Semi-analytical Vibration Characteristics of Rotating Timoshenko Beams Made of Functionally Graded Materials

Ebrahimia, Farzad; Mokhtaria, Mohadese

doi:10.1590/1679-78251446

Farzad Ebrahimia ^**Corresponding Author email: febrahimy@eng.ikiu.ac.ir

Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran, P.O.B. 16818-34149

Mohadese Mokhtaria

Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran, P.O.B. 16818-34149

*Corresponding Author email: febrahimy@eng.ikiu.ac.ir

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Figure 1
Typical functionally graded beam with Cartesian coordinates.

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Figure 2
Convergence study for the first frequency of rotating FG beam, q=0, η=8.

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Figure 3
Convergence study for the second frequency of rotating FG beam, q=0, η=8.

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Figure 4
Convergence study for the third frequency of rotating FG beam, q=0, η=8.

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Figure 5
The effect of power-law exponent (q<10) on first natural frequency with different rotational speed parameters, L/h=10.

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Figure 6
The effect of power-law exponent (q>10) on first natural frequency with different rotational speed parameters, L/h=10.

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Figure 8
The effect of power-law exponent on third natural frequency with different rotational speed parameters, L/h=10.

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Figure 9
Variation of the fundamental frequency of FG rotating beam with L/h ratio for various power-law exponent.

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Figure 10
Variation of the first three natural frequencies of FG rotating beam with L/h ratio, q=2، η=2

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Figure 11
The effect of rotational speed on first three natural frequencies, L/h=10, q=0.5.

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Figure 12
Effect of hub radius parameter on first three natural frequencies, L/h =10، η=2 ، q=0.

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Figure 13
The normalized first mode shape of rotating FG beam for different slenderness ratios.

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Figure 14
The normalized second mode shape of rotating FG beam for different slenderness ratios.

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Figure 15:
The normalized third mode shape of rotating FG beam for different slenderness ratios.

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Table 1
Some of the transformation rules of the one-dimensional DTM.

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Table 2
Transformed boundary conditions (B.C.) based on DTM.

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Table 3
material properties of the FGM constituents (Simsek, 2010Şimşek, M., "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories." Nuclear Engineering and Design 240 (2010): 697-705.).

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Table 4
material properties of the FGM constituents (Sina et al., 2009Sina, S. A., H. M. Navazi, and H. Haddadpour. "An analytical method for free vibration analysis of functionally graded beams." Materials & Design 30.3 (2009): 741-747.).

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Table 5
Comparison of fundamental frequencies of FG rotating beams for different L/h ratios.

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Table 6
Comparison of fundamental frequencies of FG rotating beams for different constituent volume fraction exponent.

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Table 7
Variation of the first three natural frequencies of FG rotating beam with L/h ratio for various values of the power-law exponent, η = 8.

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Table 8
The effect of rotational speed parameter on natural frequencies of FG rotating beam with different power law exponent for the case: L/h=10.

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Table 9
The effect of hub radius parameter on the first three natural frequencies, L/h =10 ، η=2 ، q=0.

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properties	unit	aluminum	Alumina( Al2O3)
E	GPa	70	380
ρ	Kg/m³	2702	3960
ν	-	0.3	0.3

properties	unit	aluminum	Alumina (Al2O3)
E	GPa	70	380
ρ	Kg/m³	2700	3800
ν	-	0.23	0.23

	L/h =10	L/h =30	L/h =100
(Sina et al. 2009Sina, S. A., H. M. Navazi, and H. Haddadpour. "An analytical method for free vibration analysis of functionally graded beams." Materials & Design 30.3 (2009): 741-747.)	0.9960	1.0030	1.0030
Present	0.9858	0.9922	0.9930
% Difference	1.0346	1.0884	1.0070

q	Present	(Simsek 2010Şimşek, M., "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories." Nuclear Engineering and Design 240 (2010): 697-705.)	% Difference
ceramic	1.9495	1.9496	0.00359
0.2	1.8286	1.8145	0.7711
0.5	1.7214	1.66046	3.5413
10	1.3555	1.26495	6.6802
metal	1.0198	1.01297	0.6697

q	mode	L/h = 5	L/h=10	L/h=20	L/h =30	L/h =50	L/h=100
0	λ1	4.9877	5.098	5.1312	5.1377	5.1410	5.1424
	λ2	14.6826	16.0314	16.4938	16.5877	16.6369	16.6578
	λ3	29.8114	35.4915	38.0107	38.5744	38.8779	39.0092
0.2	λ1	4.6815	4.7829	4.8132	4.8191	4.8222	4.8235
	λ2	13.8066	15.0516	15.475	15.5607	15.6056	15.6247
	λ3	28.0963	33.3684	35.6797	36.1944	36.4711	36.5907
0.5	λ1	4.4077	4.5028	4.5312	4.5367	4.5396	4.5408
	λ2	13.0029	14.1718	14.5688	14.6491	14.6911	14.709
	λ3	26.4718	31.4259	33.593	34.0752	34.3344	34.4464
1	λ1	4.1853	4.2779	4.3057	4.3111	4.3139	4.3151
	λ2	12.4432	13.4523	13.8403	13.9191	13.9603	13.978
	λ3	25.0154	29.7817	31.8956	32.3686	32.6233	32.7335
2	λ1	4.0066	4.1008	4.1296	4.1353	4.1382	4.1395
	λ2	11.7309	12.8662	13.2655	13.3473	13.3902	13.4086
	λ3	23.6558	28.3609	30.525	31.0167	31.2827	31.3980
5	λ1	3.7400	3.8348	3.8648	3.8707	3.8738	3.8751
	λ2	10.8707	11.9929	12.4027	12.4877	12.5325	12.5517
	λ3	21.7191	26.2714	28.476	28.9882	29.2671	29.3884
10	λ1	3/4514	3.5396	3.5676	3.5731	3.5760	3.5772
	λ2	10.0238	11.0652	11.4476	11.5270	11.5689	11.5868
	λ3	20.003	24.2188	26.2751	26.7543	27.0154	27.129
50	λ1	2.8686	2.9364	2.9573	2.9614	2.9635	2.9644
	λ2	8.3948	9.2106	9.4988	9.5579	9.5889	9.6022
	λ3	16.9114	20.2897	21.8525	22.2084	22.4011	22.4846
100	λ1	2.7417	2.8047	2.8239	2.8277	2.8296	2.8304
	λ2	8.0442	8.8073	9.0734	9.1277	9.1562	9.1684
	λ3	16.2600	19.4438	20.8899	21.2166	21.3931	21.4696
metal	λ1	2.6086	2.6666	2.6840	2.6874	2.6892	2.6899
	λ2	7.6763	8.3841	8.6272	8.6766	8.7024	8.7135
	λ3	15.5780	18.5555	19.8796	20.1762	20.3359	20.4051

q	mode	η=0	η=2	η=4	η=6	η=8
0	λ1	1.9381	2.2815	3.0801	4.0570	5.0980
	λ2	11.6155	11.9408	12.8669	14.2743	16.0314
	λ3	30.5505	30.8865	31.8701	33.4349	35.4915
0.2	λ1	1.8183	2.1405	2.8896	3.8061	4.7829
	λ2	10.9114	11.2163	12.0844	13.4039	15.0516
	λ3	28.7438	29.0582	29.9784	31.4429	33.3684
0.5	λ1	1.7118	2.0151	2.7204	3.5832	4.5028
	λ2	10.2747	10.5617	11.3787	12.6208	14.1718
	λ3	27.0743	27.3701	28.236	29.6140	31.4259
1	λ1	1.6263	1.9145	2.5846	3.4043	4.2779
	λ2	9.7468	10.0198	10.7969	11.9779	13.4523
	λ3	25.6356	25.9176	26.7429	28.0559	29.7817
2	λ1	1.5589	1.8353	2.4779	3.2636	4.1008
	λ2	9.3053	9.5680	10.3153	11.4502	12.8662
	λ3	24.3533	24.6263	25.4249	26.6944	28.3609
5	λ1	1.4577	1.7165	2.3176	3.1245	3.8348
	λ2	8.6492	8.8962	9.5987	10.6646	11.9929
	λ3	22.474	22.7334	23.4913	24.6945	26.2714
10	λ1	1.3454	1.5843	2.1392	2.8173	3.5396
	λ2	7.9766	8.2049	8.8539	9.8384	11.0652
	λ3	20.7062	20.9461	21.6475	22.7605	24.2188
50	λ1	1.1162	1.3142	1.7744	2.3370	2.9364
	λ2	6.6589	6.8472	7.3828	8.1960	9.2106
	λ3	17.4142	17.6101	18.1832	19.0941	20.2897
100	λ1	1.0662	1.2552	1.6947	2.2321	2.8047
	λ2	6.3735	6.5530	7.0636	7.8393	8.8073
	λ3	16.7097	17.5121	17.4405	18.3065	19.4438
Metal	λ1	1.0137	1.1934	1.6111	2.1220	2.6666
	λ2	6.0738	6.2442	6.7286	7.4649	8.3841
	λ3	15.9694	16.1453	16.6601	17.4791	18.5555

mode	δ=0	δ=0.02	δ=0.04	δ=0.06	δ=0.08	δ=0.1	δ=0.12
λ1	2.2815	2.2899	2.2983	2.3066	2.3149	2.3231	2.3314
λ2	11.9408	11.9495	11.9581	11.9667	11.9753	11.9839	11.9925
λ3	30.8865	30.896	30.9054	30.9149	30.9243	30.9338	30.9432

[1] *Corresponding Author email: febrahimy@eng.ikiu.ac.ir

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Semi-analytical Vibration Characteristics of Rotating Timoshenko Beams Made of Functionally Graded Materials

Abstract