## 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.

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.

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 continuous.

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 uni-directionally 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 available in the literature, which validates the accuracy and reliability of the proposed technique.

## 2. Formulation

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:

Where

(2) |

The flexural rigidities

Where

(4) |

Where

Or in contraction form:

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

(7) |

Equation (7) may be written as:

(8) |

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

(9) |

Where

## 3. Method of solution

The displacement

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:

Where

at ξ=0 and ξ=1

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

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

The different value of

(15) |

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

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

(17) |

Where

and

From the orthogonality of the beam Eigen function,

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

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

Where the frame denotes differentiation with respect to η.

Where:

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

where the coefficients of the matrix

Where:

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,

Where

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:

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

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

At η=0 and η=1

Or in contraction form:

Where

The solution is found using 2N initial vectors

Where

## 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

Where

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.

[i] - *Y.K. Cheung and D. Zhou [15].

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

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.

## 5. Concluding remarks

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 angle-ply 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

## 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.

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

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

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

Where

and

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