Skip Navigation Links
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


Nonlinear Dynamics of a Reduced Cracked Rotor

Journal of Vcibration Testing and System Dynamics 2(3) (2018) 257--269 | DOI:10.5890/JVTSD.2018.09.006

K. Lu$^{1}$,$^{3}$,$^{4}$, Y. Lu$^{1}$, B. C. Zhou$^{1}$, W. Jian$^{1}$, Y. F. Yang$^{1}$, Y. L. Jin$^{2}$, Y. S. 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, P. R. China

$^{3}$ 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

Download Full Text PDF



The transient proper orthogonal decomposition (TPOD) method is applied for order reduction in the rotor system in this paper. A 26-DOFs rotor model with crack is established by the Newton’s second law, and the dynamical behaviors (bifurcation diagram, mplitude frequency curve, etc.) of the crack fault are discussed. The optimal reduced model can be provided by the proper orthogonal mode (POM) energy method, the TPOD method is applied to reduce the original system to a three-DOFs one at a certain speed corresponds to the maximum energy. The efficiency of the TPOD method is verified via comparing with the bifurcation diagram of the original and reduced rotor system. The order reduction method provides qualitative analysis to study the reduced model of the high-dimensional rotor system.


  1. [1]  Alho, A. and Uggla, C. (2015), Global dynamics and inflationary center manifold and slow-roll approximants, J. Math. Phys., 56, 012502.
  2. [2]  Valls, C. (2015), Center problem in the center manifold for quadratic and cubic differential systems in R3, Appl. Math. Comput., 251, 180-191.
  3. [3]  Kazufumi, I. and Karl, K. (2008), Reduced-order optimal control based on approximate inertial manifolds for nonlinear dynamical systems, SIAM J. Numer. Anal., 46, 2867-2891.
  4. [4]  Marion, M. (1989), Approximate inertial manifolds for reaction-diffusion equations in high space dimension, Dyn. Differ. Equ., 1 245-267.
  5. [5]  Guo, S.J. and Yan, S.L. (2016), Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, J. Differential Equations, 260, 781-817.
  6. [6]  Marion, M. and Temam, R. (1989), Nonlinear Galerkin methods, SIAM J. Numer Anal., 5, 1139-1157.
  7. [7]  Stefanescu, R., Sandu, A., and Navon, I.M. (2015), POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation, J. Comput. Physi., 295, 569-595.
  8. [8]  Benaissa, B., Hocine, N.A., Belaidi, I., Hamrani, A., and Pettarin, V. (2016), Crack identification using model reduction based on proper orthogonal decomposition coupled with radial basis functions, Struc. Multidisc. Optim., DOI: 10.1007/s00158-016-1400-y.
  9. [9]  Yang, H.L. and Radons, G. (2012), Geometry of inertial manifolds probed via a Lyapunov projection method, Phys. Rev. Lett., 108, 154101.
  10. [10]  Rega, G. and Troger, H. (2005), Dimension reduction of dynamical systems: methods, models, applications, Nonlinear Dynamics, 41, 1-15.
  11. [11]  Steindl, A. and Troger, H. (2001), Methods for dimension reduction and their application in nonlinear dynamics, Int. J. Solids Struct., 38, 2131-2147.
  12. [12]  Holmes, P.J., Lumley, L., and Berkooz, G. (2012), Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press.
  13. [13]  Lumley, J.L. (2001), Early work on fluid mechanics in the IC engine, Ann. Rev. Fluid Mech., 33, 319-38.
  14. [14]  Vaccaro, R. (1991), SVD and Signal Processing II: Algorithms, analysis and applications, Elsevier Science Inc..
  15. [15]  Rosenfeld, A. and Kak, A.C. (1982), Digital Picture Processing, Academic Press, New York.
  16. [16]  Oliveira, I. and Patera, A. (2007), Reduced-basis techniques for rapid reliable optimization of systems described by affnely parametrized coercive elliptic partial differential equations, Optimization and Engineering, 8, 43-65.
  17. [17]  Georgiou, I.T. and Schwartz, I.B. (1999), Dynamics of large scale coupled structural mechanical systems: a singular perturbation proper orthogonal decomposition approach, SIAM J. Appl. Math., 59, 1178-1207.
  18. [18]  Kerschen, G., Feeny, B.F., and Golinval, J.C. (2003), On the exploitation of chaos to build reduced-order models, Computer Methods in Applied Mechanics and Engineering, 192, 1785-1795.
  19. [19]  Gay, D.H. and Ray, W.H. (1988), Application of singular value methods for identification and model based control of distributed parameter systems, In Proc. IFAC Workshop on Model Based Process Control, Atlanta, GA, June 1988, 95-102.
  20. [20]  Preisendorfer, R.W. (1988), Principal component analysis in meteorology and oceanography, Elsevier, Amsterdam.
  21. [21]  Hinze, M. and Volkwein, S. (2008), Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition, Computational Optimization and Applications, 39, 319-345.
  22. [22]  Fossati, M. and Habashi, W.G. (2013), Multiparameter Analysis of aero-icing problems using proper orthogonal decomposition and multidimensional interpolation, AIAA Journal, 51, 946-960.
  23. [23]  Terragni, F., Valero, E., and Vega, J.M. (2011), Local POD plus Galerkin projection in the unsteady lid-driven cavity problem, SIAM J. Sci. Comput., 33, 3538-3561.
  24. [24]  Lieu, T. and Farhat, C. (2007), Adaptation of aeroelastic reduced-ordermodels and application to an F-16 configuration, AIAA Journal, 45, 1244-1257.
  25. [25]  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.
  26. [26]  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.
  27. [27]  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.
  28. [28]  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.
  29. [29]  Lu, K., Lu, Z.Y., and Chen, Y.S. (2017), Comparative study of two order reduction methods for high-dimensional rotor systems, International Journal of Non-Linear Mechanics, DOI: 10.1016/j.ijnonlinmec.2017.09.006.
  30. [30]  Adiletta, G., Guido, A.R., and Rossi, C. (1996), Chaotic motions of a rigid rotor in short journal bearings, Nonlinear Dynamics, 10, 251-269.
  31. [31]  Riaz, M.S. and Feeny, B.F. (2003), Proper orthogonal modes of a beam sensed with strain gages, Journal of Vibration and Acoustics, 125, 129-131.
  32. [32]  Han, S. and Feeny, B.F. (2003), Application of proper orthogonal decomposition to structural vibration analysis, Mech. Syst. Signal Process, 17, 989-1001.
  33. [33]  Feeny, B.F. and Kappagantu, R. (1998), On the physical interpretation of proper orthogonal modes in vibrations, Journal of Sound Vibration, 211, 607-616.
  34. [34]  Sipcic, S.R. and Bengeudouar, A. (1996), Karhunen-Loeve decomposition in dynamical modeling, Sixth conference on Nonlinear Vibrations, Stability, and Dynamics of Structures, Blacksburg, VA.
  35. [35]  Han, S. and Feeny, B.F. (2002), Enhanced proper orthogonal decomposition for the modal analysis of homogeneous structures, Journal of Sound and Vibration, 8, 19-40.
  36. [36]  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.
  37. [37]  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.