Nonlinear Dynamics

, Volume 42, Issue 2, pp 113–136 | Cite as

Control of a Nonlinear System Using the Saturation Phenomenon



In this paper, a theoretical investigation of nonlinear vibrations of a 2 degrees of freedom system when subjected to saturation is studied. The method has been especially applied to a system that consists of a DC motor with a nonlinear controller and a harmonic forcing voltage. Approximate solutions are sought using the method of multiple scales. It is shown that the closed-loop system exhibits different response regimes. The nature and stability of these regimes are studied and the stability boundaries are obtained. The effects of the initial conditions on the response of the system have also been investigated. Furthermore, the second-order solution is presented and the corresponding results are compared with those of the first-order solution. It is shown that by increasing the amplitude of the excitation voltage, the higher-order term in the solution becomes significant and causes a drift in the response. In order to verify the obtained theoretical results, they are compared with the corresponding numerical results. Good agreement between the two sets of results is observed.


energy transfer internal resonance method of multiple scales saturation stability vibration control 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sethna, P. R., ‘Vibrations of dynamical systems with quadratic nonlinearities’, Journal of Applied Mechanics 32, 1965, 576–582.Google Scholar
  2. 2.
    Froude, W., ‘Remarks on Mr. Scott Russell's paper on rolling’, Transactions of the Institute of Naval Architects 4, 1863, 232–275.Google Scholar
  3. 3.
    Nayfeh, A. H., Mook, D. T., and Marshall, L. R., ‘Nonlinear coupling of pitch and roll modes in ship motion’, Journal of Hydronautics 7, 1973, 145–152.Google Scholar
  4. 4.
    Tuer, K. L., Golnaraghi, M. F., and Wang, D., ‘Development of a generalized active vibration suppression strategy for a cantilever beam using internal resonance’, Nonlinear Dynamics 5, 1994, 131–151.Google Scholar
  5. 5.
    Golnaraghi, M. F., ‘Regulation of flexible structures via nonlinear coupling’, Journal of Dynamics and Control 1, 1991, 405–428.CrossRefGoogle Scholar
  6. 6.
    Khajepour, A., Golnaraghi, M. F., and Morris, K. A., ‘Theoretical development of a nonlinear modal coupling controller for forced vibration applications using normal forms’, in Proceedings of the 32nd Annual Technical Meeting of the Society of Engineering Science, New Orleans, LA, 1995, pp. 263–364.Google Scholar
  7. 7.
    Salemi, P., Golnaraghi, M. F., and Heppler, G. R., ‘Control of forced and unforced structural vibration using a linear coupling strategy’, in Proceedings of the 32nd Annual Technical Meeting of the Society of Engineering Science, New Orleans, LA, 1995, pp. 221–222.Google Scholar
  8. 8.
    Oueini, S. S., Nayfeh, A. H., and Golnaraghi, M. F., ‘A theoretical and experimental implementation of a control method based on saturation’, Nonlinear Dynamics 13, 1997, 189–202.CrossRefGoogle Scholar
  9. 9.
    Shoeybi, M. and Ghorashi, M., ‘Saturation and its applications in the vibration control of non-linear systems’, in Proceedings of ESDA, 7th Biennial Conference of Engineering Systems, Design, and Analysis, Manchester, UK, July 19–22, 2004, ESDA 2004-58297 (on CD).Google Scholar
  10. 10.
    Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981.Google Scholar
  11. 11.
    Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley-Interscience, New York, 1979.Google Scholar
  12. 12.
    Nayfeh, A. H., Nonlinear Interactions, Wiley-Interscience, New York, 2000.Google Scholar
  13. 13.
    Ogata, K., Modern Control Engineering, Prentice-Hall, New Jersey, 2002.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentSharif University of TechnologyTehranIran
  2. 2.Flow Physics and Computation DivisionDepartment of Mechanical Engineering, Stanford UniversityStanfordU.S.A.
  3. 3.The Rotorcraft Research Group, Department of Mechanical and Aerospace EngineeringCarleton UniversityOttawaCanada

Personalised recommendations