Vibration Analysis of FG Plate

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Vibration Analysis of FG Plate

Narayanan. N. I 1,2,* , Sauvik Banerjee1, Akshay Prakash Kalgutkar1, T. Rajanna3 1Department of Civil Engineering, Indian Institute of Technology, Bombay 400076, India 2 Government College of Engineering, Kannur, Kerala -670563, India

3 B.M. S. College of Engineering, Bengaluru, Karnataka-560019, India

Abstract-In this article, vibration response of func- tionally graded material(FGM) plates are investigated by finite element formulation. By applying the Hamiltons principle, the governing equations of the FGM plates are derived based on the first-order shear deformation the- ory. The FGM plate is modelled by using 9-noded heter- osis element by incorporating the effect of rotary inertia and shear deformation. 9- noded heterosis plate element is used to formulate the elastic stiffness matrix and mass matrix. The results are also extracted from Abaqus CAE by using S8R5 shell elements. Free vibration analysis is done to obtain the different modes as well as the frequen- cies. Harmonic sine load is applied at the centre of the FGM plate to obtain a forced vibration response. Im- pulse forces of rectangular, triangular, and half-cycle sine shapes are applied on the top of the plate at the cen- tre and the Response spectra of C-Si C FGM plate is plot- ted.

Keywords-FGMs, Finite element method, heterosis plate element, Response spectra

  1. INTRODUCTION

    The diverse and potential applications of FGMs in aero- space, medicine, defence, energy, and other industries have attracted a lot of attention recently. The concept of function- ally graded materials (FGMs) were first demonstrated by a group of scientists in Japan in 1984during a space plane pro- ject[1]. Combination of materials used here served the pur- pose of a thermal barrier system capable of withstanding a surface temperature of 2000 K with a temperature gradient of 1000 K across a 10 mm thick section (Jha et al. [2]). Later, its applications have been expanded to also the components of chemical plants, solar energy generators, heat exchangers, nu- clear reactors, and high-efficiency combustion systems. The concept of FGMs has been successfully applied in thermal barrier coatings where requirements are aimed to improve thermal, oxidation and corrosion resistance. FGMs can also find application in communication and information tech- niques. Abrasive tools for metal and stone cutting are other important examples where the gradation of the surface layer has improved performance.

    It has been found from the literature that not many studies are done to the vibration analysis of functionally graded plates.

    B. Sidda Reddy et al. [3] carried out the free vibration analysis of functionally graded plates. The variations of the volume

    fractions through the thickness are assumed to follow a power-law function. The Reissener-Mindlin first-order shear deformation theory is very much appropriate for thick plates [4]. It was taken to analyze the behaviour of the plate sub- jected to free and forced vibration. They have developed ana- lytical formulations and solutions for the free vibration anal- ysis of functionally graded plates using higher-order shear de- formation theory (HSDT). The principle of virtual work was used to derive the equations of equilibrium and boundary con- ditions. Naviers technique was used to obtain the solutions for FGM plates. Jyoti Vimal et al. [5] have studied the free vibration analysis of functionally graded skew plates using the finite element method. The first-order shear deformation plate theory is used to consider the transverse shear effect and rotary inertia. The properties of functionally graded skew plates are assumed to vary through the thickness according to a power law. It is found that when the length to thickness ratio of functionally graded skew plates increases beyond 25, the variation in the frequency parameter is very negligible and also found that a volume fraction exponent that ranges be- tween 0 and 5 has a significant influence on the frequency. M.

    N. Gulshan Taj et al. [6] carried out a free vibration analysis of functionally graded material (FGM) skew plates subjected to the thermal environment. It was concluded that the volume fraction index and skew angle plays an important role in pre- dicting the vibration of FGM skew plate subjected to thermal load.

    J. N. Reddy [7] have studied theoretical formulation and FEM model based on TSDT for FGM plate. The formulation ac- counted for thermo-mechanical effects combining change with time and geometric nonlinearity. In this higher-order the- ory, transverse shear stress was expressed as a quadratic func- tion along with the depth. Hence this theory requires no shear correction factor. The plate was considered as the homoge- nous and material composition was varied along with the thickness. The Young's modulus was assumed to vary as per rule of the mixture in terms of the volume fractions of the ma- terial constituents. Hughes and Cohen [8] developed the het- erosis element and elemental equation. They derived lumped positive definite mass matrix, element matrix and load vector and method for finding critical time step. High-accuracy finite element for thick and thin plate bending is developed, based upon Mindlin plate theory.

    It has been found from the literature survey that not many re- searchers attempted to the vibration analysis of functionally graded plates. Further, we observed that many authors could

    model such problems with a stepped variation in material properties instead of continuous variation. This would have happened because of the limitations of the commercial soft-

    x

    x

    x

    x

    ware available. In this context, we felt that MATLAB code could be used for tailoring the continuous variation in material

    y

    y

    y

    (3)

    properties in FE Modelling. Hence MATLAB code was de- veloped for vibration analysis of FG plate. The analysis was carried out for C-Si C FGM plate with different volume frac- tion indices. The results are compared with Abaqus CAE by using S8R5 shell elements.

  2. PROBLEM FORMULATION

First-order shear deformation theory is used for plate for- mulation. Displacement variation is linear, across the plate thickness. But there is no change in plate thickness during de- formation. A further assumption is that the normal stress across the thickness is neglected. Properties are graded through the thickness direction which follows a volume frac- tion power-law distribution. The different elements of the plate are expected to undergo translational and rotational dis- placement. In the present work 9- noded heterosis element is used to discretize the plate.

    1. Strain-Displacement Relations

      The displacement field at any arbitrary distance z from the midplane based on the first-order shear deformation plate the- ory is given by

      up (x, y, z), vp (x, y, z), wp (x,y,z) u0 (x, y), v0 (x, y), w0 (x, y) z x (x, y), y (x, y), 0

      (1)

      where, u p, v p, w p are displacements in x, y and z directions respectively, u0, v0 and w0 are the associated midplane dis- placements along x, y and z axes respectively. and x and y are the rotations about y and x-axes respectively.

      The linear strain displacement relations are given by

      xl u0,x zx

      yl v0,y zy

      xy x y

      y x

      The strain-displacement field at any distance z as shown i Figure.1.

      zx

      u

      A

      x

      z B A

      C B w

      D

      C y

      D

      Figure 1. Deformed and un-deformed beam

    2. Finite element formulation

      In the current work, the FGM plate has been discretized using 9-noded heterosis element with 5-degree of freedom (dofs) at all the edge nodes and 4 dofs at the internal node as shown in the Figure 2. The serendipity shape functions have been used for the transverse dofs, w, and Lagrange shape function are used in the remaining dofs, u, v, x, and y

      8-N SE 9-N HE 9- N LE

      Node with u, v, w, x, and y degrees of free-

      dom

      Node with u, v, x, and y degrees of freedom

      Figure 2. Nodal configuration of the plate element

    3. Resultant Forces and moments.

      xyl u0,y v0,x zxy

      xzl w0,x x

      (2)

      The analysis of FG plate is carried out to establish the relation between the forces and strains by considering transverse shear

      yzl w0,y y

      terms.

      Constitutive matrix of the isotropic plate is

      where, xl, yl and xyl are the linear in-plane normal and

      Q11 Q12 0

      shear strains, xzl and xzl are transverse shear strains, z is the

      Q Q Q 0

      (4)

      distance of any layer from the middle plane of the plate and are the curvatures.

      12 11

      0 0 Q66

      where,

      Q11

      E 1 2

      , Q12

      E 1 2

      , Q66

      21

      The different participating element-level matrices such as elastic stiffness matrix ke , and consistent mass matrix me

      E

      E

      The material properties PZ (Elastic constants E, , density) at distance, z from the middle surface of the plate is

      have been derived using corresponding energy expression. The element elastic stiffness matrix and element mass matrix are derived using the following relations

      P P P P (z / h) 0.5n P P P Vf (5) 1 1

      z b t b b t b

      where, h is the plate thickness, t and b denotes the top and the bottom surface z / 2 ,n is material volume fraction

      index, Vf is volume fraction.

      ke

      T

      T

      1 1

      1 1

      BT CBJ dd

      (11)

      Stress-strain relationship is

      Q

      (6)

      me N IN J dd

      1 1

      (12)

      x y xy 0

      x y xy 0

      where, = , , T , = z

      The in-plane resultant forces and moments in the kth layer are evaluated as

      N zk

      In which, [I] is the inertia matrix

    4. Computer coding and Implementation

      A computer program is developed using MATLAB to imple- ment the finite element formulation and include all the neces- sary parameters to investigate the vibration behaviour of the

      N, M = 1, z dz

      k 1 zk1

      (7)

      FGM plate. In the present code, selective integration scheme is incorporated for the generation of the element stiffness ma-

      Resultant Transverse Shear Force on the kth layer is given by

      trix. The 3×3 Gauss quadrature rule is adopted to get the bend- ing terms and 2×2 Gauss rule is used to solve shear terms to

      Q N zk N zk Q Q

      xz xz

      xz xz

      avoid possible shear locking. The mass matrix is evaluated by

      dz

      44 45 xz

      using 3×3 Gauss rule [12].

      dz

      dz

      Qyz

      k1 z

      yz k1 z Q45

      Q55 yz

      (8)

      k1 k1

      2z hk

    5. Formulation of Dynamic problems

      Stiffness matrix is validated by bending problems and mass

      Q44 G13t G13b

      2h

      G13b

      matrix is validated through vibration problems. In order to

      2z hk

      validate the formulation of mass matrix, one has to solve a free vibration problem by incorporating the validated elastic

      Q55 G23t G23b

      2h

      G23b

      stiffness matrix. The standard governing equation in matrix

      Q45 0

      (9)

      form for the deflection problem is

      Ke q F

      (13)

      The constitutive relation for FGM plate is given by

      F is the nodal load vector, Ke is the system elastic stiff-

      N

      C

      (10)

      ness matrix. For a given set of loads, the displacement q

      Where,

      N Nx, Ny, Nxy, Mx, My, Mxy, Qxz, QyzT rep- resents the in-plane stress resultants (N), out of plane bending moments (M) and shear resultants (Q). Here, [C] is the con- stitutive matrix [9] of the FGM plate. To compensate for the parabolic shear stress variation across the thickness of the

      plate, a correction factor of 5/6 is used in the shear-shear cou-

      can be

      determined using the above equation. If the displacement vector is validated, it ensures the correctness of formulation and coding of the stiffness matrix.

      The standard governing equation in matrix form for the free vibration problem is

      pling components of the constitutive matrix [10]. Using

      Mq Ke q F

      (14)

      Green-Lagranges strain-displacement expression [11], the linear strain-displacement matrix[B] have been worked out.

      The standard governing equation in matrix form for the force vibration problem is

      Mq Cq Ke q F

      (15)

      M ,Ke and C represents global mass matrix, global

      Table2. Variation of fundamental frequency with n values Cantilever FGM plate-comparison

      stiffness matrix and damping matrix respectively.

      C M Ke

      n

      9-NHE

      Simulation

      He 2001)

      0

      25.37

      27.21

      25.58

      0.2

      29.14

      31.45

      29.87

      0.5

      32.27

      33.86

      32.84

      1

      33.90

      36.79

      35.33

      5

      38.48

      42.14

      40.97

      15

      42.16

      45.34

      43.97

      100

      45.64

      48.12

      46.12

      1000

      46.08

      48.94

      46.55

      n

      9-NHE

      Simulation

      He 2001)

      0

      25.37

      27.21

      25.58

      0.2

      29.14

      31.45

      29.87

      0.5

      32.27

      33.86

      32.84

      1

      33.90

      36.79

      35.33

      5

      38.48

      42.14

      40.97

      15

      42.16

      45.34

      43.97

      100

      45.64

      48.12

      46.12

      1000

      46.08

      48.94

      46.55

      (16)

      where, and are the Rayleigh damping coefficients. From

      this, we can solve the forced vibration problem. From this, we can solve the force vibration problem using Newmark-beta method.

  1. RESULTS AND DISCUSSION

The properties of FGM plates are graded through the thick- ness direction according to a volume fraction power law dis- tribution (Figure 3).

    1. Free vibration analysis

      The heterosis element is used in the code for free vibration analysis. For validation of the present code, the data available for the functionally graded plate aluminium oxide titanium alloy of size 0.4m x 0.4m x 0.005m available in the literature of He et al.[13] is used. In numerical simulation by Abaqus, S8R5 element has been used. Table 1. shows the material properties. Table 2. validated the code with literature and sim- ulation.

      Table1. Material properties of Aluminium Oxide Titanium alloy FGM plate

      Material

      E(N/m2)

      (kg/m3)

      Ti-6A1-4V

      (ceramic)

      122.56 x 109

      4429

      0.2884

      Aluminium

      oxide

      349.55 x 109

      3750

      0.26

      Figure 3. Variation of volume fraction with the non-dimensional thick- ness

      The present code is validated with results of He et al. (2001). The simulation results are also in good agreement with results obtained from FEM coding. This ensures the correctness of the formulation of the stiffness and mass matrix.

    2. Free Vibration Analysis of C-Si C Plate

      The analysis is done for C-Si C plate (0.5×0.5×0.001m). Ma- terial properties are given in Table 3. Convergence results are shown in Figure 4. First four mode of vibration shown in Figure. 5 by Abaqus using S8R5 shell element .Frequency of Vibration is minimum for carbon plate as shown in Table 4.

      Table3. Material Properties C-Si C FGM plate

      Material

      E(G

      Pa)

      (Kg/m3)

      Si-C (Ce-

      ramic)

      320

      0.3

      3220

      C(Metal)

      28

      0.3

      1780

    3. Forced vibration analysis

      Forced vibration analysis was carried out at the centre of the plate using harmonic sine loading and different impulse load- ings.

      1. Harmonic Sine Wave Loading.

        A harmonic force P(t)= P0 sin( t ) load is applied at the centre

        of the plate(Figure6(a)-6(b), where P0 is the amplitude or peak value of the force and is the forcing frequency.

        T 2/ is the forcing period of the FGM plate P0 =1N and

        = 2f where is circular frequency and f is natural fre-

        quency of the Plate.. Table 5 compares the un-damped and damped cases. Figure 6(c) for simply supported plate. The maximum displacement at the center of the fixed plate as shown in Figure 6(d) is less than that of simply supported plate. Fig. 6 shows that maximum displacement at the centre of the plate increases with the material index (n value)

        Figure 4. Convergence of fundamental frequency of simply supported C-Si C FGM plate for first 4 modes (n=2)

        Figure 5. First 4 mode shapes of simply supported C-Si C FGM plate (n=2)-Simulation

        Table 4. Variation of the natural frequencies (Hz) of FGM simply supported Square Plate for different k values. 22×22 mesh-Heterosis element(FEM)

        Table5. Displacement of simply supported C-Si C plate for differ- ent n values

        Power-law in- dex

        Maximum Displacement at Centre(m)

        CCCC

        SSSS

        Undamped

        Damped

        Undamped

        Damped

        n=0 (Si C)

        4.759x

        10-5

        4.73x

        10-5

        1.165 x 10-4

        1.006 x 10-4

        n=2

        1.722 x10-

        4

        1.71 x 10-4

        3.142 x 10-4

        3.035 x 10-4

        n=15

        2.796 x10-

        4

        2.68 x 10-4

        5.222x

        10-4

        5.201x

        10-4

        n=1000(C)

        5.442 x10-

        4

        5.41 x 10-4

        1.229×10-3

        1.199x

        10-3

        Mode no

        k=0 Si C

        n=2

        k=15

        k=1000

        Carbon

        1

        37.91

        27.39

        22.2114

        15.08

        2

        94.77

        60.84

        51.9549

        37.7

        3

        94.77

        60.84

        51.9549

        37.7

        4

        151.63

        98.77

        83.7656

        60.33

        5

        189.55

        121.26

        103.718

        75.41

        6

        189.55

        123.04

        104.522

        75.41

        7

        246.41

        158.98

        135.449

        98.03

        8

        246.41

        158.98

        135.449

        98.03

        9

        322.27

        202.97

        174.955

        128.21

        10

        322.27

        202.97

        174.955

        128.21

        Figure 6. Response of C-Si C FGM plate for different k values using heterosis element.

      2. Impulse loading

A very large force that acts for a very short time but with a time integral that is finite is called an impulse force. Impulse forces of rectangular, half-cycle sine, triangular shapes each with the same value of maximum force 1N is applied at the centre of the plate. The response behavior of FGM plate is studied for material index(k) value=2. td is pulse duration. The Response spectra of the FGM plate with material index n=2 is shown in Figure 7. Tn is the natural time period of vibration of the plate and u st0 is the static deflection of the plate. Static deflection is 1.438e-4m and Natural time period is 0.0365 sec. Table 6 presents the variation of deformation response factor (Rd) with td/T n values (n=2).The present results are in good agreement with the available literature [14].

Figure 7. Response spectra of simply supported C-Si C FGM Plate (material index n=2)

Table7. Variation of deformation response factor (Rd) with td/T n values (n=2)

Rd= u0/ u st0

td/T n

Rectangular Loading

Half Sine

Loading

Trian- gular Loading

0

0

0

0

0.5

1.569

0.982

0.863

0.75

1.876

1.469

1.3

1

1.901

1.701

1.52

1.5

1.901

1.5

1.298

2

1.901

1.25

0.93

2.5

1.901

1.071

0.996

3

1.901

1.15

1.148

3.5

1.901

1.111

1.103

4

1.901

1.108

1.003

4. CONCLUSIONS

In the present study, an FE solution is obtained for free and forced vibration analysis of FG plates using heterosis element. The analysis is carried out by developing a computer program in MATLAB. A 9- noded heterosis element is used to model the FGM plate. The heterosis element exhibits improved char- acteristics as compared to the 8- noded serendipity and 9- noded Lagrange elements. It offers a high level of accuracy for extremely thin plate configurations. Convergence study has been carried out for ensuring the convergence of the nu- merical results. The results are also extracted from Abaqus CAE by using S8R5 shell elements andare in very good agreement with the developed elements. Free vibration anal- ysis is done to study the different modes as well as frequen- cies. It is observed that free vibration response is minimum

for carbon and maximum for Silicon carbide plate. The cen- tral deflection of the plate increases with increase in volume fraction index for all types of boundary conditions. From the Response spectra, it is clearly understood that if the pulse du- ration (td) is longer than Tn/2, the overall maximum defor- mation occurs during the pulse. Then the pulse shape is of great significance. For the larger value of td/Tn, the overall maximum deformation is influenced by the rapidity of the loading. The rectangular pulse in which the force increases suddenly from zero to maximum show the large deformation. The triangular pulse in which the increase in the force is ini- tially slowest among the three pulses produces the smallest deformation. The half-cycle sine pulse in which the force ini- tially increases at an intermediate rate causes deformation that for many values of td/Tn is larger than the response of the tri- angular pulse. Sufficient duration steep loading produces a magnification factor of 2 and gradual loading increase results in a magnification factor of 1.

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