Nonlinear Dynamics

, Volume 86, Issue 3, pp 2023–2034 | Cite as

Characterization of the nonlinear response of defective multi-DOF oscillators using the method of phase space topology (PST)

  • M. SamadaniEmail author
  • C. A. Kitio Kwuimy
  • C. Nataraj
Original Paper


Most engineered systems are nonlinear and exhibit phenomena that can only be predicted by nonlinear models. However, the application of model-based approaches for diagnostics has been constrained mostly to linearized or simplified models. This paper introduces a fundamentally new approach (“PST”) for characterization of nonlinear response of systems based on the topology of the phase space trajectory. The method uses the density distribution of the system states to quantify this topology and extracts features that can be used for system diagnostics. The various parameters of the PST method have been analyzed to explore the effectiveness of the method, and it has been employed in the non-trivial diagnostics problem of a multiple degree of freedom oscillator system with various defects occurring simultaneously.


Phase space topology Diagnostics Fault detection Feature extraction Parameter estimation Time series Nonlinear systems 



This work is supported by the US Office of Naval Research under the grant ONR N00014-13-1-0485 with Mr. Anthony Seman III and Captain Lynn Petersen as the Program Managers. We deeply appreciate this support and are humbled by ONR’s enthusiastic recognition of the importance of this research.


  1. 1.
    Nelson, H.D., Nataraj, C.: Dynamics of a rotor system with a cracked shaft. J. Vib. Acoust. Stress Reliab Des. 108(2), 189–196 (1986)CrossRefGoogle Scholar
  2. 2.
    Sekhar, A.: Crack identification in a rotor system: a model-based approach. J. Sound Vib. 270(4), 887–902 (2004)CrossRefGoogle Scholar
  3. 3.
    Jain, J., Kundra, T.: Model based online diagnosis of unbalance and transverse fatigue crack in rotor systems. Mech. Res. Commun. 31(5), 557–568 (2004)CrossRefGoogle Scholar
  4. 4.
    Rabiei, E., Droguett, E.L., Modarres, M., Amiri, M.: Damage precursor based structural health monitoring and damage prognosis framework. In: Podofillini, L., Sudret, B., Stojadinovic, B., Zio, E., Kröger, W. (eds.) Safety and Reliability of Complex Engineered Systems, vol. ch. 304, pp. 2441–2449. CRC Press, Boca Raton (2015)Google Scholar
  5. 5.
    Kwuimy, C.K., Samadani, M., Nataraj, C.: Preliminary diagnostics of dynamic systems from time series. In: ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. V006T10A029–V006T10A029, American Society of Mechanical Engineers (2014)Google Scholar
  6. 6.
    Isermann, R.: Process fault detection based on modeling and estimation methods-a survey. Automatica 20(4), 387–404 (1984)CrossRefzbMATHGoogle Scholar
  7. 7.
    Isermann, R.: Fault diagnosis of machines via parameter estimation and knowledge processing-tutorial paper. Automatica 29(4), 815–835 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wu, X., Bellgardt, K.H.: On-line fault detection of flow-injection analysis systems based on recursive parameter estimation. Anal. Chimica Acta 313(3), 161–176 (1995)CrossRefGoogle Scholar
  9. 9.
    Dey, S., Biron, Z.A., Tatipamula, S., Das, N., Mohon, S., Ayalew, B., Pisu, P.: On-board thermal fault diagnosis of lithium-ion batteries for hybrid electric vehicle application, IFAC-PapersOnLine, vol. 48, no. 15, pp. 389 – 394, 2015. 4th IFAC Workshop on Engine and Powertrain Control, Simulation and Modeling E-COSM 2015Columbus, Ohio, USA, 23-26 August (2015)Google Scholar
  10. 10.
    Mevel, B., Guyader, J.L.: Routes to chaos in ball bearings. J. Sound Vib. 162(3), 471–487 (1993)CrossRefzbMATHGoogle Scholar
  11. 11.
    Muller, P., Bajkowski, J., Soffker, D.: Chaotic motions and fault detection in a cracked rotor. Nonlinear Dyn. 5(2), 233–254 (1994)Google Scholar
  12. 12.
    Sankaravelu, A., Noah, S.T., Burger, C.P.: Bifurcation and chaos in ball bearings. ASME Appl. Mech. Div. Publ. 192, 313–313 (1994)Google Scholar
  13. 13.
    Yang, Y., Ren, X., Qin, W., Wu, Y., Zhi, X.: Analysis on the nonlinear response of cracked rotor in hover flight. Nonlinear Dyn. 61(1–2), 183–192 (2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kappaganthu, K., Nataraj, C.: Nonlinear modeling and analysis of a rolling element bearing with a clearance. Commun. Nonlinear Sci. Numer. Simul. 16(10), 4134–4145 (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    Zhang, B., Li, Y.: Six degrees of freedom coupled dynamic response of rotor with a transverse breathing crack. Nonlinear Dyn. 78(3), 1843–1861 (2014)Google Scholar
  16. 16.
    Wang, W., Wu, Z., Chen, J.: Fault identification in rotating machinery using the correlation dimension and bispectra. Nonlinear Dyn. 25(4), 383–393 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dubey, C., Kapila, V.: Detection and characterization of cracks in beams via chaotic excitation and statistical analysis. In: Banerjee, S., Mitra, M., Rondoni, L. (eds.) Applications of Chaos and Nonlinear Dynamics in Engineering, vol. 1, pp. 137–164. Springer, Berlin (2011). doi: 10.1007/978-3-642-21922-1_5
  18. 18.
    Iwaniec, J., Uhl, T., Staszewski, W.J., Klepka, A.: Detection of changes in cracked aluminium plate determinism by recurrence analysis. Nonlinear Dyn. 70(1), 125–140 (2012)CrossRefGoogle Scholar
  19. 19.
    Kwuimy, C.K., Samadani, M., Nataraj, C.: Bifurcation analysis of a nonlinear pendulum using recurrence and statistical methods: applications to fault diagnostics. Nonlinear Dyn. 76(4), 1963–1975 (2014)Google Scholar
  20. 20.
    Ng, S.S., Cabrera, J., Tse, P., Chen, A., Tsui, K.: Distance-based analysis of dynamical systems reconstructed from vibrations for bearing diagnostics. Nonlinear Dyn. 80(1–2), 147–165 (2015)CrossRefGoogle Scholar
  21. 21.
    Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)Google Scholar
  22. 22.
    Letellier, C., Le Sceller, L., Dutertre, P., Gouesbet, G., Fei, Z., Hudson, J.: Topological characterization and global vector field reconstruction of an experimental electrochemical system. J. Phys. Chem. 99(18), 7016–7027 (1995)Google Scholar
  23. 23.
    Letellier, C., Gouesbet, G., Rulkov, N.: Topological analysis of chaos in equivariant electronic circuits. Int. J. Bifurc. Chaos 6(12b), 2531–2555 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lefranc, M.: The topology of deterministic chaos: stretching, squeezing and linking. NATO Secur. Through Sci. Ser D Inf. Commun. Secur. 7, 71 (2007)MathSciNetGoogle Scholar
  25. 25.
    Carroll, T.: Attractor comparisons based on density. Chaos Interdiscip. J. Nonlinear Sci. 25(1), 013111 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tufillaro, N., Holzner, R., Flepp, L., Brun, E., Finardi, M., Badii, R.: Template analysis for a chaotic NMR laser. Phys. Rev. A 44(8), R4786 (1991)CrossRefGoogle Scholar
  27. 27.
    King, G.P., Jones, R., Broomhead, D.: Phase portraits from a time series: a singular system approach. Nucl Phys B-Proc. Suppl. 2, 379–390 (1987)CrossRefGoogle Scholar
  28. 28.
    Samadani, M., Kwuimy, C.K., Nataraj, C.: Model-based fault diagnostics of nonlinear systems using the features of the phase space response. Commun. Nonlinear Sci. Numer. Simul. 20(2), 583–593 (2015)CrossRefGoogle Scholar
  29. 29.
    Samadani, M., Kwuimy, C.K., Nataraj, C.: Diagnostics of a nonlinear pendulum using computational intelligence. In: ASME 2013 Dynamic Systems and Control Conference, American Society of Mechanical Engineers (2013)Google Scholar
  30. 30.
    “ECP rectilinear plant.” (Online; accessed 29-June-2015)

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.The Villanova Center for Analytics of Dynamic Systems (VCADS), College of EngineeringVillanova UniversityVillanovaUSA
  2. 2.Department of Engineering Education, College of Engineering and Applied ScienceUniversity of CincinnatiCincinnatiUSA

Personalised recommendations