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

$$\frac{\partial^2}{\partial \mathbf{x}^2} \left[ E(\mathbf{x}) I(\mathbf{x}) \frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x}^2} \right] + \rho(\mathbf{x}) A(\mathbf{x}) \frac{\partial^2 w(\mathbf{x}, t)}{\partial t^2} = \mathbf{0}, \qquad \mathbf{0} \le \mathbf{x} \le L \tag{1}$$

where w xð Þ ; t is the transverse deflection at position x and time t; L is the length of the beams; E xð ÞI xð Þ is the bending stiffness, which is determined by Young's modulus E xð Þ and the area moment of inertia I xð Þ; and ρð Þ x A xð Þ is the unit mass

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

Figure 1. The geometry and coordinate system of an AFG beam.

length of beam, which is determined by volume mass density ρð Þ x and variable cross-sectional area A xð Þ.

Because of the particularity of AFG beam, bending stiffness E xð ÞI xð Þ and unit mass ρð Þ x A xð Þ 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<sup>0</sup> and reference mass ρA<sup>0</sup> are introduced, and the above two dimensionless parameters are invariant. Suppose E xð ÞI xð Þ¼ EI<sup>0</sup> þ E xð ÞI xð Þ and ρð Þ x A xð Þ¼ ρA<sup>0</sup> þ ρð Þ x A xð Þ, where EI<sup>0</sup> and ρA<sup>0</sup> are the invariant parts and E xð ÞI xð Þ and ρð Þ x A xð Þ represent flexural stiffness and mass per unit length, respectively, and vary with the axial coordinates. Here, we introduce a dimensionless space variable ξ ¼ x=L and a dimensionless time variable <sup>τ</sup> <sup>¼</sup> <sup>t</sup> L2 ffiffiffiffiffiffi EI<sup>0</sup> ρA<sup>0</sup> q ; Eq. (1) can be rewritten in the dimensionless form:

$$\frac{\partial^2}{\partial \xi^2} \left\{ \left[ \mathbf{1} + f\_1(\xi) \right] \frac{\partial^2 w(\xi, \tau)}{\partial \xi^2} \right\} + \left[ \mathbf{1} + f\_2(\xi) \right] \frac{\partial^2 w(\xi, \tau)}{\partial \tau^2} = \mathbf{0}, \qquad \mathbf{0} \le \xi \le \mathbf{1} \tag{2}$$

where

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

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

method to examine the free vibration of nonuniform circular beam.

Mechanics of Functionally Graded Materials and Structures

the asymptotic perturbation method to analyze the finite deformation of

analysis.

conclusions are presented.

∂2

4

<sup>∂</sup>x<sup>2</sup> E xð ÞI xð Þ <sup>∂</sup><sup>2</sup>

2. Governing equation of the AFG beam

differential equation of the beam can be written as

w xð Þ ; t ∂x<sup>2</sup>

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

The present topic focus on the free vibration of AFG beams with uniform

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

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

w xð Þ ; t

where w xð Þ ; t is the transverse deflection at position x and time t; L is the length of the beams; E xð ÞI xð Þ is the bending stiffness, which is determined by Young's modulus E xð Þ and the area moment of inertia I xð Þ; and ρð Þ x A xð Þ is the unit mass

<sup>∂</sup>t<sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>0</sup>≤x≤<sup>L</sup> (1)

<sup>þ</sup> <sup>ρ</sup>ð Þ <sup>x</sup> A xð Þ <sup>∂</sup><sup>2</sup>

or nonuniform cross sections using analytical method: the asymptotic

$$f\_1(\xi) = \frac{\overline{E(\xi)I(\xi)}}{EI\_0} \text{ and} \\ f\_2(\xi) = \frac{\overline{\rho(\xi)A(\xi)}}{\rho A\_0} \tag{3}$$

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

$$w(\xi,\tau) = W(\xi)\sin(a\tau) \tag{4}$$

where Wð Þξ 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): Mechanics of Functionally Graded Materials and Structures

$$\frac{d^2}{d\xi^2} \left\{ \left[ \mathbf{1} + f\_1(\xi) \right] \frac{d^2 W}{d\xi^2} \right\} - \alpha^2 \left[ \mathbf{1} + f\_2(\xi) \right] W = \mathbf{0}, \qquad \mathbf{0} \le \xi \le \mathbf{1} \tag{5}$$

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

$$f\_1(\xi) \to \mathfrak{e}f\_1(\xi),\ f\_2(\xi) \to \mathfrak{e}f\_2(\xi) \tag{6}$$

and the frequency equation is

is satisfied. As a result,

Integrating by parts, we obtain

dξ d2 W<sup>0</sup> <sup>d</sup>ξ<sup>2</sup> <sup>W</sup><sup>0</sup> <sup>þ</sup> <sup>f</sup> <sup>1</sup>

d3 W<sup>0</sup> <sup>d</sup>ξ<sup>3</sup> <sup>W</sup><sup>0</sup> � <sup>f</sup> <sup>1</sup>

d2 W<sup>0</sup> <sup>d</sup>ξ<sup>2</sup> <sup>W</sup><sup>0</sup> <sup>þ</sup>

dW<sup>0</sup> <sup>d</sup><sup>ξ</sup> � <sup>d</sup><sup>3</sup>

!�

W<sup>0</sup> <sup>d</sup>ξ<sup>3</sup> <sup>W</sup><sup>0</sup>

þ ð1 0 f 1 d2 W<sup>0</sup> dξ<sup>2</sup> !<sup>2</sup>

2 4

<sup>f</sup> <sup>1</sup>ð Þ¼ <sup>ξ</sup> <sup>E</sup>ð Þ<sup>ξ</sup> <sup>I</sup>ð Þ<sup>ξ</sup>

d2 W<sup>0</sup> dξ<sup>2</sup>

> d3 W<sup>0</sup>

> > ð1 0

Eð Þξ Ið Þξ EI<sup>0</sup>

)� � � � �

1

0 þ 1 EI<sup>0</sup>

<sup>h</sup>1ð Þ<sup>ξ</sup> <sup>W</sup>0d<sup>ξ</sup> <sup>¼</sup> df <sup>1</sup>

By definition we have

geneous beam.

ð1 0

so that

(

þ d2 W<sup>0</sup> dξ<sup>2</sup>

7

<sup>¼</sup> d E½ � ð Þ<sup>ξ</sup> <sup>I</sup>ð Þ<sup>ξ</sup> EI0dξ

df <sup>1</sup> dξ d2 W<sup>0</sup> <sup>d</sup>ξ<sup>2</sup> <sup>W</sup><sup>0</sup> <sup>þ</sup> <sup>f</sup> <sup>1</sup>

The spatial mode shape can be obtained as

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

> ð1 0

> > ω<sup>1</sup> ¼

Ð 1

2ω<sup>0</sup> Ð 1 <sup>0</sup> <sup>W</sup><sup>2</sup> <sup>0</sup>dξ

<sup>0</sup> h1ð Þξ W0dξ

Because h1ð Þξ is linearly correlated with W0, the former equations indicate that the arbitrary amplitude of W<sup>0</sup> does not impact ω1. This finding yields the first-order correction of the natural frequency ω<sup>0</sup> corresponding to a nonuniform and homo-

> d3 W<sup>0</sup> <sup>d</sup>ξ<sup>3</sup> <sup>W</sup><sup>0</sup> � <sup>f</sup> <sup>1</sup>

!�

� <sup>ω</sup><sup>2</sup>

<sup>E</sup>0<sup>I</sup> <sup>¼</sup> <sup>E</sup>ð Þ<sup>ξ</sup> <sup>I</sup>ð Þ� <sup>ξ</sup> EI<sup>0</sup>

dW<sup>0</sup> dξ

EI<sup>0</sup>

� � � �

1

0 þ ð1 0 f 1 d2 W<sup>0</sup> dξ<sup>2</sup> !<sup>2</sup>

<sup>d</sup>ξ<sup>3</sup> <sup>W</sup><sup>0</sup> � <sup>E</sup>ð Þ<sup>ξ</sup> <sup>I</sup>ð Þ<sup>ξ</sup>

<sup>E</sup>ð Þ<sup>ξ</sup> <sup>I</sup>ð Þ<sup>ξ</sup> <sup>d</sup><sup>2</sup>

EI<sup>0</sup>

W<sup>0</sup> dξ<sup>2</sup> !<sup>2</sup>

d2 W<sup>0</sup> dξ<sup>2</sup>

> dξ � ð1 0

<sup>0</sup> f <sup>2</sup>W<sup>2</sup> 0

<sup>W</sup><sup>0</sup> <sup>¼</sup> cosh ð Þ� <sup>k</sup><sup>ξ</sup> cosð Þþ <sup>k</sup><sup>ξ</sup> <sup>C</sup>

Now, the solution of the first-order equation is analyzed. In Eq. (9), both h1ð Þξ and W<sup>1</sup> are linearly correlated with W0. 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:

cos k cosh k þ 1 ¼ 0 (16)

½ � h1ð Þ� ξ 2ω1ω0W<sup>0</sup> W0dξ ¼ 0 (18)

d2 W<sup>0</sup> dξ<sup>2</sup>

> 3 5dξ

dW<sup>0</sup> dξ

� � � �

1

0

dξ

dW<sup>0</sup> dξ

> d2 W<sup>0</sup> dξ<sup>2</sup> !<sup>2</sup>

<sup>D</sup> ½ � sinh ð Þ� <sup>k</sup><sup>ξ</sup> sin ð Þ <sup>k</sup><sup>ξ</sup> (17)

(19)

(20)

(21)

dξ

(22)

According to the Poincaré-Lindstedt method [78–82], we assume the expansion for ω and Wð Þξ as

$$\begin{aligned} \omega &= \omega\_0 + \varepsilon \alpha\_1 + \varepsilon^2 \alpha\_2 + \dots \\ \mathcal{W}(\xi) &= \mathcal{W}\_0(\xi) + \varepsilon \mathcal{W}\_1(\xi) + \varepsilon^2 \mathcal{W}\_2(\xi) + \dots \end{aligned} \tag{7}$$

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<sup>0</sup> and ε<sup>1</sup> to zero, yielding a sequence of problems for the unknowns ω<sup>i</sup> and Wið Þξ :

$$a\frac{d^4W\_0}{d\xi^4} - a\_0^2W\_0 = 0\tag{8}$$

$$\frac{d^4W\_1}{d\xi^4} - a\_0^2W\_1 + h\_1(\xi) - 2a\_1a\_0W\_0 = 0\tag{9}$$

where

$$h\_1(\xi) = 2\frac{d\mathcal{f}\_1(\xi)}{d\xi}\frac{d^3\mathcal{W}\_0}{d\xi^3} + \frac{d^2\mathcal{f}\_1(\xi)}{d\xi^2}\frac{d^2\mathcal{W}\_0}{d\xi^2} + w\_0^2[\mathcal{f}\_1(\xi) - \mathcal{f}\_2(\xi)]\mathcal{W}\_0 \tag{10}$$

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

$$W\_0 = A\sin\left(k\xi\right) + B\cos\left(k\xi\right) + C\sinh\left(k\xi\right) + D\cosh\left(k\xi\right) \tag{11}$$

where

$$k = \sqrt{a\_0} \tag{12}$$

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

$$\mathcal{W}\_0 = \frac{dW\_0}{d\xi} = \mathbf{0}, \ \xi = \mathbf{0} \tag{13}$$

$$\frac{d^2W\_0}{d\xi^2} = \frac{d^3W\_0}{d\xi^3} = 0, \ \xi = 1\tag{14}$$

Then, the following equations from equation can be obtained:

$$\begin{aligned} A + C &= 0\\ B + D &= 0\\ \frac{C}{D} &= \frac{\sin k - \sinh k}{\cos k + \cosh k} \end{aligned} \tag{15}$$

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

and the frequency equation is

d2

for ω and Wð Þξ as

Wið Þξ :

where

where

6

h1ð Þ¼ ξ 2

boundary conditions are

<sup>d</sup>ξ<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>ξ</sup> � � <sup>d</sup><sup>2</sup>

( )

Mechanics of Functionally Graded Materials and Structures

W dξ<sup>2</sup>

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

<sup>ω</sup> <sup>¼</sup> <sup>ω</sup><sup>0</sup> <sup>þ</sup> εω<sup>1</sup> <sup>þ</sup> <sup>ε</sup>2ω<sup>2</sup> <sup>þ</sup> …,

d4 W<sup>0</sup> <sup>d</sup>ξ<sup>4</sup> � <sup>ω</sup><sup>2</sup>

d2 f <sup>1</sup>ð Þξ dξ<sup>2</sup>

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

d2 W<sup>0</sup> <sup>d</sup>ξ<sup>2</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup>

<sup>k</sup> <sup>¼</sup> ffiffiffiffiffiffi ω0

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

W<sup>0</sup>

<sup>D</sup> <sup>¼</sup> sin <sup>k</sup> � sinh <sup>k</sup> cos k þ cosh k

<sup>W</sup><sup>0</sup> <sup>¼</sup> dW<sup>0</sup>

Then, the following equations from equation can be obtained:

C

A þ C ¼ 0 B þ D ¼ 0

d2 W<sup>0</sup> <sup>d</sup>ξ<sup>2</sup> <sup>¼</sup> <sup>d</sup><sup>3</sup>

W<sup>0</sup> ¼ A sin ð Þþ kξ B cosð Þþ kξ Csinh ð Þþ kξ D cosh ð Þ kξ (11)

d4 W<sup>1</sup> <sup>d</sup>ξ<sup>4</sup> � <sup>ω</sup><sup>2</sup>

df <sup>1</sup>ð Þξ dξ

d3 W<sup>0</sup> <sup>d</sup>ξ<sup>3</sup> <sup>þ</sup>

According to the Poincaré-Lindstedt method [78–82], we assume the expansion

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<sup>0</sup> and ε<sup>1</sup> to zero, yielding a sequence of problems for the unknowns ω<sup>i</sup> and

� <sup>ω</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>2</sup>ð Þ<sup>ξ</sup> � �<sup>W</sup> <sup>¼</sup> <sup>0</sup>, <sup>0</sup>≤ξ<sup>≤</sup> 1 (5)

<sup>0</sup>W<sup>0</sup> ¼ 0 (8)

<sup>0</sup> <sup>f</sup> <sup>1</sup>ð Þ� <sup>ξ</sup> <sup>f</sup> <sup>2</sup>ð Þ<sup>ξ</sup> � �W<sup>0</sup> (10)

p (12)

<sup>d</sup><sup>ξ</sup> <sup>¼</sup> <sup>0</sup>, <sup>ξ</sup> <sup>¼</sup> <sup>0</sup> (13)

<sup>d</sup>ξ<sup>3</sup> <sup>¼</sup> <sup>0</sup>, <sup>ξ</sup> <sup>¼</sup> <sup>1</sup> (14)

(15)

<sup>0</sup>W<sup>1</sup> þ h1ð Þ� ξ 2ω1ω0W<sup>0</sup> ¼ 0 (9)

f <sup>1</sup>ð Þ!ξ εf <sup>1</sup>ð Þξ , f <sup>2</sup>ð Þ!ξ εf <sup>2</sup>ð Þξ (6)

<sup>W</sup>ð Þ¼ <sup>ξ</sup> <sup>W</sup>0ð Þþ <sup>ξ</sup> <sup>ε</sup>W1ð Þþ <sup>ξ</sup> <sup>ε</sup>2W2ð Þþ <sup>ξ</sup> …: (7)

$$k\cos k\cosh k + 1 = 0\tag{16}$$

The spatial mode shape can be obtained as

$$W\_0 = \cosh\left(k\xi\right) - \cos\left(k\xi\right) + \frac{C}{D} [\sinh\left(k\xi\right) - \sin\left(k\xi\right)] \tag{17}$$

Now, the solution of the first-order equation is analyzed. In Eq. (9), both h1ð Þξ and W<sup>1</sup> are linearly correlated with W0. 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:

$$\int\_{0}^{1} [h\_1(\xi) - 2\alpha\_1 \alpha\_0 \mathcal{W}\_0] \mathcal{W}\_0 d\xi = 0 \tag{18}$$

is satisfied. As a result,

$$\rho\_1 = \frac{\int\_0^1 h\_1(\xi) \, W\_0 d\xi}{2\alpha\_0 \int\_0^1 W\_0^2 d\xi} \tag{19}$$

Because h1ð Þξ is linearly correlated with W0, the former equations indicate that the arbitrary amplitude of W<sup>0</sup> does not impact ω1. This finding yields the first-order correction of the natural frequency ω<sup>0</sup> corresponding to a nonuniform and homogeneous beam.

Integrating by parts, we obtain

$$\begin{split} \int\_{0}^{1} h\_{1}(\xi) \mathcal{W}\_{0} d\xi &= \left( \frac{df\_{1}}{d\xi} \frac{d^{2} \mathcal{W}\_{0}}{d\xi^{2}} \mathcal{W}\_{0} + f\_{1} \frac{d^{3} \mathcal{W}\_{0}}{d\xi^{3}} \mathcal{W}\_{0} - f\_{1} \frac{d^{2} \mathcal{W}\_{0}}{d\xi^{2}} \frac{d \mathcal{W}\_{0}}{d\xi} \right) \bigg|\_{0}^{1} \\ &+ \int\_{0}^{1} \left[ f\_{1} \left( \frac{d^{2} \mathcal{W}\_{0}}{d\xi^{2}} \right)^{2} - a\_{0}^{2} f\_{2} \mathcal{W}\_{0}^{2} \right] d\xi \end{split} \tag{20}$$

By definition we have

$$f\_1(\xi) = \frac{\overline{E(\xi)I(\xi)}}{E\_0 I} = \frac{E(\xi)I(\xi) - EI\_0}{EI\_0} \tag{21}$$

so that

$$
\begin{split}
&\left(\frac{d\mathcal{f}\_{1}}{d\tilde{\xi}}\frac{d^{2}W\_{0}}{d\xi^{2}}W\_{0} + f\_{1}\frac{d^{3}W\_{0}}{d\xi^{3}}W\_{0} - f\_{1}\frac{d^{2}W\_{0}}{d\xi^{2}}\frac{dW\_{0}}{d\xi}\right)\bigg|\_{0}^{1} + \int\_{0}^{1}f\_{1}\left(\frac{d^{2}W\_{0}}{d\xi^{2}}\right)^{2}d\xi\\&= \left\{\frac{d[E(\xi)I(\xi)]}{EI\_{0}d\xi}\frac{d^{2}W\_{0}}{d\xi^{2}}W\_{0} + \frac{E(\xi)I(\xi)}{EI\_{0}}\frac{d^{3}W\_{0}}{d\xi^{3}}W\_{0} - \frac{E(\xi)I(\xi)}{EI\_{0}}\frac{d^{2}W\_{0}}{d\xi^{2}}\frac{dW\_{0}}{d\xi}\right.\\&\left.+\frac{d^{2}W\_{0}}{d\xi^{2}}\frac{dW\_{0}}{d\xi} - \frac{d^{3}W\_{0}}{d\xi^{3}}W\_{0}\right\}\bigg|\_{0}^{1} + \frac{1}{EI\_{0}}\int\_{0}^{1}E(\xi)I(\xi)\left(\frac{d^{2}W\_{0}}{d\xi^{2}}\right)^{2}d\xi - \int\_{0}^{1}\left(\frac{d^{2}W\_{0}}{d\xi^{2}}\right)^{2}d\xi.\tag{22}
\end{split}
$$

Choosing the reference bending stiffness

$$EI\_{0} = \frac{\left. \frac{d[E(\xi)I(\xi)]}{d\xi} \frac{d^{3}W\_{0}}{d\xi^{2}} \mathcal{W}\_{0} + E(\xi)I(\xi) \frac{d^{3}W\_{0}}{d\xi^{3}} \mathcal{W}\_{0} - E(\xi)I(\xi) \frac{d^{2}W\_{0}}{d\xi^{2}} \frac{dW\_{0}}{d\xi} \right|\_{0}^{1} + \int\_{0}^{1} E(\xi)I(\xi) \left(\frac{d^{3}W\_{0}}{d\xi^{2}}\right)^{2} d\xi}{\left(\frac{d^{3}W\_{0}}{d\xi} \mathcal{W}\_{0} - \frac{d^{2}W\_{0}}{d\xi} \frac{dW\_{0}}{d\xi}\right) \Big|\_{0}^{1} + \int\_{0}^{1} \left(\frac{d^{3}W\_{0}}{d\xi^{2}}\right)^{2} d\xi} \tag{23}$$

$$\text{we have } \left(\frac{d\xi\_1}{d\xi}\frac{d^2W\_0}{d\xi^2}W\_0 + f\_1\frac{d^3W\_0}{d\xi^3}W\_0 - f\_1\frac{d^2W\_0}{d\xi^2}\frac{dW\_0}{d\xi}\right)\Big|\_{0}^{1} + \int\_{0}^{1} f\_1\left(\frac{d^2W\_0}{d\xi^2}\right)^2 d\xi = 0.$$

Analogously, we choose

$$
\rho A\_0 = \frac{\int\_0^1 \rho(\xi) A(\xi) \mathcal{W}\_0^2 d\xi}{\int\_0^1 \mathcal{W}\_0^2 d\xi} \tag{24}
$$

giving Ð <sup>1</sup> <sup>0</sup> <sup>f</sup> <sup>2</sup>W<sup>2</sup> <sup>0</sup>dξ ¼ 0. Then, we obtain ω<sup>1</sup> ¼ 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

$$
\lambda\_n = \frac{1}{L^2} \sqrt{\frac{EI\_0}{\rho A\_0}} \rho\_0 \tag{25}
$$

cross-sectional area A xð Þ and moment of inertia I xð Þ along the beam axis can be

Geometry and coordinate system of an AFG beam for different taper ratios: (a) case 1, cb ¼ ch ¼ 0; (b) case 2,

� �, Ixð Þ¼ IL <sup>1</sup> � cb

respectively. AL and IL are cross-sectional area and area moment of inertia of the beam left sides, respectively. It is instructive to remember that if cb ¼ ch ¼ 0, the beam would be uniform; if ch ¼ 0, cb 6¼ 0, the beam would be tapered with constant height; if cb ¼ 0, ch 6¼ 0, the beam would be tapered with constant width; and if

corresponding to Figure 2(a)–(d), respectively. Moreover, the material properties such as Young's modulus E xð Þ and mass density ρð Þ x along the beam axis are

� �, <sup>ρ</sup>ð Þ¼ <sup>x</sup> <sup>ρ</sup><sup>L</sup> <sup>1</sup> <sup>þ</sup>

where EL and ρ<sup>L</sup> are Young's modulus and mass density of the beam left sides,

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

nonuniform AFG beams with different boundary configurations were obtained. The results were listed in Tables 2–7, respectively, and it also was compared with

x L � � <sup>1</sup> � ch

HL are the breadth and height taper ratios,

x L þ

ρLAL=ELIL

x L

� �<sup>2</sup> � � (27)

p ) of the four cases of

x L � �<sup>3</sup>

(26)

x L

cb 6¼ 0, ch 6¼ 0, the beam would be double tapered. These four cases are

x L

Based on the introduced analytical equation, the first third-order

nondimensional natural frequencies (Ω<sup>n</sup> <sup>¼</sup> <sup>λ</sup>nL<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

clearly expressed as follows:

A xð Þ¼ AL 1 � cb

where cb <sup>¼</sup> <sup>1</sup> � BR

assumed as

Figure 2.

respectively.

9

x L � � <sup>1</sup> � ch

BL and ch <sup>¼</sup> <sup>1</sup> � HR

ch ¼ 0, cb 6¼ 0; (c) case 3, cb ¼ 0, ch 6¼ 0; and (d) case 4, cb 6¼ 0, ch 6¼ 0.

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

E xð Þ¼ EL 1 þ

those of published work by Shahba et al. [38].

agreement with those given by Ref. [38].

Each order of frequency of ω<sup>0</sup> 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.

#### 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, BL and BR are the width of the left and right ends of the beams, respectively, and HL and HR 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


Table 1.

Frequency equations and mode shapes for various beams.

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

Figure 2.

Choosing the reference bending stiffness

<sup>d</sup>ξ<sup>2</sup> <sup>W</sup><sup>0</sup> <sup>þ</sup> <sup>E</sup>ð Þ<sup>ξ</sup> <sup>I</sup>ð Þ<sup>ξ</sup> <sup>d</sup><sup>3</sup>

Mechanics of Functionally Graded Materials and Structures

first order) as the given nonuniform AFG beam.

omitted, and the final results are shown in Table 1.

equation

Simply supported (S-S) sin k ¼ 0 W<sup>0</sup> ¼ sin ð Þ kξ

Frequency equations and mode shapes for various beams.

d3 W<sup>0</sup> <sup>d</sup>ξ<sup>3</sup> <sup>W</sup><sup>0</sup> � <sup>d</sup><sup>2</sup>

> d3 W<sup>0</sup> <sup>d</sup>ξ<sup>3</sup> W<sup>0</sup> � f <sup>1</sup>

� ��

ρA<sup>0</sup> ¼

Ð 1

W<sup>0</sup>

� ��

n o�

<sup>d</sup>ξ<sup>3</sup> <sup>W</sup><sup>0</sup> � <sup>E</sup>ð Þ<sup>ξ</sup> <sup>I</sup>ð Þ<sup>ξ</sup> <sup>d</sup><sup>2</sup>

d2 W<sup>0</sup> dξ<sup>2</sup> dW<sup>0</sup> dξ

<sup>0</sup> ρ ξð ÞAð Þ<sup>ξ</sup> <sup>W</sup><sup>2</sup>

ffiffiffiffiffiffiffiffi EI<sup>0</sup> ρA<sup>0</sup>

Ð 1 <sup>0</sup> <sup>W</sup><sup>2</sup> <sup>0</sup>dξ

of the equivalent homogeneous beam having the same frequency (at least up to the

s

Each order of frequency of ω<sup>0</sup> 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

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

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

In Figure 2, BL and BR are the width of the left and right ends of the beams, respectively, and HL and HR 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

Mode shape

Clamped-pinned (C-P) tan <sup>k</sup> � tanh<sup>k</sup> <sup>¼</sup> <sup>0</sup> <sup>W</sup><sup>0</sup> <sup>¼</sup> cosh ð Þ� <sup>k</sup><sup>ξ</sup> cosð Þ� <sup>k</sup><sup>ξ</sup> cosh <sup>k</sup>� cos <sup>k</sup> sinh <sup>k</sup>� sin <sup>k</sup> ½ � sinh ð Þ� <sup>k</sup><sup>ξ</sup> sin ð Þ <sup>k</sup><sup>ξ</sup>

Clamped-clamped (C-C) cos <sup>k</sup> cosh <sup>k</sup> � <sup>1</sup> <sup>¼</sup> <sup>0</sup> <sup>W</sup><sup>0</sup> <sup>¼</sup> cosh ð Þ� <sup>k</sup><sup>ξ</sup> cosð Þþ <sup>k</sup><sup>ξ</sup> sin <sup>k</sup><sup>þ</sup> sinh <sup>k</sup>

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

<sup>λ</sup><sup>n</sup> <sup>¼</sup> <sup>1</sup> L2 � � 1 <sup>0</sup> <sup>þ</sup> <sup>Ð</sup> <sup>1</sup> 0 d2 W<sup>0</sup> dξ<sup>2</sup> � �<sup>2</sup>

W<sup>0</sup> dξ<sup>2</sup> dW<sup>0</sup> dξ

W<sup>0</sup> dξ<sup>2</sup> dW<sup>0</sup> dξ

� � 1 0 þ ð1 0 f 1 d2 W<sup>0</sup> dξ<sup>2</sup> !<sup>2</sup>

<sup>0</sup>dξ

<sup>0</sup>dξ ¼ 0. Then, we obtain ω<sup>1</sup> ¼ 0. These values are the properties

� � 1 0 <sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

dξ

ω<sup>0</sup> (25)

cos <sup>k</sup>� cosh <sup>k</sup> ½ � sinh ð Þ� <sup>k</sup><sup>ξ</sup> sin ð Þ <sup>k</sup><sup>ξ</sup>

<sup>0</sup> <sup>E</sup>ð Þ<sup>ξ</sup> <sup>I</sup>ð Þ<sup>ξ</sup> <sup>d</sup><sup>2</sup>

W<sup>0</sup> dξ<sup>2</sup> � �<sup>2</sup>

(23)

(24)

dξ ¼ 0.

dξ

EI<sup>0</sup> ¼

d E½ � ð Þξ Ið Þξ dξ

we have df <sup>1</sup>

giving Ð <sup>1</sup>

d2 W<sup>0</sup>

dξ d2 W<sup>0</sup> <sup>d</sup>ξ<sup>2</sup> W<sup>0</sup> þ f <sup>1</sup>

Analogously, we choose

<sup>0</sup> <sup>f</sup> <sup>2</sup>W<sup>2</sup>

variable cross-sectional AFG beams.

3.2 Numerical results and discussion

validity of the proposed method.

Boundary conditions Frequency

Table 1.

8

Geometry and coordinate system of an AFG beam for different taper ratios: (a) case 1, cb ¼ ch ¼ 0; (b) case 2, ch ¼ 0, cb 6¼ 0; (c) case 3, cb ¼ 0, ch 6¼ 0; and (d) case 4, cb 6¼ 0, ch 6¼ 0.

cross-sectional area A xð Þ and moment of inertia I xð Þ along the beam axis can be clearly expressed as follows:

$$A(\boldsymbol{\kappa}) = A\_L \left(\mathbf{1} - c\_b \frac{\boldsymbol{\kappa}}{L}\right) \left(\mathbf{1} - c\_h \frac{\boldsymbol{\kappa}}{L}\right), \quad I(\boldsymbol{\kappa}) = I\_L \left(\mathbf{1} - c\_b \frac{\boldsymbol{\kappa}}{L}\right) \left(\mathbf{1} - c\_h \frac{\boldsymbol{\kappa}}{L}\right)^3 \tag{26}$$

where cb <sup>¼</sup> <sup>1</sup> � BR BL and ch <sup>¼</sup> <sup>1</sup> � HR HL are the breadth and height taper ratios, respectively. AL and IL are cross-sectional area and area moment of inertia of the beam left sides, respectively. It is instructive to remember that if cb ¼ ch ¼ 0, the beam would be uniform; if ch ¼ 0, cb 6¼ 0, the beam would be tapered with constant height; if cb ¼ 0, ch 6¼ 0, the beam would be tapered with constant width; and if cb 6¼ 0, ch 6¼ 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 xð Þ and mass density ρð Þ x along the beam axis are assumed as

$$E(\mathbf{x}) = E\_L \left( \mathbf{1} + \frac{\mathbf{x}}{L} \right), \quad \rho(\mathbf{x}) = \rho\_L \left[ \mathbf{1} + \frac{\mathbf{x}}{L} + \left( \frac{\mathbf{x}}{L} \right)^2 \right] \tag{27}$$

where EL and ρ<sup>L</sup> 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<sup>n</sup> <sup>¼</sup> <sup>λ</sup>nL<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρLAL=ELIL p ) of the four cases of nonuniform AFG beams with different boundary configurations were obtained. The results were listed in Tables 2–7, respectively, and it also was compared with those of published work by Shahba et al. [38].

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].


#### Table 2.

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


#### Table 3.

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


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

cb 0.2 0.4 0.6 0.8

0.2 Present 2.6873 2.9380 3.3113 3.9455

0.4 Present 2.8226 3.0877 3.4796 4.1377

0.6 Present 3.0640 3.3506 3.7700 4.4625

0.8 Present 3.5271 3.8475 4.3081 5.0458

0.2 Present 17.7225 18.3289 19.1598 20.3725

0.4 Present 16.7822 17.4061 18.2458 19.4418

0.6 Present 16.1771 16.8214 17.6687 18.8380

0.8 Present 16.0947 16.7493 17.5836 18.6877

0.2 Present 50.2194 51.1534 52.4114 54.1995

0.4 Present 46.1970 47.2734 48.6925 50.6520

0.6 Present 43.2042 44.4117 45.9613 48.0269

0.8 Present 41.7065 42.9817 44.5636 46.5828

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

Ref. [38] 2.6863 2.9336 3.2993 3.9219

Ref. [38] 2.7987 3.0486 3.4181 4.0471

Ref. [38] 2.9699 3.2237 3.5985 4.2355

Ref. [38] 3.2794 3.5401 3.9232 4.5695

Ref. [38] 17.7501 18.2379 18.9501 20.2432

Ref. [38] 16.4092 16.8571 17.5139 18.7164

Ref. [38] 14.9567 15.3627 15.9616 17.0694

Ref. [38] 13.3850 13.7466 14.2848 15.2955

Ref. [38] 50.3934 50.8645 51.6029 53.1332

Ref. [38] 44.9504 45.4003 46.0957 47.5129

Ref. [38] 39.0605 39.4844 40.1304 41.4236

Ref. [38] 32.4229 32.8123 33.3986 34.5521

ch First mode

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

ch Second mode

ch Third mode

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

found that the natural frequencies of AFG beams at low order are in good

natural frequencies of three boundary conditions (free clamping, simply supported, and clamping) are studied in Tables 5–7. From the data in the table, it can be clearly

agreement with Ref. [38], while at high order, there are some errors in the natural

a certain fractional error in the AFG-tapered beam.

frequencies.

11

Table 5.

#### Table 4.

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


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

#### Table 5.

Boundary condition First mode Second mode Third mode C-F Present 2.439 18.437 54.339

S-S Present 9.053 35.834 80.470

C-C Present 20.585 56.251 109.869

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

cb C-F S-S C-C

First 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

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

ch C-F S-S C-C

First 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

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

Second mode

Third mode

First mode Second mode

Third mode

Third mode Second mode

Third mode

First mode Second mode

Third mode

Third mode

Table 2.

Table 3.

Table 4.

10

boundary conditions.

boundary conditions.

First mode

First mode Second mode

Second mode

Mechanics of Functionally Graded Materials and Structures

Ref. [38] 2.426 18.604 55.180

Ref. [38] 9.029 36.372 81.732

Ref. [38] 20.472 56.549 110.947

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

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 5–7. 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.

### Mechanics of Functionally Graded Materials and Structures


#### Table 6.

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

E xð Þ¼ EL 1 � f <sup>E</sup>

, <sup>ρ</sup><sup>E</sup> <sup>¼</sup> <sup>1</sup> � <sup>ρ</sup><sup>R</sup>

ρL

<sup>1</sup> � <sup>f</sup> <sup>E</sup><sup>x</sup> <sup>w</sup><sup>00</sup> <sup>00</sup>

2 t and β<sup>n</sup>

where <sup>f</sup> <sup>E</sup> <sup>¼</sup> <sup>1</sup> � ER

Table 7.

ch First mode

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

ch Second mode

ch Third mode

qnðÞ¼ t An cos β<sup>n</sup>

13

EL

beam. Eq. (2) is then rewritten as

2

t þ Bn sin β<sup>n</sup>

The governing equation is then derived as

x L

cb 0.2 0.4 0.6 0.8

0.2 Present 18.2779 18.3231 18.3818 18.4612

0.4 Present 16.4975 16.7396 17.0484 17.4563

0.6 Present 15.2512 15.6622 16.1771 16.8423

0.8 Present 14.6662 15.2004 15.8587 16.6925

0.2 Present 50.4430 50.7713 51.2035 51.7981

0.4 Present 45.6257 46.3346 47.2495 48.4763

0.6 Present 42.0890 43.1214 44.4245 46.1234

0.8 Present 40.2151 41.4614 42.9975 44.9420

0.2 Present 98.2992 99.7466 100.8219 102.313

0.4 Present 88.4345 90.9806 92.8200 95.3023

0.6 Present 80.9747 84.4598 86.8967 90.0855

0.8 Present 76.7690 80.8426 83.5867 87.0576

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

Ref. [38] 18.1996 18.1286 17.9437 17.4566

Ref. [38] 15.8498 15.8350 15.7367 15.4025

Ref. [38] 13.2896 13.3319 13.3238 13.1529

Ref. [38] 10.3229 10.4255 10.5168 10.5339

Ref. [38] 50.4565 50.3599 50.1017 49.3728

Ref. [38] 44.0553 44.0370 43.9027 43.4066

Ref. [38] 37.0509 37.1137 37.1104 36.8678

Ref. [38] 28.8912 29.0409 29.1842 29.2402

Ref. [38] 99.1474 99.0414 98.7543 97.9046

Ref. [38] 86.6608 86.6414 86.4932 85.9176

Ref. [38] 72.9681 73.0382 73.0375 72.7615

Ref. [38] 56.9674 57.1341 57.2991 57.3787

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

Based on the vibration theory, we assume w xð Þ¼ ; t ϕð Þ x q tð Þ, where

, <sup>ρ</sup>ð Þ¼ <sup>x</sup> <sup>ρ</sup><sup>L</sup> <sup>1</sup> � <sup>f</sup> <sup>ρ</sup>

þ 1 � f <sup>ρ</sup>x

x L

<sup>w</sup>€ <sup>¼</sup> <sup>0</sup> (29)

<sup>2</sup> is the modal frequency for dimensionless.

. EL and ER are Young's modulus at the left/right

(28)
