Volume 3, Issue 6, December 2015, Page: 265-270
Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems
Md. Mahafujur Rahaman, Department of Computer Science & Engineering, Z.H. Sikder University of Science & Technology, Shariatpur, Bangladesh
Received: Sep. 15, 2015;       Accepted: Nov. 2, 2015;       Published: Nov. 17, 2015
DOI: 10.11648/j.ajam.20150306.15      View  3196      Downloads  63
Abstract
With a view to obtaining the transient response of the system where triply eigenvalues are equal and another is distinct, we have considered a fourth order more critically damped nonlinear systems, and enquired into analytical approximate solution in this paper. We have also suggested that the results obtained by the proposed method correspond to the numerical solutions obtained by the fourth order Runge-Kutta method satisfactorily.
Keywords
KBM, Eigenvalues, More Critically Damped System, Nonlinearity, Runge-Kutta Method
To cite this article
Md. Mahafujur Rahaman, Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems, American Journal of Applied Mathematics. Vol. 3, No. 6, 2015, pp. 265-270. doi: 10.11648/j.ajam.20150306.15
Reference
[1]
Bogoliubov, N. N. and Mitropolskii, Y. A., Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York, 1961.
[2]
Krylov, N. N. and Bogoliubov, N. N., Introduction to Nonlinear Mechanics, Princeton University Press, New Jersey, 1947.
[3]
Popov, I. P., “A Generalization of the Bogoliubov Asymptotic Method in the Theory of Nonlinear Oscillations (in Russian)”, Dokl. Akad. USSR, vol. 3, pp. 308-310, 1956.
[4]
Mendelson, K. S., “Perturbation Theory for Damped Nonlinear Oscillations”, J. Math. Physics, vol. 2, pp. 3413-3415, 1970.
[5]
Murty, I. S. N., and Deekshatulu, B. L., “Method of Variation of Parameters for Over-Damped Nonlinear Systems”, J. Control, vol. 9(3), pp. 259-266, 1969.
[6]
Sattar, M. A., “An asymptotic Method for Second Order Critically Damped Nonlinear Equations”, J. Frank. Inst., vol. 321, pp. 109-113, 1986.
[7]
Murty, I. S. N., “A Unified Krylov-Bogoliubov Method for Solving Second Order Nonlinear Systems”, Int. J. Nonlinear Mech. vol. 6, pp. 45-53, 1971.
[8]
Osiniskii, Z., “Longitudinal, Torsional and Bending Vibrations of a Uniform Bar with Nonlinear Internal Friction and Relaxation,” Nonlinear Vibration Problems, vol. 4, pp. 159-166, 1962.
[9]
Mulholland, R. J., “Nonlinear Oscillations of Third Order Differential Equation,” Int. J. Nonlinear Mechanics, vol. 6. Pp. 279-294, 1971.
[10]
Bojadziev, G. N., “Damped Nonlinear Oscillations Modeled by a 3-dimensional Differential System”, Acta Mechanica, vol. 48, pp. 193-201, 1983.
[11]
Sattar, M. A., “An Asymptotic Method for Three-dimensional Over-damped Nonlinear Systems”, Ganit, J. Bangladesh Math. Soc., vol. 13, pp. 1-8, 1993.
[12]
Shamsul, M. A., “Asymptotic Methods for Second Order Over-damped and Critically Damped Nonlinear Systems”, Soochow Journal of Math., vol. 27, pp. 187-200, 2001.
[13]
Shamsul, M. A., “On Some Special Conditions of Third Order Over-damped Nonlinear Systems”, Indian J. Pure Appl. Math., vol. 33, pp. 727-742, 2002.
[14]
Akbar, M. A., Paul, A. C. and Sattar, M. A., “An Asymptotic Method of Krylov-Bogoliubov for Fourth Order Over-damped Nonlinear Systems”, Ganit, J. Bangladesh Math. Soc., vol. 22, pp. 83-96, 2002.
[15]
Murty, I. S. N., “Deekshatulu, B. L. and Krishna, G., “On an Asymptotic Method of Krylov-Bogoliubov for Over-damped Nonlinear Systems”, J. Frank. Inst., vol. 288, pp. 49-65, 1969.
[16]
Akbar, M. A., Shamsul, M. A. and Sattar M. A. “Asymptotic Method for Fourth Order Damped Nonlinear Systems”, Ganit, J. Bangladesh Math. Soc., Vol. 23, pp. 41-49, 2003.
[17]
Akbar, M. A., Shamsul, M. A. and Sattar M. A., “Krylov-Bogoliubov-Mitropolskii Unified Method for Solving n-th Order Nonlinear Differential Equations Under Some Special Conditions Including the Case of Internal Resonance”, Int. J. Non-linear Mech, Vol. 41, pp. 26-42, 2006.
[18]
Akbar, M. A., Uddin, M. S., Islam, M. R. and Soma, A. A. “Krylov-Bogoliubov-Mitropolskii (KBM) Method for Fourth Order More Critically Damped Nonlinear Systems”, J. Mech. of Continua and Math. Sciences, Vol. 2(1), pp. 91-107, 2007.
[19]
Islam, M. R., Rahman, M. H. and Akbar, M. A. “An Analytical Approximate Solution of Fourth Order More Critically Damped Nonlinear Systems”, Indian Journal of Mathematics, Vol. 50(3), pp. 611-626, 2009.
[20]
Hakim, M. A. “On Fourth Order More Critically Damped Nonlinear Differential Systems”, Journal of Physical Science, Vol. 15, pp.113-127, 2011.
[21]
Akbar, M. A., Shamsul, M. A. and Sattar, M. A. “A Simple Technique for Obtaining Certain Over-damped Solutions of an n-th Order Nonlinear Differential Equation”, Soochow Journal of Mathematics, Vol. 31(2), pp. 291-299, 2005.
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