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Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions

Published: 30 December 2012
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Abstract

Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The solution obtained for different initial conditions for a second order nonlinear system show a good coincidence with those obtained by numerical method. The method is illustrated by an example.

Published in Applied and Computational Mathematics (Volume 1, Issue 1)
DOI 10.11648/j.acm.20120101.11
Page(s) 1-5
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2012. Published by Science Publishing Group

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Keywords

Nonlinear System, Varying Coefficient, Unperturbed Equation, Damped Oscillatory System

References
[1] N.N, Krylov and N.N., Bogoliubov, Introduction to Nonlinear Mechanics. Princeton University Press, New Jersey, 1947.
[2] N. N, Bogoliubov and Yu. Mitropolskii, Asymptotic Methods in the Theory of nonlinear Oscillations, Gordan and Breach, New York, 1961.
[3] Yu.,Mitropolskii, "Problems on Asymptotic Methods of Non-stationary Oscillations" (in Russian), Izdat, Nauka, Moscow, 1964.
[4] I. P. Popov, "A generalization of the Bogoliubov asymptotic method in the theory of nonlinear oscillations", Dokl.Akad. Nauk SSSR 111, 1956, 308-310 (in Russian).
[5] G.,Bojadziev, and J. Edwards, "On Some Asymptotic Methods for Non-oscillatory and Oscillatory Processes", Nonlinear Vibration Problems, 20, 1981, pp69-79.
[6] I.S.N. Murty, "A Unified Krylov-Bogoliubov Method for Second Order Nonlinear Systems", Int. J. nonlinear Mech., 6, 1971, pp45-53.
[7] M.,Shamsul Alam, "Unified Krylov-Bogoliubov-Mitropolskii Method for Solving n-th Order Nonlinear System with Slowly Varying Coefficients", Journal of Sound and Vibration, 256, 2003, pp987-1002.
[8] Hung Cheng and Tai Tsun Wu, "An Aging Spring, Studies in Applied Mathematics", 49, 1970, pp183-185.
[9] K.C, Roy, M. Shamsul Alam, "Effects of Higher Approximation of Krylov- Bogoliubov-Mitropolskii Solution and Matched Asymptotic Solution of a Differential System with Slowly Varying Coefficients and Damping Near to a Turning Point", Vietnam Journal of Mechanics, VAST, 26, 2004, pp182-192.
[10] Pinakee Dey., Harun or Rashid, M. Abul Kalam Azad and M S Uddin, "Approximate Solution of Second Order Time Dependent Nonlinear Vibrating Systems with Slowly Varying Coefficients", Bull. Cal. Math. Soc, 103, (5), 2011, pp371-38.
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  • APA Style

    Pinakee Dey, Babul Hossain, Musa Miah, Mohammad Mokaddes Ali. (2012). Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions. Applied and Computational Mathematics, 1(1), 1-5. https://doi.org/10.11648/j.acm.20120101.11

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    ACS Style

    Pinakee Dey; Babul Hossain; Musa Miah; Mohammad Mokaddes Ali. Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions. Appl. Comput. Math. 2012, 1(1), 1-5. doi: 10.11648/j.acm.20120101.11

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    AMA Style

    Pinakee Dey, Babul Hossain, Musa Miah, Mohammad Mokaddes Ali. Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions. Appl Comput Math. 2012;1(1):1-5. doi: 10.11648/j.acm.20120101.11

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  • @article{10.11648/j.acm.20120101.11,
      author = {Pinakee Dey and Babul Hossain and Musa Miah and Mohammad Mokaddes Ali},
      title = {Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions},
      journal = {Applied and Computational Mathematics},
      volume = {1},
      number = {1},
      pages = {1-5},
      doi = {10.11648/j.acm.20120101.11},
      url = {https://doi.org/10.11648/j.acm.20120101.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20120101.11},
      abstract = {Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The solution obtained for different initial conditions for a second order nonlinear system show a good coincidence with those obtained by numerical method. The method is illustrated by an example.},
     year = {2012}
    }
    

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    AB  - Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The solution obtained for different initial conditions for a second order nonlinear system show a good coincidence with those obtained by numerical method. The method is illustrated by an example.
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Author Information
  • Department of Mathematics, Mawlana Bhashani Science and Technology University, Santosh, Tangail-1902, Bangladesh.

  • Department of Mathematics, Mawlana Bhashani Science and Technology University, Santosh, Tangail-1902, Bangladesh.

  • Department of Mathematics, Mawlana Bhashani Science and Technology University, Santosh, Tangail-1902, Bangladesh.

  • Department of Mathematics, Mawlana Bhashani Science and Technology University, Santosh, Tangail-1902, Bangladesh.

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