Abstract
This paper focuses on the nonlinear response characteristics of a dual-rotor system with a breathing transverse crack in the hollow shaft of the high-pressure rotor (rotor 1). A finite element model of the system is set up, and the motion equations of the system are formulated, in which the unbalance excitations of the rotor 1 and rotor 2 (low-pressure rotor) and the time-varying stiffness of the cracked shaft are considered. By using the harmonic balance method, the motion equations are analytically solved to obtain the dynamic responses of the two rotors. Accordingly, the effects of the crack depth and location on the vibration amplitudes are discussed in detail. The results indicate that when a transverse crack appears, it may bring super-harmonic responses to the rotor system, and the resonance peaks at the second, third and even fourth subcritical whirling speeds of the two rotors can be observed. The deeper the crack is, the larger the resonances amplitudes are, especially when the crack is located in the middle of the shaft or around the disks. In addition, the super-harmonic responses of rotor 1 where the crack located, can also be observed in rotor 2, which means that the crack signals can be detected in the entire system. Moreover, the numerical computations are carried out by using the Newmark-\(\beta \) method, which shows great agreement with the previous analytical results. The results obtained in this paper will contribute to the modeling and the fault diagnosis of dual-rotor systems with hollow-shaft crack.
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Abbreviations
- \(\alpha \) :
-
Crack angle of the outer circle cross section
- \(\alpha _1 \) :
-
Crack angle of the inner circle cross section
- \(A_\mathrm{ce}\) :
-
The area of the uncracked segment
- \(\mathbf{A}_{s}\) :
-
Coefficient of Fourier series
- \(\beta \) :
-
Speed ratio
- \(\mathbf{B}_{s}\) :
-
Coefficient of Fourier series
- c :
-
The new centroid of the cross section
- \(\mathbf{C}_{1}, \mathbf{C}_{2}\) :
-
Damping matrixes
- \(\mathbf{C}1_{s}\) :
-
Coefficient of Fourier series
- \(\mathbf{D}_{s}\) :
-
Coefficient of Fourier series
- e :
-
The distance between o and c
- E :
-
Young’s modulus
- \(\mathbf{F}_{i}\) :
-
Unbalance excitation vectors
- \(\mathbf{F}_{g}\) :
-
Gravity vector
- \(\mathbf{G}_{i}\) :
-
Gyroscopic matrixes
- G :
-
Shear modulus
- h :
-
Depth of the transverse crack
- \(I_{1}, I_{2}\) :
-
The reduction in the moment of inertias of the cracked element for c-xy
- \(I_x^o ,I_y^o \) :
-
The reduction in the moment of inertias of the cracked element for o-xy
- \(J_{\mathrm{d}}^{d}\) :
-
Diametral moment of inertia of the disk
- \(J_{\mathrm{p}}^{d} \) :
-
Polar moment of inertia of the disk
- \(\alpha _1 ,\beta _1 ,\alpha _2 ,\beta _2 \) :
-
Damping proportional coefficients
- \(k_{xx}, k_{yy}\) :
-
Bearing stiffness
- \(\mathbf{K}_{i}\) :
-
Stiffness matrix
- \(\mathbf{k}_c^k \) :
-
The reduced stiffness matrix by crack in element k
- L :
-
Length of the rotor 1
- \(m^{\mathrm{d}}\) :
-
Mass of disk
- \(\mathbf{M}_{i}\) :
-
Mass matrixes
- N :
-
Number of the element of shaft
- o :
-
Centroid of the cross section without crack
- o-xy :
-
The fixed coordinate system
- q :
-
Displacement vector
- \({\dot{\mathbf{q}}}\) :
-
Velocity vector
- \({\ddot{\mathbf{q}}}\) :
-
Acceleration vector
- r :
-
Inner radius of the rotor 1
- \(\mathbf{Q}_{1}, \mathbf{Q}_{2}\) :
-
Vectors of the unbalance excitation
- R :
-
Outer radius of the rotor 1
- t :
-
Time
- \(\mu \) :
-
Non-dimensional crack depth
- \(\mu _{\mathrm{s}}\) :
-
Shear coefficient of section
- \(\upsilon \) :
-
Poisson’s ratio
- \({\varvec{\Omega }}_{1}, \varvec{\Omega }_{2}\) :
-
The rotation speeds of rotor 1, rotor 2
- \(c_{xx}\),\(c_{yy}\) :
-
Bearing damping
- \(c_{xy} =c_{yx}\) :
-
Coupling damping of the bearing
- \(k_{xy}=k_{yx}\) :
-
Coupling stiffness of the bearing
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Acknowledgments
The authors would like to acknowledge the financial supports from the National Basic Research Program (973 Program) of China (Grant No. 2015CB057400).
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Appendix
Finite element matrices:
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Lu, Z., Hou, L., Chen, Y. et al. Nonlinear response analysis for a dual-rotor system with a breathing transverse crack in the hollow shaft. Nonlinear Dyn 83, 169–185 (2016). https://doi.org/10.1007/s11071-015-2317-5
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DOI: https://doi.org/10.1007/s11071-015-2317-5