Journal of Materials Science: Materials in Engineering

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
τ _h	0.0	0.2	0.4	0.6	0.8
	(a) Fundamental frequency
0.0	3.15601	3.76285	4.09697	4.58531	5.39758
	3.51602^a	3.76286^a	4.09698^a	4.58531^a	5.39759^a
0.2	3.60827	3.85511	4.18932	4.67815	5.49265
	3.60828^a	3.85512^a	4.18932^a	4.67816^a	5.49266^a
0.4	3.73707	3.98419	4.31877	4.80842	5.62551
	3.73708^a	3.98419^a	4.31878^a	4.80842^a	5.62552^a
0.6	3.93427	4.18234	4.51799	5.00903	5.82882
	3.93428^a	4.18235^a	4.51799^a	5.00903^a	5.82882^a
0.8	4.29250	4.54369	4.88244	5.37615	6.19641
	4.29249^a	4.54368^a	4.88244^a	5.37614^a	6.19639^a
	(b) Second natural frequency
0.0	22.0344	22.5018	23.1186	24.0210	25.6558
	22.0345^a	22.5018^a	23.1186^a	24.0211^a	25.6558^a
0.2	20.6210	21.0567	21.6327	22.4774	24.0153
	20.6210^a	21.0567^a	21.6327^a	22.4774^a	24.0153^a
0.4	19.1138	19.5166	20.0500	20.8343	22.2710
	19.1138^a	19.5166^a	38.4920^a,b	20.8343^a	22.2710^a
0.6	17.4879	17.8557	18.3441	19.0649	20.3953
	17.4879^a	17.8557^a	18.3441^a	19.0649^a	20.3952^a
0.8	15.7473	16.0731	16.5132	17.1668	18.3871
	15.7427^a	16.0725^a	16.5123^a	17.1657^a	18.3855^a
	(c) Third natural frequency
0.0	61.6972	62.1525	62.7763	63.7515	65.7470
	61.6972^a	62.1525^a	62.7763^a	63.7515^a	65.7470^a
0.2	56.1923	56.6304	57.2258	58.1467	60.0096
	56.1923^a	56.6303^a	57.2257^a	58.1466^a	60.0094^a
0.4	50.3537	50.7715	51.3348	52.1965	53.9177
	50.3537^a	50.7714^a	51.3346^a	52.1963^a	53.9173^a
0.6	44.0252	44.4180	44.9437	45.7391	47.3066
	44.0248^a	44.4175^a	44.9432^a	45.7384^a	47.3051^a
0.8	36.8955	37.2558	37.7358	38.4555	39.8568
	36.8846^a	37.2439^a	37.7223^a	38.4392^a	39.8336^a

^aResults of Ozgumus and Kaya (2006)
^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

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