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International Journal of Current Engineering and Technology ©2018 INPRESSCO®, All Rights Reserved E-ISSN 2277 – 4106, P...

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International Journal of Current Engineering and Technology ©2018 INPRESSCO®, All Rights Reserved

E-ISSN 2277 – 4106, P-ISSN 2347 – 5161 Available at http://inpressco.com/category/ijcet

Research Article

Free Vibration Analysis of Functionally Graded Material Plates with Various Types of Cutouts using Finite Element Method Mohammad Mahdi Khamsi* Gilan, Rasht Golsar Street, 102 street, No. 32, 4165643795, Iran Received 05 Aug 2018, Accepted 08 Oct 2018, Available online 11 Oct, Vol.8, No.5 (Sept/Oct 2018)

Abstract This paper presents free vibration analysis of functionally graded plates of different shapes with different cutouts. In order to investigate the free vibration of functionally graded plates, subspace method is used. The material properties of the plates are assumed to vary according to a power law distribution in terms of the volume fraction of the constituents. In this paper, rectangular, trapezoidal and circular plates with cutouts are studied and the effects of volume fraction index, thickness ratio and different external boundary conditions on the natural frequencies of plates are studied in detail. Keywords: Different cutouts, Free vibration, Subspace method. 1. Introduction 1 In

functionally graded materials, the volume fraction of two or more constitute materials are varied continuously as a function of position along certain dimension of the structure. The continuous change in the microstructure of the functionally graded materials advances them from the composite materials. Mismatch of mechanical properties across an interface, de-bonding of constituents due to high thermal loading, presence of residual stresses due to difference in coefficient of thermal expansion of fiber and matrix of composite materials are common problems seeing in the use of composite materials. In FGM’s these problems are avoided or gradually reduced by varying the volume fraction of constituents of FGM. The concept of FGM was first considered in Japan in 1984 during a space plane project, where a combination of materials used would serve the purpose of a thermal barrier capable of withstanding a surface temperature of 2000 K and a temperature gradient of 1000 K across a 10 mm section (Azom.com). Science the concept of FGM was first proposed, FGM’s are extensively studied by researchers, who have mainly focused on their static, dynamic and thermal behavior. The problems of free vibrations, wave propagation and static deformations in FGM beams were solved using an especially developed finite element method accounting for power law and other alternative variations of elastic and thermal properties in the thickness direction (A. *Corresponding author’s ORCID ID: 0000-0000-0000-0000 DOI: https://doi.org/10.14741/ijcet/v.8.5.12

Chakraborty, 2003). The modal employed a first order shear deformation theory of beam. Three methods were used for the static and dynamic analyses of square thick FGM plates with simply supported edges (L. F. Qian, 2005). The methods employed in the paper included a higher order shear deformation theory and two novel solutions for FGM structures. According to this paper, the application of the normal deformation theory may be justified if the in-plane size to thickness is equal to or smaller than 5. Researchers have also turned their attention to the vibration and dynamicJ. Yang,, 2003 response of FGM’s structures (H.L. Dai, 2005; X. L. Huang, 2004). Chen et al presented exact solutions for free vibration analysis of rectangular plates using Bessel functions with three edges conditions. Liew et al studied the free vibration analysis of functionally graded plates using the element-free Kp-Ritz method. They studied the free vibration analysis of four types of functionally graded rectangular and skew plates. Hiroyuki Matsunaga presented in his paper, the analysis of natural frequencies and buckling of FGM’s plates by taking into account the effects of transverse shear and normal deformations and rotary inertia. Atashipour et al presented a new exact closed- form procedure to solve free vibration analysis of FGM’s rectangular thick plates based on the Reddy’s third-order shear deformation plate theory. For plates with cutouts, Chai presented finite element and some experimental results on the free vibration of symmetric composite plates with central hole. Sakiyama and Huang proposed an approximate

1293| International Journal of Current Engineering and Technology, Vol.8, No.5 (Sept/Oct 2018)

Mohammad Mahdi Khamsi

Free Vibration Analysis of Functionally Graded Material Plates with Various Types of Cutouts..

method for analyzing the free vibration of square plate with different cutouts. Liu, et al studied static and free vibration analyses of composite plates with different cutouts via a linearly conforming radial point interpolation method. Maziar and Iman studied the effect of relative distance of cutouts and size of cutouts on natural frequencies of FG plates with cutouts. From the review of the above literature it is observed that very little research and analysis work has been done yet on the natural frequencies of the FG plates with cutouts. The study presents here, the effect of volume fraction index, thickness ratio and different external boundary conditions on the natural frequencies of FG (Al/Al₂O₃) plates (such as rectangular, trapezoidal and circular) with cutouts.

More than 2000 nodes might be used in calculation work.

2. Functionally Graded Material Assets

4.1 Over-all Explanations

A functionally graded material plate as shown in fig. 1 is considered to be a plate of uniform thickness that is made of ceramic and metal. The material property is to be graded through the thickness according to a PowerLaw distribution that is

The subspace iteration method was developed by K. J. Bathe for the solution of frequencies and mode shapes of structures and in particular for the earthquake analysis of buildings and bridges. The original development of subspace method is based on vector simultaneous iterations, as proposed by Bauer and Rutishauser. The subspace method uses a starting subspace close to the subspace of interest that can lead to a very fast solution. The subspace method was revisited by K. J. Bathe in 2012.

( )

) ,

( ( ) (

(1a) (1b)

),

Where P represents the effective material property, denotes the ceramic and metal properties respectively, is the volume fraction of the ceramic, h is the thickness of the plate, and n is the volume fraction index.

Fig. 2 SOLID 185 4. Hypothetical Design

4.2 The Elementary Calculations Let K and M be the stiffness and mass matrices of a finite element system with n degree of freedom, and consider the generalized symmetric eigenvalue problem =λ

(2)

We seek the smallest p eigenvalues λᵢ, i=1,…..,p, and corresponding eigenvectors φᵢ, i=1,…..p, with the ordering …..

(3)

which satisfies =

, i=1,…..,p

(4)

and the Kronecker Delta relationships Fig. 1 Functionally Graded Plate 3. Functionally Graded Plate Rudiments The finite element software (ANSYS) is used with the aim of analyzing. In addition SOLID 185 is used for modeling general 3-D solid structures. It allows for prism and tetrahedral degenerations when used in irregular regions. The element is defined by eight nodes having three degree of freedom at each node.

(5) 5. Arithmetical Consequences and Debate The material properties used in the present study are as follows: ,

,

,

,

1294| International Journal of Current Engineering and Technology, Vol.8, No.5 (Sept/Oct 2018)

Mohammad Mahdi Khamsi

,

Free Vibration Analysis of Functionally Graded Material Plates with Various Types of Cutouts..

,

,

,

,

,

,

,

,

, ,

In order to show the accuracy of methodology used for free vibration analysis of FG plates with cutouts, the fundamental natural frequencies of different plates (such as rectangular, trapezoidal and circular) with different cutouts (such as circular and non-circular) are compared with the solutions presented by J. Maziar and R. Iman . The material properties used in the convergence study are as follows:

, 5.1 Isotropic Plates Free vibration analysis of a simply supported square plate with a square hole is analyzed in table 1. The geometry and material properties of plate are a=10, size ratio d/a=0.5, thickness ratio h/a=0.01, Young modulus E=200 GPa and Density .

Table 1 Comparison of non-dimensional frequencies of isotropic square plate with square cut-out at the center (simply supported for external boundaries, Mode 1 2 3 4 5 6 7 8 9 10

Present 4.9955 6.6133 6.6284 8.8010 9.0946 11.147 11.149 12.577 13.987 14.836

Liu et al [20] 4.9717 6.4810 6.4821 8.5509 8.8656 10.720 10.767 12.045 13.370 14.180

⁄ (

)

)

Huang et al [21] 4.839 6.435 6.440 8.492 8.875 10.81 10.83 12.29 13.53 14.11

Iman et al [15] 4.9214 6.4347 6.4347 8.5859 8.6863 10.734 10.734 12.235 13.329 14.399

Free vibration analysis of a simply supported square plate with a circular hole is analyzed in table 2. The geometry and material properties of plate are a=10, radius to length ratio r/a=0.1, thickness ratio h/a=0.01, Young modulus E=200 GPa and Density . Table 2 Comparison of non-dimensional frequencies of isotropic square plate with circular cut-out at the center (simply supported for external boundaries, Mode 1 2 3 4 5 6 7 8 9 10

Present 4.540 7.223 7.231 9.124 10.23 10.56 11.79 11.80 13.54 13.56

Liu et al 6.149 8.577 8.634 10.42 11.41 11.84 12.83 12.84 -

⁄ (

Huang et al 6.240 8.457 8.462 10.23 11.72 12.30 13.04 13.04 -

)

) Iman et al 6.1666 8.6197 8.6300 10.478 11.521 12.017 12.964 12.964 -

5.2 Rectangular Plates Rectangular plates with multiple cutouts are shown in fig.2

Fig. 3 Rectangular plates with multiple cutouts

Fig.2 Rectangular plates

Table 3 shows the comparison of natural frequencies of FG rectangular plate of side ratio a/b=2 having two holes of radius ratio r/b=0.15 and center to center distance ratio e/b=0.5, 0.6 and 0.7. Thickness of the

1295| International Journal of Current Engineering and Technology, Vol.8, No.5 (Sept/Oct 2018)

Mohammad Mahdi Khamsi

Free Vibration Analysis of Functionally Graded Material Plates with Various Types of Cutouts..

plate is h=0.05. Comparison of FG rectangular plate with three holes having same side ratio, radius ratio of holes, thickness of the plate and e/b=0.7 is shown in table 4. Table 5 shows the comparison of natural frequencies of FG rectangular plate having same side

ratio and thickness with four holes of radius ratio r/b=0.1 and center to center distance ratio e/b=0.7 and f/b=0.4. Table 6 shows the comparison of natural frequencies of FG rectangular plate of same side ratio, thickness with two square cutouts of side ratio d/b=0.1 and e/b=0.7.

Table 3 Comparison of natural frequencies of FG rectangular plate with two circular cutouts (fully clamped for external boundaries)

e/b 0.5 Iman[15] 0.6 Iman[15] 0.7 Iman[15]

1

2

3

4

Mode 5

6

7

8

9

512.55 516.59 507.54 510.71 503.58 503.90

648.11 642.85 650.74 654.29 642.15 663.07

805.35 826.80 792.53 823.05 785.82 832.46

1098.0 1147.6 1107.1 1153.4 1091.9 1157.3

1251.9 1287.4 1203.2 1247.1 1110.9 1183.6

1287.6 1288.8 1213.2 1278.8 1165.3 1269.4

1466.3 1526.9 1447.1 1525.7 1440.4 1519.3

1560.2 1598.1 1584.9 1671.5 1598.7 1725.6

1801.6 1817.7 1748.7 1818.6 1735.0 1817.4

10 1979.4 1904.3 1903.6 -

Table 4 Comparison of natural frequencies of FG rectangular plate with three circular cutouts (fully clamped for external boundaries) Mode 1

2

3

4

5

6

7

8

9

10

0.7

496.99

607.47

931.21

1135.2

1221.3

1268.9

1436.0

1719.8

1727.1

1850.5

Iman[15]

500.45

614.97

931.33

1161.6

1249.9

1286.0

1458.8

1777.2

1800.5

2103.7

e/b

Table 5 Comparison of natural frequencies of FG rectangular plate with four circular cutouts (fully clamped for external boundaries) Mode 1

2

3

4

5

6

7

8

9

10

0.7

487.08

620.94

756.67

1081.0

1250.0

1348.1

1387.3

1538.4

1765.9

1879.2

Iman[15]

486.40

633.55

801.23

1136.3

1251.1

1435.4

1489.5

1679.3

1890.0

2035.2

e/b

Table 6 Comparison of natural frequencies of FG rectangular plate with two square cutouts (clamped-free for external boundaries) Mode 1

2

3

4

5

6

7

8

9

10

0.7

473.62

496.04

602.99

769.94

1017.0

1242.4

1275.0

1364.4

1402.5

1435.0

Iman[15]

470.74

476.14

541.35

837.44

922.06

1087.4

1095.6

1136.3

1245.3

1375.9

e/b

The variation of natural frequencies with the volume fraction exponent for Al/Al₂O₃ FG rectangular plates of side ratio a/b=2 and thickness ratio h/b=0.04, 0.06 and 0.08, having two holes of radius ratio r/b=0.1 and center to center distance ratio e/b=0.8 are shown in tables 7, 8 and 9 respectively. The results for first ten modes are computed. For the FG plates with fully clamped external boundary condition, the frequencies in all ten modes decreases as the volume fraction index increases. This is expected, because a large volume fraction index means that a plate has a smaller ceramic component and thus its stiffness is reduced. Similar

results are also observed for the clamped-free and clamped-simply supported external boundary conditions. From tables 7, 8 and 9 it is also observed that natural frequencies slowly improve as the thickness ratio h/b increases from 0.04 to 0.06. Table 10 shows that the variation of natural frequencies for first five modes of Al/Al₂O₃ FG rectangular plate (h/b=0.04) with two holes (r/b=0.1) for fully clamped, clamped-free and clamped-simply supported external boundary conditions are quite close to each other.

1296| International Journal of Current Engineering and Technology, Vol.8, No.5 (Sept/Oct 2018)

Mohammad Mahdi Khamsi

Free Vibration Analysis of Functionally Graded Material Plates with Various Types of Cutouts..

Table 7 Variation of natural frequencies with the volume fraction index n for Al/Al₂O₃ FG rectangular plates with two circular holes (fully clamped for external boundaries)

1

2

3

4

Mode 5

6

7

8

9

10

h/b 0.04

n 0 0.5 1 2

947.42 820.79 656.11 667.44

1225.6 1017.3 838.38 819.91

1627.8 1352.4 1102.9 1066.8

2258.9 1872.3 1529.8 1449.8

2270.6 1909.3 1568.4 1521.0

2492.6 2023.5 1694.0 1614.8

2906.7 2460.9 2002.3 2032.1

3125.1 2597.4 2126.4 2103.1

3475.6 2938.5 2450.5 2390.5

3920.0 3252.9 2677.3 2423.1

0.06

0 0.5 1 2

1323.9 1077.4 887.06 729.59

1707.5 1349.1 1082.7 918.40

2231.2 1764.8 1435.1 1204.7

3069.7 2425.3 1971.8 1666.3

3103.8 2516.1 2028.5 1690.1

3312.5 2644.4 2156.3 1787.9

3940.6 3211.7 2610.6 2140.3

4114.2 3335.5 2739.5 2343.7

4231.2 3736.2 3125.4 2613.6

4685.3 3778.9 3383.6 2827.6

0.08

0 0.5 1 2

1675.9 1292.7 1005.6 878.81

2132.4 1621.6 1274.2 1081.5

2796.0 2120.1 1683.8 1423.5

3710.6 2910.3 2316.4 1947.1

3855.2 2934.5 2333.1 1961.4

4067.8 3131.2 2516.1 2101.3

4133.9 3743.8 2974.8 2495.3

4829.2 3782.8 3217.0 2693.0

5179.5 4035.7 3412.0 2884.1

5663.5 4504.5 3557.2 2966.0

Table 8 Variation of natural frequencies with the volume fraction index n for Al/Al₂O₃ FG rectangular plates with two circular holes (clamped-free for external boundaries) 1

2

3

4

Mode 5

6

7

8

9

10

h/b 0.04

n 0 0.5 1 2

865.00 716.37 582.40 567.73

915.63 742.00 607.83 582.81

1063.9 919.51 720.84 743.11

1378.6 1155.7 637.92 943.25

1783.1 1490.2 1208.1 1195.0

2216.2 1826.6 1504.5 1421.6

2310.9 1869.8 1553.2 1488.0

2398.9 1986.8 1616.4 1549.3

2484.0 2108.8 1712.3 1786.9

2767.8 2297.8 1899.8 1878.5

0.06

0 0.5 1 2

1210.5 969.48 767.75 634.85

1271.8 1006.3 786.47 660.96

1458.7 1184.3 967.83 816.72

1888.7 1514.0 1219.9 1046.9

2427.5 1943.6 1575.1 1339.7

2993.1 2406.5 1943.8 1577.5

3067.9 2448.9 1988.0 1620.9

3088.0 2587.6 2096.6 1770.4

3290.8 2774.7 2238.3 1828.5

3378.8 2788.4 2440.9 2052.6

0.08

0 0.5 1 2

1530.6 1154.4 889.35 784.29

1607.5 1198.5 924.58 802.90

1869.0 1421.8 1101.6 951.64

2400.0 1818.3 1419.8 1211.8

3082.3 2328.3 1834.8 1579.1

3088.5 2791.9 2242.7 1915.3

3645.6 2836.6 2304.4 1961.5

3728.2 2939.6 2453.7 2091.5

3825.4 3132.6 2518.4 2154.9

4132.8 3261.8 2558.8 2176.5

Table 9 Variation of natural frequencies with the volume fraction index n for Al/Al₂O₃ FG rectangular plates with two circular holes (clamped-simply supported for external boundaries) 1

2

3

4

Mode 5

6

7

8

9

10

h/b 0.04

n 0 0.5 1 2

947.24 820.68 656.15 657.42

1224.8 1016.9 838.38 814.09

1626.3 1351.7 1102.7 1065.0

2258.2 1871.3 1529.6 1446.6

2268.8 1908.9 1568.3 1544.4

2491.6 2022.8 1693.9 1588.6

2904.2 2459.4 2001.8 2025.9

3121.6 2595.7 2126.1 2141.1

3471.2 2935.5 2449.1 2375.7

3915.7 3250.9 2676.4 2440.4

0.06

0 0.5 1 2

1323.4 1077.2 886.91 720.36

1705.4 1348.3 1082.2 909.95

2227.5 1763.3 1434.1 1191.0

3068.9 2423.2 1970.5 1648.5

3099.0 2515.5 2028.0 1671.2

3310.3 2643.2 2155.2 1773.8

3935.5 3209.5 2608.4 2135.5

4103.8 3332.1 2737.3 2297.3

4223.3 3726.4 3121.3 2560.3

4675.4 3774.7 3379.1 2804.4

0.08

0 0.5 1 2

1675.1 1292.8 1005.4 884.57

2129.4 1620.9 1273.3 1094.8

2790.8 2118.5 1682.3 1448.1

3709.6 2910.1 2315.7 1978.4

3848.1 2932.2 2331.2 1991.1

4064.1 3129.5 2514.7 2133.8

4124.2 3734.0 2971.7 2513.1

4819.7 3776.9 3213.9 2773.8

5168.5 4032.5 3403.9 2878.7

5341.2 4493.6 3551.8 3008.4

1297| International Journal of Current Engineering and Technology, Vol.8, No.5 (Sept/Oct 2018)

Mohammad Mahdi Khamsi

Free Vibration Analysis of Functionally Graded Material Plates with Various Types of Cutouts..

Table 10 Variation of natural frequencies with the volume fraction index n for Al/Al₂O₃ FG rectangular plates with two circular holes (h/b=0.04) n 0

0.5

1

2

1 2 3 4 5

947.42 1225.6 1627.8 2258.9 2270.6

820.79 1017.3 1352.4 1872.3 1909.3

656.11 838.38 1102.9 1529.8 1568.4

667.44 819.91 1066.8 1449.8 1521.0

CFCF

1 2 3 4 5

856.00 915.63 1063.9 1378.6 1783.1

716.37 742.00 919.51 1155.7 1490.2

582.40 607.83 720.84 637.92 1208.1

567.73 582.81 743.11 943.25 1195.0

CSCS

1 2 3 4 5

947.24 1224.8 1626.3 2258.2 2268.2

820.68 1016.9 1351.7 1871.3 1908.9

656.15 838.38 1102.7 1529.6 1568.3

657.42 814.09 1065.0 1446.6 1544.4

Boundary condition CCCC

Mode

5.3 Trapezoidal Plates

Fig. 4 Trapezoidal plates with multiple cutouts

Trapezoidal plates with multiple cutouts are shown in fig. 4

Table 11 shows the comparison of natural frequencies of FG trapezoidal plate whose side ratio is b/a=0.7 and height is 1. Thickness of the plate is h=0.05. The plate has two holes of radius ratio r/a=0.05 at location A(x/a=0.25, y=0.45) and B(x/a=0.70, y=0.45) respectively. Table 12 shows the comparison of natural frequencies of FG trapezoidal plate of same side ratio, height and thickness with three holes of radius ratio r/a=0.05, 0.1 and 0.15 at location A(x/a=0.25, y/a=0.35), B(x/a=0.65, y/a=0.35) and C(x/a=0.45, y/a=0.75) respectively.

Table 11 Comparison of natural frequencies of FG trapezoidal plate with two circular cutouts (fully clamped for external boundaries)

r/a of left hole 0.05 Iman[15] 0.1 Iman[15] 0.15 Iman[15]

1

2

3

4

Mode 5

6

7

8

9

10

852.11 852.84 842.86 840.27 819.27 817.18

1449.5 1489.5 1418.5 1469.6 1390.5 1444.3

1792.1 1867.0 1835.8 1869.8 1891.7 1897.7

2368.8 2405.1 2353.4 2367.4 2276.5 2332.4

2511.7 2536.4 2413.6 2501.3 2328.1 2463.6

2971.5 3156.6 2902.9 3245.1 2826.8 3277.3

3030.2 3542.1 2985.2 3549.3 3015.4 3648.8

3089.8 3775.3 3105.7 3710.7 3191.8 3736.3

3529.2 3945.0 3615.6 3951.8 3581.9 3982.1

3686.3 3643.5 3614.7 -

Table 12 Comparison of natural frequencies of FG trapezoidal plate with three circular cutouts (fully clamped for external boundaries)

r/a 0.05 Iman[15] 0.1 Iman[15] 0.15 Iman[15]

1

2

3

4

Mode 5

6

7

8

9

10

427.44 438.87 444.41 447.66 461.74 462.49

539.52 541.99 545.51 537.26 564.00 534.93

1007.6 985.51 1010.2 936.57 994.79 881.02

1109.5 1166.8 1131.8 1167.4 1111.0 1206.3

1295.3 1320.7 1233.0 1301.8 1184.5 1301.6

1330.8 1829.2 1322.2 1769.7 1337.5 1686.2

1861.4 1899.5 1882.8 1874.7 1828.4 1805.1

1963.1 2237.9 2065.0 2132.0 2068.7 1892.9

2173.8 2279.4 2201.5 2135.4 2137.3 2122.0

2335.5 2252.5 2165.8 -

1298| International Journal of Current Engineering and Technology, Vol.8, No.5 (Sept/Oct 2018)

Mohammad Mahdi Khamsi

Free Vibration Analysis of Functionally Graded Material Plates with Various Types of Cutouts..

Table 13 Variation of natural frequencies with the volume fraction index n for Al/Al₂O₃ FG trapezoidal plates with two circular holes (fully clamped for external boundaries) 1

2

3

4

Mode 5

6

7

8

9

10

h/a 0.04

n 0 0.5 1 2

2149.8 1591.6 1218.9 1040.9

3907.4 2966.8 2300.2 1936.5

4528.9 3412.9 2636.0 2196.0

5656.4 4282.2 3362.1 2833.2

6388.7 4854.7 3853.5 3163.7

6965.2 5794.1 4589.1 3783.2

7508.9 6281.3 5190.6 4390.5

7806.4 6611.3 5473.8 4445.7

8508.8 6843.6 5730.2 4826.3

8796.2 7060.2 6025.4 5044.1

0.06

0 0.5 1 2

2742.2 2110.7 1663.4 1358.7

4890.8 3800.9 3033.1 2470.0

5695.5 4471.6 3532.2 2824.9

6933.5 5591.9 4431.5 3510.0

7047.8 6239.1 4938.1 3968.3

7777.9 6354.5 5748.5 4725.2

7894.2 7114.3 5873.2 4819.2

9030.7 7410.6 6442.2 5322.6

9206.0 8113.1 6765.3 5457.9

10311. 8410.6 6944.4 5482.7

0.08

0 0.5 1 2

3274.1 2558.4 2083.6 1629.3

5776.2 4480.1 3752.3 2911.3

6701.4 5217.4 4281.4 3344.0

7140.6 6386.9 5328.5 4074.5

7988.8 6439.3 5803.7 4609.6

8050.8 7144.4 6009.7 4848.6

8887.1 7218.7 6480.4 5349.2

9207.9 8191.9 7045.8 5438.3

10702. 8458.6 7398.6 6070.8

11519. 9740.6 7998.7 6163.4

Table 14 Variation of natural frequencies with the volume fraction index n for Al/Al₂O₃ FG trapezoidal plates with two circular holes (clamped-free for external boundaries) 1

2

3

4

Mode 5

6

7

8

9

10

h/a 0.04

n 0 0.5 1 2

1349.8 998.16 764.14 646.20

1585.4 1197.8 912.76 773.78

2392.9 1803.8 1392.8 1175.8

3266.1 2483.6 1923.2 1601.9

3522.2 2639.1 2071.3 1713.1

3692.9 3331.2 2626.5 2189.6

4537.1 3405.0 2674.7 2251.4

4581.4 3492.1 3035.7 2562.8

6048.0 4572.4 3610.7 3018.4

6297.7 4592.4 3808.0 3181.9

0.06

0 0.5 1 2

1772.9 1348.7 1042.5 861.09

2096.7 1617.1 1253.4 1018.4

3094.1 2417.6 1883.2 1536.1

3745.6 3194.9 2528.8 2084.0

4122.0 3366.1 2743.9 2232.6

4424.3 3473.3 3050.6 2567.4

5738.9 4485.0 3493.2 2851.1

5832.7 4517.0 3533.1 2896.1

6374.2 5718.8 4639.2 3796.2

6416.3 5764.2 4983.7 4010.1

0.08

0 0.5 1 2

2159.3 1622.2 1329.5 1058.9

2583.7 1916.9 1556.9 1246.2

3785.2 2863.3 2332.5 1837.8

3792.8 3398.1 3077.8 2459.8

4917.9 3741.7 3153.1 2573.7

5335.6 4005.0 3383.6 2629.4

6541.5 5191.7 4321.7 3380.6

6616.3 5314.9 4396.7 3425.7

6824.2 5772.7 5204.9 4250.6

6945.2 5830.6 5260.7 4324.3

Table 15 Variation of natural frequencies with the volume fraction index n for Al/Al₂O₃ FG trapezoidal plates with two circular holes (clamped-simply supported for external boundaries) 1

2

3

4

Mode 5

6

7

8

9

10

h/a 0.04

n 0 0.5 1 2

2137.5 1581.1 1213.9 1035.7

3888.6 2956.3 2294.9 1931.1

4489.3 3381.0 2616.8 2179.7

5591.4 4249.1 3342.3 2813.5

6346.1 4831.1 3835.7 3147.1

6612.5 5756.8 4563.0 3764.5

7048.6 5965.9 5144.5 4353.4

7454.2 6365.0 5400.3 4401.8

7561.0 6537.0 5480.4 4586.3

8406.7 6779.1 5800.6 4891.1

0.06

0 0.5 1 2

2716.8 2097.2 1640.5 1350.7

4861.4 3785.4 2987.4 2459.8

5622.0 4427.5 3454.8 2799.9

6709.5 5537.2 4348.0 3477.2

6848.4 6036.1 4879.5 3941.7

7130.4 6199.2 5459.9 4585.9

7627.8 6421.1 5767.2 4691.1

7726.7 6892.1 5839.2 4896.5

8731.4 7346.8 6252.0 5249.9

9117.9 7859.9 6643.5 5323.3

0.08

0 0.5 1 2

3241.9 2533.2 2070.4 1616.5

5743.5 4452.6 3733.9 2898.5

6625.7 5150.2 4240.1 3308.3

6819.6 6080.4 5266.4 4034.6

7190.7 6346.6 5510.2 4578.9

7725.4 6457.9 5845.2 4606.5

7955.9 6932.7 5972.7 4893.3

8843.2 7141.5 6285.2 5265.2

8933.1 7951.8 6982.7 5331.4

9474.2 8373.3 7161.5 5943.9

1299| International Journal of Current Engineering and Technology, Vol.8, No.5 (Sept/Oct 2018)

Mohammad Mahdi Khamsi

Free Vibration Analysis of Functionally Graded Material Plates with Various Types of Cutouts..

Table 16 Variation of natural frequencies with the volume fraction index n for Al/Al₂O₃ FG trapezoidal plates with two circular holes (h/a=0.04) n 0

0.5

1

2

1 2 3 4 5

2149.8 3907.4 4528.9 5656.4 6388.7

1591.6 2966.8 3412.9 4282.2 4854.7

1218.9 2300.2 2636.0 3362.1 3853.5

1040.9 1936.5 2196.0 2833.2 3163.7

CFCF

1 2 3 4 5

1349.8 1585.4 2392.9 3266.1 3522.2

998.16 1197.8 1803.8 2483.6 2639.1

764.14 912.76 1392.8 1923.2 2071.3

646.20 773.78 1175.8 1601.9 1713.1

CSCS

1 2 3 4 5

2137.5 3888.6 4489.3 5591.4 6346.1

1518.1 2956.3 3381.0 4249.1 4831.1

1213.9 2294.9 2616.8 3342.3 3835.7

1035.7 1931.1 2179.7 2813.5 3147.1

Boundary condition CCCC

Mode

The variation of natural frequencies with the volume fraction exponent for Al/Al₂O₃ FG trapezoidal plates whose side ratio is b/a=0.6 and height is 1. Thickness ratio of the plate is h/a=0.04, 0.06 and 0.08. The plate has two holes of radius ratio r/b=0.1 at locations A(x/a=0.25, y/a=0.45) and B(x/a=0.70, y/a=0.45) respectively are shown in tables 13, 14 and 15 respectively. The results for first ten modes are computed. For the FG plates with fully clamped external boundary condition, the frequencies in all ten modes decreases as the volume fraction index increases. This is expected, because a large volume fraction index means that a plate has a smaller ceramic component and thus its stiffness is reduced.

Similar results are also observed for the clamped-free and clamped-simply supported external boundary conditions. From tables 13, 14 and 15 it is also observed that natural frequencies slowly improve as the thickness ratio h/b increases from 0.04 to 0.06. Table 16 shows that the variation of natural frequencies for first five modes of Al/Al₂O₃ FG trapezoidal plate (h/a=0.04) with two holes (r/a=0.1) for fully clamped, clamped-free and clamped-simply supported external boundary conditions are quite close to each other. 5.4 Circular Plates or Discs Circular plates with different cutouts are shown in fig. 4

Fig. 5 Circular plates with altered cutouts Table 17 shows the comparison of natural frequencies of FG circular disc of radius R=1 with circular holes of radius r=0.1 at locations A(x/R=-0.55, y/R=0) and B(x/R=0.55, y/R=0) respectively. Thickness of the plate is h=0.05. Table 18 shows the comparison of natural frequencies of FG circular disc of same radius and thickness with four holes of radius r=0.1 at location A(x/R=-0.7, y/R=0), B(x/R=0.7, y/R=0), C(x/R=0, y/R=0.7) and D(x/R=0, y/R=-0.7) respectively. 1300| International Journal of Current Engineering and Technology, Vol.8, No.5 (Sept/Oct 2018)

Mohammad Mahdi Khamsi

Free Vibration Analysis of Functionally Graded Material Plates with Various Types of Cutouts..

Table 17 Comparison of natural frequencies of FG circular disc with two circular cutouts (fully clamped for external boundaries)

r/R 0.1 Iman[15] 0.15 Iman[15] 0.2 Iman[15]

1

2

3

4

Mode 5

6

7

8

9

10

202.04 206.01 209.07 207.64 210.75 204.84

413.77 420.34 430.69 420.05 420.98 411.34

421.03 427.10 446.52 437.29 460.54 444.64

668.48 684.40 695.70 678.39 676.99 659.38

684.35 694.61 741.85 720.89 775.70 727.09

782.04 782.63 808.34 795.19 836.75 823.53

954.79 991.48 961.35 977.63 959.11 946.89

973.43 1009.4 1071.9 1051.3 1105.6 1069.6

1123.9 1173.5 1131.8 1161.2 1178.8 1176.0

1248.7 1265.5 1254.4 -

Table 18 Comparison of natural frequencies of FG circular disc with four circular cutouts (fully clamped for external boundaries)

e/b 0.7 Iman[15]

1

2

3

4

Mode 5

6

7

8

9

10

209.46 201.00

435.34 418.02

437.12 418.72

699.64 671.82

718.95 688.25

847.69 780.02

1032.8 987.20

1038.8 988.24

1267.6 1180.6

1278.5 1184.8

Table 19 shows the variation of natural frequencies of Al/Al₂O₃ FG circular disc or plate of radius R=1 having two holes of radius r=0.1 at locations A(x/R=-0.5, y/R=0) and B(X/r=0.5, y/R=0) respectively. Thickness ratio of the plate is h/R=0.04, 0.06 and 0.08 respectively. The results for first ten modes are computed. For the FG plates with fully clamped external boundary condition, the frequencies in all ten modes decreases as the volume fraction index increases. This is expected, because a large volume fraction index means that a plate has a smaller ceramic component and thus its stiffness is reduced. Table 19 Variation of natural frequencies with the volume fraction index n for Al/Al₂O₃ FG circular plates with two circular holes (fully clamped for external boundaries) 1

2

3

4

Mode 5

6

7

8

9

10

h/a 0.04

n 0 0.5 1 2

464.14 410.78 336.07 291.81

946.79 832.36 685.58 568.65

959.26 870.92 692.46 582.18

1541.2 1357.0 1129.2 936.01

1566.2 1377.4 1154.5 992.84

1732.9 1562.2 1265.5 1037.0

2239.6 1911.7 1600.2 1345.5

2286.1 1944.6 1625.3 1388.8

2554.0 2269.0 1848.2 1513.8

2643.8 2466.2 1985.5 1727.1

0.06

0 0.5 1 2

576.64 502.96 401.20 343.23

1144.5 1000.4 809.53 681.15

1167.6 1031.8 825.37 682.30

1837.6 1656.6 1343.2 1105.3

1883.2 1683.2 1357.3 1121.9

2079.2 1850.3 1522.4 1239.0

2626.3 2378.7 1881.5 1552.3

2677.9 2419.1 1937.5 1594.4

3026.9 2714.3 2237.7 1778.8

3125.2 2895.6 2357.0 1902.7

0.08

0 0.5 1 2

757.34 584.96 468.48 392.02

1481.6 1167.9 913.55 764.37

1507.7 1195.9 941.98 792.53

2367.5 1909.6 1493.7 1248.7

2384.6 1924.6 1512.4 1267.0

2638.5 2189.9 1647.8 1388.4

3244.0 2673.0 2123.0 1766.3

3291.6 2741.2 2188.6 1795.6

3307.3 2939.1 2394.1 1990.1

3348.0 2996.8 2505.6 2105.6

Conclusion The free vibration analysis of FG plates with cutouts is carried out using subspace method. The elastic properties of FG plates are assumed to vary through the thickness according to a power law as shown in eq. 1. The results derived with subspace method compared with the solutions presented by Iman et al to validate their accuracy. It is found that a volume fraction exponent that ranges between 0 and 2 has a significant influence on the natural frequency of FG plates with cutouts. For rectangular, trapezoidal and circular FG plates with cutouts natural frequency decreases as the volume fraction index increases. Natural frequency of FG plates with cutouts increases as the thickness ratio

increases. For rectangular and trapezoidal plates with cutouts variation of natural frequencies for fully clamped and clamped-simply supported external boundaries are very close to each other. References Functionally Graded Materials (FGM) and Their Production Methods, Azom.com. A. Chakraborty, S. Gopalakrishnan and J.N. Reddy (2003), A New Beam Finite Element for the Analysis of Functionally Graded Materials, Int. J. of Mechanical Science 45, pp. 519539. A. Chakraborty and S. Gopalakrishnan (2003), A Spectrally Formulated Finite Element for Wave Propagation Analysis in Functionally Graded Beam, Int. J. of Solids and Structures, 40, pp. 2421-2448.

1301| International Journal of Current Engineering and Technology, Vol.8, No.5 (Sept/Oct 2018)

Mohammad Mahdi Khamsi

Free Vibration Analysis of Functionally Graded Material Plates with Various Types of Cutouts..

L. F. Qian, R.C. Batra and L. M. Chen (2005), Static and Dynamic Deformation of Thick Functionally Graded Elastic Plates by Using Higher Order Shear and Normal Deformable Plate Theory and Meshless Local PetrovGalerkin Method, Composites, Part B 35, pp. 685-697. H.L. Dai, X. Wang (2005), Thermo-Electro-Elastic Transient Responses in Piezoelectric Hollow Structures, Int. J. of Solids and Structures 42, pp. 1151-1171. X. L. Huang, S. H. Shen (2004), Nonlinear Vibration and Dynamic Response of Functionally Graded Plates in Thermal Environments, Int. J. of Solids and Structures 41, pp. 2403-2427. J. Yang, S. H. Sheen (2003), Free Vibration and Parametric Response of Shear Deformable Functionally Graded Cylindrical Panels, Journal of Sound and Vibration 261, pp. 871-893. Jiu Hui Wu, A.Q. Liu and H. L. Chen (2007), Exact Solutions for Free Vibration Analysis of Rectangular Plate using Bessel Functions, Journal of Applied Mechanics 74, pp. 12471251. X. Zhao, Y. Y. Lee and K. M. Liew (2009), Free Vibration Analysis of Functionally Graded Plates using the ElementFree Kp-Ritz Method, Int. J. of Sound and Vibration 319, pp. 918-939. Hiroyuki Matsunaga (2008), Free Vibration and Stability of Functionally Graded Plates According to a 2-D HigherOrder Deformation Theory, Int. J. of Composite Structures 82, pp. 499-512. Ah. Hosseini- Hashemi, M. Fadaee and S. R. Atashipour (2011), Study on the Free Vibration of Thick Functionally Graded Rectangular Plates According to a New Exact Closed-Form Procedure, Int. J. of Composite Structures 93, pp. 722-735.

B. G. Chai (1996), Free Vibration of Laminated Plates with a Central Circular Hole, Journal of Composite Structures 35, pp. 357-368. T. Sakiyama, M. Huang (1996), Free Vibration Analysis of Rectangular Plates with Variously Shape-Hole, Journal of Sound and Vibration 226, pp. 769-786. G. R. Liu, X. Zhao, K.Y. Dai, Z.H. Zhong, G. Y. Li and X. Han (2008), Static and Free Vibration Analysis of Laminated Composite Plates using the Conforming Radial Point Interpolation Method, Int. J. of Composites Science and Technology 68, pp. 354-366. J. Maziar and R. Iman (2011), Free Vibration Analysis of Functionally Graded Plates with Multiple Circular and Noncircular Cutouts, Chinese Journal of Mechanical Engineering 24 K. J. Bathe (1971), Solution Methods for Large Generalized Eigen Value Problems in Structural Engineering, Report UCSESM, Department of Civil Engineering, University of California, Berkeley. H. Rutishauser (1970),Simultaneous Iteration Method for Symmetric Matrices, Int. J. of Numerische Mathematik 16, pp. 205-223. H. Rutishauser (1969), Computational Aspects of F. L. Bauer’s Simultaneous Iteration Method, Int. J. of Numerische Mathematik 13, pp. 4-13. K. J. Bathe (2012), The subspace Iteration Method-Revisited, Int. J. of Computers and Structures. G. R. Liu, X. Zhao, K.Y. Dai (2008), Static and Free Vibration Analysis of Laminated Composite Plates using the Conforming Radial Point Interpolation Method, Journal 68, pp. 354-366. M. Huang, T. Sakiyama (1999), Free Vibration Analysis of Rectangular Plates with Various Shape Hole, Journal of Sound and Vibration 226, pp. 769-786.

1302| International Journal of Current Engineering and Technology, Vol.8, No.5 (Sept/Oct 2018)