Vibration Analysis of Rotating Functionally Graded Cylindrical Shells with Orthogonal Stiffeners

Tu, Tran Minh; Loi, Nguyen Van

doi:10.1590/1679-78252934

Tran Minh Tu

National University of Civil Engineering, 55 Giai Phong Road, Ha noi, Viet nam. E-mail address: tpnt2002@yahoo.com

Nguyen Van Loi

National University of Civil Engineering, 55 Giai Phong Road, Ha noi, Viet nam. E-mail address: nguyenvanloi.n.v@gmail.com

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Figure 1:
Coordinate system and geometry of rotating stiffened FGM cylindrical shell.

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Figure 2:
Variation of the natural fundamental frequencies of the orthogonal stiffened and unstiffened FG cylindrical shells with the rotating speed (power law index p = 20).

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Figure 3:
Variation of the fundamental frequencies for the orthogonal stiffened FG cylindrical shells with the stiffener height-to-width ratio (power law index p = 20).

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Figure 4:
Variation of the fundamental frequencies for the orthogonal stiffened FG cylindrical shells with the rotating speed for the various power law indexes.

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Figure 5:
Variation of natural frequencies for the rotating orthogonally stiffened FG cylindrical shell with the circumferential wave number n (power law index p = 20).

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Figure 6:
Variation of the fundamental frequencies for the rotating orthogonally stiffened FG cylindrical shell with L / R ratio (power law index p = 20).

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Figure 7:
Variation of the fundamental frequencies for the rotating orthogonally stiffened FG cylindrical shell with R / h ratio (power law index p = 20).

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Table 1:
The geometrical and material properties for the stiffened FG cylindrical shells.

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Table 2:
Natural frequencies of vibration of the non-rotating stiffened isotropic cylindrical shells (Hz).

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Table 3:
Natural frequencies of un-stiffened non-rotating FG cylindrical shell (Hz).

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Table 4:
Comparison of frequency parameter h/R=0.002, L/R=200, v = 0.3).

for a very long rotating isotropic cylindrical shell (

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Table 5:
Natural fundamental frequencies (Hz) of stiffened and unstiffened FG cylindrical shell with various rotating speeds.

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Table 6:
Natural fundamental frequencies (Hz) of rotating stiffened FG cylindrical shell with the various stiffener height-to-width ratios.

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Table 7:
Natural fundamental frequencies (Hz) of stiffened FGM cylindrical shell with the various power law indexes p.

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Table 8:
Natural fundamental frequencies (Hz) of stiffened FG cylindrical shell with various the circumferential wave numbers n.

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Table 9:
Natural fundamental frequencies (Hz) of stiffened FG cylindrical shell with various L / R ratios.

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Table 10:
Natural fundamental frequencies (Hz) of stiffened FG cylindrical shell with various R / h ratios.

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Property				Geometric Data
Type of stiffeners Model	Stringers	Rings	Stringers	Orthogonal rings/stringers	Unstiffened	Orthogonal rings/stringers
	M1	M2	M3	M4	M5	M6
No. of stiffeners	60	19	04	13/20		05/10
Radius (m)	0.242	0.49759	0.1945	0.203	1	0.2
Thickness (m)	6E-4	1.65E-3	4.64E-4	2.04E-3	0.05	0.002
Length (m)	0.6096	0.3945	0.9868	0.813	20	1
Height of stiffeners (m)	0.00702	0.005334	0.0101	0.006/0.006		0.002/0.012
Width of stiffeners (m)	0.002554	0.003175	0.00104	0.004/0.008		0.002/0.012
E₀ (N / m²)	6.9E+10	6.9E+10	2.00E+11	2.07E+11		2.00E+11
υ₀	0.3	0.3	0.3	0.3		0.3
ρ₀ (kg / m³)	2714	2762	7770	7430		5700
E₁ (N / m²)					2.07788E+11	2.00E+11
υ₁					0.317756	0.3
ρ₁ (kg / m³)					8166	5700
E₂ (N / m²)					2.05098E+11	7.0E+10
υ₂					0.31	0.3
ρ₂ (kg / m³)					8900	2702
Stiffening type	External	External	Internal	Internal		External

Mode number		Natural frequencies (Hz)
		M1 (stringers)		M2 (rings)		M3 (stringers)		M4 (orth. stiffened)
m	n	Mustafa and Ali (1989Mustafa, B.A.J., and Ali,R. (1989). An Energy Method for Free Vibration Analysis of Stiffened Circular Cylindrical Shells. Computers & Structures 32.2: 355-363.)	Present	Mustafa and Ali (1989Mustafa, B.A.J., and Ali,R. (1989). An Energy Method for Free Vibration Analysis of Stiffened Circular Cylindrical Shells. Computers & Structures 32.2: 355-363.)	Present	Mustafa and Ali (1989Mustafa, B.A.J., and Ali,R. (1989). An Energy Method for Free Vibration Analysis of Stiffened Circular Cylindrical Shells. Computers & Structures 32.2: 355-363.)	Present	Mustafa and Ali (1989Mustafa, B.A.J., and Ali,R. (1989). An Energy Method for Free Vibration Analysis of Stiffened Circular Cylindrical Shells. Computers & Structures 32.2: 355-363.)	Present
1	1	1141	1143.1	1204	1225.7	778	778.1	942	936.3
	2	674	672.4	1587	1597.1	317	317.2	439	436.9
	3	427	425.3	4462	4476.4	159	159.3	337	331.0
	4	296	294.5	8559	8593.9	99.6	101.6	482	477.5
	5	225	223.3	13,780	13,874.6	91.5	91.4	740	739.8

Mode number					Natural frequencies (Hz)
		p = 0		p = 1		p = 2		p = 30
m	n	Loy et al.(1999Loy, C.T., Lam,K.Y., and Reddy,J.N. (1999).Vibration of Functionally Graded Cylindrical Shells. International Journal of Mechanical Sciences 41.3: 309-324.)	Present	Loy et al.(1999Loy, C.T., Lam,K.Y., and Reddy,J.N. (1999).Vibration of Functionally Graded Cylindrical Shells. International Journal of Mechanical Sciences 41.3: 309-324.)	Present	Loy et al.(1999Loy, C.T., Lam,K.Y., and Reddy,J.N. (1999).Vibration of Functionally Graded Cylindrical Shells. International Journal of Mechanical Sciences 41.3: 309-324.)	Present	Loy et al.(1999Loy, C.T., Lam,K.Y., and Reddy,J.N. (1999).Vibration of Functionally Graded Cylindrical Shells. International Journal of Mechanical Sciences 41.3: 309-324.)	Present
1	1	13.572	13.572	13.235	13.235	13.127	13.127	12.937	12.936
	2	33.296	33.296	32.430	32.430	32.170	32.170	31.730	31.670
	3	93.001	93.001	90.553	90.553	89.828	89.828	88.614	88.449
	4	178.06	178.06	173.36	173.36	171.98	171.98	169.66	169.34
	5	287.79	287.79	280.20	280.20	277.95	277.95	274.21	273.70
	6	422.05	422.05	410.91	410.91	407.62	407.62	402.13	401.38
	7	580.78	580.78	565.46	565.46	560.93	560.93	553.37	552.35
	8	763.98	763.98	743.82	743.82	737.86	737.86	727.92	726.57
	9	971.62	971.62	945.98	945.98	938.40	938.40	925.76	924.05
	10	1,203.7	1,203.7	1,171.9	1,171.9	1,162.5	1,162.5	1,146.9	1,144.8

Ω (rev/s)	Mode number			Natural frequencies (Hz)
			ω'_b		ω'_f
	m	n	Chen et al.(1993Chen, Y., Zhao, H. B., Shea, Z. P. (1993). Vibrations of High Speed Rotating Shells With Calculations for Cylindrical Shells. Journal of Sound and Vibration 160: 137-160.)	Present	Chen et al.(1993Chen, Y., Zhao, H. B., Shea, Z. P. (1993). Vibrations of High Speed Rotating Shells With Calculations for Cylindrical Shells. Journal of Sound and Vibration 160: 137-160.)	Present
0.05	1	2	0.00167	0.00167	0.00144	0.00145
		3	0.00447	0.00448	0.00430	0.00431
		4	0.00847	0.00848	0.00834	0.00835
		5	0.01364	0.01366	0.01353	0.01355
0.1	1	2	0.00179	0.00182	0.00135	0.00138
		3	0.00456	0.00461	0.00422	0.00427
		4	0.00854	0.00859	0.00827	0.00833
		5	0.01370	0.01376	0.01348	0.01354

Type of shell				Ω (rev/s)
		0	10	20	30	50
Stiffened shell	ω_b	344.93	353.24	362.11	371.53	391.96
	ω_f	344.93	337.16	329.95	323.28	311.55
Unstiffened shell	ω_b	193.84	201.55	212.50	226.46	262.07
	ω_f	193.84	189.46	188.32	190.19	201.63

Ω (rev/s)					Stiffener height-to-width ratios (h/b)
		0.5	1	2	3	4	5	7	9
Ω = 10	ω_b	201.67	204.38	221.37	255.74	305.20	344.63	364.49	394.09
	ω_f	189.58	192.28	209.24	243.56	292.95	328.58	348.37	377.88
Ω = 50	ω_b	262.17	264.48	279.06	309.40	354.51	383.46	403.07	432.33
	ω_f	201.71	203.96	218.39	248.48	293.26	303.20	322.47	351.31

Power law index						Ω (rev/s)
		0	10	20	30	50	60	70	80	100
p = 0	ω_b	369.13	376.11	384.81	395.21	420.86	435.94	448.43	460.07	484.66
	ω_f	369.13	363.91	360.41	358.52	346.48	341.18	336.34	331.96	324.52
p = 3	ω_b	356.04	364.35	373.18	382.55	402.85	413.76	425.17	437.06	462.22
	ω_f	356.04	348.27	341.04	334.33	322.48	317.31	312.64	308.45	301.45
p = 020	ω_b	344.93	353.24	362.11	371.53	391.96	402.97	414.48	426.49	451.92
	ω_f	344.93	337.16	329.95	323.28	311.55	306.47	301.89	297.81	291.05

Ω (rev/s)					n
		1	2	3	4	5	6	7	8
Ω = 50	ω_b	794.02	405.84	505.37	855.01	1337.79	1935.81	2645.59	3466.04
	ω_f	697.76	326.60	446.62	810.15	1302.46	1907.38	2622.44	3447.11
Ω = 100	ω_b	841.22	464.76	583.75	935.80	1417.63	2014.24	2722.61	3541.76
	ω_f	648.73	306.27	466.19	846.03	1346.92	1957.35	2676.29	3503.88

Ω (rev/s)					L / R
		2	4	6	12	16	18	20
Ω = 10	ω_b	970.641	506.664	284.693	132.437	108.847	95.867	80.265
	ω_f	959.431	490.879	268.814	116.486	92.880	76.118	60.476
Ω = 50	ω_b	1001.098	542.781	324.830	182.476	156.150	134.928	119.402
	ω_f	945.043	463.852	245.427	102.718	57.668	36.185	20.457

Ω (rev/s)					R / h
		100	150	200	300	400	450	500
Ω = 10	ω_b	672.235	351.174	239.728	125.005	87.800	78.195	71.402
	ω_f	660.673	339.344	227.794	115.608	78.358	68.741	61.938
Ω = 50	ω_b	706.956	396.529	294.480	221.795	195.024	187.476	181.949
	ω_f	649.141	337.363	234.782	161.659	134.684	127.061	121.468

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Vibration Analysis of Rotating Functionally Graded Cylindrical Shells with Orthogonal Stiffeners

Abstract