ASYMPTOTIC SOLUTION OF FOURTH ORDER OVER DAMPED SYMMETRICAL NONLINEAR SYSTEMS

Authors:

M.Ali Akbar,Anup Kumar Datta,Md.Eliyas Karim,

DOI NO:

https://doi.org/10.26782/jmcms.2009.07.00005

Keywords:

over-damped symmetrical system ,transient response,forth order non-linear differential equation,eigen values,

Abstract

A fourth order nonlinear differention equation modeling an over-damped symmetrical system is considered. A perturbation technique is developed in this artical for obtaining the transient responsewhen the eigenvalues are in integral multiple. The results obtained by the presented technique agree with those results obtained by the numerical method nicely. An example is solved to illustrated method.

Refference:

I. Ali Akber, M., A.C. Paul and M.A. Sattar, An Asymptotic Method of krylov-Bogoliubov for fourth order over-damped nonlinear system, Geint, J. Bangladesh Math. Soc., vol.22, pp. 83-96, 2002.

II. All  Akbar, M.M. Shamsul Alam and M.A. Sattar, Asymptotic Method for Forth order Damped Nonliner System, Ganit, J. Bangladesh Math. Soc., vol.23, pp. 83-96, 2002.

III. Ali Akbar, M.M. Shamsul Alam and M.A. Satter,  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.

IV. Bogoliubov, N.N. and Yu. Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York, 1961.

V. Krylov, N.N. and N.N. Bogoliubov, Introduction to Nonlinear Mechanics, Princeton University Press, New Jersey, 1947.

VI. Mulholland, R.J., Nonlinear Oscillations of Third Order Differential Equation, Int.J. Nonlinear Mechanics, vol.6, pp.279-294, 1971.

VII. Murty, I.S.N., B.L. Deekshatulu and G.Krishna, on an Asymptotic Method of Krylov-Bogoliubov for Over-damped Nonlinear Systems, J.Frank. Inst., vol.288, pp.49-65, 1969.

VIII. 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.

IX. Osiniskii. Z., Vibration of a one degree Freedom system with Nonlinear Internal Friction and Rrlaxation, Proceedings of intermations Symposium of Nonlinear Vibrations, vio.111, pp. , . 314-325 Kiew, Lazst, Akand, Nauk Ukr. SSR, 1963.

X. Popov, I.P., A Generalization of the Bogoliubov Asymptotic Method in the Theory of Nonlinear Oscillations (in Russian), Doll. Akad. USSR Vol.3, pp.308-310, 1956.

XI. Sattar, M.A., An asymmetric method for second Order Critically Damped Nonlinear Equations, J.Frank.Inst., vol. 321, pp.109-113, 1986.

XII. Sattar, M.A., An Asymptotic Method for Three-dimensional Over-damped Nonlinear Systems, Habit, J. Bangladesh Math. Soc., Vol.13, pp.1-8, 1993.

XII. Shamsul Alam, M., Asymptotic Methods for second-order Over-damped and Critically Damped Nonlinear Systems, Soochow J. Of Math., Vol.27, pp.187-200, 2001.

XIII. Sattar, M.A., An Asymptotic Method for Three-dimensional Over-damped Nonlinear Systems, Ganit, J. Bangladesh Math. Soc., Vol.13, pp.1-8, 1993.

XIV. Shamsul Alam, M., A Unified Krylov-Bogoliubov-Mitropolskii Method for Solving n-th Order Nonlinear Systems, J.Frank.Inst., vol.339, pp. 239-248, 2002.

XV.Shamsul Alam,M., Some special conditions of third order Over-damped Nonlinear Systems, Indian J. Pure APPL. Math., Vol. 33, pp.727-742, 2002.

 

M. Ali Akbar, Anup Kumar Dutta,Md. Eliyas Karim View Download