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Quasi-linearization of non-linear systems under random vibration by probabilistic method

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Abstract

Vibration of a non-linear system under random parametric excitations was evaluated by probabilistic methods. The non-linear characteristic terms of a system were quasi-linearized and excitation terms remained as they were. An analytical method where the square mean of error was minimized was used. An alternative method was an energy method where the damping energy and restoring energy of the linearized system were equalized to those of the original nonlinear system. The numerical results were compared with those obtained by Monte Carlo simulation. A new method was proposed from the comparison of those results. Finally the results obtained by the combined method showed good agreement with those obtained by Monte Carlo simulation.

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References

  1. J. B. Roberts and P. D. Spanos, Random Vibration and Statistical Linearization, New York: Dover Publications, Inc., (1999).

    Google Scholar 

  2. J. B. Chen and J. Li, Dynamic response and reliability analysis of non-linear stochastic structure, Probabilistic Engineering Mechanics, 20 (2005) 33–44.

    Article  Google Scholar 

  3. G. Deodatis, Stochastic FEM sensitivity analysis of nonlinear dynamic problems, Probabilistic Engineering Mechanics, 4 (1989) 135–141.

    Article  Google Scholar 

  4. C. H. S. To, Nonlinear Random Vibration: Analytical Techniques and Applications, the Netherlands: Swets & Zeitlinger B.V., (2000).

    MATH  Google Scholar 

  5. Y. K. Lin and G. Q. Cai, Probabilistic Structural Dynamics: Advanced Theory and Applications, New York: McGraw-Hill, (2004).

    Google Scholar 

  6. W. D. Iwan and C. T. Huang, On the dynamic response of non-linear systems with parameter uncertainty, International Journal of Non-Linear Mechanics 31 (1996) 631–645.

    Article  MATH  Google Scholar 

  7. S. Y. Lee and G. Q. Cai, Random vibration of nonlinear system with multiple degrees of freedom, Trans. of the KSMTE, Korea 15 (2006) 21–28.

    Google Scholar 

  8. S. Y. Lee, Vibration of non-linear system under random parametric excitations by probabilistic method, Journal of the KSPE, Korea 23(12) (2006) 72–79.

    Google Scholar 

  9. S. Y. Lee and G. Q. Cai, Linearization of nonlinear random vibration beam by equivalent energy method, Trans. of the KSMTE, Korea 17 (2008) 71–76.

    Google Scholar 

  10. P. Bernard, Stochastic linearization: what is available and what is not, Computers and Structures, 67 (1988) 9–18.

    Article  Google Scholar 

  11. S. T. Ariaratnam and T. K. Srikantaiah, Oarametric instabilities in elastic structures under stochastic loading, Journal of Structural Mechanics, 6 (1978) 349–365.

    Google Scholar 

  12. P. Marek, J. Brozzetti and M. Gustar, Prabobilistic Assessment of Structures using Monte Carlo Simulation, Praha Czech: Institute of Theoretical and Applied Mechanics (2001).

    Google Scholar 

  13. D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge: Cambridge University Press (2000).

    MATH  Google Scholar 

  14. P. D. Spanos and B. A. Zeldin, Monte Carlo treatment of random fields: a broad perspective, Applied Mechanics Review, 51 (1998) 219–237.

    Article  Google Scholar 

  15. L. Meirovitch, Introduction to Dynamics and Control, New York: John Wiley & Sons, (1985).

    Google Scholar 

  16. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in C, New York: Cambridge University Press, (1988).

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mechanical and Automotive Engineering, Kunsan National University, Kunsan, 573-701, Korea

    Sin-Young Lee

  2. Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, 33431, USA

    Guoqiang Cai

Authors
  1. Sin-Young Lee
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  2. Guoqiang Cai
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Correspondence to Sin-Young Lee.

Additional information

This paper was recommended for publication in revised form by Associate Editor Seokhyun Kim

Sin-Young Lee received a B.S. degree in Mechanical Engineering from Seoul National University in 1979. He worked in a company for 5 years and returned to receive his M.S. and Ph.D. degrees from Seoul National University in 1986 and 1989, respectively. Dr. Lee is currently a Professor at the School of Mechanical & Automotive Engineering at Kunsan National University in Kunsan, Korea. Dr. Lee’s research interests are in the area of machine tools, vibration and noise.

Guoqiang Cai is currently a Professor at the department of Mechanical Engineering at Florida Atlantic University in Florida, USA. Prof. Cai’s research interests are in the area of stochastics, structural engineering, dynamics, and reliability analysis.

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Lee, SY., Cai, G. Quasi-linearization of non-linear systems under random vibration by probabilistic method. J Mech Sci Technol 23, 2022–2029 (2009). https://doi.org/10.1007/s12206-009-0110-4

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  • DOI: https://doi.org/10.1007/s12206-009-0110-4

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