Approximate Solution of Strongly Forced Nonlinear Vibrating Systems Which Vary With Time


Pinakee dey,Nasir Uddin,Md Asaduzzaman,Sanjay kumar saha,M. A. Sattar,



Asymptotic solution,Forced nonlinear oscillation, Varying coefficient,Unperturbed equation, KBM method, HB method,


Based on the combined work of extended Krylov-Bogoliubov-Mitropolskii method and harmonic balance (HB) method an analytical technique is presented to determine approximate solutions of nonlinear differential systems whose coefficients change slowly and periodically with time. Furthermore, a non-autonomous case also investigated in which an external force acts in this system. Formulation as well as determination of the solution is systematic and easier than the existing procedures. The method is illustrated by suitable examples.


I.C. W. Lim and B. S. Wu. “A new analytical approach to the Duffing-harmonic oscillator”, Physics Letters A, 311, 365-373, 2003.

II.I. S. N. Murty, “A unified Krylov-Bogoliubov method for second order nonlinear systems”, Int. J. nonlinear Mech. 6. 45-53. 1971.

III.I. P. Popov. “A generalization of the Bogoliubov asymptotic method in the theory of nonlinear oscillations”, Dokl.Akad. Nauk SSSR 111, 308-310 (in Russian) 1956.

IV.J. C. Arya and G. N. Bojadziev, “Damped oscillating systems modeled by hyperbolic differential equations with slowly varying coefficients”, Acta Mechanica, 35, 215-221, 1980.

V.K.C. Roy and M. Shamsul Alam. “Effect 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, 26,182-192, 2004.

VI.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. 987-1002, 2003.

VII.N. N. Bogoliubov and Yu. Mitropolskii.“Asymptotic methods in the theory of nonlinear oscillations”, Gordan and Breach, New York, 1961.

VIII.N. N. Krylov and N. N. Bogoliubov, “Introduction to nonlinear mechanics”. Princeton University Press, New Jersey, 1947.

IX.R. E. Mickens. “Oscillation in Planar Dynamic Systems”, World Scientific, Singapore, (1996).

X.Yu. Mitropolskii. “Problems on asymptotic method of non-stationary oscillations” (in Russian), Izdat, Nauka, Moscow, 1964.

Author(s): Pinakee dey, Nasir Uddin, Md Asaduzzaman, Sanjay kumar saha, M. A. Sattar View Download