Nonlinear dynamic behavior of rotating blade with breathing crack

doi:10.1007/s11465-020-0609-z

Front. Mech. Eng.

2021, Vol. 16

Issue (1) : 196-220 https://doi.org/10.1007/s11465-020-0609-z

RESEARCH ARTICLE

Nonlinear dynamic behavior of rotating blade with breathing crack

Laihao YANG¹(

), Zhu MAO², Shuming WU¹, Xuefeng CHEN¹, Ruqiang YAN¹

¹. The State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China; School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China
². The Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, Lowell, MA 01854, USA

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Abstract

This study aims at investigating the nonlinear dynamic behavior of rotating blade with transverse crack. A novel nonlinear rotating cracked blade model (NRCBM), which contains the spinning softening, centrifugal stiffening, Coriolis force, and crack closing effects, is developed based on continuous beam theory and strain energy release rate method. The rotating blade is considered as a cantilever beam fixed on the rigid hub with high rotating speed, and the crack is deemed to be open and close continuously in a trigonometric function way with the blade vibration. It is verified by the comparison with a finite element-based contact crack model and bilinear model that the proposed NRCBM can well capture the dynamic characteristics of the rotating blade with breathing crack. The dynamic behavior of rotating cracked blade is then investigated with NRCBM, and the nonlinear damage indicator (NDI) is introduced to characterize the nonlinearity caused by blade crack. The results show that NDI is a distinguishable indicator for the severity level estimation of the crack in rotating blade. It is found that severe crack (i.e., a closer crack position to blade root as well as larger crack depth) is expected to heavily reduce the stiffness of rotating blade and apparently result in a lower resonant frequency. Meanwhile, the super-harmonic resonances are verified to be distinguishable indicators for diagnosing the crack existence, and the third-order super-harmonic resonances can serve as an indicator for the presence of severe crack since it only distinctly appears when the crack is severe.

Keywords rotating blade breathing crack nonlinear vibration nonlinear damage indicator

Corresponding Author(s): Laihao YANG

Just Accepted Date: 18 December 2020 Online First Date: 18 January 2021 Issue Date: 11 March 2021

Cite this article:

Laihao YANG,Zhu MAO,Shuming WU, et al. Nonlinear dynamic behavior of rotating blade with breathing crack[J]. Front. Mech. Eng., 2021, 16(1): 196-220.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-020-0609-z
https://academic.hep.com.cn/fme/EN/Y2021/V16/I1/196

Fig.1 (a) Schematic of the rotating blade; (b) motion of the blade; (c) schematic diagram of the blade deformation.

Fig.2 Schematic of the centrifugal force for a micro-unit of the blade.

Fig.3 Schematic diagram of the cracked blade.

Fig.4 FE model of the cantilever blade with breathing crack.

Modal order	Rotating speed, n/(r·min⁻¹)	Natural frequency/Hz		Error/%
Modal order	Rotating speed, n/(r·min⁻¹)	$f u F F M i$	$f u i$	Error/%
The first-order natural frequency	0	254.2	254.5	0.118
	5000	277.1	277.4	0.108
	10000	336.0	336.3	0.089
The second-order natural frequency	0	1577.1	1594.8	1.122
	5000	1607.5	1625.2	1.101
	10000	1695.1	1713.0	1.056
The third-order natural frequency	0	4347.9	4465.6	2.707
	5000	4380.1	4497.9	2.689
	10000	4475.2	4593.4	2.641

Tab.1 Natural frequency of uncracked blade under different rotating speed

Fig.5 Campbell diagram of the uncracked blade.

Fig.6 Comparison of the harmonic vibration responses for uncracked blade, harmonic vibration responses at (a) 0 r/min and (b) 4800 r/min, spectrums at (c) 0 r/min and (d) 4800 r/min. FECCM: Finite element-based contact crack model; NRCBM: Nonlinear rotating cracked blade model.

Fig.7 The transient vibration responses and AFRs under different rotating speed: (a) 0 r/min, (b) 5000 r/min, and (c) 10000 r/min. AFR: Amplitude–frequency responses.

Fig.8 The impact vibration response with mean removed under different rotating speeds: (a) 0 r/min, (b) 5000 r/min, and (c) 10000 r/min. FECCM: Finite element-based contact crack model; NRCBM: Nonlinear rotating cracked blade model.

Fig.9 The harmonic vibration responses and spectrums under different rotating speeds: (a) 0 r/min, (b) 4800 r/min, and (c) 9600 r/min. FECCM: Finite element-based contact crack model; NRCBM: Nonlinear rotating cracked blade model.

Rotating speed, $n$ /(r·min⁻¹)	Crack depth, $a c$ /m	Natural frequency/Hz		Error/%
Rotating speed, $n$ /(r·min⁻¹)	Crack depth, $a c$ /m	$f c F F M i$	$f c i$	Error/%
0	0.001	252.0	253.8	0.7143
	0.002	250.0	249.7	–0.1200
	0.003	246.0	242.5	–1.6260
5000	0.001	274.0	276.0	0.7299
	0.002	271.0	273.0	0.5535
	0.003	264.0	263.5	–0.3788
10000	0.001	332.0	336.0	1.2048%
	0.002	330.0	332.5	1.0638
	0.003	326.0	327.0	0.6079
15000	0.001	405.0	414.0	2.2222
	0.002	404.0	413.3	2.3020
	0.003	401.0	409.0	1.9950

Tab.2 Natural frequency of the cracked blade

Fig.10 Relative errors of the natural frequency obtained by FECCM and NRCBM, respectively. FECCM: Finite element-based contact crack model; NRCBM: Nonlinear rotating cracked blade model.

Relative crack depth $γ$	Natural frequency/Hz
Relative crack depth $γ$	$ζ$ = 0.1	$ζ$ = 0.2	$ζ$ = 0.3	$ζ$ = 1/3	$ζ$ = 0.4	$ζ$ = 0.5	$ζ$ = 0.6	$ζ$ = 0.7	$ζ$ = 0.8	$ζ$ = 0.9
0.00	336.3	336.3	336.3	336.3	336.3	336.3	336.3	336.3	336.3	336.3
0.05	335.5	336.0	336.0	336.0	336.0	336.0	336.0	336.0	336.0	336.0
0.10	335.0	335.5	335.5	335.5	335.5	336.0	336.0	336.0	336.0	336.0
0.15	333.5	334.5	335.0	335.0	335.5	335.5	336.0	336.0	336.0	336.0
0.20	331.5	333.0	334.5	334.5	335.0	335.5	336.0	336.0	336.0	336.0
0.25	329.0	331.5	333.5	334.0	334.5	335.0	335.5	336.0	336.0	336.0
0.30	325.0	329.5	332.0	333.0	334.0	335.0	335.5	336.0	336.0	336.0
0.35	320.0	326.5	330.0	331.0	333.0	334.5	335.0	335.5	336.0	336.0
0.40	313.0	323.0	327.5	329.0	331.5	333.5	334.5	335.5	336.0	336.0
0.45	302.5	317.5	323.0	325.0	329.0	332.5	334.0	335.0	336.0	336.0
0.50	284.5	309.5	315.5	318.0	324.5	330.5	332.5	334.0	335.5	336.0

Tab.3 Natural frequency of the cracked blade with different relative crack depths and locations obtained by NRCBM

Relative crack depth, $γ$	Natural frequency/Hz
Relative crack depth, $γ$	$ζ$ = 0.1	$ζ$ = 0.2	$ζ$ = 0.3	$ζ$ = 0.4	$ζ$ = 0.5	$ζ$ = 0.6	$ζ$ = 0.7	$ζ$ = 0.8	$ζ$ = 0.9
0.00	336.3	336.3	336.3	336.3	336.3	336.3	336.3	336.3	336.3
0.05	336.0	336.2	336.2	336.3	336.3	336.3	336.3	336.3	336.3
0.10	335.2	335.6	335.9	336.1	336.2	336.3	336.3	336.3	336.3
0.15	333.8	334.8	335.4	335.8	336.1	336.2	336.3	336.3	336.3
0.20	331.8	333.6	334.7	335.4	335.9	336.1	336.2	336.3	336.3
0.25	328.9	331.9	333.7	334.9	335.6	335.9	336.2	336.3	336.3
0.30	324.9	329.7	332.2	334.1	335.2	335.7	336.1	336.3	336.3
0.35	319.0	326.6	330.0	332.9	334.6	335.4	335.9	336.2	336.3
0.40	310.1	322.3	326.7	331.1	333.7	334.8	335.7	336.2	336.3
0.45	295.2	316.0	321.0	328.1	332.3	333.8	335.1	336.1	336.3
0.50	265.8	306.0	309.5	322.1	329.8	331.4	333.6	336.0	336.3

Tab.4 Natural frequency of the cracked blade with different relative crack depths and locations estimated by BM

Fig.11 Comparison of the first-order resonant frequency obtained by NRCBM and BM, respectively. NRCBM: Nonlinear rotating cracked blade model; BM: Bilinear model.

Influence factors	Invariant parameters	Varying parameters
EO (Case 1)	$(γ, ζ, F E O) = (0.3, 1 / 3, 2000 N)$	$E O = 1, 2, ..., 4$
	$(γ, ζ, F E O) = (0.5, 1 / 3, 2000 N)$	$E O = 1, 2, ..., 4$
	$(γ, ζ, F E O) = (0.5, 2 / 3, 2000 N)$	$E O = 1, 2, ..., 4$
$γ$ (Case 2)	$(ζ, E O, F E O) = (1 / 3, 1, 2000 N)$	$γ = 0.10, ? 0.15, ..., ? 0.50$
	$(ζ, E O, F E O) = (1 / 3, 2, 2000 N)$	$γ = 0.10, ? 0.15, ..., ? 0.50$
	$(ζ, E O, F E O) = (1 / 3, 3, 2000 N)$	$γ = 0.10, ? 0.15, ..., ? 0.50$
	$(ζ, E O, F E O) = (1 / 3, 4, 2000 N)$	$γ = 0.10, ? 0.15, ..., ? 0.50$
$ζ$ (Case 3)	$(γ, E O, F E O) = (0.5, 1, 2000 N)$	$ζ = 0.10, 0.15, ..., 0.90$
	$(γ, E O, F E O) = (0.5, 2, 2000 N)$	$ζ = 0.10, 0.15, ..., 0.90$
	$(γ, E O, F E O) = (0.5, 3, 2000 N)$	$ζ = 0.10, 0.15, ..., 0.90$
	$(γ, E O, F E O) = (0.5, 4, 2000 N)$	$ζ = 0.10, 0.15, ..., 0.90$

Tab.5 Condition and crack parameters for the numerical simulation of NRCBM

Fig.12 Spectrum cascades under different rotating speeds when (a)

γ = 0.3

ζ = 1 / 3

; (b)

γ = 0.5

ζ = 1 / 3

; and (c)

γ = 0.5

ζ = 2 / 3

Fig.13 Synchronous vibration responses under different excitation frequencies: (a)

E O = 1

, (b)

E O = 2

, (c)

E O = 3

, and (d)

E O = 4

Fig.14 Vibration responses under different multi-frequency excitation: (a)

E O + 2 E O

, (b)

E O + 0.5 E O

, (c)

E O + 3 E O

, (d)

E O + 1.7 E O

, (e)

E O + 5 E O

, and (f)

E O + 4.7 E O

Fig.15 NDIs of displacement and velocity in sub-critical region under different crack parameters: (a)

γ = 0.3

ζ = 1 / 3

; (b)

γ = 0.5

ζ = 1 / 3

; and (c)

γ = 0.5

ζ = 2 / 3

. NDI: Nonlinear damage indicator.

Fig.16 AFRs for different relative crack depths. (a) Primary resonant region when

ζ = 1 / 3

; (b) sub-critical region when

ζ = 1 / 3

; (c) primary resonant region when

ζ = 2 / 3

; (d) sub-critical region when

ζ = 2 / 3

. AFR: Amplitude–frequency responses.

Fig.17 Spectrum cascades for rotating blade with crack in different relative crack depths when (a)

E O = 1

; (b)

E O = 2

; (c)

E O = 3

; and (d)

E O = 4

Fig.18 Phase portraits under different relative crack depths (

ζ = 1 / 3

). (a)

E O = 1

; (b)

E O = 2

; (c)

E O = 3

; and (d)

E O = 4

Fig.19 The variation of NDIs with the change of relative crack depth

γ

. NDI: Nonlinear damage indicator.

Fig.20 AFRs for different relative crack locations: (a) Primary resonant region when

ζ = 0. 30

, (b) sub-critical region when

ζ = 0. 30

, (c) primary resonant region when

ζ = 0. 50

, and (d) sub-critical region when

ζ = 0. 50

. AFR: Amplitude–frequency responses.

Fig.21 Spectrum cascades for rotating blade with crack in different locations when (a)

E O = 1

, (b)

E O = 2

, (c)

E O = 3

, and (d)

E O = 4

Fig.22 Phase portraits under different relative crack locations. (a)

E O = 1

; (b)

E O = 2

; (c)

E O = 3

; and (d)

E O = 4

Fig.23 NDIs of the displacement under different relative crack locations. NDI: Nonlinear damage indicator.

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