RD, Scremin A. Accurate assessment of natural frequencies for uniform and nonuniform EulerBernoulli beams and frames by adaptive generalized finite element method. Engineering Computations. 2016; 33 (5):15861609. DOI: 10.1108/EC0520150116  62.
Zhernakov VS, Pavlov VP, Kudoyarova VM. Splinemethod for numerical calculation of naturalvibration frequency of beam with variable crosssection. Procedia Engineering. 2017; 206 :710715. DOI: 10.1016/j.proeng.2017.10.542  63.
Wang CY. Vibration of a tapered cantilever of constant thickness and linearly tapered width. Archive of Applied Mechanics. 2013; 83 (1):171176. DOI: 10.1007/s0041901206371  64.
Silva CJ, Daqaq MF. Nonlinear flexural response of a slender cantilever beam of constant thickness and linearlyvarying width to a primary resonance excitation. Journal of Sound and Vibration. 2017; 389 (2017):438453. DOI: 10.1016/j.jsv.2016.11.029  65.
Rajasekaran S, Khaniki HB. Bending, buckling and vibration of smallscale tapered beams. International Journal of Engineering Science. 2017; 120 :172188. DOI: 10.1016/j.ijengsci.2017.08.005  66.
Çalım FF. Free and forced vibrations of nonuniform composite beams. Composite Structures. 2009; 88 (3):413423. DOI: 10.1016/j.compstruct.2008.05.001  67.
Yang X, Wang S, Zhang W, Qin Z, Yang T. Dynamic analysis of a rotating tapered cantilever Timoshenko beam based on the power series method. Applied Mathematics and Mechanics. 2017; 38 (10):14251438. DOI: 10.1007/s1048301722496  68.
Clementi F, Demeio L, Mazzilli CEN, Lenci S. Nonlinear vibrations of nonuniform beams by the MTS asymptotic expansion method. Continuum Mechanics and Thermodynamics. 2015; 27 (4):703717. DOI: 10.1007/s0016101403683  69.
Wang X, Wang Y. Free vibration analysis of multiplestepped beams by the differential quadrature element method. Applied Mathematics and Computation. 2013; 219 (11):58025810. DOI: 10.1016/j.amc.2012.12.037  70.
Abdelghany SM, Ewis KM, Mahmoud AA, Nassar MM. Vibration of a circular beam with variable cross sections using differential transformation method. BeniSuef University Journal of Basic and Applied Sciences. 2015; 4 (3):185191. DOI: 10.1016/j.bjbas.2015.05.006  71.
Chen LQ, Chen H. Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model. Journal of Engineering Mathematics. 2010; 67 (3):205218. DOI: 10.1007/s1066500993169  72.
Yan Q, Ding H, Chen L. Nonlinear dynamics of axially moving viscoelastic Timoshenko beam under parametric and external excitations. Applied Mathematics and Mechanics. 2015; 36 (8):971984. DOI: 10.1007/s1048301519667  73.
Ding H, Tang YQ, Chen LQ. Frequencies of transverse vibration of an axially moving viscoelastic beam. Journal of Vibration and Control. 2015; 23 (20):111. DOI: 10.1177/1077546315600311  74.
Ding H, Huang L, Mao X, Chen L. Primary resonance of traveling viscoelastic beam under internal resonance. Applied Mathematics and Mechanics. 2016; 38 (1):114. DOI: 10.1007/s1048301621526  75.
Chen RM. Some nonlinear dispersive waves arising in compressible hyperelastic plates. International Journal of Engineering Science. 2006; 44 (18–19):11881204. DOI: 10.1016/j.ijengsci.2006.08.003  76.
Hao YX, Zhang W, Yang J. Nonlinear oscillation of a cantilever FGM rectangular plate based on thirdorder plate theory and asymptotic perturbation method. Composites Part B: Engineering. 2011; 42 (3):402413. DOI: 10.1016/j.compositesb.2010.12.010  77.
Andrianov IV, Danishevs’Kyy VV. Asymptotic approach for nonlinear periodical vibrations of continuous structures. Journal of Sound and Vibration. 2002; 249 (3):465481. DOI: 10.1006/jsvi.2001.3878  78.
Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: John Wiley and Sons; 1979  79.
Lenci S, Clementi F, Mazzilli CEN. Simple formulas for the natural frequencies of nonuniform cables and beams. International Journal of Mechanical Sciences. 2013; 77 (4):155163. DOI: 10.1016/j.ijmecsci.2013.09.028  80.
Cao DX, Gao YH, Wang JJ, Yao MH, Zhang W. Analytical analysis of free vibration of nonuniform and nonhomogenous beams: Asymptotic perturbation approach. Applied Mathematical Modelling. 2019; 65 :526534. DOI: 10.1016/j.apm.2018.08.026  81.
Cao D, Gao Y. Free vibration of nonuniform axially functionally graded beams using the asymptotic development method. Applied Mathematics and Mechanics (English Edition). 2018; 40 (1):8596. DOI: 10.1007/s1048301924029  82.
Cao DX, Gao YH, Yao MH, Zhang W. Free vibration of axially functionally graded beams using the asymptotic development method. Engineering Structures. 2018; 173 :442448. DOI: 10.1016/j.engstruct.2018.06.111  83.
Tarnopolskaya T, de Hoog F, Fletcher NH, Thwaites S. Asymptotic analysis of the free inplane vibrations of beams with arbitrarily varying curvature and crosssection. Journal of Sound and Vibration. 1996; 196 (5):659680. DOI: 10.1006/jsvi.1996.0507  84.
Kryzhevich SG, Volpert VA. Different types of solvability conditions for differential operators. Electronic Journal of Differential Equations. 2006; 2006 (100):124. DOI: 10.1142/9789812772992_0015  85.
Wang B, Han J, Du S. Dynamic response for functionally graded materials with pennyshaped cracks. Acta Mechanica Solida Sinica. 1999; 12 (2):106113