Table 13 Variation of the first three dimensionless frequency parameters of a uniform (2D-FGM) (CF) beam with gradation indexes \(n_x\) and \(n_y\).

From: A discrete model for analyzing the free vibrations of a non-uniform 2D-FGM beam under elastic foundations and different support conditions

First Mode for CF

\(n_x\)

\(n_y\)

0.2

0.4

0.8

1

2

4

6

8

0.2

2.94

2.93

2.89

2.87

2.67

2.12

1.62

1.26

0.4

2.93

2.92

2.88

2.86

2.66

2.11

1.61

1.25

0.6

2.92

2.91

2.87

2.85

2.65

2.11

1.61

1.25

0.8

2.91

2.90

2.86

2.84

2.65

2.10

1.60

1.24

2

2.87

2.86

2.83

2.80

2.62

2.08

1.58

1.23

4

2.88

2.87

2.84

2.81

2.62

2.08

1.59

1.23

8

3.00

3.00

2.96

2.93

2.74

2.17

1.65

1.29

Second Mode for CF

\(n_x\)

\(n_y\)

0.2

0.4

0.8

1

2

4

6

8

0.2

22.47

22.41

22.14

21.95

20.47

16.25

12.39

9.63

0.4

22.41

22.34

22.07

21.88

20.41

16.20

12.35

9.60

0.6

22.34

22.27

22.01

21.81

20.34

16.15

12.31

9.57

0.8

22.27

22.20

21.94

21.75

20.28

16.10

12.28

9.55

2

21.87

21.80

21.55

21.36

19.92

15.81

12.06

9.37

4

21.33

21.27

21.02

20.83

19.43

15.42

11.76

9.14

8

20.86

20.80

20.55

20.37

19.00

15.08

11.50

8.94

Third Mode for CF

\(n_x\)

\(n_y\)

0.2

0.4

0.8

1

2

4

6

8

0.2

66.52

66.32

66.54

66.52

66.52

66.52

66.52

66.52

0.4

66.35

66.15

66.37

66.35

66.35

66.35

66.35

66.35

0.6

66.17

65.97

66.19

66.17

66.17

66.17

66.17

66.17

0.8

65.99

65.79

65.61

65.99

65.99

65.99

65.99

65.99

2

64.81

64.62

64.86

64.81

64.81

64.81

64.81

64.81

4

62.96

62.77

62.03

62.96

62.96

62.96

62.96

62.96

8

60.77

60.59

60.88

60.77

60.77

60.77

60.77

60.77

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