Statistical Linearization of Duffing Oscillator Using Constrained Optimization Technique

  • Sabarethinam Kameshwar
  • Arunasis ChakrabortyEmail author
Conference paper


The present work aims to evaluate the response of Duffing oscillator using equivalent linearization. The stiffness and damping forces are proportional to the cube power of the displacement and velocity, respectively. The oscillator is excited by stationary process. The method suggested in this work aims to replace the original nonlinear system with an equivalent linear system by minimizing the difference in the displacement between the nonlinear system and the equivalent linear system in a least square sense using different constraints (e.g., restoring force, potential energy, complementary energy). Numerical results are presented to show the efficiency of the proposed linearization scheme. For this purpose, instantaneous mean square values of the displacement are evaluated and compared with simulation. A close agreement between the simulations and the proposed model is observed which, in turn, shows the efficiency and applicability of the proposed model. A discussion on the use of different constraint conditions and their relative importance is also presented.


Equivalent linearization Constrained optimization Duffing oscillator Stationary process 


  1. 1.
    Bulsara AR, Katja L, Shuler KE, Rod F, Cole WA (1982) Analog computer simulation of a duffing Oscillator and comparison with statistical linearization. Int J Non-Linear Mech 17(4):237–253CrossRefGoogle Scholar
  2. 2.
    Caughey T (1963) Derivation and application of Fokker-Planck equation to discrete dynamics system subjected to white random excitation. J Acoust Soc Am 35:1683–1692MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caughey T (1963) Equivalent linearization techniques. J Acoust Soc Am 35:1706–1711MathSciNetCrossRefGoogle Scholar
  4. 4.
    Elishakoff I, Zhang XT (1992) An appraisal of different stochastic linearization criteria. J Sound Vib 153:370–375MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Elishakoff I, Bert CW (1999) Complementary energy criterion in nonlinear stochastic dynamics. In: Melchers RL, Stewart MG (eds) Application of stochastic and probability. A. A. Balkema, Rotterdam, pp 821–825Google Scholar
  6. 6.
    Elishakoff I (2000) Multiple combinations of the stochastic linearization criteria by the moment approach. J Sound Vib 237(3):550–559CrossRefGoogle Scholar
  7. 7.
    Grigoriu M (1995) Equivalent linearization for Poisson White noise input. Probab Eng Mech 10:45–51CrossRefGoogle Scholar
  8. 8.
    Grigoriu M (2000) Equivalent linearization for systems driven by Levy White noise. Probab Eng Mech 15:185–190CrossRefGoogle Scholar
  9. 9.
    Hammond J (1973) Evolutionary spectra in random vibrations. J R Stat Soc 35:167–188MathSciNetzbMATHGoogle Scholar
  10. 10.
    Iyengar RN (1988) Stochastic response and stability of the Duffing oscillator under narrow band excitation. J Sound Vib 126(2):255–263MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Mickens R (2003) A combined equivalent linearization and averaging perturbation method for non-linear oscillator equations. J Sound Vib 264:1195–1200MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Noori M, DavoodI H (1990) Comparison between equivalent linearization and Gaussian closure for random vibration analysis of several nonlinear systems. Int J Eng Sci 28(9):897–905MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Proppe C (2003) Stochastic linearization of dynamical systems under parametric Poisson White noise excitation. Int J Non-Linear Mech 38:543–555MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Ricciardi G (2007) A non-Gaussian stochastic linearization method. Probab Eng Mech 22:1–11CrossRefGoogle Scholar
  15. 15.
    Roberts J, Spanos P (1990) Random vibration and statistical linearization. Wiley, New YorkzbMATHGoogle Scholar
  16. 16.
    Sobiechowski C, Socha L (2000) Statistical linearization of the duffing oscillator under non-Gaussian external excitation. J Sound Vib 231:19–35MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Wu W-F (1987) Comparison of Gaussian closure technique and equivalent linearization method. Probab Eng Mech 2(1):2–8CrossRefGoogle Scholar

Copyright information

© Springer India 2013

Authors and Affiliations

  1. 1.Department of Civil EngineeringIndian Institute of Technology GuwahatiGuwahatiIndia

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