Dynamics of Systems with Randomly Disordered Periodic Excitations

  • M. Dimentberg
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


A model of a periodic process with random phase modulation, or disorder, is described. It can be easily incorporated into Stochastic Differential Equations Calculus, thereby providing potential for analytical solution to dynamic problems where it represents the forcing function, or excitation. Thus, the corresponding method of moments had been applied to a linear system subject to external and parametric excitation with preliminary reduction of the equation of motion by asymptotic stochastic averaging in the latter case; boundaries for parametric instability had been derived both in the mean square and in the almost sure sense. Solution for a strongly nonlinear system with impacts had also been obtained illustrating potentially strong influence of imperfect periodicity of excitation on response subharmonics. Examples of application from engineering mechanics are presented.


Bounded noise Non-Gaussian processes Stochastic differential equations Stochastic mechanics Engineering mechanics Parametric resonance Subharmonics 


  1. 1.
    Dimentberg, M.: Statistical Dynamics of Nonlinear and Time-Varying Systems. Research Studies Press, Taunton (1988)MATHGoogle Scholar
  2. 2.
    Dimentberg, M.: A stochastic model of parametric excitation of a straight pipe due to slug flow of a two-phase fluid.: In: Proceedings of the 5th International Symposium on Flow-Induced Vibrations, pp. 207–209, Brighton, UK (1991)Google Scholar
  3. 3.
    Dimentberg, M.: Probab. Eng. Mech. 7, 131–134 (1992)CrossRefGoogle Scholar
  4. 4.
    Dimentberg, M., Bucher, C.: J. Sound Vib. 331, 4373 (2012)CrossRefGoogle Scholar
  5. 5.
    Dimentberg, M., Hou, Z., Noori, M., Zhang W.: Non-Gaussian response of a single-degree-of-freedom system to a periodic excitation with random phase modulation. In: ASME Special Volume: Recent Developments in the Mechanics of Continua, ASME-AMD, 160, pp. 27–33 (1993)Google Scholar
  6. 6.
    Dimentberg, M., Hou, Z., Noori, M.: Stability of a SDOF system under periodic parametric excitation with a white-noise phase modulation. In: Kliemann, W., Sri Namachivaya, N. (eds.) Nonlinear Dynamics and Stochastic Mechanical, CRC Press (1995)Google Scholar
  7. 7.
    Dimentberg, M., Hou, Z., Noori, M., Zhang, W.: J. Sound Vib. 192(3), 621–627 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dimentberg, M., Iourtchenko, D.: Probab. Eng. Mech. 14, 323–328 (1999)CrossRefGoogle Scholar
  9. 9.
    Dimentberg, M., Iourtchenko, D.: Int. J. Bifurc. Chaos 15, 2057–2061 (2005)CrossRefGoogle Scholar
  10. 10.
    Dimentberg, M., Iourtchenko, D., van-Ewijk, O.: Nonlinear Dyn. 17, 173–168 (1998)Google Scholar
  11. 11.
    Dimentberg, M., Iourtchenko, D., van Ewij, O.: Subharmonic response of moored systems to ocean waves. In: Spencer, B., Johnson, E. (eds.) Stochastic Structural Dynamics, pp. 495–498. Balkema, Rotterdam (1999)Google Scholar
  12. 12.
    Dimentberg, M., Mo, E., Naess, A.J.: Eng. Mech., 133, 1037–1041 (2007)
  13. 13.
    Hou, Z., Wang, Y., Dimentberg, M., Noori, M.: Probab. Eng. Mech. 14, 83–95 (1999)CrossRefGoogle Scholar
  14. 14.
    Hou, Z., Zhou, Y., Dimentberg, M., Noori, M.: Probab. Eng. Mech. 10, 73–81 (1995)CrossRefGoogle Scholar
  15. 15.
    Hou, Z., Zhou, Y., Dimentberg, M., Noori, M.: J. Eng. Mech. 122, 1101–1109 (1996)CrossRefGoogle Scholar
  16. 16.
    Hou, Z., Zhou, Y., Dimentberg, M., Noori, M.: Stochastic models for disordered periodic processes and their applications. In: Shlesinger, M., Swean, T. (eds.) Stochastically Excited Nonlinear Ocean Structures, pp. 225–251. World Scientific, Singapore (1998)CrossRefGoogle Scholar
  17. 17.
    Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics. Advanced Theory and Applications. McGraw-Hill, New York (1995)Google Scholar
  18. 18.
    Naess, A., Dimentberg, M., Gaidai, O.: Phys. Rev. E 78, 021126 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nayfeh, A., Mook, D.: Nonlinear Oscillations. Wiley, New York (1979)MATHGoogle Scholar
  20. 20.
    Stratonovich, R.L.: Topics in the Theory of Random Noise, vol. II. Gordon and Breach, New York (1967)MATHGoogle Scholar
  21. 21.
    Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos (Chapters 14, 15). Wiley, Chichester (1986)Google Scholar
  22. 22.
    Wedig, W.V.: Analysis and simulation of nonlinear stochastic systems. In: Schielen, W. (ed.) Nonlinear Dynamics in Engineering Systems, pp. 337–344. Springer, New York (1989)Google Scholar
  23. 23.
    Zhuravlev, V.F., Klimov, D.M.: Applied Methods in Vibration Theory (in Russian). Nauka, Moscow (1988)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringWorcester Polytechnic InstituteWorcesterUSA

Personalised recommendations