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Journal of Applied Nonlinear Dynamics 9(1) (2020) 71--91 | DOI:10.5890/JAND.2020.03.007

Kuan Lu$^{1}$,$^{3}$,$^{4}$, Yongfeng Yang$^{1}$, Hai Yu$^{3}$, Yulin Jin$^{2}$, Yushu Chen$^{3}$

$^{1}$ Institute of Vibration Engineering, Northwestern Polytechnical University, Xi’an, 710072, P. R. China

$^{2}$ School of Aeronautics and Astronautics, Sichuan University, 610065

$^{3}$ School of Astronautics, Harbin Institute of Technology, Harbin 150001, P. R. China

$^{4}$ College of Engineering, The University of Iowa, Iowa City, IA 52242, USA

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Abstract

The transient proper orthogonal decomposition (TPOD) method is generalized to high-dimensional rotor system models with common faults and the efficiency of the reduced models is discussed in this paper. The method to confirm the optimal reduced rotor model for order reduction is proposed based on the physical significance of the TPOD method. The physical significance of the TPOD method can be provided by the proper orthogonalmode (POM). Three rotor models with faults are established by the Newton’s second law: the first is crack fault, the second is looseness fault and the third is the model with coupling faults. The model with coupling faults contains more complex characteristics than the other two models with single fault (looseness, crack). The TPOD method is applied to obtain the relatively optimal reduced model based on the POM energy. The efficiency of order reduction method is verified via the energy curves of POM and many other dynamical behaviors (the bifurcation diagrams, the amplitude-frequency curves, the phase curves, etc.). The optimal reduced models of the rotor systems can be obtained via applying the TPOD method on the basis of the POM energy.

References

1. [1]	Muszynska, A. (1989), Rotor-to-stationary element rub-related vibration phenomena in rotating machineryliterature survey, Shock and Vibration Digest, 21, 3–11

2. [2]	Hou, L., Chen, Y.S., and Cao, Q.J. (2014), Nonlinear vibration phenomenon of an aircraft rub-impact rotor system due to hovering flight, Communications in Nonlinear Science and Numerical Simulation, 19, 286–297.

3. [3]	Chu, F.L. and Tang, Y. (2001), Stability and nonlinear responses of a rotor-bearing system with pedestal looseness, Journal Sound and Vibration, 241, 879-893.

4. [4]	Ma, H., Zhao, X.Y., Teng, Y.N., and Wen, B.C. (2011), Analysis of dynamic characteristics for a rotor system with pedestal looseness. Shock and Vibration, 18, 13-27

5. [5]	Lu, K., Yu, H., Chen, Y.S., Cao, Q.J., and Hou, L. (2015), A modified nonlinear POD method for order reduction based on transient time series, Nonlinear Dynamics, 79, 1195-1206

6. [6]	Lu, K., Jin, Y.L., Chen, Y.S., Cao, Q.J., and Zhang, Z.Y. (2015), Stability analysis of reduced rotor pedestal looseness fault model, Nonlinear Dynamics, 82, 1611-1622

7. [7]	Lu, K., Chen, Y.S., Jin, Y.L., and Hou, L. (2016), Application of the transient proper orthogonal decomposition method for order reduction of rotor systems with faults, Nonlinear Dynamics, 86, 1913-1926

8. [8]	Lu, Z.Y., Hou, L., Chen, Y.S., and Sun, C.Z. (2016), Nonlinear response analysis for a dual-rotor system with a breathing transverse crack in the hollow shaft, Nonlinear Dynamics, 83, 169–185

9. [9]	Adewusi, S. and Al-Bedoor, B. (2002), Experimental study on the vibration of an overhung rotor with a propagating transverse crack, Shock and Vibration, 9, 91–104

10. [10]	Sekhar, A.S. (2003), Crack detection and monitoring in a rotor supported on fluid film bearings: start-up vs run-down, Mechanical Systems and Signal Processing, 17, 897–901

11. [11]	Li, Y.B., etc. (2018), Early fault diagnosis of rolling bearings based on hierarchical symbol dynamic entropy and binary tree support vector machine, Journal of Sound and Vibration, 428, 72-86

12. [12]	Sinou, J.J. and Lees, A.W. (2007), A non-linear study of a cracked rotor, European Journal of Mechanics A/Solids, 26, 152–170

13. [13]	Sinou, J.J. and Faverjon, B. (2012), The vibration signature of chordal cracks in a rotor system including uncertainties, Journal Sound and Vibration, 331, 138–154

14. [14]	Verdugo, A. and Rand, R. (2008), Center manifold analysis of a DDE model of gene expression, Communications in Nonlinear Science and Numerical Simulation, 13, 1112–1120.

15. [15]	Gentile, G., Mastropietro, V., and Procesi, M. (2005), Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions, Communications in Mathematical Physics, 256, 437–490

16. [16]	Marion, M. and Temam, R. (1989), Nonlinear Galerkin methods, SIAM Journal on Numerical Analysis, 5, 1139–1157

17. [17]	Kerschen, G., Golinval, J.C., Vakakis, A.F., and Bergman, L.A. (2005), The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview, Nonlinear Dynamics, 41, 147–169

18. [18]	Marion, M. (1989), Approximate inertial manifolds for reaction–diffusion equations in high space dimension, Dynamics and Differential Equations, 1, 245–267

19. [19]	Yang, H.L. and Radons, G. (2012), Geometry of inertial manifolds probed via a Lyapunov projection method, Physical Review Letters, 108, 154101

20. [20]	Rega, G. and Troger, H. (2005), Dimension reduction of dynamical systems: methods, models, applications, Nonlinear Dynamics, 41, 1–15

21. [21]	Lu, K., Chen, Y.S., Cao, Q.J., Hou, L., and Jin, Y.L. (2017), Bifurcation analysis of reduced rotor model based on nonlinear transient POD method, International Journal of Non-Linear Mechanics, 89, 83-92

22. [22]	Lu, K. (2018), Statistical moment analysis of multi-degree of freedom dynamic system based on polynomial dimensional decomposition method, Nonlinear Dynamics, 10.1007/s11071-018-4303-1.

23. [23]	Lu, K., Yang, Y.F., Xia, Y.S., and Fu, C. (2019), Statistical moment analysis of nonlinear rotor system with multi uncertain variables, Mechanical Systems and Signal Processing, 116, 1029-1041

24. [24]	Lu, K., Lu, Z.Y., and Chen, Y.S. (2017), Comparative study of two order reduction methods for highdimensional rotor systems, International Journal of Non-Linear Mechanics, DOI:10.1016/j.ijnonlinmec.2017. 09.006.

25. [25]	Kerschen, G., Golinval, J.C., Vakakis, A.F., and Bergman, L.A. (2005), The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An overview, Nonlinear Dynamics, 41, 147-169.

26. [26]	Smith, T.R., Moehlis, J., and Holmes, P. (2005), Low-dimensional modeling of turbulence using the proper orthogonal decomposition: a tutorial, Nonlinear Dynamics, 41, 275-307.

27. [27]	Feeny, B.F. and Kappagantu, R. (1998), On the physical interpretation of proper orthogonal modes in vibration, Journal of Sound and Vibration, 211, 607-616.

28. [28]	Lin, Y.L. and Chu, F.L. (2010), The dynamical behavior of a rotor system with a slant crack on the shaft, Mechanical Systems and Signal Processing, 24, 522–545.

29. [29]	Adiletta, G., Guido, A.R., and Rossi, C. (1996), Chaotic motions of a rigid rotor in short journal bearings, Nonlinear Dynamics, 10, 251-269.