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 xð Þ and material mass density ρð Þ 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:


Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

#### Table 7.

4. The method of Meijer G-function

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 xð Þ and material mass density ρð Þ 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

cb 0.2 0.4 0.6 0.8

Mechanics of Functionally Graded Materials and Structures

0.2 Present 8.1682 8.2018 8.2456 8.3051

0.4 Present 7.3172 7.3647 7.4262 7.5089

0.6 Present 6.5357 6.5960 6.6732 6.7754

0.8 Present 5.8537 5.9240 6.0128 6.1283

0.2 Present 32.4133 32.7007 33.0819 33.6118

0.4 Present 29.2971 29.7076 30.2419 30.9665

0.6 Present 26.7834 27.2965 27.9493 28.8091

0.8 Present 25.1032 25.6683 26.3683 27.2590

0.2 Present 72.8179 73.5237 74.4625 75.7732

0.4 Present 65.9158 66.9202 68.2291 70.0069

0.6 Present 60.4922 61.7392 63.3243 65.4089

0.8 Present 57.0969 58.4547 60.1303 62.2530

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

Ref. [38] 8.1462 8.1498 8.1336 8.0646

Ref. [38] 7.1455 7.1254 7.0794 6.9703

Ref. [38] 6.0082 5.9638 5.8868 5.7351

Ref. [38] 4.6046 4.5355 4.4264 4.2283

Ref. [38] 32.5123 32.5079 32.5164 32.5326

Ref. [38] 28.4822 28.5003 28.5370 28.5928

Ref. [38] 24.1371 24.1791 24.2469 24.3497

Ref. [38] 19.1803 19.2509 19.3590 19.5300

Ref. [38] 73.0959 73.0903 73.1116 73.1855

Ref. [38] 64.0054 64.0350 64.1007 64.2374

Ref. [38] 54.1330 54.1992 54.3126 54.5207

Ref. [38] 42.7677 42.8742 43.0436 43.3451

4.1 Equation deriving

Table 6.

ch First mode

ch Second mode

ch Third mode

parameter equation:

12

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

$$E(\mathbf{x}) = E\_L \left(\mathbf{1} - f\_E \frac{\mathbf{x}}{L}\right), \qquad \rho(\mathbf{x}) = \rho\_L \left(\mathbf{1} - f\_\rho \frac{\mathbf{x}}{L}\right) \tag{28}$$

where <sup>f</sup> <sup>E</sup> <sup>¼</sup> <sup>1</sup> � ER EL , <sup>ρ</sup><sup>E</sup> <sup>¼</sup> <sup>1</sup> � <sup>ρ</sup><sup>R</sup> ρL . EL and ER are Young's modulus at the left/right end of the beam, and ρ<sup>L</sup> and ρ<sup>R</sup> are the mass density at the left/right end of the beam. Eq. (2) is then rewritten as

$$\left[ (\mathbf{1} - f\_E \mathbf{x}) \boldsymbol{\omega}'' \right]'' + \left( \mathbf{1} - f\_\rho \mathbf{x} \right) \ddot{\boldsymbol{\omega}} = \mathbf{0} \tag{29}$$

Based on the vibration theory, we assume w xð Þ¼ ; t ϕð Þ x q tð Þ, where qnðÞ¼ t An cos β<sup>n</sup> 2 t þ Bn sin β<sup>n</sup> 2 t and β<sup>n</sup> <sup>2</sup> is the modal frequency for dimensionless. The governing equation is then derived as

$$\left[\left(\mathbf{1} - f\_E \mathbf{x}\right) \boldsymbol{\phi}\_n^{\prime\prime}\right]^\prime - \boldsymbol{\beta}\_n^{\prime\prime} \left(\mathbf{1} - f\_{\boldsymbol{\rho}} \mathbf{x}\right) \boldsymbol{\phi}\_n = \mathbf{0} \tag{30}$$

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

Case b<sup>1</sup> b<sup>2</sup> b<sup>3</sup> b<sup>4</sup> 1/2 1/4 0 1/4 1/4 1/4 0 1/2 0 1/4 1/2 1/4 1/2 0 1/4 1/4 0 1/2 1/4 1/4 0 1/4 1/4 1/2 1/4 0 1/4 1/2 1/4 1/2 0 1/4 1/4 0 1/2 1/4

One set of data can be selected from Table 8 and expressed in the form of closed

� � � � �

!

!

!

!

ϕnð Þ¼ x C1<sup>n</sup>φ1<sup>n</sup>ð Þþ x C2<sup>n</sup>φ2<sup>n</sup>ð Þþ x C3<sup>n</sup>φ3<sup>n</sup>ð Þþ x C4<sup>n</sup>φ4<sup>n</sup>ð Þ x , n ¼ 1, 2, 3, … (41)

In order to determine the undetermined coefficients Ci and βn, the boundary

<sup>3</sup><sup>n</sup>ð Þ 0 φ<sup>0</sup>

<sup>3</sup><sup>n</sup>ð Þ1 φ<sup>00</sup>

<sup>3</sup><sup>n</sup>ð Þ1 φ<sup>000</sup>

<sup>4</sup><sup>n</sup>ð Þ 0

1

8 >>><

>>>:

C1<sup>n</sup> C2<sup>n</sup> C3<sup>n</sup> C4<sup>n</sup> 9 >>>=

9 >>>=

>>>;

8 >>><

>>>:

>>>; ¼

CCCA

<sup>4</sup><sup>n</sup>ð Þ1

<sup>4</sup><sup>n</sup>ð Þ1

<sup>4</sup>ð Þ <sup>1</sup> � Fx <sup>4</sup> 256F<sup>4</sup>

> <sup>4</sup>ð Þ <sup>1</sup> � Fx <sup>4</sup> 256F<sup>4</sup>

> <sup>4</sup>ð Þ <sup>1</sup> � Fx <sup>4</sup> 256F<sup>4</sup>

> <sup>4</sup>ð Þ <sup>1</sup> � Fx <sup>4</sup> 256F<sup>4</sup>

(37)

(38)

(39)

(40)

(42)

shown in Table 8.

Table 8.

solutions of Meijer G-function:

Modal modes of beams:

1. C-F:

15

conditions of beams need to be considered:

<sup>1</sup><sup>n</sup>ð Þ 0 φ<sup>0</sup>

<sup>1</sup><sup>n</sup>ð Þ1 φ<sup>00</sup>

<sup>1</sup><sup>n</sup>ð Þ1 φ<sup>000</sup>

φ0

0

BBB@

φ00

φ<sup>000</sup>

<sup>φ</sup>1<sup>n</sup>ð Þ¼ <sup>x</sup> <sup>G</sup><sup>2</sup>,<sup>0</sup>

Possible case of unknown constant bk of Meijer G-function equation.

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

<sup>φ</sup>2<sup>n</sup>ð Þ¼ <sup>x</sup> <sup>G</sup><sup>1</sup>,<sup>0</sup>

<sup>φ</sup>3<sup>n</sup>ð Þ¼ <sup>x</sup> <sup>G</sup><sup>1</sup>,<sup>0</sup>

<sup>φ</sup>4<sup>n</sup>ð Þ¼ <sup>x</sup> <sup>G</sup><sup>1</sup>,<sup>0</sup>

0,4 1 4 ; 1 <sup>4</sup> ; <sup>0</sup>; 1 2 βn

0,4 1 <sup>4</sup> ; <sup>0</sup>; 1 2 ; 1 <sup>4</sup> j � <sup>β</sup><sup>n</sup>

<sup>0</sup>,<sup>4</sup> 0; 1 2 ; 1 4 ; 1 <sup>4</sup> j � <sup>β</sup><sup>n</sup>

0,4 1 2 ; 1 4 ; 1 <sup>4</sup> ; <sup>0</sup>j � <sup>β</sup><sup>n</sup>

φ1<sup>n</sup>ð Þ 0 φ2<sup>n</sup>ð Þ 0 φ3<sup>n</sup>ð Þ 0 φ4<sup>n</sup>ð Þ 0

<sup>2</sup><sup>n</sup>ð Þ 0 φ<sup>0</sup>

<sup>2</sup><sup>n</sup>ð Þ1 φ<sup>00</sup>

<sup>2</sup><sup>n</sup>ð Þ1 φ<sup>000</sup>

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

$$\left[ (-\mathbf{1})^{(p-m-n)} z \prod\_{l=1}^{p} \left( \mathbf{z} \frac{d}{dz} + \mathbf{1} - a\_l \right) - \prod\_{k=1}^{q} \left( \eta \frac{d}{dz} - b\_k \right) \right] \mathbf{G}(\mathbf{z}) = \mathbf{0} \tag{31}$$

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 al and bk 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:

$$\left| G\_{p,q}^{m,n} \left\langle \begin{matrix} a\_1...a\_n, a\_{n+1}...a\_p\\ b\_1...b\_m, b\_{m+1}...b\_q \end{matrix} \bigg| z \right\rangle = \frac{1}{2\pi i} \int\_{\mathbf{r}} \frac{\prod\_{k=1}^m \Gamma(\xi - b\_k) \prod\_{k=1}^n \Gamma(\mathbf{1} - a\_k + \xi)}{\prod\_{k=1}^p \Gamma(\xi - a\_k) \prod\_{k=m+1}^q \Gamma(\mathbf{1} - b\_k + \xi)} \mathbf{z}^{-\xi} d\xi \tag{32}$$

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 <sup>Γ</sup> <sup>b</sup><sup>j</sup> � <sup>ξ</sup> � �, jð Þ <sup>¼</sup> <sup>1</sup>…<sup>m</sup> coincides with the poles of <sup>Γ</sup> <sup>1</sup> � aj <sup>þ</sup> <sup>ξ</sup> � �, jð Þ <sup>¼</sup> <sup>1</sup>…<sup>n</sup> . Where <sup>i</sup> is a complex number such that <sup>i</sup> <sup>2</sup> ¼ �1.

A special case of Eq. (31) can be expanded by assuming n ¼ p ¼ 0 and q ¼ 4. We can get that

$$\begin{aligned} &z^4 \frac{d^4 G}{dz^4} + \left(6 - \sum\_{k=1}^4 b\_k\right) z^3 \frac{d^3 G}{dz^4} + \left(7 - 3 \sum\_{n=1}^4 b\_k + \sum\_{k,l=1}^4 b\_k b\_l\right) z^2 \frac{d^2 G}{dz^2} \\ &+ \left[1 - \sum\_{k=1}^4 b\_k + \sum\_{k,l=1}^4 b\_k b\_l - (b\_1 b\_2 + b\_1 b\_3 + b\_2 b\_3) b\_4\right] z \frac{dG}{dz} \\ &- \left[(-1)^{-m} z - \prod\_{k=1}^4 b\_k\right] G = 0 \end{aligned} \tag{33}$$

where k 6¼ l. Although Eq. (30) is not similar to Eq. (33), the two equations can be similar by introducing some transformations:

$$\phi\_n(\mathbf{x}) = \mathbf{G}(\mathbf{z}\_n(\mathbf{x})),\\ z\_n(\mathbf{x}) = \left(\frac{\beta\_n}{4f\_E}\right)^4 \left(\mathbf{1} - f\_E \mathbf{x}\right)^4 \tag{34}$$

Eq. (30) is transformed into

$$
\eta\_n{}^3 G^{\prime\prime\prime} + 5\eta\_n{}^2 G^{\prime\prime} + \frac{69}{16}\eta\_n{}^6 G^{\prime} + \frac{9}{32}G^{\prime} - \frac{1 - f\_{\rho}\varkappa}{1 - f\_E\varkappa} G = 0\tag{35}
$$

Because of the difficulty of solving the differential equation with variable coefficients, we can simplify Eq. (35). Let 1 � f <sup>E</sup>x ¼ 1 � f <sup>ρ</sup>x ¼ 1 � Fx; it can be rewritten as

$$
\eta\_n{}^3 G^{\prime\prime\prime} + 5\eta\_n{}^2 G^{\prime\prime} + \frac{69}{16}\eta\_n{}^6 \overline{G}^{\prime} + \frac{9}{32}G^{\prime} - G = 0\tag{36}
$$


Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

#### Table 8.

<sup>1</sup> � <sup>f</sup> <sup>E</sup><sup>x</sup> � �ϕ<sup>n</sup> <sup>00</sup> h i<sup>00</sup>

Mechanics of Functionally Graded Materials and Structures

ten as

Gm,n p, <sup>q</sup>

can get that

ten as

14

<sup>z</sup><sup>4</sup> <sup>d</sup><sup>4</sup> G

þ 1 � ∑ 4 k¼1

ð Þ �<sup>1</sup> ð Þ <sup>p</sup>�m�<sup>n</sup> <sup>z</sup>

and al and bk are real numbers.

a1…an, anþ1…ap b1…bm, bmþ1…bq

� �

dz<sup>4</sup> <sup>þ</sup> <sup>6</sup> � <sup>∑</sup>

� �ð Þ<sup>1</sup> �mz � <sup>Y</sup>

Eq. (30) is transformed into

ηn 3 G<sup>0000</sup>

> ηn 3 G<sup>0000</sup>

" #

Y p

z d

dz <sup>þ</sup> <sup>1</sup> � al � �

" # � �

l¼1

the complex plane, called the Mellin-Barnes type:

� � �z

4 k¼1 bk � �

bk þ ∑ 4 k,l¼<sup>1</sup>

4

k¼1 bk

be similar by introducing some transformations:

þ 5η<sup>n</sup> 2 G<sup>000</sup> þ 69 <sup>16</sup> <sup>η</sup>nG<sup>00</sup>

> þ 5η<sup>n</sup> 2 G<sup>000</sup> þ 69 <sup>16</sup> <sup>η</sup>nG<sup>00</sup>

¼ 1 2πi ð τ

> z3 <sup>d</sup><sup>3</sup> G

G ¼ 0

ϕnð Þ¼ x Gð Þ znð Þ x , znð Þ¼ x

" #

� β<sup>n</sup>

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

> � <sup>Y</sup> q

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,

A definition of the Meijer G-function is given by the following path integral in

<sup>k</sup>¼<sup>1</sup> <sup>Γ</sup>ð Þ <sup>ξ</sup> � bk

<sup>k</sup>¼<sup>1</sup> <sup>Γ</sup>ð Þ <sup>ξ</sup> � ak

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 <sup>Γ</sup> <sup>b</sup><sup>j</sup> � <sup>ξ</sup> � �, jð Þ <sup>¼</sup> <sup>1</sup>…<sup>m</sup> coincides with the poles of <sup>Γ</sup> <sup>1</sup> � aj <sup>þ</sup> <sup>ξ</sup> � �, jð Þ <sup>¼</sup> <sup>1</sup>…<sup>n</sup> . Where <sup>i</sup> is a complex number such that <sup>i</sup>

A special case of Eq. (31) can be expanded by assuming n ¼ p ¼ 0 and q ¼ 4. We

4 n¼1

bk þ ∑ 4 k,l¼<sup>1</sup> bkbl

> z dG dz

!

dz<sup>4</sup> <sup>þ</sup> <sup>7</sup> � <sup>3</sup> <sup>∑</sup>

bkbl � ð Þ b1b<sup>2</sup> þ b1b<sup>3</sup> þ b2b<sup>3</sup> b<sup>4</sup>

where k 6¼ l. Although Eq. (30) is not similar to Eq. (33), the two equations can

þ 9 32

> þ 9

Because of the difficulty of solving the differential equation with variable coefficients, we can simplify Eq. (35). Let 1 � f <sup>E</sup>x ¼ 1 � f <sup>ρ</sup>x ¼ 1 � Fx; it can be rewrit-

βn 4f <sup>E</sup> � �<sup>4</sup>

> <sup>G</sup><sup>0</sup> � <sup>1</sup> � <sup>f</sup> <sup>ρ</sup><sup>x</sup> 1 � f <sup>E</sup>x

Q<sup>n</sup>

Q<sup>q</sup>

<sup>k</sup>¼<sup>1</sup> <sup>Γ</sup>ð Þ <sup>1</sup> � ak <sup>þ</sup> <sup>ξ</sup>

<sup>k</sup>¼mþ<sup>1</sup> <sup>Γ</sup>ð Þ <sup>1</sup> � bk <sup>þ</sup> <sup>ξ</sup>

z2 d2 G dz<sup>2</sup>

<sup>1</sup> � <sup>f</sup> <sup>E</sup><sup>x</sup> � �<sup>4</sup> (34)

<sup>32</sup> <sup>G</sup><sup>0</sup> � <sup>G</sup> <sup>¼</sup> <sup>0</sup> (36)

G ¼ 0 (35)

Q<sup>m</sup>

Q<sup>p</sup>

k¼1

η d dz � bk

<sup>4</sup> <sup>1</sup> � <sup>f</sup> <sup>ρ</sup><sup>x</sup> � �

ϕ<sup>n</sup> ¼ 0 (30)

Gð Þ¼ z 0 (31)

z�<sup>ξ</sup>

dξ (32)

<sup>2</sup> ¼ �1.

(33)

Possible case of unknown constant bk of Meijer G-function equation.

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.

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

$$\rho\_{1n}(\mathbf{x}) = G\_{0,4}^{2,0}\left(\frac{\mathbf{1}}{4}, \frac{\mathbf{1}}{4}, \mathbf{0}, \frac{\mathbf{1}}{2} \middle| \frac{\beta\_n{}^4 (\mathbf{1} - F\mathbf{x})^4}{256F^4}\right) \tag{37}$$

$$\rho\_{2n}(\mathbf{x}) = G\_{0,4}^{1,0}\left(\frac{1}{4}, 0, \frac{1}{2}, \frac{1}{4} \vert -\frac{\beta\_n{}^4(\mathbf{1} - F\mathbf{x})^4}{256F^4}\right) \tag{38}$$

$$\rho\_{3n}(\mathbf{x}) = \mathbf{G}\_{0,4}^{1,0}\left(\mathbf{0}, \frac{\mathbf{1}}{2}, \frac{\mathbf{1}}{4}, \frac{\mathbf{1}}{4} \big| -\frac{\beta\_n^{-4}(\mathbf{1} - F\mathbf{x})^4}{256F^4}\right) \tag{39}$$

$$\varphi\_{4n}(\mathbf{x}) = G\_{0,4}^{1,0}\left(\frac{1}{2}, \frac{1}{4}, \frac{1}{4}, 0 \middle| -\frac{\beta\_n{}^4 (1 - F\mathbf{x})^4}{256F^4}\right) \tag{40}$$

Modal modes of beams:

$$\phi\_n(\mathbf{x}) = \mathbf{C}\_{1n}\rho\_{1n}(\mathbf{x}) + \mathbf{C}\_{2n}\rho\_{2n}(\mathbf{x}) + \mathbf{C}\_{3n}\rho\_{3n}(\mathbf{x}) + \mathbf{C}\_{4n}\rho\_{4n}(\mathbf{x}), \ \mathbf{n} = 1, 2, 3, \dots \tag{41}$$

In order to determine the undetermined coefficients Ci and βn, the boundary conditions of beams need to be considered:

1. C-F:

$$
\begin{pmatrix}
\rho\_{1n}(\mathbf{0}) & \rho\_{2n}(\mathbf{0}) & \rho\_{3n}(\mathbf{0}) & \rho\_{4n}(\mathbf{0}) \\
\rho\_{1n}'(\mathbf{0}) & \rho\_{2n}'(\mathbf{0}) & \rho\_{3n}'(\mathbf{0}) & \rho\_{4n}'(\mathbf{0}) \\
\rho\_{1n}''(\mathbf{1}) & \rho\_{2n}''(\mathbf{1}) & \rho\_{3n}''(\mathbf{1}) & \rho\_{4n}''(\mathbf{1}) \\
\rho\_{1n}'''(\mathbf{1}) & \rho\_{2n}'''(\mathbf{1}) & \rho\_{3n}'''(\mathbf{1}) & \rho\_{4n}'''(\mathbf{1})
\end{pmatrix}
\begin{Bmatrix}
\mathbf{C}\_{1n} \\
\mathbf{C}\_{2n} \\
\mathbf{C}\_{3n} \\
\mathbf{C}\_{4n}
\end{Bmatrix} = \begin{Bmatrix}
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0}
\end{Bmatrix} \tag{42}
$$

Mechanics of Functionally Graded Materials and Structures

1. C-P:

$$
\begin{pmatrix}
\rho\_{1n}(\mathbf{0}) & \rho\_{2n}(\mathbf{0}) & \rho\_{3n}(\mathbf{0}) & \rho\_{4n}(\mathbf{0}) \\
\rho\_{1n}'(\mathbf{0}) & \rho\_{2n}'(\mathbf{0}) & \rho\_{3n}'(\mathbf{0}) & \rho\_{4n}'(\mathbf{0}) \\
\rho\_{1n}(\mathbf{1}) & \rho\_{2n}(\mathbf{1}) & \rho\_{3n}(\mathbf{1}) & \rho\_{4n}(\mathbf{1}) \\
\rho\_{1n}''(\mathbf{1}) & \rho\_{2n}''(\mathbf{1}) & \rho\_{3n}''(\mathbf{1}) & \rho\_{4n}''(\mathbf{1})
\end{pmatrix}
\begin{Bmatrix}
\mathbf{C}\_{1n} \\
\mathbf{C}\_{2n} \\
\mathbf{C}\_{3n} \\
\mathbf{C}\_{4n}
\end{Bmatrix} = \begin{Bmatrix}
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0}
\end{Bmatrix} \tag{43}
$$

are given in detail in Table 9. We choose the sizes of commonly used beams which

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

In order to verify the correctness of this method, some finite element simulation

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

are L = 0.2 m, B = 0.02 m, and H = 0.001 m.

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

meshes.

Table 10.

Figure 3.

Table 9.

17

Material properties of the AFG beam [44].

Variation of the material properties defined by Eq. (46) with YR ¼ 3YL.

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

1. S-S:

$$
\begin{pmatrix}
\rho\_{1n}(\mathbf{0}) & \rho\_{2n}(\mathbf{0}) & \rho\_{3n}(\mathbf{0}) & \rho\_{4n}(\mathbf{0}) \\
\rho\_{1n}''(\mathbf{0}) & \rho\_{2n}''(\mathbf{0}) & \rho\_{3n}''(\mathbf{0}) & \rho\_{4n}''(\mathbf{0}) \\
\rho\_{1n}(\mathbf{1}) & \rho\_{2n}(\mathbf{1}) & \rho\_{3n}(\mathbf{1}) & \rho\_{4n}(\mathbf{1}) \\
\rho\_{1n}''(\mathbf{1}) & \rho\_{2n}''(\mathbf{1}) & \rho\_{3n}''(\mathbf{1}) & \rho\_{4n}''(\mathbf{1})
\end{pmatrix}
\begin{Bmatrix}
\mathbf{C}\_{1n} \\
\mathbf{C}\_{2n} \\
\mathbf{C}\_{3n} \\
\mathbf{C}\_{4n}
\end{Bmatrix} = \begin{Bmatrix}
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0}
\end{Bmatrix} \tag{44}
$$

1. C-C:

$$
\begin{pmatrix}
\rho\_{1n}(\mathbf{0}) & \rho\_{2n}(\mathbf{0}) & \rho\_{3n}(\mathbf{0}) & \rho\_{4n}(\mathbf{0}) \\
\rho\_{1n}''(\mathbf{0}) & \rho\_{2n}''(\mathbf{0}) & \rho\_{3n}''(\mathbf{0}) & \rho\_{4n}''(\mathbf{0}) \\
\rho\_{1n}(\mathbf{1}) & \rho\_{2n}(\mathbf{1}) & \rho\_{3n}(\mathbf{1}) & \rho\_{4n}(\mathbf{1}) \\
\rho\_{1n}'(\mathbf{1}) & \rho\_{2n}'(\mathbf{1}) & \rho\_{3n}'(\mathbf{1}) & \rho\_{4n}'(\mathbf{1}) \\
\end{pmatrix}
\begin{Bmatrix}
C\_{1n} \\
C\_{2n} \\
C\_{3n} \\
C\_{4n} \\
\end{Bmatrix} = \begin{Bmatrix}
0 \\
0 \\
0 \\
0 \\
\end{Bmatrix} \tag{45}
$$

### 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 xð Þ and mass density ρð Þ x are listed in detail:

$$Y(\mathbf{x}) = \begin{cases} Y\_L \left( \mathbf{1} - \frac{e^{\mathbf{x} \times \boldsymbol{L}} - \mathbf{1}}{e^a - \mathbf{1}} \right) + Y\_R \frac{e^{\mathbf{x} \times \boldsymbol{L}} - \mathbf{1}}{e^a - \mathbf{1}}, & a \neq \mathbf{0}, \\\ Y\_L \left( \mathbf{1} - \frac{\boldsymbol{x}}{L} \right) + Y\_R \frac{\boldsymbol{x}}{L}, & a = \mathbf{0}. \end{cases} \tag{46}$$

where YL and YR 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 xð Þ along the axis direction of the beam can be shown in Figure 3 for YR ¼ 3YL. 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 (ZrO2).The left and right ends of the beam are pure aluminum and pure zirconia, respectively. The material properties of AFG beams

1. C-P:

1. S-S:

1. C-C:

16

φ0

0

BBB@

φ00

φ00

0

BBB@

φ00

φ00

0

BBB@

φ0

Y xð Þ¼

8 >><

>>:

change with the length direction of the beam.

4.2 Numerical results and discussion

<sup>1</sup>nð Þ 0 φ<sup>0</sup>

Mechanics of Functionally Graded Materials and Structures

<sup>1</sup>nð Þ1 φ<sup>00</sup>

<sup>1</sup>nð Þ 0 φ<sup>00</sup>

<sup>1</sup><sup>n</sup>ð Þ1 φ<sup>00</sup>

<sup>1</sup><sup>n</sup>ð Þ 0 φ<sup>00</sup>

<sup>1</sup><sup>n</sup>ð Þ1 φ<sup>0</sup>

φ1nð Þ 0 φ2nð Þ 0 φ3nð Þ 0 φ4nð Þ 0

φ1nð Þ1 φ2nð Þ1 φ3nð Þ1 φ4nð Þ1

φ1nð Þ 0 φ2nð Þ 0 φ3nð Þ 0 φ4nð Þ 0

φ1<sup>n</sup>ð Þ1 φ2<sup>n</sup>ð Þ1 φ3<sup>n</sup>ð Þ1 φ4<sup>n</sup>ð Þ1

φ1<sup>n</sup>ð Þ 0 φ2<sup>n</sup>ð Þ 0 φ3<sup>n</sup>ð Þ 0 φ4<sup>n</sup>ð Þ 0

φ1<sup>n</sup>ð Þ1 φ2<sup>n</sup>ð Þ1 φ3<sup>n</sup>ð Þ1 φ4<sup>n</sup>ð Þ1

sions of Young's modulus E xð Þ and mass density ρð Þ x are listed in detail:

e<sup>α</sup> � 1 � �

þ YR x

where YL and YR 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

The variation of Y xð Þ along the axis direction of the beam can be shown in Figure 3 for YR ¼ 3YL. 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 (ZrO2).The left and right ends of the beam are pure aluminum and pure zirconia, respectively. The material properties of AFG beams

YL <sup>1</sup> � <sup>e</sup><sup>α</sup>x=<sup>L</sup> � <sup>1</sup>

YL <sup>1</sup> � <sup>x</sup> L � �

<sup>2</sup><sup>n</sup>ð Þ 0 φ<sup>00</sup>

<sup>2</sup><sup>n</sup>ð Þ1 φ<sup>0</sup>

<sup>2</sup>nð Þ 0 φ<sup>00</sup>

<sup>2</sup><sup>n</sup>ð Þ1 φ<sup>00</sup>

<sup>3</sup>nð Þ 0 φ<sup>0</sup>

<sup>3</sup>nð Þ1 φ<sup>00</sup>

<sup>3</sup>nð Þ 0 φ<sup>00</sup>

<sup>3</sup><sup>n</sup>ð Þ1 φ<sup>00</sup>

<sup>3</sup><sup>n</sup>ð Þ 0 φ<sup>00</sup>

<sup>3</sup><sup>n</sup>ð Þ1 φ<sup>0</sup>

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 expres-

<sup>4</sup>nð Þ 0

1

8 >>><

>>>:

C1<sup>n</sup> C2<sup>n</sup> C3<sup>n</sup> C4<sup>n</sup> 9 >>>=

9 >>>=

>>>;

8 >>><

>>>:

9 >>>=

>>>;

8 >>><

>>>:

9 >>>=

>>>;

(43)

(44)

(45)

(46)

8 >>><

>>>:

>>>; ¼

CCCA

1

8 >>><

>>>:

C1<sup>n</sup> C2<sup>n</sup> C3<sup>n</sup> C4<sup>n</sup> 9 >>>=

>>>; ¼

CCCA

1

8 >>><

>>>:

<sup>e</sup><sup>α</sup>x=<sup>L</sup> � <sup>1</sup>

<sup>L</sup> , <sup>α</sup> <sup>¼</sup> <sup>0</sup>:

<sup>e</sup><sup>α</sup> � <sup>1</sup> , <sup>α</sup> 6¼ <sup>0</sup>,

C1<sup>n</sup> C2<sup>n</sup> C3<sup>n</sup> C4<sup>n</sup> 9 >>>=

>>>; ¼

CCCA

<sup>4</sup>nð Þ1

<sup>4</sup>nð Þ 0

<sup>4</sup><sup>n</sup>ð Þ1

<sup>4</sup><sup>n</sup>ð Þ 0

<sup>4</sup><sup>n</sup>ð Þ1

þ YR

<sup>2</sup>nð Þ 0 φ<sup>0</sup>

<sup>2</sup>nð Þ1 φ<sup>00</sup>

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.

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 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.

Figure 3.

Variation of the material properties defined by Eq. (46) with YR ¼ 3YL.


Table 9. Material properties of the AFG beam [44].

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

FGMs are innovative materials and are very important in engineering and other

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 gradi-

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

The authors gratefully acknowledge the support of the National Natural Science

efficient and can be used to analyze the free vibration of AFG beams.

Foundation of China (Grant Nos. 11672008, 11702188, and 11272016).

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

can be obtained.

5. Conclusions

ent parameter.

Acknowledgements

19

analyze the free vibration of the AFG beams.

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

Figure 4.

Finite segment model of the AFG beam.


#### Table 10.

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

#### 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.

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

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.
