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Table 1 Comparison of the non-dimensional lowest three flapwise bending natural frequencies of a homogenous stationary beam with different taper ratios (δ = 0)

From: Flapwise bending vibration analysis of functionally graded rotating double-tapered beams

τ h

τ b

0.2

0.4

0.6

0.8

0.0

 

(a) Fundamental frequency

0.0

3.15601

3.76285

4.09697

4.58531

5.39758

 

3.51602a

3.76286a

4.09698a

4.58531a

5.39759a

0.2

3.60827

3.85511

4.18932

4.67815

5.49265

 

3.60828a

3.85512a

4.18932a

4.67816a

5.49266a

0.4

3.73707

3.98419

4.31877

4.80842

5.62551

 

3.73708a

3.98419a

4.31878a

4.80842a

5.62552a

0.6

3.93427

4.18234

4.51799

5.00903

5.82882

 

3.93428a

4.18235a

4.51799a

5.00903a

5.82882a

0.8

4.29250

4.54369

4.88244

5.37615

6.19641

 

4.29249a

4.54368a

4.88244a

5.37614a

6.19639a

 

(b) Second natural frequency

0.0

22.0344

22.5018

23.1186

24.0210

25.6558

 

22.0345a

22.5018a

23.1186a

24.0211a

25.6558a

0.2

20.6210

21.0567

21.6327

22.4774

24.0153

 

20.6210a

21.0567a

21.6327a

22.4774a

24.0153a

0.4

19.1138

19.5166

20.0500

20.8343

22.2710

 

19.1138a

19.5166a

38.4920a,b

20.8343a

22.2710a

0.6

17.4879

17.8557

18.3441

19.0649

20.3953

 

17.4879a

17.8557a

18.3441a

19.0649a

20.3952a

0.8

15.7473

16.0731

16.5132

17.1668

18.3871

 

15.7427a

16.0725a

16.5123a

17.1657a

18.3855a

 

(c) Third natural frequency

0.0

61.6972

62.1525

62.7763

63.7515

65.7470

 

61.6972a

62.1525a

62.7763a

63.7515a

65.7470a

0.2

56.1923

56.6304

57.2258

58.1467

60.0096

 

56.1923a

56.6303a

57.2257a

58.1466a

60.0094a

0.4

50.3537

50.7715

51.3348

52.1965

53.9177

 

50.3537a

50.7714a

51.3346a

52.1963a

53.9173a

0.6

44.0252

44.4180

44.9437

45.7391

47.3066

 

44.0248a

44.4175a

44.9432a

45.7384a

47.3051a

0.8

36.8955

37.2558

37.7358

38.4555

39.8568

 

36.8846a

37.2439a

37.7223a

38.4392a

39.8336a

  1. aResults of Ozgumus and Kaya (2006)
  2. bThe trends in the second frequency reported by Ozgumus and Kaya (2006) should exhibit decreasing trend with an increase in taper ratio. However, at τ b  = τ h  = 0.4, a higher value is reported. It is opened that it is a typographical error