**Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique and MATLAB Verifications**

Wael A. Al-Tabey

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/57384

#### **1. Introduction**

In the past, the model of thin plate on the elastic foundation was mainly used in structural applications. Currently, thin films of metal, ceramic or synthetic materials deposited on the surface of the structural parts of the electronic devices are used to improve their mechanical, thermal, electrical and tribological properties. These thin films of material are considered as thin plates and in these applications, the substrate of thin film can be simulated as an elastic foundation [1-2].

The laminated composite rectangular plate is very common in many engineering fields such as aerospace industries, civil engineering and marine engineering. The ability to conduct an accurate free vibration analysis of plates with variable thickness is absolutely essential if the designer is concerned with possible resonance between the plate and driving force [3].

Ungbhakorn and Singhatanadgid [4] investigated the buckling problem of rectangular laminated composite plates with various edge supports by using an extended Kantorovich method is employed.

Setoodeh, Karami [5] investigated A three-dimensional elasticity approach to develop a general free vibration and buckling analysis of composite plates with elastic restrained edges.

Luura and Gutierrez [6] studied the vibration of rectangular plates by a non-homogenous elastic foundation using the Rayleigh-Ritz method.

Ashour [7] investigated the vibration analysis of variable thickness plates in one direction with edges elastically restrained against both rotation and translation using the finite strip transition matrix technique.

© 2014 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Grossi, Nallim [8] investigated the free vibration of anisotropic plates of different geometrical shapes and generally restrained boundaries. An analytical formulation, based on the Ritz method and polynomial expressions as approximate functions for analyzing the free vibrations of laminated plates with smooth and non-smooth boundary with non classical edge supports is presented.

available in the literature, which validates the accuracy and reliability of the proposed

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

**b** 

**F** 

**(i-1)** 

**a S S** 

**F S-S-F-F** 

**Strip (i)** 

**y,η**

**ho h1** 

**h2 hk-1 hk hn-1 hn** **(i)** 

**y,η**

http://dx.doi.org/10.5772/57384

585

**hb**

which validates the accuracy and reliability of the proposed technique.

**x,ξ**

**F** 

**F** 

**ho**

**1 2** 

**n** 

**k k-1** 

**k+1** 

**Free-Free Beam Strip** 

Figure 1. A rectangular laminated plate with variable thickness

**Tk= hk - hk -1** 

2 2

(1)

the classical deformation theory in terms of the plate deflection W(x, y, t) is given by:

The equation of motion governing the vibration of rectangular plate under the assumption of

*<sup>w</sup> yh*

bending and the twisting moments in terms of displacements are given by:

terms of the plate deflection W(x, y, t) is given by:

2 2

*yx M*

2

*y M*

*M <sup>X</sup> XY <sup>Y</sup> <sup>o</sup>*

**Figure 1.** A rectangular laminated plate with variable thickness

<sup>2</sup> )( *<sup>t</sup>*

**1. 2. Formulation** 

**z** 

**ho /2** 

**ho /2** 

2 2

*x*

**2. Formulation**

This chapter presents the finite strip transition matrix technique (FSTM) and a semi-analytical method to obtain the natural frequencies and mode shapes of symmetric angle-ply laminated composite rectangular plate with classical boundary conditions (S-S-F-F). The plate has a uniform thickness in x direction and varying thickness h(y) in y direction, as shown in Figure 1. The boundary conditions in the variable thickness direction are simply supported and they are satisfied identically and the boundary conditions in the other direction are free and are approximated. Numerical results for simple-free (S-S-F-F) boundary conditions at the plate edges are presented. The illustrated results are in excellent agreement compared with solutions available in the literature,

The equation of motion governing the vibration of rectangular plate under the assumption of the classical deformation theory in

**h (y)** 

Where *W* is the transverse deflection, ρ = the density per unit area of the plate and h(y) is the plate thickness at any point. The

technique.

**1.1. The chapter aims** 

LU, et al [9] presented the exact analysis for free vibration of long-span continuous rectangular plates based on the classical Kirchhoff plate theory, using state space approach associated with joint coupling matrices.

Chopra [10] studied the free vibration of stepped plates by analytical method. Using the solutions to the differential equations for each region of the plate with uniform thickness, he formulated the overall Eigen value problem by introducing the boundary conditions and continuity conditions at the location of abrupt change of thickness. However this method suffers from the drawback of excessive continuity, as in theory the second and third derivatives of the deflection function at the locations of abrupt change of thickness should not be contin‐ uous.

Cortinez and Laura [11] computed the natural frequencies of stepped rectangular plates by means of the Kantorovich extended method, whereby the accuracy was improved by inclusion of an exponential optimization parameter in the formulation.

Bambill et al. [12] subsequently obtained the fundamental frequencies of simply supported stepped rectangular plates by the Rayleigh–Ritz method using a truncated double Fourier expansion.

Laura and Gutierrez [13] studied the free vibration problem of uniform rectangular plates supported on a non-homogeneous elastic foundation based on the Rayleigh–Ritz method using polynomial coordinate functions which identically satisfy the governing boundary conditions.

Harik and Andrade [14] used the "analytical strip method" to the stability analysis of unidirectionally stepped plates. In essence, the stepped plate is divided into rectangular regions of uniform thickness. The differential equations of stability for each region are solved and the continuity conditions at the junction lines as well as the boundary conditions are then imposed.

#### **1.1. The chapter aims**

This chapter presents the finite strip transition matrix technique (FSTM) and a semi-analytical method to obtain the natural frequencies and mode shapes of symmetric angle-ply laminated composite rectangular plate with classical boundary conditions (S-S-F-F). The plate has a uniform thickness in x direction and varying thickness h(y) in y direction, as shown in Figure 1. The boundary conditions in the variable thickness direction are simply supported and they are satisfied identically and the boundary conditions in the other direction are free and are approximated. Numerical results for simple-free (S-S-F-F) boundary conditions at the plate edges are presented. The illustrated results are in excellent agreement compared with solutions S-F-F). The plate has a uniform thickness in x direction and varying thickness h(y) in y direction, as shown in Figure 1. The

available in the literature, which validates the accuracy and reliability of the proposed technique. boundary conditions in the variable thickness direction are simply supported and they are satisfied identically and the boundary conditions in the other direction are free and are approximated. Numerical results for simple-free (S-S-F-F) boundary conditions at the plate edges are presented. The illustrated results are in excellent agreement compared with solutions available in the literature,

which validates the accuracy and reliability of the proposed technique.

**Figure 1.** A rectangular laminated plate with variable thickness

terms of the plate deflection W(x, y, t) is given by:

2

*y M*

Figure 1. A rectangular laminated plate with variable thickness

**1.1. The chapter aims** 

#### 2 2 2 2 <sup>2</sup> )( *<sup>t</sup> M x M <sup>X</sup> XY <sup>Y</sup> <sup>o</sup>* **2. Formulation**

**1. 2. Formulation** 

*yx*

Grossi, Nallim [8] investigated the free vibration of anisotropic plates of different geometrical shapes and generally restrained boundaries. An analytical formulation, based on the Ritz method and polynomial expressions as approximate functions for analyzing the free vibrations of laminated plates with smooth and non-smooth boundary with non classical edge supports

LU, et al [9] presented the exact analysis for free vibration of long-span continuous rectangular plates based on the classical Kirchhoff plate theory, using state space approach associated with

Chopra [10] studied the free vibration of stepped plates by analytical method. Using the solutions to the differential equations for each region of the plate with uniform thickness, he formulated the overall Eigen value problem by introducing the boundary conditions and continuity conditions at the location of abrupt change of thickness. However this method suffers from the drawback of excessive continuity, as in theory the second and third derivatives of the deflection function at the locations of abrupt change of thickness should not be contin‐

Cortinez and Laura [11] computed the natural frequencies of stepped rectangular plates by means of the Kantorovich extended method, whereby the accuracy was improved by inclusion

Bambill et al. [12] subsequently obtained the fundamental frequencies of simply supported stepped rectangular plates by the Rayleigh–Ritz method using a truncated double Fourier

Laura and Gutierrez [13] studied the free vibration problem of uniform rectangular plates supported on a non-homogeneous elastic foundation based on the Rayleigh–Ritz method using polynomial coordinate functions which identically satisfy the governing boundary

Harik and Andrade [14] used the "analytical strip method" to the stability analysis of unidirectionally stepped plates. In essence, the stepped plate is divided into rectangular regions of uniform thickness. The differential equations of stability for each region are solved and the continuity conditions at the junction lines as well as the boundary conditions are then imposed.

This chapter presents the finite strip transition matrix technique (FSTM) and a semi-analytical method to obtain the natural frequencies and mode shapes of symmetric angle-ply laminated composite rectangular plate with classical boundary conditions (S-S-F-F). The plate has a uniform thickness in x direction and varying thickness h(y) in y direction, as shown in Figure 1. The boundary conditions in the variable thickness direction are simply supported and they are satisfied identically and the boundary conditions in the other direction are free and are approximated. Numerical results for simple-free (S-S-F-F) boundary conditions at the plate edges are presented. The illustrated results are in excellent agreement compared with solutions

of an exponential optimization parameter in the formulation.

is presented.

uous.

expansion.

conditions.

**1.1. The chapter aims**

joint coupling matrices.

584 MATLAB Applications for the Practical Engineer

Where *W* is the transverse deflection, ρ = the density per unit area of the plate and h(y) is the plate thickness at any point. The bending and the twisting moments in terms of displacements are given by: The equation of motion governing the vibration of rectangular plate under the assumption of the classical deformation theory in terms of the plate deflection W(x, y, t) is given by:

2 2

(1)

*<sup>w</sup> yh*

The equation of motion governing the vibration of rectangular plate under the assumption of the classical deformation theory in

$$\frac{\partial^2 M\_X}{\partial \mathbf{x}^2} - 2\frac{\partial^2 M\_{XY}}{\partial \mathbf{x} \partial \mathbf{y}} + \frac{\partial^2 M\_Y}{\partial \mathbf{y}^2} = -\rho \,\, h(\mathbf{y}) \frac{\partial^2 w\_o}{\partial \mathbf{t}^2} \tag{1}$$

4 4 4 44 2 <sup>11</sup> <sup>4</sup> <sup>16</sup> <sup>3</sup> 12 66 2 2 <sup>26</sup> 3 4 <sup>22</sup> <sup>2</sup> 4 2( 2 ) 4

*x xy x y xy y <sup>t</sup>* (5)

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

( ) *D W D W D D W D W D W h yW xxxx xxxy xxyy xyyy yyyy tt* (6)

*xx xy xy*

<sup>ï</sup> ( ) <sup>ý</sup> = ïþ *o tt o h y m W h*

The substitution of equation (3) into equation (6) given the governing Partial differential

( ) ( ) ( ) 2( 2 )

ì ü ì üì ü ï ï ï ïï ï ¶ ¶¶ æ ö æö æö í ý í ýí ý ç ÷ + + ç÷ ç÷ <sup>+</sup> ¶ ¶ ¶ ¶ ï ï ï ïï ï î þ î þî þ è ø èø èø

*o o o*

 ( ) ¶ ¶ ¶ ¶¶ ¶ + + + + + =- ¶ ¶¶ ¶ ¶ ¶¶ ¶ ¶ *w w w ww w o o o oo o D D DD D D h y*

<sup>11</sup> <sup>16</sup> 12 66 <sup>26</sup> <sup>22</sup> + + + + + =- 4 2( 2 ) 4

2 3 2 3 2 3 11 2 3 , 12 66 3 , 16 2 3 ,

3 3 3 12 66 12 66 3 11 3 3 3 16 3

*h y D D h y D D h y D W <sup>W</sup> h yW D W hh h h <sup>y</sup>*

( ) 2( 2 ) ( ) 2( 2 ) ( ) ( )

*xxxx xxy xxyy xxxy*

*xy xyyy xyy yy*

The equation of motion (8) can be normalized using the non-Dimensional variables ξ and η as

xhh

> h

> > *D*<sup>11</sup> *D*<sup>22</sup> , *ψ*<sup>2</sup> =

*<sup>b</sup>* , *ψ*<sup>1</sup> <sup>=</sup>

*o o tt*

xxhh

yyy

h

xxxh

h

(*D*<sup>12</sup> + 2*D*66) *D*<sup>22</sup>

 h hh

, *ψ*<sup>3</sup> =

*D*<sup>16</sup> *D*<sup>22</sup>

and *ψ*<sup>4</sup> =

xhhh

+ ++ ç ÷ç ÷ ¶ è øè ø

*o o o o*

2 3 3 2 3 26 26 3 26 22 32 3 3 3 2

*D DD h y hy D hy W h yW <sup>W</sup> <sup>W</sup> hy h h y h y*

1 4 3 2 234 2 2 3 3 2 3 3 2 3

 h

*hh h WW W*

h

¶ ¶ ¶

*<sup>a</sup>* , *<sup>η</sup>* <sup>=</sup> *<sup>y</sup>*

1 2 1 () 1 <sup>1</sup> <sup>1</sup> 2 4

+ + ++

*<sup>h</sup> W W WW W a h ab a b a b ab*

22

3 2 3 2 23 2 3 2

¶ ¶¶

1 4 () 8 1 () 1 1 () ( ) ( ) ( )

xxh

( ) ( )

hhh

æ öæ ö + + ¶

4 48 ( ) ( ) ( ) ( )

*o oo o*

æ ö ¶ ¶ ¶ æ ö + ++ + ç ÷ ç ÷ ¶ ¶ ¶ è ø è ø

*h y h y h y D W DD W D W x h x y h x h*

2 3 2 3 26 ,2 2 3 2 , 2 3

+ + í ýí ç÷ ç÷ ï ïï ¶ ¶ î þî èø èø

2 () () ( ) ¶ + = - ¶ *yyyy yyy o tt*

3

¶

¶

++ +

 h

h

 y

h

*<sup>h</sup> m h W WW*

h

 h

*ab h h ab b h*

4 33 2

¶ + + = - ¶

*b hb D h*

*<sup>b</sup>* is the aspect ratio, *<sup>ξ</sup>* <sup>=</sup> *<sup>x</sup>*

*o o o D h y hy h yW W mW h h y h*

2

y

( )

h

1 2 1 ()

h

4 4

xh

*h y h y D WD W y h y h*

ì üì ü ï ïï ¶ ¶ æö æö

*o o*

*xy yy*

( ) ( ) <sup>4</sup>

Equation (7) may be written as:

*D* <sup>3</sup> 3 22 3 3

> xxxx

yh

hh

hhhh

Or in contraction form:

equation:

22

follows :

y

Where *<sup>β</sup>* <sup>=</sup> *<sup>a</sup>*

+

r

http://dx.doi.org/10.5772/57384

r

(7)

587

(8)

(9)

*D*<sup>26</sup> *D*<sup>22</sup> .

Where *W* is the transverse deflection, ρ = the density per unit area of the plate and h(y) is the plate thickness at any point. The bending and the twisting moments in terms of displacements are given by:

$$\begin{aligned} M\_X &= -D\_{11} \frac{\hat{\sigma}^2 w\_o}{\partial \mathbf{x}^2} - D\_{12} \frac{\hat{\sigma}^2 w\_o}{\partial \mathbf{y}^2} - 2D\_{16} \frac{\hat{\sigma}^2 w\_o}{\partial \mathbf{x} \partial \mathbf{y}} \\\ M\_Y &= -D\_{12} \frac{\hat{\sigma}^2 w\_o}{\partial \mathbf{x}^2} - D\_{22} \frac{\hat{\sigma}^2 w\_o}{\partial \mathbf{y}^2} - 2D\_{26} \frac{\hat{\sigma}^2 w\_o}{\partial \mathbf{x} \partial \mathbf{y}} \\\ M\_{XY} &= -D\_{16} \frac{\hat{\sigma}^2 w\_o}{\partial \mathbf{x}^2} - D\_{26} \frac{\hat{\sigma}^2 w\_o}{\partial \mathbf{y}^2} - 2D\_{66} \frac{\hat{\sigma}^2 w\_o}{\partial \mathbf{x} \partial \mathbf{y}} \end{aligned} \tag{2}$$

The flexural rigidities *Dij* of the plate are given by:

$$D\_{\bar{y}} = \frac{1}{3} \frac{h^3(\mathcal{Y})}{h\_o^3} \sum\_{k=1}^n \left[ (\overline{\mathcal{Q}}\_{\bar{y}}) \right]\_k (h\_{ok}^3 - h\_{ok-1}^3), \qquad i, j = 1, 2, 3, \dots \tag{3}$$

Where *hok* is the distance from the middle-plane of the plate according to *ho* to the bottom of the *hoth* layer as shown in Figure 1. And *Qij* ¯*<sup>k</sup>* are the plane stress transformed reduced stiffness coefficients of the lamina in the laminate Cartesian coordinate system. They are related to reduced stiffness coefficients of the lamina in the material axes of lamina *Qij k* by proper coordinate relationships they can be expressed in terms of the engineering notations as:

$$Q\_{ij} = \begin{bmatrix} Q\_{11} & Q\_{12} & Q\_{13} \\ Q\_{12} & Q\_{22} & Q\_{23} \\ Q\_{13} & Q\_{23} & Q\_{66} \end{bmatrix} = \begin{bmatrix} \frac{E\_{11}}{\left(1 - \nu\_{12}\nu\_{21}\right)} & \frac{\nu\_{21}E\_{11}}{\left(1 - \nu\_{12}\nu\_{21}\right)} & 0 \\\\ \frac{\nu\_{21}E\_{11}}{\left(1 - \nu\_{12}\nu\_{21}\right)} & \frac{E\_{22}}{\left(1 - \nu\_{21}\nu\_{12}\right)} & 0 \\\\ 0 & 0 & G\_{12} \end{bmatrix} \tag{4}$$

Where *E*11, *E*22 are the longitudinal and transverse young's moduli parallel and perpendicular to the fiber orientation, respectively and *G*<sup>12</sup> is the plane shear modulus of elasticity, *υ*12 and *υ*21 are the poisson's ratios. Thus, the governing partial differential equation of laminated composite rectangular plate with variable thickness as shown in Figure 1 is reduced to:

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique… http://dx.doi.org/10.5772/57384 587

$$D\_{11}\frac{\partial^4 w\_o}{\partial x^4} + 4D\_{16}\frac{\partial^4 w\_o}{\partial x^3 \partial y} + 2(D\_{12} + 2D\_{66})\frac{\partial^4 w\_o}{\partial x^2 \partial y^2} + 4D\_{26}\frac{\partial^4 w\_o}{\partial x \partial y^3} + D\_{22}\frac{\partial^4 w\_o}{\partial y^4} = -\rho \text{ } h(\mathbf{y})\frac{\partial^2 w\_o}{\partial t^2} \tag{5}$$

Or in contraction form:

2 22 2 2 22 2

Where *W* is the transverse deflection, ρ = the density per unit area of the plate and h(y) is the plate thickness at any point. The bending and the twisting moments in terms of displacements

> 22 2 11 2 2 12 16

¶¶ ¶ <sup>ü</sup> =- - - <sup>ï</sup> ¶ ¶ ¶ ¶ <sup>ï</sup>

*oo o <sup>X</sup>*

*ww w MD D D*

*ww w MD D D*

*ww w MD D D*

The flexural rigidities *Dij* of the plate are given by:

3

the *hoth* layer as shown in Figure 1. And *Qij*

*ij*

1

*ij ij ok ok <sup>k</sup> <sup>o</sup> <sup>k</sup>*

=

11 12 13

12 22 23

13 23 66

ë û

*QQQ*

*n*

*oo o <sup>Y</sup>*

*oo o XY*

3 3

1 () ( ) ( ), , 1,2,3,......... <sup>3</sup> -

Where *hok* is the distance from the middle-plane of the plate according to *ho* to the bottom of

coefficients of the lamina in the laminate Cartesian coordinate system. They are related to

( ) ( )

*E E*

1 1

uu

ê ú - - é ù

uu

Where *E*11, *E*22 are the longitudinal and transverse young's moduli parallel and perpendicular to the fiber orientation, respectively and *G*<sup>12</sup> is the plane shear modulus of elasticity, *υ*12 and *υ*21 are the poisson's ratios. Thus, the governing partial differential equation of laminated composite rectangular plate with variable thickness as shown in Figure 1 is reduced to:

1 1

11 21 11 12 21 12 21

u

é ù

 uu

 uu

( ) ( )

0 0

12 21 21 12

ë û

21 11 22

3 1

= -= é ù åë û

reduced stiffness coefficients of the lamina in the material axes of lamina *Qij*

coordinate relationships they can be expressed in terms of the engineering notations as:

u

ê ú <sup>=</sup> <sup>=</sup> ê ú - - ê ú

*QQQ E E Q QQQ*

<sup>ï</sup> ¶¶ ¶ <sup>ï</sup> =- - - <sup>ý</sup> ¶ ¶ ¶ ¶ <sup>ï</sup>

<sup>ï</sup> ¶¶ ¶ =- - - <sup>ï</sup>

22 2 12 2 2 22 26

*x y x y*

22 2 16 2 2 26 66

*x y x y*

¶ ¶ ¶ ¶ ï

*x y x y*

 ( ) ¶ ¶¶ ¶ - + =- ¶ ¶¶ ¶ ¶ *M MM <sup>X</sup> XY Y wo h y x yt x y*

are given by:

586 MATLAB Applications for the Practical Engineer

r

2

2

2

*h y <sup>D</sup> Q h h ij <sup>h</sup>* (3)

þ

¯*<sup>k</sup>* are the plane stress transformed reduced stiffness

12

*G*

0

0

*k*

by proper

(4)

(1)

(2)

$$(D\_{11}W\_{\rm xxx} + 4D\_{16}W\_{\rm xxy} + 2(D\_{12} + 2D\_{66})W\_{\rm xyy} + 4D\_{26}W\_{\rm yyy} + D\_{22}W\_{\rm yyy} = -\rho \text{ } h(\mathbf{y})W\_{\rm x} \tag{6}$$

The substitution of equation (3) into equation (6) given the governing Partial differential equation:

$$\begin{aligned} D\_{11} \left\{ \frac{\hat{\sigma}^2}{\partial \mathbf{x}^2} \left( \frac{h^3(\mathbf{y})}{h\_o^3} W\_{,\mathbf{x}\mathbf{y}} \right) \right\} &+ 2(D\_{12} + 2D\_{66}) \left\{ \frac{\hat{\sigma}^2}{\partial \mathbf{x} \partial \mathbf{y}} \left( \frac{h^3(\mathbf{y})}{h\_o^3} W\_{,\mathbf{y}\mathbf{y}} \right) \right\} + D\_{16} \left\{ \frac{\hat{\sigma}^2}{\partial \mathbf{x}^2} \left( \frac{h^3(\mathbf{y})}{h\_o^3} W\_{,\mathbf{y}\mathbf{y}} \right) \right\} \\ + 4D\_{26} \left\{ \frac{\hat{\sigma}^2}{\partial \mathbf{y}^2} \left( \frac{h^3(\mathbf{y})}{h\_o^3} W\_{,\mathbf{y}\mathbf{y}} \right) \right\} + D\_{22} \left\{ \frac{\hat{\sigma}^2}{\partial \mathbf{y}^2} \left( \frac{h^3(\mathbf{y})}{h\_o^3} W\_{,\mathbf{y}\mathbf{y}} \right) \right\} = -m\_o \frac{h(\mathbf{y})}{h\_o} W\_{\mathbf{z}} \end{aligned} \tag{7}$$

Equation (7) may be written as:

$$\begin{aligned} &D\_{11}\frac{h^{3}(\mathbf{y})}{h\_{o}^{3}}W\_{\text{xxx}}+\left(\frac{2(D\_{12}+2D\_{66})}{h\_{o}^{3}}\right)\frac{\partial h^{3}(\mathbf{y})}{\partial\mathbf{y}}W\_{\text{xy}}+\left(\frac{2(D\_{12}+2D\_{66})}{h\_{o}^{3}}\right)h^{3}(\mathbf{y})W\_{\text{xyy}}+D\_{16}\frac{h^{3}(\mathbf{y})}{h\_{o}^{3}}W\_{\text{xxx}}\\ &+\left(\frac{4D\_{26}}{h\_{o}^{3}}\frac{\partial^{2}h^{3}(\mathbf{y})}{\partial\mathbf{y}^{2}}\right)W\_{\text{xy}}+\frac{4D\_{26}}{h\_{o}^{3}}h^{3}(\mathbf{y})W\_{\text{xyy}}+\frac{8D\_{26}}{h\_{o}^{3}}\frac{\partial h^{3}(\mathbf{y})}{\partial\mathbf{y}}W\_{\text{xy}}+\left(\frac{D\_{22}}{h\_{o}^{3}}\frac{\partial^{2}h^{3}(\mathbf{y})}{\partial\mathbf{y}^{2}}\right)W\_{\text{yy}}\\ &+\frac{D\_{22}}{h\_{o}^{3}}h^{3}(\mathbf{y})W\_{\text{yy}\mathbf}+\frac{2D\_{22}}{h\_{o}^{3}}\frac{\partial h^{3}(\mathbf{y})}{\partial\mathbf{y}}W\_{\text{yy}}=-m\_{o}\frac{h(\mathbf{y})}{h\_{o}}W\_{\text{z}}\end{aligned} \tag{8}$$

The equation of motion (8) can be normalized using the non-Dimensional variables ξ and η as follows :

$$\begin{split} &\nu\_{1}\frac{1}{a}W\_{\xi\xi\xi\xi\xi}+\frac{2\nu\_{2}}{h^{3}(\eta)}\frac{1}{a^{2}b}\frac{\partial h^{3}(\eta)}{\partial\eta}W\_{\xi\xi\eta}+2\nu\_{2}\frac{1}{a^{2}b^{2}}W\_{\xi\xi\eta\eta}+\nu\_{3}\frac{1}{a^{3}b}W\_{\xi\xi\xi\xi\eta}+4\nu\_{4}\frac{1}{ab^{3}}W\_{\xi\eta\eta\eta} \\ &+\frac{1}{ab}\frac{4\nu\_{4}}{h^{3}(\eta)}\frac{\hat{c}^{2}h^{3}(\eta)}{\hat{c}\eta^{2}}W\_{\xi\eta}+\frac{8\nu\_{4}}{h^{3}(\eta)}\frac{1}{ab^{2}}\frac{\partial h^{3}(\eta)}{\partial\eta}W\_{\xi\eta\eta}+\frac{1}{b^{2}}\frac{1}{h^{3}(\eta)}\frac{\hat{c}^{2}h^{3}(\eta)}{\hat{c}\eta^{2}}W\_{\eta\eta} \\ &+\frac{1}{b^{4}}W\_{\eta\eta\eta\eta}+\frac{2}{h^{3}(\eta)}\frac{1}{b^{3}}\frac{\partial h^{3}(\eta)}{\hat{c}\eta}W\_{\eta\eta\eta}=-\frac{m\_{o}}{D\_{22}}\frac{h\_{o}^{2}}{h^{2}(\eta)}W\_{\eta} \end{split} \tag{9}$$

Where *<sup>β</sup>* <sup>=</sup> *<sup>a</sup> <sup>b</sup>* is the aspect ratio, *<sup>ξ</sup>* <sup>=</sup> *<sup>x</sup> <sup>a</sup>* , *<sup>η</sup>* <sup>=</sup> *<sup>y</sup> <sup>b</sup>* , *ψ*<sup>1</sup> <sup>=</sup> *D*<sup>11</sup> *D*<sup>22</sup> , *ψ*<sup>2</sup> = (*D*<sup>12</sup> + 2*D*66) *D*<sup>22</sup> , *ψ*<sup>3</sup> = *D*<sup>16</sup> *D*<sup>22</sup> and *ψ*<sup>4</sup> = *D*<sup>26</sup> *D*<sup>22</sup> .

#### **3. Method of solution**

The displacement *W* (*ξ*, *η*, *t*)=*W* (*ξ*, *η*)*ei<sup>ω</sup> <sup>t</sup>* can be expressed in terms of the shape function *Xi* (*ξ*), chosen a prior; and the unknown function *Yi* (*η*)as:

$$W(\xi,\eta,t) = \sum\_{i=0}^{N} X\_i(\xi)Y\_i(\eta) \text{ e}^{i\alpha \cdot t} \tag{10}$$

In this paper, the beam shape function in ξ-direction is considered as a strip element of the

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

One can obtain the following system of homogenous linear equations by satisfying the

 m


 m

*i i*

 mx

sinh sin cosh cos <sup>1</sup> ( ) sin( ) sinh( ) cos( ) cosh( )

ï =- + -

22 2 2 2cos( )cosh( ) cos ( ) sin ( ) sinh ( ) cosh ( ) <sup>0</sup> sinh( )cosh( ) sin( )cosh( ) sinh( )cos( ) sin( )cos( )


 m

> mm

3 12 12 2 2 2 ,, 4 ,

*i i i*

*<sup>f</sup> c b f f Y aY a aY*

 yb

y b

> h

1 *h* 3 (*η*)

() 2 ( ) ( ) 2 (8 ) ( ) () () () ()

+ ++ +

3

*ij ij*

h

*i*

∂<sup>2</sup> *h* <sup>3</sup> (*η*) <sup>∂</sup>*<sup>η</sup>* <sup>2</sup> , *<sup>f</sup>* <sup>3</sup>(*η*)=

h

 h

*i*

m

m

*iii i i*

*i i <sup>i</sup>*

 mx

are the roots of equation:

 m

> m

The substitution of equation (10) into equation (9), multiplying both sides by *Xj*

0 0 3 33 3 3

 yb

*f f f a fa f*

h

 h

 m

The roots of equation (15) are represented in the recurrence form:

 p  mx

=+ + + (13)

( )

F ï

 m

= + ( 0.5) , 0, 1, 2, 3 = , ....... .. . *<sup>i</sup> ii* (16)

 mx

 mm

 mx

ý

þ

 m

)*D*22 and for *υ* =0.3, it can

http://dx.doi.org/10.5772/57384

(14)

589

(15)

(*x*) and after

(17)

hh

h

 h

*ho* 2

*h* 2 (*η*) ,

b

plate and the flexural rigidity *EI* of the beam can replaced by (1−*υ* <sup>2</sup>

 mx

 mx

x

boundary conditions (12) at ξ=0 and ξ=1.

x

mm

m

b

y b

+ ++ +

, *f* <sup>1</sup>(*η*)=

hh

4 2

h

1 3 2 2 4 2 4, 33 3 3

y b

hhh

*i j ij ij*

*f f cd b b a aY f af a f a f a*

hh

*ij ij ij ij*

*ij ij ij ij*

∂*h* <sup>3</sup> (*η*) <sup>∂</sup>*<sup>η</sup>* , *<sup>f</sup>* <sup>2</sup>(*η*)=

( ) () 4 (2 <sup>4</sup> ) () () () ()

1 *h* 3 (*η*)

 m

*X*

The different value of *μi*

m

some manipulation, we can find:

hhhh

<sup>2</sup> - = l ) 0 *ij i*

> *mo<sup>ω</sup>* <sup>2</sup> *a* 4 *D*<sup>22</sup>

h

1 3

h

b

h

*e*

*ij <sup>Y</sup> <sup>a</sup>*

( ( )

Where *λ* <sup>2</sup> =

*f*

y

+

y b

= =

åå *N M*

be just approximated by *E* ≈*D*22. The solution of the beam equation is given as:

12 3 4 *XA A A A i ii i i* ( ) sin( ) cos( ) sinh( ) cosh( )

 mx

The most commonly used is the Eigen function obtained from the solution of beam free vibration under the prescribed boundary conditions at ξ=0 and ξ=1.

The free vibration of a beam of length a can be described by the non-Dimensional differential equation:

$$\left(\frac{2F\_{T1}\mu\_i\left(F\_{R1}F\_{R2}-\mu\_i^2\right)}{\left(\mu\_i^4+F\_{R1}F\_{T1}\right)}\sin\mu\_i+\frac{2F\_{T1}\mu\_i\left(F\_{R1}+F\_{R2}\right)}{\left(\mu\_i^4+F\_{R1}F\_{T1}\right)}\mu\_i\cos\mu\_i\right)A\_3\tag{11}$$

Where *EI* is the flexural rigidity of the beam. The boundary conditions for free edges beam as shown in Fig. 2 are:

at ξ=0 and ξ=1

$$\begin{cases} \frac{\partial^2 X\_i(\xi)}{\partial \xi^2} = 0\\ \frac{\partial^3 X\_i(\xi)}{\partial \xi^3} = 0 \end{cases} \tag{12}$$

**Figure 2.** The two free edges beam strip in ξ-direction

In this paper, the beam shape function in ξ-direction is considered as a strip element of the plate and the flexural rigidity *EI* of the beam can replaced by (1−*υ* <sup>2</sup> )*D*22 and for *υ* =0.3, it can be just approximated by *E* ≈*D*22. The solution of the beam equation is given as:

$$X\_j(\xi) = A\_1 \sin(\mu\_j \xi) + A\_2 \cos(\mu\_j \xi) + A\_3 \sinh(\mu\_j \xi) + A\_4 \cosh(\mu\_j \xi) \tag{13}$$

One can obtain the following system of homogenous linear equations by satisfying the boundary conditions (12) at ξ=0 and ξ=1.

$$\Phi\_i = \frac{-\sinh\mu\_i + \sin\mu\_i}{\cosh\mu\_i - \cos\mu\_i}$$

$$X\_i(\xi) = \sin(\mu\_i \xi) - \sinh(\mu\_i \xi) + \frac{1}{\Phi\_i} \left(\cos(\mu\_i \xi) - \cosh(\mu\_i \xi)\right)\tag{14}$$

The different value of *μi* are the roots of equation:

**3. Method of solution**

588 MATLAB Applications for the Practical Engineer

*Xi*

equation:

shown in Fig. 2 are:

at ξ=0 and ξ=1

The displacement *W* (*ξ*, *η*, *t*)=*W* (*ξ*, *η*)*ei<sup>ω</sup> <sup>t</sup>*

(*ξ*), chosen a prior; and the unknown function *Yi*

xh

vibration under the prescribed boundary conditions at ξ=0 and ξ=1.

2

1 12 1 12

*i RT i RT*

2

3

2

x

*i*

*X*

x

¶ <sup>ü</sup> <sup>=</sup> <sup>ï</sup> ¶ <sup>ï</sup>

x

¶ <sup>ï</sup> <sup>=</sup> <sup>ï</sup> ¶ <sup>þ</sup>

( ) <sup>0</sup>

( ) <sup>0</sup>

ý

3

x

*i*

*X*

*Ti RR i Ti R R*

*F FF F FF*

m

1 1 1 1

æ ö - <sup>+</sup> ç ÷ <sup>+</sup> + + è ø

( )

2 2

 m

( )

m

m

**Figure 2.** The two free edges beam strip in ξ-direction

can be expressed in terms of the shape function

(*η*)as:

w

( )

*i i i*

( )

sin cos

*W t XY e* (10)

m

*F F F F* (11)

 m *A*

(12)

0 ( , ,) ( ) ( )

= <sup>=</sup> å

*i*

 x h

The most commonly used is the Eigen function obtained from the solution of beam free

The free vibration of a beam of length a can be described by the non-Dimensional differential

3 4 4

Where *EI* is the flexural rigidity of the beam. The boundary conditions for free edges beam as

 m

m

*<sup>N</sup> i t i i*

$$\frac{-2\cos(\mu\_i)\cosh(\mu\_i) + \cos^2(\mu\_i) + \sin^2(\mu\_i) - \sinh^2(\mu\_i) + \cosh^2(\mu\_i)}{\sinh(\mu\_i)\cosh(\mu\_i) + \sin(\mu\_i)\cosh(\mu\_i) - \sinh(\mu\_i)\cos(\mu\_i) - \sin(\mu\_i)\cos(\mu\_i)} = 0\tag{15}$$

The roots of equation (15) are represented in the recurrence form:

$$\mu\_i = (i+0.5)\pi, \qquad i = 0, 1, 2, 3, \dots \tag{16}$$

The substitution of equation (10) into equation (9), multiplying both sides by *Xj* (*x*) and after some manipulation, we can find:

$$\begin{split} &\sum\_{i=0}^{N} \sum\_{j=0}^{M} \frac{\beta^{4}}{f\_{3}(\eta)} Y\_{i,\eta\eta\eta\eta} + 2\beta^{3} a \frac{f\_{1}(\eta)}{f\_{3}(\eta)} Y\_{i,\eta\eta\eta} + (\frac{2\nu\_{2}\beta^{2}}{f\_{3}(\eta)} \frac{a\_{\bar{\eta}}}{a\_{\bar{\eta}}} + 8\nu\_{4}\beta^{2} a \frac{f\_{1}(\eta)}{f\_{3}(\eta)} \frac{b\_{\bar{\eta}}}{a\_{\bar{\eta}}} + \beta^{2} a^{2} \frac{f\_{2}(\eta)}{f\_{3}(\eta)} Y\_{i,\eta\eta} \\ &+ (2\nu\_{2}\beta a \frac{f\_{1}(\eta)}{f\_{3}(\eta)} \frac{c\_{\bar{\eta}}}{a\_{\bar{\eta}}} + \frac{\nu\_{3}\beta}{f\_{3}(\eta)} \frac{d\_{\bar{\eta}}}{a\_{\bar{\eta}}} + 4\nu\_{4}\beta a^{2} \frac{f\_{2}(\eta)}{f\_{3}(\eta)} \frac{b\_{\bar{\eta}}}{a\_{\bar{\eta}}} + \frac{4\nu\_{4}\beta^{3}}{f\_{3}(\eta)} \frac{b\_{\bar{\eta}}}{a\_{\bar{\eta}}}) Y\_{i,\eta} \\ &+ (\frac{\nu\_{1}}{f\_{3}(\eta)} \frac{e\_{\bar{\eta}}}{a\_{\bar{\eta}}} - \lambda^{2}) Y\_{i} = 0 \end{split} \tag{17}$$

$$\text{Where } \lambda^2 = \frac{m\_o \omega^2 a^4}{D\_{22}}, f\_1(\eta) = \frac{1}{h^3(\eta)} \frac{\partial h^{\;3}(\eta)}{\partial \eta}, f\_2(\eta) = \frac{1}{h^3(\eta)} \frac{\partial^2 h^{\;3}(\eta)}{\partial \eta^2}, f\_3(\eta) = \frac{h\_o^{\;2}}{h^2(\eta)}.$$

$$\begin{aligned} a\_{ij} &= \bigcup\_{0} \mathbf{X}\_{j} \mathbf{X}\_{j} d\boldsymbol{\xi}\_{\prime} \\ b\_{ij} &= \bigcup\_{0} \mathbf{X}\_{j} \mathbf{X}\_{i,\zeta} d\boldsymbol{\xi}\_{\prime} \\ c\_{ij} &= \bigcup\_{0} \mathbf{X}\_{j} \mathbf{X}\_{i,\zeta\zeta} d\boldsymbol{\xi}\_{\prime} \\ d\_{ij} &= \bigcup\_{0} \mathbf{X}\_{j} \mathbf{X}\_{i,\zeta\zeta\zeta} d\boldsymbol{\xi} \\ \vdots \\ d\_{ij} &= \bigcap\_{0} \mathbf{X}\_{j} \mathbf{X}\_{i,\zeta\zeta\zeta\zeta} d\boldsymbol{\xi} \\ \text{and } c\_{ij} &= \bigcap\_{0} \mathbf{X}\_{j} \mathbf{X}\_{i,\zeta\zeta\zeta\zeta} d\boldsymbol{\xi} .\end{aligned}$$

From the orthogonality of the beam Eigen function, *aij* =*eij* =0 for *i* ≠ *j*, this is true for all boundary conditions except for plates having free edges in the ξ-direction.

The system of fourth order partial differential equations in equation (17) can be reduced to a system of first order homogeneous ordinary differential equations:

$$\frac{d}{d\eta} \{ Y\_k \}\_{ij} = \left[ A\_i \right]\_k \{ Y\_k \}\_{ij} \tag{18}$$

where the coefficients of the matrix *Ai <sup>k</sup>* in equation (18), in general, are functions of η and the

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

Solving the above system of first order ordinary differential equations using the transition

the method of system linear differential equations of the strip element (i) in equation (18) (the

Following the same procedure, the above boundary conditions (equations (12)) can be written.

is called the transition matrix of the strip element (i), which can be obtained using

matrix technique yields, at any strip element (i) with boundaries (i-1) and (i) to,

The simple boundary conditions at η=0 and η=1 as shown in Figure 3 are:

**Figure 3.** The two edges clamped variable thickness beam strip in η-direction

The boundary conditions at η=0 and η=1 can be expressed as:

{ } *Y BY i ii j j* = [ ]*<sup>j</sup>*

1 2 <sup>=</sup> é ù ë û *Y YY Y Y <sup>k</sup>* K K *i N* (20)

/ // /// <sup>=</sup> é ù ë û *<sup>i</sup> Y YY Y Y ii i i* (21)

{ } -<sup>1</sup> (22)

http://dx.doi.org/10.5772/57384

591

Eigen value parameter λ. The vector *Yk* is given by:

Where:

Where *Bi <sup>j</sup>*

exact solution of (ODE)).

And after some manipulation, the governing differential equation (17) will become:

$$\sum\_{i=0}^{N} \sum\_{j=0}^{M} E\_{\bar{y}} Y\_i^{\prime \prime \prime \prime} + \frac{\left(O\_1\right)\_{\bar{y}}}{\left(O\_0\right)\_{\bar{y}}} Y\_i^{\prime \prime \prime} + \frac{\left(O\_2\right)\_{\bar{y}}}{\left(O\_0\right)\_{\bar{y}}} Y\_i^{\prime \prime \prime} + \frac{\left(O\_3\right)\_{\bar{y}}}{\left(O\_0\right)\_{\bar{y}}} Y\_i^{\prime} + \frac{\left(O\_4\right)\_{\bar{y}} - \lambda^2}{\left(O\_0\right)\_{\bar{y}}} Y\_i = 0 \tag{19}$$

Where the frame denotes differentiation with respect to η.

$$\begin{split} \text{Where:} & \{\mathcal{O}\_{0}\}\_{ij} = \beta^{4} t\_{1} \langle \eta \rangle E\_{ij\prime} \{ \mathcal{O}\_{1\prime j} \} = 2\beta^{3} a t\_{2} \langle \eta \rangle E\_{ij\prime} \{ \mathcal{O}\_{2} \} = (2\psi\_{2}\beta^{2} t\_{1} \langle \eta \rangle \frac{c\_{ij}}{a\_{ij}} + 8\psi\_{4}\beta^{2} a t\_{2} \langle \eta \rangle \frac{b\_{ij}}{a\_{ij}} + \beta^{2} a^{2} t\_{3} \langle \eta \rangle \frac{c\_{ij}}{a\_{ij}} \\ \{\mathcal{O}\_{3}\}\_{ij} = (2\psi\_{2}\beta a t\_{2} \langle \eta \rangle \frac{c\_{ij}}{a\_{ij}} + \psi\_{3}\beta^{4} t\_{1} \langle \eta \rangle \frac{d\_{ij}}{a\_{ij}} + 4\psi\_{4}\beta a^{2} t\_{3} \langle \eta \rangle \frac{b\_{ij}}{a\_{ij}} + 4\psi\_{4}\beta^{3} t\_{1} \langle \eta \rangle \frac{b\_{ij}}{a\_{ij}}), \{\mathcal{O}\_{4}\}\_{ij} = \psi\_{1} t\_{1} \langle \eta \rangle \frac{c\_{ij}}{a\_{ij}}, \\ \{\mathcal{E}\_{ij}\} = i \times j \text{ Unit matrix}, \end{split}$$

i= 0, 1, 2, 3, ……….,N, j= 0, 1, 2, 3, ……….,M

where the coefficients of the matrix *Ai <sup>k</sup>* in equation (18), in general, are functions of η and the Eigen value parameter λ. The vector *Yk* is given by:

$$Y\_k = \begin{bmatrix} \overline{Y}\_1 & \overline{Y}\_2 & \mathbf{K} & \overline{Y}\_i & \mathbf{K} & \overline{Y}\_N \end{bmatrix} \tag{20}$$

Where:

*aij* =*∫* 0

*bij* =*∫* 0

*cij* =*∫* 0

*dij* =*∫* 0

1

1

1

1

and *eij* =*∫*

0

1

*XiXj dξ*,

*XjXi*,*ξdξ*,

590 MATLAB Applications for the Practical Engineer

*XjXi*,*ξξdξ*,

*XjXi*,*ξξξdξ*

*XjXi*,*ξξξξdξ*.

= =

Where: (*O*0)*ij* <sup>=</sup>*<sup>β</sup>* <sup>4</sup>

(*O*3)*ij* =(2*ψ*2*βat*2(*η*)

*Eij* =*i* × *j* Unit matrix,

*N M*

From the orthogonality of the beam Eigen function, *aij* =*eij* =0 for *i* ≠ *j*, this is true for all

The system of fourth order partial differential equations in equation (17) can be reduced to a

{ } [ ] { }

And after some manipulation, the governing differential equation (17) will become:

( ) ( )

*i j ij ij ij ij*

+ 4*ψ*4*β<sup>a</sup>* <sup>2</sup>

1 2 3 4 //// /// // / 0 0 0 00 0

åå + + ++ =

*OO O O EY Y Y Y Y*

*k ik* = *ij <sup>k</sup> ij <sup>d</sup> Y AY*

> ( ) ( )

*ij ij ij ij ij i i i i i*

*at*2(*η*)*Eij*, (*O*2)*ij* =(2*ψ*2*<sup>β</sup>* <sup>2</sup>

*t*3(*η*) *bij aij*

*<sup>d</sup>* (18)

( ) ( )

*O OO O* (19)

*t*1(*η*) *cij aij*

*t*1(*η*) *bij aij*

+ 4*ψ*4*<sup>β</sup>* <sup>3</sup>

2

+ 8*ψ*4*<sup>β</sup>* <sup>2</sup>

l


0

*at*2(*η*) *bij aij* + *β* <sup>2</sup> *a* 2 *t*3(*η*))

> *eij aij* ,

), (*O*4)*ij* =*ψ*1*t*1(*η*)

boundary conditions except for plates having free edges in the ξ-direction.

system of first order homogeneous ordinary differential equations:

h

( ) ( )

Where the frame denotes differentiation with respect to η.

*dij aij*

*<sup>t</sup>*1(*η*)*Eij*, (*O*1)*ij* =2*<sup>β</sup>* <sup>3</sup>

+ *ψ*3*β t*1(*η*)

*cij aij*

i= 0, 1, 2, 3, ……….,N, j= 0, 1, 2, 3, ……….,M

$$
\overline{Y}\_i = \begin{bmatrix} Y\_i & Y\_i^{\prime} & Y\_i^{\prime\prime\prime} & Y\_i^{\prime\prime\prime\prime} \end{bmatrix} \tag{21}
$$

Solving the above system of first order ordinary differential equations using the transition matrix technique yields, at any strip element (i) with boundaries (i-1) and (i) to,

$$\left\{ \left. Y\_i \right\rangle\_j = \left[ B\_i \right]\_j \left\{ Y\_{i-1} \right\}\_j \tag{22}$$

Where *Bi <sup>j</sup>* is called the transition matrix of the strip element (i), which can be obtained using the method of system linear differential equations of the strip element (i) in equation (18) (the exact solution of (ODE)).

Following the same procedure, the above boundary conditions (equations (12)) can be written. The simple boundary conditions at η=0 and η=1 as shown in Figure 3 are:

**Figure 3.** The two edges clamped variable thickness beam strip in η-direction

The boundary conditions at η=0 and η=1 can be expressed as:

$$\mathbf{w}\_o = \mathbf{0}$$

$$-\frac{D\_{12}}{a^2} \frac{\partial^2 w\_o}{\partial \xi^2} - \frac{D\_{22}}{b^2} \frac{\partial^2 w\_o}{\partial \eta^2} - \frac{2D\_{26}}{ab} \frac{\partial^2 w\_o}{\partial \xi \partial \eta} = \mathbf{0} \tag{23}$$

and free in the other direction. The designation (S-S-F-F) means that the edges x=0, x=a, y=0, y=b are free, free, simple supported and simple supported respectively. The plates are made up of five laminates with the fiber orientations [θ, - θ, θ, - θ, θ] and the composite material is Graphite/Epoxy, of which mechanical properties are given in Table 1. The Eigen frequencies obtained are expressed in terms of non-dimensional frequency parameter

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

 h

Where *Δ* is the tapered ratio of plate given by *Δ* =(*hb* −*h*0) / *h*0, (*h*0) is the thickness of the plate at η=0 and (*hb*) is the thickness of the plate at η=1. A convergence investigation is carried out for a uniform plate and for plate of variable thickness (*Δ* =0.5) with aspect ratio *β* =(0.5, 1.0). By varying the harmonic numbers of the series solution in equation (10). The results are shown in Table 2. It is found that excellent agreement and stable and fast convergence can be achieved

**Material E1, (GPa) E2, (GPa) G12, (GPa) υ12 E2/ E1 G12/ E1** Graphite/Epoxy 138 8.96 7.1 0.3 25 0.8

In order to validate the proposed technique, a comparison of the results with some results available for other numerical methods [15] for uniform laminated plates with simple support in the y-direction and free in the other direction. The first six natural frequencies of such

**Δ = 0.0 N λ1 λ2 λ3 λ4 λ5 λ6** 70.4212 70.7012 140.4421 173.5211 180.6231 235.6753 70.4212 70.7012 140.4421 173.5211 180.6231 235.6753 70.2882 70.5827 140.2496 173.2098 180.2833 235.3197 70.2882 70.5827 140.2496 173.2098 180.2833 235.3197 70.2882 70.5827 140.2496 173.2098 180.2833 235.3197 Ref\* 70.302 70.604 140.255 173.218 180.287 235.322

**Table 2.** Comparison of the first six natural frequencies of symmetric angle-ply uniform laminated square plates

*h*() 1 h

. To illustrate the solution, a plate with linear variable thickness, *h* (*y*)is

= +D (27)

http://dx.doi.org/10.5772/57384

593

*<sup>λ</sup>* =(*<sup>ρ</sup> ho<sup>ω</sup>* <sup>2</sup>

*<sup>a</sup>* <sup>4</sup> / *<sup>D</sup>*22)

used (see Appendix A).

1/2

with only a few terms of series solution (N= 3 to 5).

**Table 1.** Material properties of unidirectional composite

uniform laminated plates are depicted in Table 2.

\*Y.K. Cheung and D. Zhou [15].

), (β =1.0)

(θ =45<sup>∘</sup>

Using the assumed solution, equation (10) the boundary conditions can be given by the following equations:

At η=0 and η=1

$$\frac{Y\_i = 0}{\left| \frac{\partial^2 Y\_i}{\partial \eta^2} \right|^2} = \sum\_{i=0}^N -\frac{2\nu\_4 b\_{ij}}{\beta a\_{ji}} \frac{\partial Y\_i}{\partial \eta} - \frac{\nu\_5 c\_{ij}}{\beta^2 a\_{ij}} Y\_i \Bigg| \tag{24}$$

Or in contraction form:

$$Y\_i = 0$$

$$Y\_i^{\prime\prime} = \sum\_{i=0}^{N} -CF\_1 \frac{b\_{\bar{y}}}{a\_{\bar{y}}} Y\_i^{\prime} - CF\_2 \frac{c\_{\bar{y}}}{a\_{\bar{y}}} Y\_i$$

Where *CF*<sup>1</sup> = 2*ψ*<sup>4</sup> *<sup>β</sup>* , *CF*<sup>2</sup> <sup>=</sup> *ψ*5 *<sup>β</sup>* <sup>2</sup> , *ψ*<sup>5</sup> <sup>=</sup> *D*<sup>12</sup> *D*<sup>22</sup>

The solution is found using 2N initial vectors *Y*0 at η=0. Equation (22) is applied across the stripped plate until the final end at η=1 is reached. Thus, 2N solutions *Si* , where i= 0, 1, 2, 3, ……….,N, can be obtained. The true solutions *S* can be written as a linear combination of these solutions [7]:

$$\begin{bmatrix} S \end{bmatrix} = \sum\_{i=1}^{2N} C\_i S\_i \tag{26}$$

Where *Ci* are arbitrary constants. These constants can be determined by satisfying 2N boun‐ dary conditions at η=1 [7]. The matrix *S* forms a standard Eigen value problem.

#### **4. Numerical results and discussion**

In this section, some numerical results are presented for symmetrically laminated, angle-ply variable thickness rectangular plate with simple support in the variable thickness direction and free in the other direction. The designation (S-S-F-F) means that the edges x=0, x=a, y=0, y=b are free, free, simple supported and simple supported respectively. The plates are made up of five laminates with the fiber orientations [θ, - θ, θ, - θ, θ] and the composite material is Graphite/Epoxy, of which mechanical properties are given in Table 1. The Eigen frequencies obtained are expressed in terms of non-dimensional frequency parameter *<sup>λ</sup>* =(*<sup>ρ</sup> ho<sup>ω</sup>* <sup>2</sup> *<sup>a</sup>* <sup>4</sup> / *<sup>D</sup>*22) 1/2 . To illustrate the solution, a plate with linear variable thickness, *h* (*y*)is used (see Appendix A).

$$h(\eta) = \mathbb{I} + \Delta\eta \tag{27}$$

Where *Δ* is the tapered ratio of plate given by *Δ* =(*hb* −*h*0) / *h*0, (*h*0) is the thickness of the plate at η=0 and (*hb*) is the thickness of the plate at η=1. A convergence investigation is carried out for a uniform plate and for plate of variable thickness (*Δ* =0.5) with aspect ratio *β* =(0.5, 1.0). By varying the harmonic numbers of the series solution in equation (10). The results are shown in Table 2. It is found that excellent agreement and stable and fast convergence can be achieved with only a few terms of series solution (N= 3 to 5).


**Table 1.** Material properties of unidirectional composite

22 2 12 22 26 22 2 2

*w D D w wDw a b ab*

<sup>ï</sup> ¶¶ ¶ <sup>ý</sup> -- - = <sup>ï</sup> ¶ ¶ ¶ ¶ <sup>þ</sup> *o*

 h

x

2

h

*ψ*5 *<sup>β</sup>* <sup>2</sup> , *ψ*<sup>5</sup> <sup>=</sup>

**4. Numerical results and discussion**

following equations:

592 MATLAB Applications for the Practical Engineer

Or in contraction form:

2*ψ*<sup>4</sup> *<sup>β</sup>* , *CF*<sup>2</sup> <sup>=</sup>

Where *CF*<sup>1</sup> =

these solutions [7]:

At η=0 and η=1

0

*oo o*

Using the assumed solution, equation (10) the boundary conditions can be given by the

4 5

 h

1 2

The solution is found using 2N initial vectors *Y*0 at η=0. Equation (22) is applied across the

……….,N, can be obtained. The true solutions *S* can be written as a linear combination of

<sup>ý</sup> =- - <sup>ï</sup> <sup>þ</sup> <sup>å</sup>

*i ii i ij ij*

*Y CF Y CF Y a a*

0

*i*

*Y*

 y

*<sup>Y</sup> <sup>a</sup> <sup>a</sup>*

= ü

 b

= ü

*ij ij*

*b c*

ï

*S CS* (26)

0

2 2

*i*

*Y Y Y b c*

> = b

<sup>ï</sup> ¶ ¶ <sup>ý</sup> =- - <sup>ï</sup> ¶ ¶ <sup>þ</sup>

*ij ij i i <sup>i</sup> i ij ij*

2y

0

// /

0

=

*D*<sup>12</sup> *D*<sup>22</sup>

stripped plate until the final end at η=1 is reached. Thus, 2N solutions *Si*

[ ]

2

*i*

dary conditions at η=1 [7]. The matrix *S* forms a standard Eigen value problem.

=1 <sup>=</sup> å *N i i*

Where *Ci* are arbitrary constants. These constants can be determined by satisfying 2N boun‐

In this section, some numerical results are presented for symmetrically laminated, angle-ply variable thickness rectangular plate with simple support in the variable thickness direction

*N*

å

*N*

<sup>2</sup> <sup>0</sup>

(23)

(24)

(25)

, where i= 0, 1, 2, 3,

= ü

x h

> In order to validate the proposed technique, a comparison of the results with some results available for other numerical methods [15] for uniform laminated plates with simple support in the y-direction and free in the other direction. The first six natural frequencies of such uniform laminated plates are depicted in Table 2.


\*Y.K. Cheung and D. Zhou [15].

**Table 2.** Comparison of the first six natural frequencies of symmetric angle-ply uniform laminated square plates (θ =45<sup>∘</sup> ), (β =1.0)

Table 3 and Table 4 shows a convergence analysis of the first six frequencies parameters of symmetrically angle-ply five laminates [45/-45/45/-45/45] variable thickness plate with tapered ratio (*Δ* =0.5) and with aspect ratio *β* =(0.5, 1.0) with simple support in the y-direction and free in the other direction (S-S-F-F).

**λi** 

**λ1**

**Z**

**0 0.25 0.5 0.75 <sup>1</sup>**

**0 0.25 0.5 0.75 <sup>1</sup>**

**0 0.25 0.5 0.75 <sup>1</sup>**

**0 0.25 0.5 0.75 <sup>1</sup>**

**0 0.25 0.5 0.75 <sup>1</sup>**

**0 0.25 0.5 0.75 <sup>1</sup>**

**λ2**

**Z**

**λ3**

**Z**

**λ4**

**Z**

**λ5**

**Z**

**λ6**

**Z**

ratio 

*a b* 0.5 , tapered ratio 0.5

**ASPECT RATIO**  *a b* 0.5 , **TAPERED RATIO** 0.5 **THE SURFACE OF MODE SHAPE THE CONTOUR OF MODE SHAPE** 

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

**-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5**

**-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5**

**-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5**

**-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5**

**-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5**

**0.25 0.5 0.75 1 1.25 1.5 1.75 2**

**0.25 0.5 0.75 1 1.25 1.5 1.75 2**

> **0.25 0.5 0.75 1 1.25 1.5 1.75 2**

> > **0.25 0.5 0.75 1 1.25 1.5 1.75 2**

> > **0.25 0.5 0.75 1 1.25 1.5 1.75 2**

> > > **0.25 0.5 0.75 1 1.25 1.5 1.75 2**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

**-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5**

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595

**-2 -1 0 1 2**

> **-2 -1 0 1 2**

> > **-2 -1 0 1 2**

> > **-2 -1 0 1 2**

**-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

Fig. 4: The mode shapes of the first six fundamental frequencies of the angle-ply symmetrically [45/-45/45/-45/45] laminated variable thickness rectangular plate with S-S-F-F edges, aspect

**Figure 4.** The mode shapes of the first six fundamental frequencies of the angle-ply symmetrically [45/-45/45/-45/45] laminated variable thickness rectangular plate with S-S-F-F edges, aspect ratioβ =*a* / *b*=0.5, tapered ratio Δ =0.5

**-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5**

Figure 4 and Figure 5 show the mode shapes of the first six fundamental frequencies of the above plate. Figure 4 and Figure 5 both are divided into two graphics. The first one shows the mode shapes of the plate in surface form and the other shows the mode shapes of the plate in surface contour form. All simulation results and graphics were obtained using MATLAB software.


**Table 3.** The first six frequencies parameter of S-S-F-F symmetrically angle-ply laminated [45/-45/45/-45/45] variable thickness plate (Δ =0.5), (β =0.5).


**Table 4.** The first six frequencies parameter of S-S-F-F symmetrically angle-ply laminated [45/-45/45/-45/45] variable thickness plate (Δ =0.5), (β =1.0)

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique… http://dx.doi.org/10.5772/57384 595

Table 3 and Table 4 shows a convergence analysis of the first six frequencies parameters of symmetrically angle-ply five laminates [45/-45/45/-45/45] variable thickness plate with tapered ratio (*Δ* =0.5) and with aspect ratio *β* =(0.5, 1.0) with simple support in the y-direction and free

Figure 4 and Figure 5 show the mode shapes of the first six fundamental frequencies of the above plate. Figure 4 and Figure 5 both are divided into two graphics. The first one shows the mode shapes of the plate in surface form and the other shows the mode shapes of the plate in surface contour form. All simulation results and graphics were obtained using MATLAB

**Δ = 0.5 β = 0.5 N λ1 λ2 λ3 λ4 λ5 λ6** 80.2177 82.5621 155.9665 188.6633 194.6253 251.7333 80.2177 82.5621 155.9665 188.6633 194.6253 251.7333 79.8625 82.0025 155.3232 188.1111 194.1002 251.2035 79.8625 82.0025 155.3232 188.1111 194.1002 251.2035 79.8625 82.0025 155.3232 188.1111 194.1002 251.2035

**Table 3.** The first six frequencies parameter of S-S-F-F symmetrically angle-ply laminated [45/-45/45/-45/45] variable

**Δ = 0.5 β = 1.0**

**N λ1 λ2 λ3 λ4 λ5 λ6** 72.7575 73.8666 143.3334 175.4963 183.7825 240.7621 72.7575 73.8666 143.3334 175.4963 183.7825 240.7621 72.1199 73.4444 142.9019 175.0024 183.1121 240.0159 72.1199 73.4444 142.9019 175.0024 183.1121 240.0159 72.1199 73.4444 142.9019 175.0024 183.1121 240.0159

**Table 4.** The first six frequencies parameter of S-S-F-F symmetrically angle-ply laminated [45/-45/45/-45/45] variable

in the other direction (S-S-F-F).

594 MATLAB Applications for the Practical Engineer

thickness plate (Δ =0.5), (β =0.5).

thickness plate (Δ =0.5), (β =1.0)

software.

 **Figure 4.** The mode shapes of the first six fundamental frequencies of the angle-ply symmetrically [45/-45/45/-45/45] laminated variable thickness rectangular plate with S-S-F-F edges, aspect ratioβ =*a* / *b*=0.5, tapered ratio Δ =0.5

Fig. 4: The mode shapes of the first six fundamental frequencies of the angle-ply symmetrically [45/-45/45/-45/45] laminated variable thickness rectangular plate with S-S-F-F edges, aspect

ratio 

*a b* 0.5 , tapered ratio 0.5

**5. Concluding remarks**

for two different aspect ratios.

**Appendix (A)**

**Plate thickness function**

in the Figure 6 is given.

**Figure 6.** The relation of the plate thickness h(y) in y-direction

A semi-analytical solution of the free vibration of angle-ply symmetrically laminated variable thickness rectangular plate with classical boundary condition (S-S-F-F) is investigated using the finite strip transition matrix technique (FSTM). The numerical results for uniform angleply symmetrically square plate with classical boundary condition (S-S-F-F) is presented and compared with some available results. The results agree very closely with other results available in the literature. It can be observed from Tables 2 and 3 that rapid convergence is achieved with small numbers of N in the series solution. Comparing to other techniques, the finite strip transition matrix (FSTM) proves to be valid enough in this kind of application. In all cases the FSTM method is easily implemented in a computer program a yields a fast convergence and reliable results. Also, the effect of the tapered ratio (*Δ*) and aspect ratio (*β*) on the fundamental natural frequencies and the mode shapes for five layers angle-ply symmetrically laminated variable thickness plates has been investigated for two cases of tapered ratio (uniform and variable thickness) and two cases of aspect ratio (square and rectangular). In fact the varying of the thickness and the increase the length (b) about a length (a) tend to increase the natural frequencies and the mode shapes of the laminated plate. The results from this investigation have been illustrated in the three dimensional surface contours

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

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597

In this appendix the derivation of the relation of the plate thickness h(y) in y-direction as shown

Fig. 5: The mode shapes of the first six fundamental frequencies of the angle-ply symmetrically [45/-45/45/-45/45] laminated variable thickness rectangular plate with S-S-F-F edges, aspect ratio *a b* 1.0 , tapered ratio 0.5 . **5. Concluding Remarks Figure 5.** The mode shapes of the first six fundamental frequencies of the angle-ply symmetrically [45/-45/45/-45/45] laminated variable thickness rectangular plate with S-S-F-F edges, aspect ratio β =*a* / *b*=1.0, tapered ratio Δ =0.5.

### **5. Concluding remarks**

**λi** 

596 MATLAB Applications for the Practical Engineer

λ1

**Z**

**0 0.25 0.5**

**0 0.25 0.5**

**0 0.25 0.5**

**0 0.25 0.5**

**0 0.25 0.5**

**0 0.25 0.5**

**0 0.5 1 -1 -0.5 0 0.5 1**

**0 0.5 1 -1 -0.5 0 0.5 1**

**0 0.5 1 -1 -0.5 0 0.5 1**

**0 0.5 1 -1 -0.5 0 0.5 1**

**0 0.5 1 -1 -0.5 0 0.5 1**

**0 0.5 1 -1 -0.5 0 0.5 1**

λ2

**Z**

λ3

**Z**

λ4

**Z**

λ5

**Z**

λ6

**Z**

ratio 

*a b* 1.0 , tapered ratio 0.5 .

**5. Concluding Remarks** 

**ASPECT RATIO**  *a b* 1.0 **TAPERED RATIO**  0.5 **. THE SURFACE OF MODE SHAPE THE CONTOUR OF MODE SHAPE** 

**0.75 <sup>1</sup>**

**0.75 <sup>1</sup>**

**0.75 <sup>1</sup>**

**0.75 <sup>1</sup>**

**0.75 <sup>1</sup>**

**0.75 <sup>1</sup>**

**-1 -0.5 0 0.5 1**

**-1 -0.5 0 0.5 1**

**-1 -0.5 0 0.5 1**

**-1 -0.5 0 0.5 1**

**-1 -0.5 0 0.5 1**

**0.25 0.5 0.75 1**

**0.25 0.5 0.75 1**

> **0.25 0.5 0.75 1**

**0.25 0.5 0.75 1**

> **0.25 0.5 0.75 1**

> > **0.25 0.5 0.75 1**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

**-1 -0.5 0 0.5 1**

**-1 -0.5 0 0.5 1**

> **-1 -0.5 0 0.5 1**

**-1 -0.5 0 0.5 1**

> **-1 -0.5 0 0.5 1**

**-1 -0.5 0 0.5 1**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

**<sup>0</sup> 0.25 0.5 0.75 <sup>1</sup> <sup>0</sup>**

Fig. 5: The mode shapes of the first six fundamental frequencies of the angle-ply symmetrically [45/-45/45/-45/45] laminated variable thickness rectangular plate with S-S-F-F edges, aspect

**Figure 5.** The mode shapes of the first six fundamental frequencies of the angle-ply symmetrically [45/-45/45/-45/45] laminated variable thickness rectangular plate with S-S-F-F edges, aspect ratio β =*a* / *b*=1.0, tapered ratio Δ =0.5.

**-1 -0.5 0 0.5 1**

A semi-analytical solution of the free vibration of angle-ply symmetrically laminated variable thickness rectangular plate with classical boundary condition (S-S-F-F) is investigated using the finite strip transition matrix technique (FSTM). The numerical results for uniform angleply symmetrically square plate with classical boundary condition (S-S-F-F) is presented and compared with some available results. The results agree very closely with other results available in the literature. It can be observed from Tables 2 and 3 that rapid convergence is achieved with small numbers of N in the series solution. Comparing to other techniques, the finite strip transition matrix (FSTM) proves to be valid enough in this kind of application. In all cases the FSTM method is easily implemented in a computer program a yields a fast convergence and reliable results. Also, the effect of the tapered ratio (*Δ*) and aspect ratio (*β*) on the fundamental natural frequencies and the mode shapes for five layers angle-ply symmetrically laminated variable thickness plates has been investigated for two cases of tapered ratio (uniform and variable thickness) and two cases of aspect ratio (square and rectangular). In fact the varying of the thickness and the increase the length (b) about a length (a) tend to increase the natural frequencies and the mode shapes of the laminated plate. The results from this investigation have been illustrated in the three dimensional surface contours for two different aspect ratios.

### **Appendix (A)**

#### **Plate thickness function**

In this appendix the derivation of the relation of the plate thickness h(y) in y-direction as shown in the Figure 6 is given.

**Figure 6.** The relation of the plate thickness h(y) in y-direction

By similarity between the triangles (ABG) and (ACF):

$$h(\underline{\mathbf{y}}) = h\_o(\mathbf{l} + \frac{\underline{\mathbf{y}}}{c}) \tag{28}$$

**Appendix (B) MATLAB Code** 

**Composite Coefficients (function programs)** 

function yEx = Ex(E1,E2,NU12,G12,theta) %Ex This function returns the elastic modulus % along the x-direction in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU12 - Poisson's ratio % G12 - shear modulus % theta - fiber orientation angle

% The angle "theta" must be given in degrees.

function yEy = Ey(E1,E2,NU21,G12,theta) %Ey This function returns the elastic modulus % along the y-direction in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU21 - Poisson's ratio % G12 - shear modulus % theta - fiber orientation angle

% The angle "theta" must be given in degrees.

function yGxy = Gxy(E1,E2,NU12,G12,theta) %Gxy This function returns the shear modulus

% coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU12 - Poisson's ratio % G12 - shear modulus % theta - fiber orientation angle

% The angle "theta" must be given in degrees.

function yNUxy = NUxy(E1,E2,NU12,G12,theta)

% Gxy is returned as a scalar m = cos(theta\*pi/180); n = sin(theta\*pi/180);

yGxy = G12/denom;

% Ey is returned as a scalar m = cos(theta\*pi/180); n = sin(theta\*pi/180);

yEy = E2/denom;

% Gxy in the global

denom = m^4 + (E1/G12 - 2\*NU12)\*n\*n\*m\*m + (E1/E2)\*n^4;

denom = m^4 + (E2/G12 - 2\*NU21)\*n\*n\*m\*m + (E2/E1)\*n^4;

denom = n^4 + m^4 + 2\*(2\*G12\*(1 + 2\*NU12)/E1 + 2\*G12/E2 - 1)\*n\*n\*m\*m;

% Ex is returned as a scalar m = cos(theta\*pi/180); n = sin(theta\*pi/180);

yEx = E1/denom;

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

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599

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

By similarity between the triangles (ABG) and (ADE):

$$\frac{h\_o}{c} = \frac{h\_b}{c+b} \tag{29}$$

From equations (28) and (29) the plate thickness relation is:

$$h(\mathbf{y}) = h\_o + \frac{(h\_b - h\_o)}{b}\mathbf{y} \tag{30}$$

Where *h* (*y*)=*ho* at *y* =0, *h* (*y*)=*hb* at *y* =*b*, *h* (*y*)=*ho* + (*hb* −*ho*) *<sup>b</sup> <sup>y</sup>* at *<sup>y</sup>* <sup>=</sup> *<sup>y</sup>*,

and *h* (*y*)=*h* at *ho* =*hb*

Using the assumed solution, equation (10) The relation between the thickness of the plate h(y) can be given by the following equation:

$$h(\eta) = h\_o + (h\_b - h\_o)\eta \tag{31}$$

#### **Appendix (B)**

#### **MATLAB code**

*Composite coefficients (function programs)*

#### **Appendix (B)**

By similarity between the triangles (ABG) and (ACF):

598 MATLAB Applications for the Practical Engineer

By similarity between the triangles (ABG) and (ADE):

From equations (28) and (29) the plate thickness relation is:

Where *h* (*y*)=*ho* at *y* =0,

and *h* (*y*)=*h* at *ho* =*hb*

**Appendix (B)**

**MATLAB code**

(*hb* −*ho*)

*<sup>b</sup> <sup>y</sup>* at *<sup>y</sup>* <sup>=</sup> *<sup>y</sup>*,

can be given by the following equation:

*Composite coefficients (function programs)*

*h* (*y*)=*hb* at *y* =*b*,

*h* (*y*)=*ho* +

( ) (1 ) = + *<sup>o</sup>*

<sup>=</sup> <sup>+</sup> *o b h h*

( ) ( ) - = + *b o <sup>o</sup>*

Using the assumed solution, equation (10) The relation between the thickness of the plate h(y)

 h

() ( )

h

*<sup>y</sup> hy h <sup>c</sup>* (28)

*c cb* (29)

*h h hy h y <sup>b</sup>* (30)

=+ - *o bo h h hh* (31)

#### **MATLAB Code**

#### **Composite Coefficients (function programs)**

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function yEx = Ex(E1,E2,NU12,G12,theta) %Ex This function returns the elastic modulus % along the x-direction in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU12 - Poisson's ratio % G12 - shear modulus % theta - fiber orientation angle % The angle "theta" must be given in degrees. % Ex is returned as a scalar m = cos(theta\*pi/180); n = sin(theta\*pi/180); denom = m^4 + (E1/G12 - 2\*NU12)\*n\*n\*m\*m + (E1/E2)\*n^4; yEx = E1/denom; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function yEy = Ey(E1,E2,NU21,G12,theta) %Ey This function returns the elastic modulus % along the y-direction in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU21 - Poisson's ratio % G12 - shear modulus % theta - fiber orientation angle % The angle "theta" must be given in degrees. % Ey is returned as a scalar m = cos(theta\*pi/180); n = sin(theta\*pi/180); denom = m^4 + (E2/G12 - 2\*NU21)\*n\*n\*m\*m + (E2/E1)\*n^4; yEy = E2/denom; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function yGxy = Gxy(E1,E2,NU12,G12,theta) %Gxy This function returns the shear modulus % Gxy in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU12 - Poisson's ratio % G12 - shear modulus % theta - fiber orientation angle % The angle "theta" must be given in degrees. % Gxy is returned as a scalar m = cos(theta\*pi/180); n = sin(theta\*pi/180); denom = n^4 + m^4 + 2\*(2\*G12\*(1 + 2\*NU12)/E1 + 2\*G12/E2 - 1)\*n\*n\*m\*m; yGxy = G12/denom; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

function yNUxy = NUxy(E1,E2,NU12,G12,theta)

%NUxy This function returns Poisson's ratio % NUxy in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU12 - Poisson's ratio % G12 - shear modulus % theta - fiber orientation angle % The angle "theta" must be given in degrees. % NUxy is returned as a scalar m = cos(theta\*pi/180); n = sin(theta\*pi/180); denom = m^4 + (E1/G12 - 2\*NU12)\*n\*n\*m\*m + (E1/E2)\*n\*n; numer = NU12\*(n^4 + m^4) - (1 + E1/E2 - E1/G12)\*n\*n\*m\*m; yNUxy = numer/denom; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function yNUyx = NUyx(E1,E2,NU21,G12,theta) %NUyx This function returns Poisson's ratio % NUyx in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU21 - Poisson's ratio % G12 - shear modulus % theta - fiber orientation angle % The angle "theta" must be given in degrees. % NUyx is returned as a scalar m = cos(theta\*pi/180); n = sin(theta\*pi/180); denom = m^4 + (E2/G12 - 2\*NU21)\*n\*n\*m\*m + (E2/E1)\*n\*n; numer = NU21\*(n^4 + m^4) - (1 + E2/E1 - E2/G12)\*n\*n\*m\*m; yNUyx = numer/denom; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function y = OrthotropicCompliance(E1,E2,E3,NU12,NU23,NU13,G12,G23,G13) %OrthotropicCompliance This function returns the compliance matrix % for orthotropic materials. There are nine % arguments representing the nine independent % material constants. The size of the compliance % matrix is 6 x 6. y = [1/E1 -NU12/E1 -NU13/E1 0 0 0 ; -NU12/E1 1/E2 -NU23/E2 0 0 0 ;-NU13/E1 -… NU23/E2 1/E3 0 0 0 ; 0 0 0 1/G23 0 0 ; 0 0 0 0 1/G13 0 ;0 0 0 0 0 1/G12]; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function C = OrthotropicStiffness(E1,E2,E3,NU12,NU23,NU13,G12,G23,G13) %OrthotropicStiffness This function returns the stiffness matrix % for orthotropic materials. There are nine % arguments representing the nine independent % material constants. The size of the stiffness % matrix is 6 x 6. x = [1/E1 -NU12/E1 -NU13/E1 0 0 0 ; -NU12/E1 1/E2 -NU23/E2 0 0 0 ;-NU13/E1 -… NU23/E2 1/E3 0 0 0 ; 0 0 0 1/G23 0 0 ; 0 0 0 0 1/G13 0 ;0 0 0 0 0 1/G12]; C = inv(x); %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function S = ReducedCompliance(E1,E2,NU12,G12)

%ReducedCompliance This function returns the reduced compliance

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

http://dx.doi.org/10.5772/57384

601

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

S = [1/E1 -NU12/E1 0 ; -NU12/E1 1/E2 0 ; 0 0 1/G12];

%ReducedStiffness This function returns the reduced stiffness

Q = [E1/(1-NU12\*NU21) NU12\*E2/(1-NU12\*NU21) 0 ;NU12\*E2/(1-NU12\*NU21) E2/(1-…

function Q = ReducedStiffness(E1,E2,NU12,G12)

%Sbar This function returns the transformed reduced % compliance matrix "Sbar" given the reduced % compliance matrix S and the orientation

% There are two arguments representing S and "theta"

T = [m\*m n\*n 2\*m\*n ; n\*n m\*m -2\*m\*n ; -m\*n m\*n m\*m-n\*n]; Tinv = [m\*m n\*n -2\*m\*n ; n\*n m\*m 2\*m\*n ; m\*n -m\*n m\*m-n\*n];

%Qbar This function returns the transformed reduced % stiffness matrix "Qbar" given the reduced % stiffness matrix Q and the orientation

% There are two arguments representing Q and "theta"

T = [m\*m n\*n 2\*m\*n ; n\*n m\*m -2\*m\*n ; -m\*n m\*n m\*m-n\*n]; Tinv = [m\*m n\*n -2\*m\*n ; n\*n m\*m 2\*m\*n ; m\*n -m\*n m\*m-n\*n];

% The angle "theta" must be given in degrees.

% The angle "theta" must be given in degrees.

% matrix for fiber-reinforced materials. % There are four arguments representing four % material constants. The size of the reduced

% matrix for fiber-reinforced materials. % There are four arguments representing four % material constants. The size of the reduced

% compliance matrix is 3 x 3.

% stiffness matrix is 3 x 3. NU21 = NU12\*E2/E1;

NU12\*NU21) 0 ; 0 0 G12];

function SBar = Sbar(S,theta)

% The size of the matrix is 3 x 3.

% angle "theta".

R=[1 0 0;0 1 0;0 0 2]; Rinv=inv(R); m = cos(theta\*pi/180); n = sin(theta\*pi/180);

SBar = Tinv\*S\*R\*T\*Rinv;

% angle "theta".

R=[1 0 0;0 1 0;0 0 2]; Rinv=inv(R); m = cos(theta\*pi/180); n = sin(theta\*pi/180);

QBar = Tinv\*Q\*R\*T\*Rinv;

% is assembled.

function y = Amatrix(A,QBar,z1,z2) %Amatrix This function returns the [A] matrix % after the layer k with stiffness [Qbar]

function QBar = Qbar(Q,theta)

% The size of the matrix is 3 x 3.

%ReducedCompliance This function returns the reduced compliance % matrix for fiber-reinforced materials. % There are four arguments representing four % material constants. The size of the reduced % compliance matrix is 3 x 3. S = [1/E1 -NU12/E1 0 ; -NU12/E1 1/E2 0 ; 0 0 1/G12]; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function Q = ReducedStiffness(E1,E2,NU12,G12) %ReducedStiffness This function returns the reduced stiffness % matrix for fiber-reinforced materials. % There are four arguments representing four % material constants. The size of the reduced % stiffness matrix is 3 x 3. NU21 = NU12\*E2/E1; Q = [E1/(1-NU12\*NU21) NU12\*E2/(1-NU12\*NU21) 0 ;NU12\*E2/(1-NU12\*NU21) E2/(1-… NU12\*NU21) 0 ; 0 0 G12]; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function SBar = Sbar(S,theta) %Sbar This function returns the transformed reduced % compliance matrix "Sbar" given the reduced % compliance matrix S and the orientation % angle "theta". % There are two arguments representing S and "theta" % The size of the matrix is 3 x 3. % The angle "theta" must be given in degrees. R=[1 0 0;0 1 0;0 0 2]; Rinv=inv(R); m = cos(theta\*pi/180); n = sin(theta\*pi/180); T = [m\*m n\*n 2\*m\*n ; n\*n m\*m -2\*m\*n ; -m\*n m\*n m\*m-n\*n]; Tinv = [m\*m n\*n -2\*m\*n ; n\*n m\*m 2\*m\*n ; m\*n -m\*n m\*m-n\*n]; SBar = Tinv\*S\*R\*T\*Rinv; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function QBar = Qbar(Q,theta) %Qbar This function returns the transformed reduced % stiffness matrix "Qbar" given the reduced % stiffness matrix Q and the orientation % angle "theta". % There are two arguments representing Q and "theta" % The size of the matrix is 3 x 3. % The angle "theta" must be given in degrees. R=[1 0 0;0 1 0;0 0 2]; Rinv=inv(R); m = cos(theta\*pi/180); n = sin(theta\*pi/180); T = [m\*m n\*n 2\*m\*n ; n\*n m\*m -2\*m\*n ; -m\*n m\*n m\*m-n\*n]; Tinv = [m\*m n\*n -2\*m\*n ; n\*n m\*m 2\*m\*n ; m\*n -m\*n m\*m-n\*n]; QBar = Tinv\*Q\*R\*T\*Rinv; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function y = Amatrix(A,QBar,z1,z2) %Amatrix This function returns the [A] matrix % after the layer k with stiffness [Qbar]

% is assembled.

%NUxy This function returns Poisson's ratio

600 MATLAB Applications for the Practical Engineer

% coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU12 - Poisson's ratio % G12 - shear modulus % theta - fiber orientation angle

% The angle "theta" must be given in degrees.

function yNUyx = NUyx(E1,E2,NU21,G12,theta) %NUyx This function returns Poisson's ratio

% coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU21 - Poisson's ratio % G12 - shear modulus % theta - fiber orientation angle

% The angle "theta" must be given in degrees.

% for orthotropic materials. There are nine % arguments representing the nine independent % material constants. The size of the compliance

% for orthotropic materials. There are nine % arguments representing the nine independent % material constants. The size of the stiffness

function S = ReducedCompliance(E1,E2,NU12,G12)

denom = m^4 + (E2/G12 - 2\*NU21)\*n\*n\*m\*m + (E2/E1)\*n\*n; numer = NU21\*(n^4 + m^4) - (1 + E2/E1 - E2/G12)\*n\*n\*m\*m;

function y = OrthotropicCompliance(E1,E2,E3,NU12,NU23,NU13,G12,G23,G13) %OrthotropicCompliance This function returns the compliance matrix

y = [1/E1 -NU12/E1 -NU13/E1 0 0 0 ; -NU12/E1 1/E2 -NU23/E2 0 0 0 ;-NU13/E1 -…

x = [1/E1 -NU12/E1 -NU13/E1 0 0 0 ; -NU12/E1 1/E2 -NU23/E2 0 0 0 ;-NU13/E1 -…

NU23/E2 1/E3 0 0 0 ; 0 0 0 1/G23 0 0 ; 0 0 0 0 1/G13 0 ;0 0 0 0 0 1/G12];

%OrthotropicStiffness This function returns the stiffness matrix

function C = OrthotropicStiffness(E1,E2,E3,NU12,NU23,NU13,G12,G23,G13)

NU23/E2 1/E3 0 0 0 ; 0 0 0 1/G23 0 0 ; 0 0 0 0 1/G13 0 ;0 0 0 0 0 1/G12];

% NUyx is returned as a scalar m = cos(theta\*pi/180); n = sin(theta\*pi/180);

yNUyx = numer/denom;

% matrix is 6 x 6.

% matrix is 6 x 6.

C = inv(x);

denom = m^4 + (E1/G12 - 2\*NU12)\*n\*n\*m\*m + (E1/E2)\*n\*n; numer = NU12\*(n^4 + m^4) - (1 + E1/E2 - E1/G12)\*n\*n\*m\*m;

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

% NUxy is returned as a scalar m = cos(theta\*pi/180); n = sin(theta\*pi/180);

yNUxy = numer/denom;

% NUyx in the global

% NUxy in the global

```
% A - [A] matrix after layer k
% is assembled.
% Qbar - [Qbar] matrix for layer k
% z1 - z(k-1) for layer k
% z2 - z(k) for layer k
for i = 1 : 3 
for j = 1 : 3 
A(i,j) = A(i,j) + QBar(i,j)*(z2-z1); 
end
end
y = A; 
%**********************************************************THE end******************************************************************* 
%*******************************************CREATED BY WAEL A. AL-TABEY*************************************************** 
%*************************************************************************************************************************************** 
function y = Bmatrix(B,QBar,z1,z2) 
%Bmatrix This function returns the [B] matrix
% after the layer k with stiffness [Qbar]
% is assembled.
% B - [B] matrix after layer k
% is assembled.
% Qbar - [Qbar] matrix for layer k
% z1 - z(k-1) for layer k
% z2 - z(k) for layer k
for i = 1 : 3 
for j = 1 : 3 
B(i,j) = B(i,j) + QBar(i,j)*(z2^2 -z1^2); 
end
end
y = B; 
%**********************************************************THE end******************************************************************* 
%*******************************************CREATED BY WAEL A. AL-TABEY*************************************************** 
%*************************************************************************************************************************************** 
function y = Dmatrix(D,QBar,z1,z2) 
%Dmatrix This function returns the [D] matrix
% after the layer k with stiffness [Qbar]
% is assembled.
% D - [D] matrix after layer k
% is assembled.
% Qbar - [Qbar] matrix for layer k
% z1 - z(k-1) for layer k
% z2 - z(k) for layer k
for i = 1 : 3 
for j = 1 : 3 
D(i,j) = D(i,j) + QBar(i,j)*(z2^3 -z1^3); 
end
end
y = D; 
%**********************************************************THE end******************************************************************* 
%*******************************************CREATED BY WAEL A. AL-TABEY*************************************************** 
%*************************************************************************************************************************************** 
%*************************************************************************************************************************************** 
%***********The Function MATLAB code for Laminate Analysis*****************
%**************calculate [A],[B]&[D] matrix for Laminate*******************
%*******************Using Orthotropic Qbar function************************
%*************************************************************************************************************************************** 
function [A,B,D]=ABDmatrix(E11,E22,NU12,G12,ho,n); 
%*************************************************************************************************************************************** 
% The reduced stiffness [Q] in GPa is calculated for this material
Q=ReducedStiffness(E11,E22,NU12,G12); 
%***************************************************************************************************************************************
```
B=zeros(3,3); D=zeros(3,3); for i=1:n

end A=A; B=B/2; D=D/3;

function

beta=a/b

%The [A],[B]&[D] matrix is calculated [A,B,D]=ABDmatrix(E11,E22,NU12,G12,ho,n)

epsy1=D(1,1)/D(2,2) %D11/D22

epsy3=D(1,3)/D(2,2) %D16/D22 epsy4=D(2,3)/D(2,2) %D26/D22 epsy5=D(1,2)/D(2,2) %D12/D22

alfa1=Nxbar/D(2,2) %Nx/D22 alfa2=Nybar/D(2,2) %Ny/D22 alfa3=Nxybar/D(2,2) %Nxy/D22

%The [A],[B]&[D] matrix is calculated [A,B,D]=ABDmatrix(E11,E22,NU12,G12,ho,n);

FR1=(R1\*a)/EI %R1\*a/EI FR2=(R2\*a)/EI %R2\*a/EI FR3=(R3\*b)/D(2,2) %R3\*b/D22 FR4=(R4\*b)/D(2,2) %R4\*b/D22 FT1=(T1\*(a^3))/EI %T1\*a^3/EI FT2=(T2\*(a^3))/EI %T2\*a^3/EI FT3=(T3\*(b^3))/D(2,2) %T3\*b^3/D22

EI=(1-(NU12^2))\*D(1,1) %(1-NU12^2)\*D11

epsy2=(D(1,2)+(2\*D(3,3)))/D(2,2) %(D12+2D66)/D22

epsy6=(D(1,2)+(4\*D(3,3)))/D(2,2) %(D12+4D66)/D22

z=-(ho/2):(ho/n):(ho/2);

QBar=Qbar(Q,theta(i)); %The [A] matrix is calculated A=Amatrix(A,QBar,z(i),z(i+1)); %The [B] matrix is calculated B=Bmatrix(B,QBar,z(i),z(i+1)); %The [D] matrix is calculated D=Dmatrix(D,QBar,z(i),z(i+1));

theta(i)=input('The angle of the layer laminate (degree)=');

%The transformed reduced stiffnesses [¯Q] in GPa for the two layers are now calculated

%\*\*\*\*\*\*\*\*The Function MATLAB code for Laminate Coefficients\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*calculate Epsy-Alfa-beta Coefficients for Laminate\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Using Orthotropic Qbar function\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*The Function MATLAB code for Laminate Coefficients\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*calculate D03,Cf2,FR3,FR4,FT3,FT4 Coefficients for Boundary Conditions\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

http://dx.doi.org/10.5772/57384

603

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

[epsy1,epsy2,epsy3,epsy4,epsy5,epsy6,beta,alfa1,alfa2,alfa3]=EpsyAlfa(E11,E22,NU12,G12,ho,n,Nxbar,Nybar,Nxybar,a,) %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

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%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function [EI,FR1,FR2,FR3,FR4,FT1,FT2,FT3,FT4]=CoeffBC(E11,E22,NU12,G12,ho,n,a,b,R1,R2,R3,R4,T1,T2,T3,T4) %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

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A=zeros(3,3);

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique… http://dx.doi.org/10.5772/57384 603

% A - [A] matrix after layer k

% z1 - z(k-1) for layer k % z2 - z(k) for layer k for i = 1 : 3 for j = 1 : 3

% Qbar - [Qbar] matrix for layer k

602 MATLAB Applications for the Practical Engineer

A(i,j) = A(i,j) + QBar(i,j)\*(z2-z1);

function y = Bmatrix(B,QBar,z1,z2) %Bmatrix This function returns the [B] matrix % after the layer k with stiffness [Qbar]

% B - [B] matrix after layer k

% z1 - z(k-1) for layer k % z2 - z(k) for layer k for i = 1 : 3 for j = 1 : 3

% Qbar - [Qbar] matrix for layer k

B(i,j) = B(i,j) + QBar(i,j)\*(z2^2 -z1^2);

function y = Dmatrix(D,QBar,z1,z2) %Dmatrix This function returns the [D] matrix % after the layer k with stiffness [Qbar]

% D - [D] matrix after layer k

% z1 - z(k-1) for layer k % z2 - z(k) for layer k for i = 1 : 3 for j = 1 : 3

% Qbar - [Qbar] matrix for layer k

D(i,j) = D(i,j) + QBar(i,j)\*(z2^3 -z1^3);

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

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%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

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%\*\*\*\*\*\*\*\*\*\*\*The Function MATLAB code for Laminate Analysis\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*calculate [A],[B]&[D] matrix for Laminate\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Using Orthotropic Qbar function\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

function [A,B,D]=ABDmatrix(E11,E22,NU12,G12,ho,n);

Q=ReducedStiffness(E11,E22,NU12,G12);

% The reduced stiffness [Q] in GPa is calculated for this material

% is assembled.

% is assembled.

% is assembled.

% is assembled.

% is assembled.

A=zeros(3,3);

end end y = D;

end end y = B;

end end y = A; B=zeros(3,3); D=zeros(3,3); for i=1:n z=-(ho/2):(ho/n):(ho/2); theta(i)=input('The angle of the layer laminate (degree)='); %The transformed reduced stiffnesses [¯Q] in GPa for the two layers are now calculated QBar=Qbar(Q,theta(i)); %The [A] matrix is calculated A=Amatrix(A,QBar,z(i),z(i+1)); %The [B] matrix is calculated B=Bmatrix(B,QBar,z(i),z(i+1)); %The [D] matrix is calculated D=Dmatrix(D,QBar,z(i),z(i+1)); end A=A; B=B/2; D=D/3; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*The Function MATLAB code for Laminate Coefficients\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*calculate Epsy-Alfa-beta Coefficients for Laminate\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Using Orthotropic Qbar function\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function [epsy1,epsy2,epsy3,epsy4,epsy5,epsy6,beta,alfa1,alfa2,alfa3]=EpsyAlfa(E11,E22,NU12,G12,ho,n,Nxbar,Nybar,Nxybar,a,) %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %The [A],[B]&[D] matrix is calculated [A,B,D]=ABDmatrix(E11,E22,NU12,G12,ho,n) %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* epsy1=D(1,1)/D(2,2) %D11/D22 epsy2=(D(1,2)+(2\*D(3,3)))/D(2,2) %(D12+2D66)/D22 epsy3=D(1,3)/D(2,2) %D16/D22 epsy4=D(2,3)/D(2,2) %D26/D22 epsy5=D(1,2)/D(2,2) %D12/D22 epsy6=(D(1,2)+(4\*D(3,3)))/D(2,2) %(D12+4D66)/D22 beta=a/b alfa1=Nxbar/D(2,2) %Nx/D22 alfa2=Nybar/D(2,2) %Ny/D22 alfa3=Nxybar/D(2,2) %Nxy/D22 %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*The Function MATLAB code for Laminate Coefficients\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*calculate D03,Cf2,FR3,FR4,FT3,FT4 Coefficients for Boundary Conditions\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* function [EI,FR1,FR2,FR3,FR4,FT1,FT2,FT3,FT4]=CoeffBC(E11,E22,NU12,G12,ho,n,a,b,R1,R2,R3,R4,T1,T2,T3,T4) %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %The [A],[B]&[D] matrix is calculated [A,B,D]=ABDmatrix(E11,E22,NU12,G12,ho,n); %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* EI=(1-(NU12^2))\*D(1,1) %(1-NU12^2)\*D11 FR1=(R1\*a)/EI %R1\*a/EI FR2=(R2\*a)/EI %R2\*a/EI FR3=(R3\*b)/D(2,2) %R3\*b/D22 FR4=(R4\*b)/D(2,2) %R4\*b/D22 FT1=(T1\*(a^3))/EI %T1\*a^3/EI FT2=(T2\*(a^3))/EI %T2\*a^3/EI FT3=(T3\*(b^3))/D(2,2) %T3\*b^3/D22


**The Beam Equations** 

echo off; clc; clear all; echo on;

echo off;

a11=FR1; a12= -m; a13= FR1; a14= m; a21= m^3; a22= FT1; a23= -m^3; a24= FT1;

b=[A4;A3;A2;A1];

A11=solve(E(1),A1);%(C.11) A12=solve(E(2),A1);%(C.10) A13=A11-A12;%(C.12) A14=solve(A13,A2);%(C.14) A15=subs(E(3),A11,A1);%(C.15) A16=simplify(A15);%(C.16) A17=subs(A16,A14,A2);%(C.17) A18=simplify(A17);%(C.18) A19=solve(A18,A3);%(C.21) fay=subs(A19,A4,1)%(C.22) A21=subs(E(4),A11,A1);%(C.23) A22=simplify(A21);%(C.24) A23=subs(A22,A14,A2);%(C.25) A24=simplify(A23);%(C.26) A25=solve(A24,A3);%(C.29) fay1=subs(A25,A4,1)%(C.30) MUEQ=fay-fay1%(C.31) Ax1=solve(E(1),A2);%(C.32) Ax2=subs(E(2),Ax1,A2);%(C.33) Ax3=subs(Ax2,A19,A3);

Ax4=solve(Ax3,A4);%(C.35)\*A4=\*\*A1

Ax6=subs(Ax5,Ax4,A4);%(C.36)\*A2=\*\*A1

Ax5=subs(A14,A19,A3);

Ax7=Ax6\*fay;%(C.37)\*A3=\*\*A1 A=1;%A1 Assumption

syms fay

E=a\*b;

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

http://dx.doi.org/10.5772/57384

605

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*The MATLAB code for Calculate Constants Of The Equation of X of The Beam\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*A1, A2, A3, A4\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*Using Normalizatio of FR1 FR2 FT1 FT2 For Each Case Separately\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*And Estimated The Constant Fay in (C.22)\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*And Beam Equation X (C.39)\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

a=[a11,a12,a13,a14;a21,a22,a23,a24;a31,a32,a33,a34;a41,a42,a43,a44];%(C.9)

syms m A1 A2 A3 A4 FR1 FR2 FT1 FT2 zeta

a31= FR2\*m\*cosh(m)+(sinh(m))\*m^2; a32= FR2\*m\*sinh(m)+(cosh(m))\*m^2; a33=FR2\*m\*cos(m)-(sin(m))\*m^2; a34= -FR2\*m\*sin(m)-(cos(m))\*m^2; a41= FT2\*sinh(m)-cosh(m)\*m^3; a42= FT2\*cosh(m)-sinh(m)\*m^3; a43= FT2\*sin(m)+cos(m)\*m^3; a44= FT2\*cos(m)-sin(m)\*m^3;

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique… http://dx.doi.org/10.5772/57384 605

#### **The Beam Equations**

FT4=(T4\*(b^3))/D(2,2) %T4\*b^3/D22

604 MATLAB Applications for the Practical Engineer

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Laminate Coefficients\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*epsy1,epsy2,epsy3,epsy4,beta,alfa1,alfa2,alfa3,Pbar,f1,f2,f3,f4\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*lambda,D03,Cf2,FR3,FR4,FT3,FT4\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

hy=input('The Thickness Equation of Layer Laminate as Function of y h(y)=');

ho=input('The Inetial thickness of Layer Laminate at y = 0.0 ho (mm)='); hb=input('The Final thickness of Layer Laminate at y = 0.0 ho (mm)='); a=input('The Dimensions of Laminate at x Direction a(mm)='); b=input('The Dimensions of Laminate at y Direction b(mm)='); n=input('The Total Number of Laminate Layer n='); Nxbar=input('The In-Plane Load in X Direction Nxbar(N)='); Nybar=input('The In-Plane Load in Y Direction Nybar(N)='); Nxybar=input('The In-Plane Load in XY Plane Nxybar(N)='); R1=input('The Rotational Stiffness of support at x=0 R1 (N.m/rad)='); R2=input('The Rotational Stiffness of support at x=a R2 (N.m/rad)='); R3=input('The Rotational Stiffness of support at y=0 R3 (N.m/rad)='); R4=input('The Rotational Stiffness of support at y=b R4 (N.m/rad)='); T1=input('The Translation Stiffness of support at x=0 T1 (N/m)='); T2=input('The Translation Stiffness of support at x=a T2 (N/m)='); T3=input('The Translation Stiffness of support at y=0 T3 (N/m)='); T4=input('The Translation Stiffness of support at y=b T4 (N/m)=');

**The main program (Call program)** 

[t1,t2,t3,t4]=VTheckniss(y,ho,hb,b,Eta,hy);

E11=input('The Modulus of Elasticity E11(GPa)='); E22=input('The Modulus of Elasticity E22(GPa)='); NU12=input('The Poisson Coefficient NU12='); G12=input('The Shear Modulus G12(GPa)=');

clc;

close all; clear all; syms y ho hb b Eta hy

t1=eval(t1) t2=eval(t2) t3=eval(t3) t4=eval(t4)

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

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%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* [epsy1,epsy2,epsy3,epsy4,epsy5,epsy6,beta,alfa1,alfa2,alfa3]=EpsyAlfa(E11,E22,NU12,G12,ho,n,Nxbar,Nybar,Nxybar,a,); %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

[EI,FR1,FR2,FR3,FR4,FT1,FT2,FT3,FT4]=CoeffBC(E11,E22,NU12,G12,ho,n,a,b,R1,R2,R3,R4,T1,T2,T3,T4);

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* echo off; clc; clear all; echo on; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*The MATLAB code for Calculate Constants Of The Equation of X of The Beam\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*A1, A2, A3, A4\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*Using Normalizatio of FR1 FR2 FT1 FT2 For Each Case Separately\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*And Estimated The Constant Fay in (C.22)\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*And Beam Equation X (C.39)\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* echo off; syms m A1 A2 A3 A4 FR1 FR2 FT1 FT2 zeta a11=FR1; a12= -m; a13= FR1; a14= m; a21= m^3; a22= FT1; a23= -m^3; a24= FT1; a31= FR2\*m\*cosh(m)+(sinh(m))\*m^2; a32= FR2\*m\*sinh(m)+(cosh(m))\*m^2; a33=FR2\*m\*cos(m)-(sin(m))\*m^2; a34= -FR2\*m\*sin(m)-(cos(m))\*m^2; a41= FT2\*sinh(m)-cosh(m)\*m^3; a42= FT2\*cosh(m)-sinh(m)\*m^3; a43= FT2\*sin(m)+cos(m)\*m^3; a44= FT2\*cos(m)-sin(m)\*m^3; a=[a11,a12,a13,a14;a21,a22,a23,a24;a31,a32,a33,a34;a41,a42,a43,a44];%(C.9) b=[A4;A3;A2;A1]; E=a\*b; A11=solve(E(1),A1);%(C.11) A12=solve(E(2),A1);%(C.10) A13=A11-A12;%(C.12) A14=solve(A13,A2);%(C.14) A15=subs(E(3),A11,A1);%(C.15) A16=simplify(A15);%(C.16) A17=subs(A16,A14,A2);%(C.17) A18=simplify(A17);%(C.18) A19=solve(A18,A3);%(C.21) fay=subs(A19,A4,1)%(C.22) A21=subs(E(4),A11,A1);%(C.23) A22=simplify(A21);%(C.24) A23=subs(A22,A14,A2);%(C.25) A24=simplify(A23);%(C.26) A25=solve(A24,A3);%(C.29) fay1=subs(A25,A4,1)%(C.30) MUEQ=fay-fay1%(C.31) Ax1=solve(E(1),A2);%(C.32) Ax2=subs(E(2),Ax1,A2);%(C.33) Ax3=subs(Ax2,A19,A3); Ax4=solve(Ax3,A4);%(C.35)\*A4=\*\*A1 Ax5=subs(A14,A19,A3); Ax6=subs(Ax5,Ax4,A4);%(C.36)\*A2=\*\*A1 syms fay Ax7=Ax6\*fay;%(C.37)\*A3=\*\*A1 A=1;%A1 Assumption

```
B=subs(Ax6,A1,1);%A2
C=subs(Ax7,A1,1);%A3
D=subs(Ax4,A1,1);%A3
X=A*cos(m*zeta)+B*sin(m*zeta)+C*cosh(m*zeta)+D*sinh(m*zeta) 
%**********************************************************THE end******************************************************************* 
%*******************************************CREATED BY WAEL A. AL-TABEY*************************************************** 
%*************************************************************************************************************************************** 
echo off; 
clc; 
clear all; 
echo on; 
%*************************************************************************************************************************************** 
%*The MATLAB code for Calculate Constants Of The Equation of X of The Beam*
%*************************A1, A2, A3, A4***********************************
%****Using Normalizatio of FR1 FR2 FT1 FT2 For Each Case Separately********
%****************And Estimated The Constant Fay in (C.22)******************
%**********************And Beam Equation X (C.39)**************************
%*************************************************************************************************************************************** 
echo off; 
syms m A1 A2 A3 A4 FR1 FR2 FT1 FT2 zeta
BC=input('The Type of The Boundary Condition ( SS, CC, FF, CF, FR, FT, SC, TT, RR, RT, RS, TS, CR, CT, ES, CE, FE, 
EE)=','s'); 
if BC=='SS'
K1=0; 
K2=0; 
K3=1; 
K4=1; 
else if BC=='CC'
K1=1; 
K2=1; 
K3=1; 
K4=1; 
else if BC=='FF'
K1=0; 
K2=0; 
K3=0; 
K4=0; 
else if BC=='CF'
K1=1; 
K2=0; 
K3=1; 
K4=0; 
else if BC=='FR'
K1=0; 
K2=1; 
K3=0; 
K4=1; 
else if BC=='FT'
K1=0; 
K2=0; 
K3=0; 
K4=1; 
else if BC=='SC'
K1=0; 
K2=1; 
K3=1; 
K4=1; 
else if BC=='TT'
K1=0; 
K2=0; 
K3=1;
```
http://dx.doi.org/10.5772/57384

607

606 MATLAB Applications for the Practical Engineer Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique… http://dx.doi.org/10.5772/57384 607

B=subs(Ax6,A1,1);%A2 C=subs(Ax7,A1,1);%A3 D=subs(Ax4,A1,1);%A3

echo off; clc; clear all; echo on;

echo off;

EE)=','s'); if BC=='SS' K1=0; K2=0; K3=1; K4=1; else if BC=='CC' K1=1; K2=1; K3=1; K4=1; else if BC=='FF' K1=0; K2=0; K3=0; K4=0; else if BC=='CF' K1=1; K2=0; K3=1; K4=0; else if BC=='FR' K1=0; K2=1; K3=0; K4=1; else if BC=='FT' K1=0; K2=0; K3=0; K4=1; else if BC=='SC' K1=0; K2=1; K3=1; K4=1; else if BC=='TT' K1=0; K2=0; K3=1;

X=A\*cos(m\*zeta)+B\*sin(m\*zeta)+C\*cosh(m\*zeta)+D\*sinh(m\*zeta)

%\*The MATLAB code for Calculate Constants Of The Equation of X of The Beam\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*A1, A2, A3, A4\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*Using Normalizatio of FR1 FR2 FT1 FT2 For Each Case Separately\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*And Estimated The Constant Fay in (C.22)\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*And Beam Equation X (C.39)\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

syms m A1 A2 A3 A4 FR1 FR2 FT1 FT2 zeta

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

BC=input('The Type of The Boundary Condition ( SS, CC, FF, CF, FR, FT, SC, TT, RR, RT, RS, TS, CR, CT, ES, CE, FE,

Ax7=Ax6\*fay;%(C.37)\*A3=\*\*A1 A=1;%A1 Assumption B=subs(Ax6,A1,1);%A2 C=subs(Ax7,A1,1);%A3 D=subs(Ax4,A1,1);%A3

echo off; clc; clear all; echo on;

echo off;

FE)=','s'); if BC=='SS'

fay2=limit(fay1,FR1,0); fay3=limit(fay2,FR2,0); fay4=limit(fay3,FT1,inf); fay=limit(fay4,FT2,inf) else if BC=='CC' fay2=limit(fay1,FR1,inf); fay3=limit(fay2,FR2,inf); fay4=limit(fay3,FT1,inf); fay=limit(fay4,FT2,inf) else if BC=='FF' fay2=limit(fay1,FR1,0); fay3=limit(fay2,FR2,0); fay4=limit(fay3,FT1,0); fay=limit(fay4,FT2,0) else if BC=='FR' fay2=limit(fay1,FR1,0); fay3=limit(fay2,FT1,0); fay=limit(fay3,FT2,inf) else if BC=='FT' fay2=limit(fay1,FR1,0); fay3=limit(fay2,FR2,0); fay=limit(fay3,FT1,0); else if BC=='SC' fay2=limit(fay1,FR1,0); fay3=limit(fay2,FR2,inf); fay4=limit(fay3,FT1,inf); fay=limit(fay4,FT2,inf) else if BC=='CF' fay2=limit(fay1,FR1,inf); fay3=limit(fay2,FR2,0); fay4=limit(fay3,FT1,inf); fay=limit(fay4,FT2,0)

syms FR1 FR2 FT1 FT2 m zeta

X=A\*cos(m\*zeta)+B\*sin(m\*zeta)+C\*cosh(m\*zeta)+D\*sinh(m\*zeta)

%\*\*\*\*The MATLAB code for Calculate The Equation of Fay of The Beam\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*For All Cases Of Boundary Condations\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*Using The Value of FR1 FR2 FT1 FT2 For Each Case Separately\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*To Used The Equation of Fay in BeamSrip Programe\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Using One Equation of Fay (C.34)\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

http://dx.doi.org/10.5772/57384

609

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

fay1=-(m^5\*sinh(m)+sin(m)\*m\*FT1\*FR1+m^4\*FR2\*cosh(m)+FR2\*m^4\*cos(m)+2\*cos(m)\*m^4\*FR1 sin(m)\*m^5+FR2\*cosh(m)\*FR1\*FT1+m\*sinh(m)\*FR1\*FT1-FR2\*cos(m)\*FT1\*FR1+2\*FR2\*sin(m)\*FR1\*m^3)/(-

FR2\*sin(m)\*m^4+2\*FR2\*m\*cos(m)\*FT1+m^4\*FR2\*sinh(m)+FR2\*sin(m)\*FR1\*FT1+cos(m)\*m\*FR1\*FT1);

BC=input('The Type of The Boundary Condition ( SS, CC, FF, CF, FR, FT, SC, TT, RR, RT, RS, TS, CR, CT, SE, CE,

2\*sin(m)\*m^2\*FT1+m^5\*cosh(m)-cos(m)\*m^5+FR2\*sinh(m)\*FR1\*FT1+m\*cosh(m)\*FR1\*FT1-

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique… http://dx.doi.org/10.5772/57384 609

608 MATLAB Applications for the Practical Engineer

```
Ax7=Ax6*fay;%(C.37)*A3=**A1
A=1;%A1 Assumption
B=subs(Ax6,A1,1);%A2
C=subs(Ax7,A1,1);%A3
D=subs(Ax4,A1,1);%A3
X=A*cos(m*zeta)+B*sin(m*zeta)+C*cosh(m*zeta)+D*sinh(m*zeta) 
%**********************************************************THE end******************************************************************* 
%*******************************************CREATED BY WAEL A. AL-TABEY*************************************************** 
%*************************************************************************************************************************************** 
echo off; 
clc; 
clear all; 
echo on; 
%*************************************************************************************************************************************** 
%****The MATLAB code for Calculate The Equation of Fay of The Beam********
%***************For All Cases Of Boundary Condations***********************
%******Using The Value of FR1 FR2 FT1 FT2 For Each Case Separately*********
%*************To Used The Equation of Fay in BeamSrip Programe*************
%*******************Using One Equation of Fay (C.34)************************
%*************************************************************************************************************************************** 
echo off; 
syms FR1 FR2 FT1 FT2 m zeta 
fay1=-(m^5*sinh(m)+sin(m)*m*FT1*FR1+m^4*FR2*cosh(m)+FR2*m^4*cos(m)+2*cos(m)*m^4*FR1-
sin(m)*m^5+FR2*cosh(m)*FR1*FT1+m*sinh(m)*FR1*FT1-FR2*cos(m)*FT1*FR1+2*FR2*sin(m)*FR1*m^3)/(-
2*sin(m)*m^2*FT1+m^5*cosh(m)-cos(m)*m^5+FR2*sinh(m)*FR1*FT1+m*cosh(m)*FR1*FT1-
FR2*sin(m)*m^4+2*FR2*m*cos(m)*FT1+m^4*FR2*sinh(m)+FR2*sin(m)*FR1*FT1+cos(m)*m*FR1*FT1); 
BC=input('The Type of The Boundary Condition ( SS, CC, FF, CF, FR, FT, SC, TT, RR, RT, RS, TS, CR, CT, SE, CE, 
FE)=','s'); 
if BC=='SS'
fay2=limit(fay1,FR1,0); 
fay3=limit(fay2,FR2,0); 
fay4=limit(fay3,FT1,inf); 
fay=limit(fay4,FT2,inf) 
else if BC=='CC'
fay2=limit(fay1,FR1,inf); 
fay3=limit(fay2,FR2,inf); 
fay4=limit(fay3,FT1,inf); 
fay=limit(fay4,FT2,inf) 
else if BC=='FF'
fay2=limit(fay1,FR1,0); 
fay3=limit(fay2,FR2,0); 
fay4=limit(fay3,FT1,0); 
fay=limit(fay4,FT2,0) 
else if BC=='FR'
fay2=limit(fay1,FR1,0); 
fay3=limit(fay2,FT1,0); 
fay=limit(fay3,FT2,inf) 
else if BC=='FT'
fay2=limit(fay1,FR1,0); 
fay3=limit(fay2,FR2,0); 
fay=limit(fay3,FT1,0); 
else if BC=='SC'
fay2=limit(fay1,FR1,0); 
fay3=limit(fay2,FR2,inf); 
fay4=limit(fay3,FT1,inf); 
fay=limit(fay4,FT2,inf) 
else if BC=='CF'
fay2=limit(fay1,FR1,inf); 
fay3=limit(fay2,FR2,0); 
fay4=limit(fay3,FT1,inf); 
fay=limit(fay4,FT2,0)
```
else if BC=='TT' fay2=limit(fay1,FR1,0); fay=limit(fay2,FR2,0) else if BC=='RR' fay2=limit(fay1,FT1,inf); fay=limit(fay2,FT2,inf) else if BC=='RT' fay2=limit(fay1,FR2,0); fay=limit(fay2,FT1,inf) else if BC=='RS' fay2=limit(fay1,FR2,0); fay3=limit(fay2,FT1,inf); fay=limit(fay3,FT2,inf) else if BC=='TS' fay2=limit(fay1,FR1,0); fay3=limit(fay2,FR2,0); fay=limit(fay3,FT2,inf) else if BC=='CR' fay2=limit(fay1,FR1,inf); fay3=limit(fay2,FT1,inf); fay=limit(fay3,FT2,inf) else if BC=='CT' fay2=limit(fay1,FR1,inf); fay3=limit(fay2,FR2,inf); fay=limit(fay3,FT1,inf) else if BC=='SE' fay2=limit(fay1,FR1,0); fay=limit(fay2,FT1,inf) else if BC=='CE' fay2=limit(fay1,FR1,inf); fay=limit(fay2,FT1,inf) else if BC=='FE'%FRT fay2=limit(fay1,FR1,0); fay=limit(fay2,FT1,0) end pretty(fay) %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* echo off; clc; clear all; echo on; %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*The MATLAB code for Calculate The Equation of MU of The Beam\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*For All Cases Of Boundary Condations\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*Using The Value of FR1 FR2 FT1 FT2 For Each Case Separately\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Using One Equation of MU (C.31)\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* echo off; syms FR1 FR2 FT1 FT2 m eMU1=-(m^5\*sinh(m) sin(m)\*m^5+m^4\*FR2\*cosh(m)+FR2\*m^4\*cos(m)+2\*cos(m)\*m^4\*FR1+FR2\*cosh(m)\*FR1\*FT1+m\*sinh(m)\*FR1\*FT1- FR2\*cos(m)\*FT1\*FR1+sin(m)\*m\*FT1\*FR1+2\*FR2\*sin(m)\*FR1\*m^3)/(-FR2\*sin(m)\*m^4- 2\*sin(m)\*m^2\*FT1+FR2\*sinh(m)\*FR1\*FT1+m^4\*FR2\*sinh(m)+m^5\*cosh(m)+2\*FR2\*m\*cos(m)\*FT1 cos(m)\*m^5+m\*cosh(m)\*FR1\*FT1+FR2\*sin(m)\*FR1\*FT1+cos(m)\*m\*FR1\*FT1)+(m^7\*cosh(m)-m^7\*cos(m) m^4\*FT2\*sinh(m)-m^4\*FT2\*sin(m)-2\*sin(m)\*FR1\*m^6- FT2\*sinh(m)\*FR1\*FT1+m^3\*cosh(m)\*FR1\*FT1+FT2\*sin(m)\*FT1\*FR1+m^3\*cos(m)\*FT1\*FR1+2\*FT2\*cos(m)\*FR1\*m^3)/(m ^7\*sinh(m)-FT2\*cos(m)\*m^4-2\*m^4\*cos(m)\*FT1-FT2\*cosh(m)\*FR1\*FT1-m^4\*FT2\*cosh(m)- 2\*m\*FT2\*sin(m)\*FT1+sin(m)\*m^7+m^3\*sinh(m)\*FR1\*FT1+FT2\*cos(m)\*FR1\*FT1-sin(m)\*m^3\*FR1\*FT1);

BC=input('The Type of The Boundary Condition ( SS, CC, FF, CF, FR, FT, SC, TT, RR, RT, RS, TS, CR, CT, SE, CE,

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

http://dx.doi.org/10.5772/57384

611

FE)=','s'); if BC=='SS'

eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FR2,0); eMU4=limit(eMU3,FT1,inf); eMU=limit(eMU4,FT2,inf) else if BC=='CC'

eMU2=limit(eMU1,FR1,inf); eMU3=limit(eMU2,FR2,inf); eMU4=limit(eMU3,FT1,inf); eMU=limit(eMU4,FT2,inf)

eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FR2,0); eMU4=limit(eMU3,FT1,0); eMU=limit(eMU4,FT2,0) else if BC=='FR'

eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FT1,0); eMU=limit(eMU3,FT2,inf) else if BC=='FT'

eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FR2,0); eMU=limit(eMU3,FT1,0); else if BC=='SC'

eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FR2,inf); eMU4=limit(eMU3,FT1,inf); eMU=limit(eMU4,FT2,inf) else if BC=='CF'

eMU2=limit(eMU1,FR1,inf); eMU3=limit(eMU2,FR2,0); eMU4=limit(eMU3,FT1,inf); eMU=limit(eMU4,FT2,0) else if BC=='TT'

eMU2=limit(eMU1,FR1,0); eMU=limit(eMU2,FR2,0) else if BC=='RR'

eMU2=limit(eMU1,FT1,inf); eMU=limit(eMU2,FT2,inf) else if BC=='RT' eMU2=limit(eMU1,FR2,0); eMU=limit(eMU2,FT1,inf) else if BC=='RS'

eMU2=limit(eMU1,FR2,0); eMU3=limit(eMU2,FT1,inf); eMU=limit(eMU3,FT2,inf)

eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FR2,0); eMU=limit(eMU3,FT2,inf) else if BC=='CR'

eMU2=limit(eMU1,FR1,inf); eMU3=limit(eMU2,FT1,inf); eMU=limit(eMU3,FT2,inf) else if BC=='CT'

eMU2=limit(eMU1,FR1,inf); eMU3=limit(eMU2,FR2,inf); eMU=limit(eMU3,FT1,inf)

else if BC=='TS'

else if BC=='SE'

else if BC=='FF'

BC=input('The Type of The Boundary Condition ( SS, CC, FF, CF, FR, FT, SC, TT, RR, RT, RS, TS, CR, CT, SE, CE, FE)=','s'); if BC=='SS' eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FR2,0); eMU4=limit(eMU3,FT1,inf); eMU=limit(eMU4,FT2,inf) else if BC=='CC' eMU2=limit(eMU1,FR1,inf); eMU3=limit(eMU2,FR2,inf); eMU4=limit(eMU3,FT1,inf); eMU=limit(eMU4,FT2,inf) else if BC=='FF' eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FR2,0); eMU4=limit(eMU3,FT1,0); eMU=limit(eMU4,FT2,0) else if BC=='FR' eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FT1,0); eMU=limit(eMU3,FT2,inf) else if BC=='FT' eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FR2,0); eMU=limit(eMU3,FT1,0); else if BC=='SC' eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FR2,inf); eMU4=limit(eMU3,FT1,inf); eMU=limit(eMU4,FT2,inf) else if BC=='CF' eMU2=limit(eMU1,FR1,inf); eMU3=limit(eMU2,FR2,0); eMU4=limit(eMU3,FT1,inf); eMU=limit(eMU4,FT2,0) else if BC=='TT' eMU2=limit(eMU1,FR1,0); eMU=limit(eMU2,FR2,0) else if BC=='RR' eMU2=limit(eMU1,FT1,inf); eMU=limit(eMU2,FT2,inf) else if BC=='RT' eMU2=limit(eMU1,FR2,0); eMU=limit(eMU2,FT1,inf) else if BC=='RS' eMU2=limit(eMU1,FR2,0); eMU3=limit(eMU2,FT1,inf); eMU=limit(eMU3,FT2,inf) else if BC=='TS' eMU2=limit(eMU1,FR1,0); eMU3=limit(eMU2,FR2,0); eMU=limit(eMU3,FT2,inf) else if BC=='CR' eMU2=limit(eMU1,FR1,inf); eMU3=limit(eMU2,FT1,inf); eMU=limit(eMU3,FT2,inf) else if BC=='CT' eMU2=limit(eMU1,FR1,inf); eMU3=limit(eMU2,FR2,inf); eMU=limit(eMU3,FT1,inf)

else if BC=='SE'

else if BC=='TT' fay2=limit(fay1,FR1,0); fay=limit(fay2,FR2,0) else if BC=='RR' fay2=limit(fay1,FT1,inf); fay=limit(fay2,FT2,inf) else if BC=='RT' fay2=limit(fay1,FR2,0); fay=limit(fay2,FT1,inf) else if BC=='RS' fay2=limit(fay1,FR2,0); fay3=limit(fay2,FT1,inf); fay=limit(fay3,FT2,inf) else if BC=='TS' fay2=limit(fay1,FR1,0); fay3=limit(fay2,FR2,0); fay=limit(fay3,FT2,inf) else if BC=='CR' fay2=limit(fay1,FR1,inf); fay3=limit(fay2,FT1,inf); fay=limit(fay3,FT2,inf) else if BC=='CT' fay2=limit(fay1,FR1,inf); fay3=limit(fay2,FR2,inf); fay=limit(fay3,FT1,inf) else if BC=='SE' fay2=limit(fay1,FR1,0); fay=limit(fay2,FT1,inf) else if BC=='CE' fay2=limit(fay1,FR1,inf); fay=limit(fay2,FT1,inf) else if BC=='FE'%FRT fay2=limit(fay1,FR1,0); fay=limit(fay2,FT1,0)

610 MATLAB Applications for the Practical Engineer

end pretty(fay)

echo off; clc; clear all; echo on;

echo off;

syms FR1 FR2 FT1 FT2 m eMU1=-(m^5\*sinh(m)-

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

sin(m)\*m^5+m^4\*FR2\*cosh(m)+FR2\*m^4\*cos(m)+2\*cos(m)\*m^4\*FR1+FR2\*cosh(m)\*FR1\*FT1+m\*sinh(m)\*FR1\*FT1-

FT2\*sinh(m)\*FR1\*FT1+m^3\*cosh(m)\*FR1\*FT1+FT2\*sin(m)\*FT1\*FR1+m^3\*cos(m)\*FT1\*FR1+2\*FT2\*cos(m)\*FR1\*m^3)/(m

%\*\*\*\*\*The MATLAB code for Calculate The Equation of MU of The Beam\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*For All Cases Of Boundary Condations\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*Using The Value of FR1 FR2 FT1 FT2 For Each Case Separately\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Using One Equation of MU (C.31)\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

FR2\*cos(m)\*FT1\*FR1+sin(m)\*m\*FT1\*FR1+2\*FR2\*sin(m)\*FR1\*m^3)/(-FR2\*sin(m)\*m^4-

^7\*sinh(m)-FT2\*cos(m)\*m^4-2\*m^4\*cos(m)\*FT1-FT2\*cosh(m)\*FR1\*FT1-m^4\*FT2\*cosh(m)-

m^4\*FT2\*sinh(m)-m^4\*FT2\*sin(m)-2\*sin(m)\*FR1\*m^6-

2\*sin(m)\*m^2\*FT1+FR2\*sinh(m)\*FR1\*FT1+m^4\*FR2\*sinh(m)+m^5\*cosh(m)+2\*FR2\*m\*cos(m)\*FT1 cos(m)\*m^5+m\*cosh(m)\*FR1\*FT1+FR2\*sin(m)\*FR1\*FT1+cos(m)\*m\*FR1\*FT1)+(m^7\*cosh(m)-m^7\*cos(m)-

2\*m\*FT2\*sin(m)\*FT1+sin(m)\*m^7+m^3\*sinh(m)\*FR1\*FT1+FT2\*cos(m)\*FR1\*FT1-sin(m)\*m^3\*FR1\*FT1);

```
eMU2=limit(eMU1,FR1,0); 
eMU=limit(eMU2,FT1,inf) 
else if BC=='CE'
eMU2=limit(eMU1,FR1,inf); 
eMU=limit(eMU2,FT1,inf) 
else if BC=='FE'%FRT
eMU2=limit(eMU1,FR1,0); 
eMU=limit(eMU2,FT1,0) 
end
pretty(eMU) 
%**********************************************************THE end******************************************************************* 
%*******************************************CREATED BY WAEL A. AL-TABEY*************************************************** 
%*************************************************************************************************************************************** 
echo off; 
clc; 
clear all; 
echo on; 
%*************************************************************************************************************************************** 
%****The MATLAB code for Calculate The Equation of X of The Beam********
%***************For All Cases Of Boundary Condations***********************
%******Using The Value of FR1 FR2 FT1 FT2 For Each Case Separately*********
%*************To Used The Equation of Fay in BeamSrip Programe*************
%*******************Using One Equation of X (C.39)************************
%*************************************************************************************************************************************** 
echo off; 
syms FR1 FR2 FT1 FT2 m zeta fay 
x1=sin(m*zeta)-((2*FR1*m*fay+FR1*FT1*(m^4)-1)/(fay-FR1*FT1*(m^4)*fay+2*FT1*(m^3)))*cos(m*zeta)-
((1+FR1*FT1*(m^4))/(fay-FR1*FT1*(m^4)*fay+2*FT1*(m^3)))*(fay*sinh(m*zeta)+cosh(m*zeta)); 
BC=input('The Type of The Boundary Condition ( SS, CC, FF, CF, FR, FT, SC, TT, RR, RT, RS, TS, CR, CT, SE, CE, 
FE)=','s'); 
if BC=='SS'
x2=limit(x1,FR1,0); 
x3=limit(x2,FR2,0); 
x4=limit(x3,FT1,inf); 
x=limit(x4,FT2,inf) 
else if BC=='CC'
x2=limit(x1,FR1,inf); 
x3=limit(x2,FR2,inf); 
x4=limit(x3,FT1,inf); 
x=limit(x4,FT2,inf) 
else if BC=='FF'
x2=limit(x1,FR1,0); 
x3=limit(x2,FR2,0); 
x4=limit(x3,FT1,0); 
x=limit(x4,FT2,0) 
else if BC=='FR'
x2=limit(x1,FR1,0); 
x3=limit(x2,FT1,0); 
x=limit(x3,FT2,inf) 
else if BC=='FT'
x2=limit(x1,FR1,0); 
x3=limit(x2,FR2,0); 
x=limit(x3,FT1,0); 
else if BC=='SC'
x2=limit(x1,FR1,0); 
x3=limit(x2,FR2,inf); 
x4=limit(x3,FT1,inf); 
x=limit(x4,FT2,inf) 
else if BC=='CF'
x2=limit(x1,FR1,inf); 
x3=limit(x2,FR2,0);
```
x4=limit(x3,FT1,inf); x=limit(x4,FT2,0) else if BC=='TT' x2=limit(x1,FR1,0); x=limit(x2,FR2,0) else if BC=='RR' x2=limit(x1,FT1,inf); x=limit(x2,FT2,inf) else if BC=='RT' x2=limit(x1,FR2,0); x=limit(x2,FT1,inf) else if BC=='RS' x2=limit(x1,FR2,0); x3=limit(x2,FT1,inf); x=limit(x3,FT2,inf) else if BC=='TS' x2=limit(x1,FR1,0); x3=limit(x2,FR2,0); x=limit(x3,FT2,inf) else if BC=='CR' x2=limit(x1,FR1,inf); x3=limit(x2,FT1,inf); x=limit(x3,FT2,inf) else if BC=='CT' x2=limit(x1,FR1,inf); x3=limit(x2,FR2,inf); x=limit(x3,FT1,inf) else if BC=='SE' x2=limit(x1,FR1,0); x=limit(x2,FT1,inf) else if BC=='CE' x2=limit(x1,FR1,inf); x=limit(x2,FT1,inf) else if BC=='FE'%FRT x2=limit(x1,FR1,0); x=limit(x2,FT1,0)

end pretty(x)

echo off; clc; clear all; echo on;

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

http://dx.doi.org/10.5772/57384

613

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*The MATLAB code for Plot The Beam Mode Shape\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*For All Cases Of Boundary Condations\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%FOR SS (FR1=0, FR2=0, FT1=inf, FT2=inf)\*\*

%FOR CC (FR1=inf, FR2=inf, FT1=inf, FT2=inf)

%FOR FF (FR1=0, FR2=0, FT1=0, FT2=0)\*\*

%FOR CF (FR1=inf, FR2=0, FT1=inf, FT2=0)\*\*

%FOR FR (FR1=0, FR2=FR, FT1=0, FT2=inf)\*\*

%FOR FT (FR1=0, FR2=0, FT1=0, FT2=FT)\*\*

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique… http://dx.doi.org/10.5772/57384 613

eMU2=limit(eMU1,FR1,0); eMU=limit(eMU2,FT1,inf) else if BC=='CE'

612 MATLAB Applications for the Practical Engineer

eMU2=limit(eMU1,FR1,inf); eMU=limit(eMU2,FT1,inf) else if BC=='FE'%FRT eMU2=limit(eMU1,FR1,0); eMU=limit(eMU2,FT1,0)

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

x1=sin(m\*zeta)-((2\*FR1\*m\*fay+FR1\*FT1\*(m^4)-1)/(fay-FR1\*FT1\*(m^4)\*fay+2\*FT1\*(m^3)))\*cos(m\*zeta)- ((1+FR1\*FT1\*(m^4))/(fay-FR1\*FT1\*(m^4)\*fay+2\*FT1\*(m^3)))\*(fay\*sinh(m\*zeta)+cosh(m\*zeta));

BC=input('The Type of The Boundary Condition ( SS, CC, FF, CF, FR, FT, SC, TT, RR, RT, RS, TS, CR, CT, SE, CE,

%\*\*\*\*The MATLAB code for Calculate The Equation of X of The Beam\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*For All Cases Of Boundary Condations\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*Using The Value of FR1 FR2 FT1 FT2 For Each Case Separately\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*To Used The Equation of Fay in BeamSrip Programe\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Using One Equation of X (C.39)\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

end pretty(eMU)

echo off; clc; clear all; echo on;

echo off;

FE)=','s'); if BC=='SS' x2=limit(x1,FR1,0); x3=limit(x2,FR2,0); x4=limit(x3,FT1,inf); x=limit(x4,FT2,inf) else if BC=='CC' x2=limit(x1,FR1,inf); x3=limit(x2,FR2,inf); x4=limit(x3,FT1,inf); x=limit(x4,FT2,inf) else if BC=='FF' x2=limit(x1,FR1,0); x3=limit(x2,FR2,0); x4=limit(x3,FT1,0); x=limit(x4,FT2,0) else if BC=='FR' x2=limit(x1,FR1,0); x3=limit(x2,FT1,0); x=limit(x3,FT2,inf) else if BC=='FT' x2=limit(x1,FR1,0); x3=limit(x2,FR2,0); x=limit(x3,FT1,0); else if BC=='SC' x2=limit(x1,FR1,0); x3=limit(x2,FR2,inf); x4=limit(x3,FT1,inf); x=limit(x4,FT2,inf) else if BC=='CF' x2=limit(x1,FR1,inf); x3=limit(x2,FR2,0);

syms FR1 FR2 FT1 FT2 m zeta fay



http://dx.doi.org/10.5772/57384

615

%FOR SC (FR1=0, FR2=inf, FT1=inf, FT2=inf)\*\*

%FOR TT (FR1=0, FR2=0, FT1=FT2=FT)\*\*

%FOR RR (FR1=FR2=FR, FT1=inf, FT2=inf)\*\*

%FOR RT (FR1=FR, FR2=0, FT1=inf, FT2=FT)\*\*

%FOR RS (FR1=FR, FR2=0, FT1=inf, FT2=inf)\*\*

%FOR TS (FR1=0, FR2=0, FT1=FT, FT2=inf)\*\*

%FOR CR (FR1=inf, FR2=FR, FT1=inf, FT2=inf)

%FOR CT (FR1=inf, FR2=0, FT1=inf, FT2=FT)

%FOR ES (FR1=FR, FR2=0, FT1=FT, FT2=inf)\*\*

%FOR CE (FR1=inf, FR2=FR, FT1=inf, FT2=FT)\*\*

FR2=input('The Rotational Stiffness of support at x=a R2 (N.m/rad)=');

FT2=input('The Translation Stiffness of support at x=a T2 (N/m)=');

%FOR FE (FR1=0, FR2=FR, FT1=0, FT2=FT)\*\*

%FOR EE (FR1=FR2=FR, FT1=FT2=FT)\*\*

echo off; format long syms m

EE)=','s'); if BC=='SS' FR1=0.00000000001; FR2=0.00000000001; FT1=10000000000; FT2=10000000000; else if BC=='CC' FR1=10000000000; FR2=10000000000; FT1=10000000000; FT2=10000000000; else if BC=='FF' FR1=0.00000000001; FR2=0.00000000001; FT1=0.00000000001; FT2=0.00000000001; else if BC=='CF' FR1=10000000000; FR2=0.00000000001; FT1=10000000000; FT2=0.00000000001; else if BC=='FR' FR1=0.00000000001;

FT1=0.00000000001; FT2=10000000000; else if BC=='FT' FR1=0.00000000001; FR2=0.00000000001; FT1=0.00000000001;

else if BC=='SC' FR1=0.00000000001;

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

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%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

BC=input('The Type of The Boundary Condition ( SS, CC, FF, CF, FR, FT, SC, TT, RR, RT, RS, TS, CR, CT, ES, CE, FE,

```
syms zeta
N=input('The Number of Plate Strips N='); 
R1=input('The Rotational Stiffness of support at x=0 R1 (N.m/rad)='); 
R2=input('The Rotational Stiffness of support at x=a R2 (N.m/rad)='); 
T1=input('The Translation Stiffness of support at x=0 T1 (N/m)='); 
T2=input('The Translation Stiffness of support at x=a T2 (N/m)='); 
NU12=input('The Poisson Coefficient NU12='); 
a=input('The Dimensions of Laminate at x Direction a(mm)='); 
D=input('enter D(1,1)='); 
EI=(1-(NU12^2))*D; 
FR1=(R1*a)/EI; %R1*a/EI
FR2=(R2*a)/EI; %R2*a/EI
FT1=(T1*(a^3))/EI; %T1*a^3/EI
FT2=(T2*(a^3))/EI; %T2*a^3/EI 
for i=1:N 
for j=1:N 
m(i)=i*pi; 
m(j)=j*pi; 
fay(i)=((((2*FR1*FR2*(m(i).^4))-(m(i).^6)+(FR1*FT1*(m(i).^2))).*sin(m(i)))+(((2*FR1*(m(i).^5))+(FR2*(m(i).^5))-
(FR1*FR2*FT1*m(i))).*cos(m(i)))+(((m(i).^6)+(FR1*FT1*(m(i).^2))).*sinh(m(i)))+((((m(i).^5)*FR2)+(FR1*FR2*FT1*m(i))).*co
sh(m(i))))./((((FR1*FR2*FT1*m(i))-(2*FT1*(m(i).^3))-
(FR2*(m(i).^5))).*sin(m(i)))+((((FR1*FT1*(m(i).^2))+(2*FR2*FT1*(m(i).^2))-
(m(i).^6))).*cos(m(i)))+(((FR2*(m(i).^5))+(FR1*FR2*FT1)).*sinh(m(i)))+(((m(i).^6)+(FR1*FT1*(m(i).^2))).*cosh(m(i)))) 
fay(j)=((((2*FR1*FR2*(m(j).^4))-(m(j).^6)+(FR1*FT1*(m(j).^2))).*sin(m(j)))+(((2*FR1*(m(j).^5))+(FR2*(m(j).^5))-
(FR1*FR2*FT1*m(j))).*cos(m(j)))+(((m(j).^6)+(FR1*FT1*(m(j).^2))).*sinh(m(j)))+((((m(j).^5)*FR2)+(FR1*FR2*FT1*m(j))).*cos
h(m(j))))./((((FR1*FR2*FT1*m(j))-(2*FT1*(m(j).^3))-
(FR2*(m(j).^5))).*sin(m(j)))+((((FR1*FT1*(m(j).^2))+(2*FR2*FT1*(m(j).^2))-
(m(i).^6))).*cos(m(j)))+(((FR2*(m(j).^5))+(FR1*FR2*FT1)).*sinh(m(j)))+(((m(j).^6)+(FR1*FT1*(m(j).^2))).*cosh(m(j)))) 
x(i)=cos(m(i)*zeta)+((((-2*m(i).*FT1*fay(i)).*((m(i).^4)-(FT1*FR1)))-(((m(i).^4)-(FT1*FR1))^2))./((((m(i).^8)-
((FT1*FR1)^2)).*fay(i))-((2*(m(i)^3)*FR1).*((m(i)^4)+(FR1*FT1))))).*sin(m(i)*zeta)-((((m(i).^4)-
(FT1*FR1))).*fay(i))./((((m(i).^4)-(FT1*FR1)))-((2*(m(i)^3)*FR1))).*(sinh(m(i)*zeta)+(fay(i).*cosh(m(i)*zeta))) 
dx(i)=diff(x(i),zeta) 
ddx(i)=diff(dx(i),zeta) 
dddx(i)=diff(ddx(i),zeta) 
ddddx(i)=diff(dddx(i),zeta) 
x(j)=cos(m(j)*zeta)+((((-2*m(j).*FT1*fay(j)).*((m(j).^4)-(FT1*FR1)))-(((m(j).^4)-(FT1*FR1))^2))./((((m(j).^8)-
((FT1*FR1)^2)).*fay(j))-((2*(m(j)^3)*FR1).*((m(j)^4)+(FR1*FT1))))).*sin(m(j)*zeta)-((((m(j).^4)-
(FT1*FR1))).*fay(j))./((((m(j).^4)-(FT1*FR1)))-((2*(m(j)^3)*FR1))).*(sinh(m(j)*zeta)+(fay(j).*cosh(m(j)*zeta))) 
f1(i,j)=x(i)*x(j) 
f2(i,j)=dx(i)*x(j) 
f3(i,j)=ddx(i)*x(j) 
f4(i,j)=dddx(i)*x(j) 
f5(i,j)=ddddx(i)*x(j) 
f6(i,j)=x(j) 
a=int(f1,zeta,0,1) 
b=int(f2,zeta,0,1) 
c=int(f3,zeta,0,1) 
d=int(f4,zeta,0,1) 
e=int(f5,zeta,0,1) 
p=int(f6,zeta,0,1) 
end
end
a=eval(a) 
b=eval(b) 
c=eval(c) 
d=eval(d) 
e=eval(e) 
p=eval(p) 
%**********************************************************THE end******************************************************************* 
%*******************************************CREATED BY WAEL A. AL-TABEY***************************************************
```
%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

http://dx.doi.org/10.5772/57384

617

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*The programe to calculate the initial value \*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*BOUNDARY CONDITIONS\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*New for B.C.against rotation\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*The programe to calculate the eigenvalue and eigenvvector\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*The eigenvalue\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

clc; clear all;

for i=1:m mi=4\*i-2 j=i yyi(mi,j)=1 end

for i=1:m for j=1:m j=i mi=4\*i

 end end

for i=1:m j=i mi=4\*i-1 yyi(mi,j)=1/R3

for i=1:m jj=i+m mii=4\*i-3 yyi(mii,jj)=1 end

for i=1:m for j=1:m j=i mii=4\*i-1 jj=j+m

 end end

end

clc; clear all;

syms k1 k3

Lampda=eig(Ak)

for i=1:m mii=4\*i jj=i+m yyi(mii,jj)=-1/T3

end

yyi(mi,j)=-a24\*(CK(i,j)/AK(i,j))

yyi(mii,jj)=-C13\*(CK(i,j)/AK(i,j))

Ak=[0 1 0 0;0 0 1 0;0 0 0 1;-k1 0 -k3 0]

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique… http://dx.doi.org/10.5772/57384 617

syms zeta

for i=1:N for j=1:N m(i)=i\*pi; m(j)=j\*pi;

dx(i)=diff(x(i),zeta) ddx(i)=diff(dx(i),zeta) dddx(i)=diff(ddx(i),zeta) ddddx(i)=diff(dddx(i),zeta)

f1(i,j)=x(i)\*x(j) f2(i,j)=dx(i)\*x(j) f3(i,j)=ddx(i)\*x(j) f4(i,j)=dddx(i)\*x(j) f5(i,j)=ddddx(i)\*x(j) f6(i,j)=x(j) a=int(f1,zeta,0,1) b=int(f2,zeta,0,1) c=int(f3,zeta,0,1) d=int(f4,zeta,0,1) e=int(f5,zeta,0,1) p=int(f6,zeta,0,1)

end end a=eval(a) b=eval(b) c=eval(c) d=eval(d) e=eval(e) p=eval(p)

D=input('enter D(1,1)='); EI=(1-(NU12^2))\*D;

N=input('The Number of Plate Strips N=');

616 MATLAB Applications for the Practical Engineer

NU12=input('The Poisson Coefficient NU12=');

sh(m(i))))./((((FR1\*FR2\*FT1\*m(i))-(2\*FT1\*(m(i).^3))-

h(m(j))))./((((FR1\*FR2\*FT1\*m(j))-(2\*FT1\*(m(j).^3))-

(FR2\*(m(i).^5))).\*sin(m(i)))+((((FR1\*FT1\*(m(i).^2))+(2\*FR2\*FT1\*(m(i).^2))-

(FR2\*(m(j).^5))).\*sin(m(j)))+((((FR1\*FT1\*(m(j).^2))+(2\*FR2\*FT1\*(m(j).^2))-

((FT1\*FR1)^2)).\*fay(i))-((2\*(m(i)^3)\*FR1).\*((m(i)^4)+(FR1\*FT1))))).\*sin(m(i)\*zeta)-((((m(i).^4)-

((FT1\*FR1)^2)).\*fay(j))-((2\*(m(j)^3)\*FR1).\*((m(j)^4)+(FR1\*FT1))))).\*sin(m(j)\*zeta)-((((m(j).^4)-

FR1=(R1\*a)/EI; %R1\*a/EI FR2=(R2\*a)/EI; %R2\*a/EI FT1=(T1\*(a^3))/EI; %T1\*a^3/EI FT2=(T2\*(a^3))/EI; %T2\*a^3/EI

R1=input('The Rotational Stiffness of support at x=0 R1 (N.m/rad)='); R2=input('The Rotational Stiffness of support at x=a R2 (N.m/rad)='); T1=input('The Translation Stiffness of support at x=0 T1 (N/m)='); T2=input('The Translation Stiffness of support at x=a T2 (N/m)=');

fay(i)=((((2\*FR1\*FR2\*(m(i).^4))-(m(i).^6)+(FR1\*FT1\*(m(i).^2))).\*sin(m(i)))+(((2\*FR1\*(m(i).^5))+(FR2\*(m(i).^5))- (FR1\*FR2\*FT1\*m(i))).\*cos(m(i)))+(((m(i).^6)+(FR1\*FT1\*(m(i).^2))).\*sinh(m(i)))+((((m(i).^5)\*FR2)+(FR1\*FR2\*FT1\*m(i))).\*co

(m(i).^6))).\*cos(m(i)))+(((FR2\*(m(i).^5))+(FR1\*FR2\*FT1)).\*sinh(m(i)))+(((m(i).^6)+(FR1\*FT1\*(m(i).^2))).\*cosh(m(i)))) fay(j)=((((2\*FR1\*FR2\*(m(j).^4))-(m(j).^6)+(FR1\*FT1\*(m(j).^2))).\*sin(m(j)))+(((2\*FR1\*(m(j).^5))+(FR2\*(m(j).^5))-

(m(i).^6))).\*cos(m(j)))+(((FR2\*(m(j).^5))+(FR1\*FR2\*FT1)).\*sinh(m(j)))+(((m(j).^6)+(FR1\*FT1\*(m(j).^2))).\*cosh(m(j)))) x(i)=cos(m(i)\*zeta)+((((-2\*m(i).\*FT1\*fay(i)).\*((m(i).^4)-(FT1\*FR1)))-(((m(i).^4)-(FT1\*FR1))^2))./((((m(i).^8)-

(FT1\*FR1))).\*fay(i))./((((m(i).^4)-(FT1\*FR1)))-((2\*(m(i)^3)\*FR1))).\*(sinh(m(i)\*zeta)+(fay(i).\*cosh(m(i)\*zeta)))

x(j)=cos(m(j)\*zeta)+((((-2\*m(j).\*FT1\*fay(j)).\*((m(j).^4)-(FT1\*FR1)))-(((m(j).^4)-(FT1\*FR1))^2))./((((m(j).^8)-

(FT1\*FR1))).\*fay(j))./((((m(j).^4)-(FT1\*FR1)))-((2\*(m(j)^3)\*FR1))).\*(sinh(m(j)\*zeta)+(fay(j).\*cosh(m(j)\*zeta)))

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

(FR1\*FR2\*FT1\*m(j))).\*cos(m(j)))+(((m(j).^6)+(FR1\*FT1\*(m(j).^2))).\*sinh(m(j)))+((((m(j).^5)\*FR2)+(FR1\*FR2\*FT1\*m(j))).\*cos

a=input('The Dimensions of Laminate at x Direction a(mm)=');


%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

#### **The Plate Mode Shape**

```
%*************************************************************************************************************************************** 
%********************** The Plate Mode Shape ***************************
%*************************************************************************************************************************************** 
echo on
clc; 
clear all; 
pause % Strike any key to continue.
clc 
%*********************Problem definition***********************************
g='squareg'; % The unit square
b='squareb3'; % 0 on the left and right boundaries and
%**********0 normal derivative on the top and bottom boundaries************
c=1; 
a=0; 
f=0; 
d=1; 
%********************************Mesh**************************************
[p,e,t]=initmesh('squareg'); 
pause % Strike any key to continue.
clc 
%**************************************************************************
x=p(1,:)'; 
y=p(2,:)'; 
%**************************************************************************
u0=atan(cos(pi/2*x)); 
ut0=3*sin(pi*x).*exp(sin(pi/2*y)); 
pause % Strike any key to continue.
clc 
%*******We want the solution at 31 points in time between 0 and 5**********
n=31; 
tlist=linspace(0,5,n); 

%*************************Solve PDE problem of Plate***********************
uu=hyperbolic(u0,ut0,tlist,b,p,e,t,c,a,f,d); 
pause % Strike any key to continue.
clc 
%*****To speed up the plotting, we interpolate to a rectangular grid*******
delta=-1:0.1:1; 
[uxy,tn,a2,a3]=tri2grid(p,t,uu(:,1),delta,delta); 
gp=[tn;a2;a3]; 
%***************************Make the animation*****************************
newplot; 
M=moviein(n); 
umax=max(max(uu)); 
umin=min(min(uu)); 
for i=1:n, 
 if rem(i,10)==0, 
 fprintf('%d ',i); 
 end
 pdeplot(p,e,t,'xydata',uu(:,i),'zdata',uu(:,i),'zstyle','continuous','mesh','on','xygrid','on','gridparam',gp,'colorbar','on'); 
 grid on
 colormap 
 axis([-1 1 -1 1 umin umax]); caxis([umin umax]); 
 title('Plate mode shape') 
 xlabel('X') 
 ylabel('Y') 
 zlabel('Z') 
 M(:,i)=getframe; 
 if i==n;
```
fprintf('done\n');

pause % Strike any key to end.

ASCE, 96 111-1185.

33(1):257–63.

tures, 73, 120-123.

Computinal Structures, 74, 78-365.

method. Compt. Geotech, 76, 20-161.

Galerkin Method. J. Sound Vibration, 135(2):263-74.

restrained edges, J. composite Structures, 60, 245-253.

translation, Thin-Walled Structures, 42, 1-24.

Applied Mathematical Modelling, 32, 2254-2273.

line-supports, J. Thin-walled Structures, 79, 33-41.

extended Kantorovich method, J. composite Structures, 73, 120-123.

with internal rigid line supports, J. Sound and Vibration, 297, 351-364.

Rayleigh-Ritz method. J. of Sound and Vibration; 101(3):307–15.

J. Mechanical Engineering. ASCE, 96 111-1185.

means of the Kantorovich extended method. J. of Sound and Vibration; 137(3):457–61.

modeled with annular plates, J. Computinal Structures, 74, 78-365.

action problem using hybrid method. Compt. Geotech, 76, 20-161.

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Show movie\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*THE end\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*CREATED BY WAEL A. AL-TABEY\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique…

http://dx.doi.org/10.5772/57384

619

[1] Utku M., Citipitioglu E. and Inceleme I, 2000, Circular plate on elastic foundation modeled with annular plates, J.

[2] Chadrashekhara K. and Antony J., 1997, Elastic analysis of an annular slab-soil interaction problem using hybrid

[3] Ng SF, Araar Y, 1989, Free vibration and buckling analysis of clamped rectangular plates of variable thickness by

[4] Ungbhakorn V and Singhatanadgid P, 2006, Buckling analysis of symmetrically laminated composite plates by the

[5] Setoodeh A.R, Karami G, 2003, A solution for the vibration and buckling of composite laminates with elastically

Department of Mechanical Engineering Faculty of Engineering, Alexandria University,

[6] Laura P. and Gutierrez R., 1985, Vibrating non uniform plates on elastic foundation. J. Mechanical Engineering.

[7] Ashour A.S, 2004, vibration of variable thickness plates with edges elastically restrained against rotational and

[8] Grossi R.O, Nallim L.G, 2008, On the existence of weak solutions of anisotropic generally restrained plates, J.

[9] LU C.F, Lee Y.Y, Lim C.W and Chen W.Q, 2006, free vibration of long-span contagious rectangular Kirchhoff plates

[12] Bambill DV, Laura PAA, Bergmann A, Carnicer R. 1991, Fundamental frequency of transverse vibration of symmetrically stepped simply supported rectangular plates. J. of Sound and Vibration; 150(1):167–9. [13] Laura PAA, Gutierrez RH. 1985, Transverse vibrations of rectangular plates on inhomogeneous foundations, Part I:

[1] Utku M., Citipitioglu E. and Inceleme I, 2000, Circular plate on elastic foundation

[2] Chadrashekhara K. and Antony J., 1997, Elastic analysis of an annular slab-soil inter‐

[14] Harik IE, Andrade MG. 1989, Stability of plates with step variation in thickness. J. Computers and Structures;

[15] Cheung Y.K and Zhou D 2001, Vibration analysis of symmetrically laminated rectangular plates with intermediate

[3] Ng SF, Araar Y, 1989, Free vibration and buckling analysis of clamped rectangular plates of variable thickness by Galerkin Method. J. Sound Vibration, 135(2):263-74.

[4] Ungbhakorn V and Singhatanadgid P, 2006, Buckling analysis of symmetrically lami‐ nated composite plates by the extended Kantorovich method, J. composite Struc‐

[5] Setoodeh A.R, Karami G, 2003, A solution for the vibration and buckling of compo‐ site laminates with elastically restrained edges, J. composite Structures, 60, 245-253.

[6] Laura P. and Gutierrez R., 1985, Vibrating non uniform plates on elastic foundation.

[7] Ashour A.S, 2004, vibration of variable thickness plates with edges elastically re‐ strained against rotational and translation, Thin-Walled Structures, 42, 1-24.

[8] Grossi R.O, Nallim L.G, 2008, On the existence of weak solutions of anisotropic gen‐ erally restrained plates, J. Applied Mathematical Modelling, 32, 2254-2273.

[10] Chopra I. 1974, Vibration of stepped thickness plates. International Journal of Mechanical Science; 16:337–44. [11] Cortinez VH, Laura PAA. 1990, Analysis of vibrating rectangular plates of discontinuously varying thickness by

 end end

nfps=5; movie(M,10,nfps);

echo off

**6. References** 

**Author details**

Wael A. Al-Tabey

Alexandria, Egypt

**References**


[1] Utku M., Citipitioglu E. and Inceleme I, 2000, Circular plate on elastic foundation modeled with annular plates, J.

[3] Ng SF, Araar Y, 1989, Free vibration and buckling analysis of clamped rectangular plates of variable thickness by

#### [2] Chadrashekhara K. and Antony J., 1997, Elastic analysis of an annular slab-soil interaction problem using hybrid method. Compt. Geotech, 76, 20-161. **Author details**

Computinal Structures, 74, 78-365.

translation, Thin-Walled Structures, 42, 1-24.

extended Kantorovich method, J. composite Structures, 73, 120-123.

with internal rigid line supports, J. Sound and Vibration, 297, 351-364.

**6. References** 

**The Plate Mode Shape** 

pause % Strike any key to continue.

618 MATLAB Applications for the Practical Engineer

g='squareg'; % The unit square

[p,e,t]=initmesh('squareg'); pause % Strike any key to continue.

u0=atan(cos(pi/2\*x));

tlist=linspace(0,5,n);

ut0=3\*sin(pi\*x).\*exp(sin(pi/2\*y)); pause % Strike any key to continue.

uu=hyperbolic(u0,ut0,tlist,b,p,e,t,c,a,f,d); pause % Strike any key to continue.

[uxy,tn,a2,a3]=tri2grid(p,t,uu(:,1),delta,delta);

axis([-1 1 -1 1 umin umax]); caxis([umin umax]);

echo on clc; clear all;

clc

c=1; a=0; f=0; d=1;

clc

clc

clc

delta=-1:0.1:1;

gp=[tn;a2;a3];

newplot; M=moviein(n); umax=max(max(uu)); umin=min(min(uu));

for i=1:n, if rem(i,10)==0, fprintf('%d ',i);

end

 grid on colormap

 xlabel('X') ylabel('Y') zlabel('Z') M(:,i)=getframe; if i==n;

title('Plate mode shape')

n=31;

x=p(1,:)'; y=p(2,:)';

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* The Plate Mode Shape \*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Problem definition\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Mesh\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*We want the solution at 31 points in time between 0 and 5\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Solve PDE problem of Plate\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*To speed up the plotting, we interpolate to a rectangular grid\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*Make the animation\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

pdeplot(p,e,t,'xydata',uu(:,i),'zdata',uu(:,i),'zstyle','continuous','mesh','on','xygrid','on','gridparam',gp,'colorbar','on');

%\*\*\*\*\*\*\*\*\*\*0 normal derivative on the top and bottom boundaries\*\*\*\*\*\*\*\*\*\*\*\*

b='squareb3'; % 0 on the left and right boundaries and

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Galerkin Method. J. Sound Vibration, 135(2):263-74. [4] Ungbhakorn V and Singhatanadgid P, 2006, Buckling analysis of symmetrically laminated composite plates by the Wael A. Al-Tabey

[5] Setoodeh A.R, Karami G, 2003, A solution for the vibration and buckling of composite laminates with elastically restrained edges, J. composite Structures, 60, 245-253. [6] Laura P. and Gutierrez R., 1985, Vibrating non uniform plates on elastic foundation. J. Mechanical Engineering. ASCE, 96 111-1185. Department of Mechanical Engineering Faculty of Engineering, Alexandria University, Alexandria, Egypt

[7] Ashour A.S, 2004, vibration of variable thickness plates with edges elastically restrained against rotational and

[8] Grossi R.O, Nallim L.G, 2008, On the existence of weak solutions of anisotropic generally restrained plates, J.

#### Applied Mathematical Modelling, 32, 2254-2273. [9] LU C.F, Lee Y.Y, Lim C.W and Chen W.Q, 2006, free vibration of long-span contagious rectangular Kirchhoff plates **References**


[14] Harik IE, Andrade MG. 1989, Stability of plates with step variation in thickness. J. Computers and Structures;


[9] LU C.F, Lee Y.Y, Lim C.W and Chen W.Q, 2006, free vibration of long-span conta‐ gious rectangular Kirchhoff plates with internal rigid line supports, J. Sound and Vi‐ bration, 297, 351-364.

**Section 4**

**Image Processing**


**Section 4**

**Image Processing**

[9] LU C.F, Lee Y.Y, Lim C.W and Chen W.Q, 2006, free vibration of long-span conta‐ gious rectangular Kirchhoff plates with internal rigid line supports, J. Sound and Vi‐

[10] Chopra I. 1974, Vibration of stepped thickness plates. International Journal of Me‐

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**Chapter 22**

**Image Processing with MATLAB and GPU**

MATLAB® (The MathWorks, Natick, MA, USA) is a software package for numerical comput‐ ing that can be used in various scientific disciplines such as mathematics, physics, electronics, engineering and biology. More than 40 toolboxes are available in the current release (R2013b released in September 2013), which include numerous built-in functions enhanced by access

Since images can be represented by 2D or 3D matrices and the MATLAB processing engine relies on matrix representation of all entities, MATLAB is particularly suitable for implemen‐ tation and testing of image processing workflows. The Image Processing Toolbox™ (IPT) includes all the necessary tools for general-purpose image processing incorporating more than 300 functions which have been optimised to offer good accuracy and high speed of processing. Moreover, the built-in Parallel Computing Toolbox™ (PCT) has recently been expanded and now supports graphics processing unit (GPU) acceleration for some functions of the IPT. However, for many image processing applications we still need to write our own code, either in MATLAB or, in the case of GPU-accelerated applications requiring specific control over

In this chapter, the first part is dedicated to some essential tools of the IPT that can be used in image analysis and assessment as well as in extraction of useful information for further processing and assessment. These include retrieving information about digital images, image adjustment and processing as well as feature extraction and video handling. The second part is dedicated to GPU acceleration of image processing techniques either by using the built-in PCT functions or through writing our own functions. Each section is accompanied by MAT‐ LAB example code. The functions and code provided in this chapter are adopted from the

> © 2014 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

GPU resources, in CUDA (Nvidia Corporation, Santa Clara, CA, USA).

MATLAB documentation [1], [2] unless otherwise stated.

Antonios Georgantzoglou, Joakim da Silva and

Additional information is available at the end of the chapter

Rajesh Jena

**1. Introduction**

http://dx.doi.org/10.5772/58300

to a high-level programming language.
