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

Written By

Wael A. Al-Tabey

Submitted: 19 September 2013 Published: 08 September 2014

DOI: 10.5772/57384

From the Edited Volume

MATLAB Applications for the Practical Engineer

Edited by Kelly Bennett

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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.

Figure 1.

A rectangular laminated plate with variable thickness

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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:

2MXx222MXYxy+2MYy2=ρh(y)2wot2E1

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:

MX=D112wox2D122woy22D162woxyMY=D122wox2D222woy22D262woxyMXY=D162wox2D262woy22D662woxy}E2

The flexural rigidities Dij of the plate are given by:

Dij=13h3(y)ho3k=1n[(Q¯ij)]k(hok3hok13),i,j=1,2,3,.........E3

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 Qijk¯ 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 Qijk by proper coordinate relationships they can be expressed in terms of the engineering notations as:

Qij=[Q11Q12Q13Q12Q22Q23Q13Q23Q66]=[E11(1υ12υ21)υ21E11(1υ12υ21)0υ21E11(1υ12υ21)E22(1υ21υ12)000G12]E4

Where E11, E22 are the longitudinal and transverse young's moduli parallel and perpendicular to the fiber orientation, respectively and G12 is the plane shear modulus of elasticity, υ12 and υ21 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:

D114wox4+4D164wox3y+2(D12+2D66)4wox2y2+4D264woxy3+D224woy4=ρh(y)2wot2E5

Or in contraction form:

D11Wxxxx+4D16Wxxxy+2(D12+2D66)Wxxyy+4D26Wxyyy+D22Wyyyy=ρh(y)WttE6

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

D11{2x2(h3(y)ho3W,xx)}+2(D12+2D66){2xy(h3(y)ho3W,xy)}+D16{2x2(h3(y)ho3W,xy)}+4D26{2y2(h3(y)ho3W,xy)}+D22{2y2(h3(y)ho3W,yy)}=moh(y)hoWttE7

Equation (7) may be written as:

D11h3(y)ho3Wxxxx+(2(D12+2D66)ho3)h3(y)yWxxy+(2(D12+2D66)ho3)h3(y)Wxxyy+D16h3(y)ho3Wxxxy+(4D26ho32h3(y)y2)Wxy+4D26ho3h3(y)Wxyyy+8D26ho3h3(y)yWxyy+(D22ho32h3(y)y2)Wyy+D22ho3h3(y)Wyyyy+2D22ho3h3(y)yWyyy=moh(y)hoWttE8

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

ψ11a4Wξξξξ+2ψ2h3(η)1a2bh3(η)ηWξξη+2ψ21a2b2Wξξηη+ψ31a3bWξξξη+4ψ41ab3Wξηηη+1ab4ψ4h3(η)2h3(η)η2Wξη+8ψ4h3(η)1ab2h3(η)ηWξηη+1b21h3(η)2h3(η)η2Wηη+1b4Wηηηη+2h3(η)1b3h3(η)ηWηηη=moD22ho2h2(η)WttE9

Where β=ab is the aspect ratio, ξ=xa, η=yb, ψ1=D11D22, ψ2=(D12+2D66)D22, ψ3=D16D22 and ψ4=D26D22.

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3. Method of solution

The displacement W(ξ,η,t)=W(ξ,η)eiωt can be expressed in terms of the shape functionXi(ξ), chosen a prior; and the unknown function Yi(η)as:

W(ξ,η,t)=i=0NXi(ξ)Yi(η)eiωtE10

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:

(2FT1μi(FR1FR2μi2)(μi4+FR1FT1)sinμi+2FT1μi(FR1+FR2)(μi4+FR1FT1)μicosμi)A3E11

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

2Xi(ξ)ξ2=03Xi(ξ)ξ3=0}E12

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 (1υ2)D22 and for υ=0.3, it can be just approximated by ED22. The solution of the beam equation is given as:

Xi(ξ)=A1sin(μiξ)+A2cos(μiξ)+A3sinh(μiξ)+A4cosh(μiξ)E13

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

Φi=sinhμi+sinμicoshμicosμiXi(ξ)=sin(μiξ)sinh(μiξ)+1Φi(cos(μiξ)cosh(μiξ))}E14

The different value of μi are the roots of equation:

2cos(μi)cosh(μi)+cos2(μi)+sin2(μi)sinh2(μi)+cosh2(μi)sinh(μi)cosh(μi)+sin(μi)cosh(μi)sinh(μi)cos(μi)sin(μi)cos(μi)=0E15

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

μi=(i+0.5)π, i= 0, 1, 2, 3, ..........E16

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

i=0Nj=0Mβ4f3(η)Yi,ηηηη+2β3af1(η)f3(η)Yi,ηηη+(2ψ2β2f3(η)cijaij+8ψ4β2af1(η)f3(η)bijaij+β2a2f2(η)f3(η))Yi,ηη+(2ψ2βaf1(η)f3(η)cijaij+ψ3βf3(η)dijaij+4ψ4βa2f2(η)f3(η)bijaij+4ψ4β3f3(η)bijaij)Yi,η+(ψ1f3(η)eijaijλ2)Yi=0E17

Where λ2=moω2a4D22, f1(η)=1h3(η)h3(η)η, f2(η)=1h3(η)2h3(η)η2, f3(η)=ho2h2(η),

aij=01XiXjdξ,

bij=01XjXi,ξdξ,

cij=01XjXi,ξξdξ,

dij=01XjXi,ξξξdξ

and eij=01XjXi,ξξξξdξ.

From the orthogonality of the beam Eigen function, aij=eij=0 for ij, 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:

ddη{Yk}ij=[Ai]k{Yk}ijE18

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

i=0Nj=0MEijYi////+(O1)ij(O0)ijYi///+(O2)ij(O0)ijYi//+(O3)ij(O0)ijYi/+(O4)ijλ2(O0)ijYi=0E19

Where the frame denotes differentiation with respect to η.

Where: (O0)ij=β4t1(η)Eij, (O1)ij=2β3at2(η)Eij, (O2)ij=(2ψ2β2t1(η)cijaij+8ψ4β2at2(η)bijaij+β2a2t3(η))(O3)ij=(2ψ2βat2(η)cijaij+ψ3βt1(η)dijaij+4ψ4βa2t3(η)bijaij+4ψ4β3t1(η)bijaij), (O4)ij=ψ1t1(η)eijaij,

[Eij]=i×j Unit matrix,

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

where the coefficients of the matrix [Ai]kin equation (18), in general, are functions of η and the Eigen value parameter λ. The vector Yk is given by:

Yk=[Y¯1Y¯2Y¯iY¯N]E20

Where:

Y¯i=[YiYi/Yi//Yi///]E21

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,

{Yi}j=[Bi]j{Yi1}jE22

Where [Bi]jis 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:

wo=0D12a22woξ2D22b22woη22D26ab2woξη=0}E23

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

At η=0 and η=1

Yi=02Yiη2=i=0N2ψ4bijβaijYiηψ5cijβ2aijYi}E24

Or in contraction form:

Yi=0Yi//=i=0NCF1bijaijYi/CF2cijaijYi}E25

Where CF1=2ψ4β, CF2=ψ5β2, ψ5=D12D22

The solution is found using 2N initial vectors Y0 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]:

[S]=i=12NCiSiE26

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

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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 λ=(ρhoω2a4/D22)1/2. To illustrate the solution, a plate with linear variable thickness, h(y)is used (see Appendix A).

h(η)=1+ΔηE27

Where Δ is the tapered ratio of plate given by Δ=(hbh0)/h0, (h0) 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).

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

Table 1.

Material properties of unidirectional composite

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.

Δ = 0.0
N λ1 λ2 λ3 λ4 λ5 λ6
1 70.4212 70.7012 140.4421 173.5211 180.6231 235.6753
2 70.4212 70.7012 140.4421 173.5211 180.6231 235.6753
3 70.2882 70.5827 140.2496 173.2098 180.2833 235.3197
4 70.2882 70.5827 140.2496 173.2098 180.2833 235.3197
5 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 (θ=45), (β=1.0)

*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 (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).

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.

Δ = 0.5
β = 0.5
N λ1 λ2 λ3 λ4 λ5 λ6
1 80.2177 82.5621 155.9665 188.6633 194.6253 251.7333
2 80.2177 82.5621 155.9665 188.6633 194.6253 251.7333
3 79.8625 82.0025 155.3232 188.1111 194.1002 251.2035
4 79.8625 82.0025 155.3232 188.1111 194.1002 251.2035
5 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 thickness plate (Δ=0.5), (β=0.5).

Δ = 0.5
β = 1.0
N λ1 λ2 λ3 λ4 λ5 λ6
1 72.7575 73.8666 143.3334 175.4963 183.7825 240.7621
2 72.7575 73.8666 143.3334 175.4963 183.7825 240.7621
3 72.1199 73.4444 142.9019 175.0024 183.1121 240.0159
4 72.1199 73.4444 142.9019 175.0024 183.1121 240.0159
5 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 thickness plate (Δ=0.5), (β=1.0)

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

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.

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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 (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.

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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(y)=ho(1+yc)E28

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

hoc=hbc+bE29

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

h(y)=ho+(hbho)byE30

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

h(y)=hb at y=b,

h(y)=ho+(hbho)by at y=y,

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(η)=ho+(hbho)ηE31
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Appendix (B)

MATLAB code

Composite coefficients (function programs)

References

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Written By

Wael A. Al-Tabey

Submitted: 19 September 2013 Published: 08 September 2014