Dynamic characteristics of a shrouded blade with impact and friction

doi:10.1007/s11465-019-0566-6

Front. Mech. Eng.

2020, Vol. 15

Issue (2) : 209-226 https://doi.org/10.1007/s11465-019-0566-6

RESEARCH ARTICLE

Dynamic characteristics of a shrouded blade with impact and friction

Xumin GUO¹, Jin ZENG¹, Hui MA^1,²(

), Chenguang ZHAO¹, Lin QU¹, Bangchun WEN¹

¹. School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China
². Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang 110819, China

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Abstract

A simplified computational model of a twisted shrouded blade with impact and friction is established. In this model, the shrouded blade is simulated by a flexible Timoshenko beam with a tip-mass, and the effects of centrifugal stiffening, spin softening, and Coriolis force are considered. Impact force is simulated using a linear spring model, and friction force is generated by a tangential spring model under sticking state and a Coulomb friction model under sliding state. The proposed model is validated by a finite element model. Then, the effects of initial gap and normal preload, coefficient of friction, and contact stiffness ratio (the ratio of tangential contact stiffness to normal contact stiffness) on system vibration responses are analyzed. Results show that resonant peaks become inconspicuous and impact plays a dominant role when initial gaps are large between adjacent shrouds. By contrast, in small initial gaps or initial normal preloads condition, resonant speed increases sharply, and the optimal initial normal preloads that can minimize resonant amplitude becomes apparent. Coefficient of friction affects the optimal initial normal preload, but it does not affect vibration responses when the contact between shrouds is under full stick. System resonant amplitude decreases with the increase of contact stiffness ratio, but the optimal initial normal preload is unaffected.

Keywords twisted shrouded blade dynamic analysis impact friction separate–stick–slip motion

Corresponding Author(s): Hui MA

Just Accepted Date: 19 January 2020 Online First Date: 02 March 2020 Issue Date: 25 May 2020

Cite this article:

Xumin GUO,Jin ZENG,Hui MA, et al. Dynamic characteristics of a shrouded blade with impact and friction[J]. Front. Mech. Eng., 2020, 15(2): 209-226.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-019-0566-6
https://academic.hep.com.cn/fme/EN/Y2020/V15/I2/209

Fig.1 Shrouded blade structures.

Fig.2 Schematic of twisted shrouded blade with impact and friction.

Tab.1 First four natural frequencies under different modal truncations at W=8500 r/min

Fig.3 Vibration responses of the bending displacement of the blade tip under D = 0.6 mm at W = 8500 r/min: (a) Displacement waveforms and (b) partial enlarged waveforms.

Tab.2 Shrouded blade parameters

Fig.4 Amplitude–frequency responses obtained from the reduced and full computational models: (a) D = 0.6 mm, (b) D = 0.1 mm, (c) D = 0.05 mm, (d) N₀ = 0 N, (e) N₀ = 10 N, and (f) N₀ = 50 N.

Fig.5 Schematics of the FE models of the shrouded blade: (a) Blade model, (b) arbitrary blade section, and (c) shroud inclination angle α.

Fig.6 Schematic of the first FE model by using ANSYS.

Fig.7 Amplitude–frequency responses: (a) Different initial gaps and (b) small initial gap (D = 1 mm) and different initial normal preloads.

Fig.8 Response comparisons among the four models: (a) Impact force on the left side of the shroud, (b) friction force on the left side of the shroud, (c) hysteresis loop on the left side of the shroud, (d) contact state on the left side of the shroud, (e) impact force on the right side of the shroud, (f) friction force on the right side of the shroud, (g) hysteresis loop on the right side of the shroud, (h) contact state on the right side of the shroud, (i) displacements in flexural direction, and (j) displacements in swing direction.

Tab.3 Calculation times of the four models

Fig.9 Schematic of the second FE model by using ANSYS.

Fig.10 Response comparison at W = 10000 r/min: (a) Impact forces, (b) friction forces, and (c) displacements in flexural direction.

Fig.11 Amplitude–frequency responses of shrouded blade: (a) Different initial gaps and (b) small initial gaps (D = 1 mm) and initial normal preloads.

Fig.12 Amplitude–frequency responses under different D, N₀, and m: (a) m = 0.1 under different initial gaps, (b) m= 0.1 under different initial normal preloads, (c) m= 0.3 under different initial gaps, (d) m= 0.3 under different initial normal preloads, (e) m= 0.5 under different initial gaps, and (f) m= 0.5 under different initial normal preloads.

Fig.13 Resonant response characteristics under different coefficients of friction: (a) Resonant rotational speed under different initial gaps, (b) resonant amplitude under different initial gaps, (c) resonant rotational speed under different initial normal preloads, and (d) resonant amplitude under different normal preloads.

Fig.14 Vibration responses at W=25500 r/min under m= 0.1: (a) Displacement

v L

, (b) friction force

F f1

, (c) impact force

N 1

, (d) frequency spectrum of displacement, (e) frequency spectrum of friction force, and (f) frequency spectrum of impact force.

Fig.15 Vibration responses at W=25500 r/min under m = 0.5: (a) Displacement

v L

, (b) friction force

F f1

, (c) impact force

N 1

, (d) frequency spectrum of displacement, (e) frequency spectrum of friction force, and (f) frequency spectrum of impact force.

Fig.16 Amplitude–frequency responses under different D, N₀, and x: (a) x= 0.2 under different initial gaps, (b) x= 0.2 under different initial normal preloads, (c) x = 0.5 under different initial gaps, (d) x = 0.5 under different initial normal preloads, (e) x = 1 under different initial gaps, and (f) x = 1 under different initial normal preloads.

Fig.17 Resonant response characteristics under different contact stiffness ratios: (a) Resonant rotational speed under different initial gaps, (b) resonant amplitude under different initial gaps, (c) resonant rotational speed under different initial normal preloads, and (d) resonant amplitude under different normal preloads.

A	Cross-sectional area of the blade
b	Blade width
D, D^*	Rayleigh damping matrices before and after dimension reduction
E	Young’s modulus
F, $F ¯$	Canonical external force vectors without and with impact and friction
F^*	Canonical external force vector after dimension reduction
F_e, F₀	Uniformly distributed aerodynamic force per unit length and aerodynamic force amplitude
F_y, F_z	Components of impact and friction force in the flexural and swing directions
F_f1, F_f2	Friction force
f_c(x)	Centrifugal force of the shrouded blade
f_e	Aerodynamic frequency
f_r	Rotational frequency
f_n1, f_n2	The first two-order natural frequencies of the shrouded blade
G, G^*	Coriolis force matrices before and after dimension reduction
h	Blade thickness
I_y, I_z	Area moment of inertias of y and z axes of the blade section
K, K^*	Stiffness matrices before and after stiffness reduction
K_e, K_c, K_s	Structural stiffness matrix, centrifugal stiffening matrix, and spin softening matrix
K_acc	Dtiffness matrix caused by angular acceleration
k_t, k_n	Tangential and normal contact stiffness
L	Blade length
M, M^*	Mass matrices before and after dimension reduction
N	Number of modal truncation
nr	Dimension of matrix after dimension reduction
n	Section number of the first FE model
N₀	Initial normal preload
N₁, N₂	Impact force
N^*	Dimension reduction matrix
m_s	Shroud mass
q, q^*	Canonical coordinate vectors of the blade before and after dimension reduction
R_d	Radius of disk
v_L, w_L	Bending and swing displacements at blade tip
v_s, w_s	Tangential and normal displacements of shroud
z_s1	Tangential displacement of active blade shroud
z_s2, z_s3	Displacements of contact point
r	Density of the blade
q	Rotational angle of the disk
$β ′$	Twist angle of shrouded blade
b₁, b_L	Stagger angle at the root and blade tip of the blade
b_n	Angle of an arbitrary cross section between z axis and z_n axis
b(x)	Twist angle of an arbitrary Point P of the blade
W	Rotational speed/(r·min⁻¹)
f₁_i(x), f₂_i(x), f₃_i(x)	ith modal shape functions in radial, flexural/swing, and rotational angle directions
D	Initial gap
m	Coefficient of friction
a	Shroud inclination angle
x	Contact stiffness ratio
x₁, x₂	First and second modal damping ratios
w	Angular velocity of the blade/(rad·s⁻¹)
FE	Finite element

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