Issue 52

N. Hebbar et alii, Frattura ed Integrità Strutturale, 52 (2020) 230-246; DOI: 10.3221/IGF-ESIS.52.18

V IBRATION ANALYSIS n this third part of our work, we studied the first three fundamental dimensionless frequencies of Al/Al 2 O 3 type functionally graded materials beams for various exponent values of the power-law p and the ratio (L/h).

I

p

L/h Mode

Theory

0

1

2

5

10

5.3781

4.1677

3.7867

3.5470 3.4014 3.4014 3.4012 3.3997 3.4012 3.4012 3.4012 3.3742 3.4311 3.5949 11.932 11.544 11.544 11.543 11.532 11.544 13.587 22.289 21.720 21.720 21.716 21.694 21.718 28.085 3.8232 3.6485 3.6485 3.6485 3.6484 3.6485 3.6484 3.6490 3.6466 3.6508 3.6628 15.046 14.374 14.374 14.374 14.373 14.374 14.595 32.998 31.579 31.579 31.578 31.570

3.4203 3.2816 3.2816 3.2816 3.2811 3.2816 3.2816 3.2962 3.2653 3.3134 3.4921 11.380 11.024 11.024 11.025 11.021 11.024 13.237 21.061 20.556 20.556 20.556 20.558 20.556 27.475 3.7083 3.5389 3.5390 3.5389 3.5389 3.5389 3.5389 3.5405 3.5378 3.5415 3.5547 14.574 13.926 13.926 13.926 13.925 13.926 14.167 31.899 30.095 30.095 30.537 30.534

Present ( ε z  0) Present ( ε z =0)

5.1527 3.9904 3.6264 5.1527 3.9904 3.6264 5.1527 3.9904 3.6264

A.S. Sayyad and Y.M.Ghugal [58]

Reddy [11] Touratier [12] Simsek [15] Thai and Vo [20] Vo et al. [29] Vo et al. [30] Timoshenko [10] Bernoulli-Euler [9]

1

5.1531

3.9906

3.6263

5.1527 3.9904 3.6264

5.1527

3.9904

3.6264

5.1526 3.9711 3.6050 5.1527 3.9716 3.5979 5.1524 3.9902 3.6343

5.3953

4.1484

3.7793

5

17.514 14.531 13.105

Present ( ε z  0) Present ( ε z =0)

17.881

14.010

12.640

A.S. Sayyad and Y.M.Ghugal [58]

17.881 14.010 12.640

2

Reddy [11] Touratier [12] Thai and Vo [20] Bernoulli-Euler [9]

17.881

14.010

12.640

17.887 14.014 12.641 17.881 14.009 12.640 20.618 15.798 14.326 34.209 27.098 24.316 34.202 27.098 24.316 35.173 27.930 25.052

Present ( ε z  0) Present ( ε z =0)

A.S. Sayyad and Y.M.Ghugal [58]

3

Reddy [11] Touratier [12] Thai and Vo [20] Bernoulli-Euler [9]

34.209

27.098

24.315

34.234 27.115 24.324

34.208

27.097

24.315

43.348 33.027 29.745

5.7222

4.4069

4.0202

Present ( ε z  0) Present ( ε z =0)

5.4603 4.2050 3.8361 5.4603 4.2050 3.8361 5.4603 4.2050 3.8361

A.S. Sayyad and Y.M.Ghugal [58]

Reddy [11] Touratier [12] Simsek [15] Thai and Vo [20] Vo et al. [29] Vo et al. [30] Timoshenko [10] Bernoulli-Euler [9]

1

5.4603

4.2051

3.8361

5.4603 4.2050 3.8361

5.4603

4.2050

3.8361

5.4603 4.2038 3.8349 5.4603 4.2038 3.8342 5.4603 4.2050 3.8367

5.4777

4.2163

3.8472

20

22.587 17.420 15.877

Present ( ε z  0) Present ( ε z =0)

21.573

16.643

15.161

A.S. Sayyad and Y.M.Ghugal [58]

21.573 16.643 15.161

2

Reddy [11] Touratier [12] Thai and Vo [20] Bernoulli-Euler [ 9]

21.573

16.634

15.162

21.574 16.635 15.162

21.573

16.643

15.161

21.843 16.810 15.333

49.761

38.458

35.003

Present ( ε z  0) Present ( ε z =0)

47.593 36.768 33.469 47.593 36.768 33.469 47.593 36.768 33.469

A.S. Sayyad and Y.M.Ghugal [58]

3

Reddy [11] Touratier [12] Thai and Vo [20] Bernoulli-Euler [ 9]

47.595

36.769

33.468

47.593 36.767 33.469 31.5789 30.537

48.899 34.295 32.6357 31.688 Table 4: Comparison of the first three dimensionless fundamental frequencies of the FG beams in regards to various power-law exponent values. 37.617

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