### International Journal of Mathematical Sciences and Computing(IJMSC)

*ISSN: *2310-9025 (Print), *ISSN: *2310-9033 (Online)

*Published By: *MECS Press

*IJMSC Vol.4, No.3, Jul. 2018*

#### Asymptotic Solutions of a Semi-submerged Sphere in a Liquid under Oscillations

Full Text (PDF, 507KB), PP.66-79

Views:43 Downloads:1

#### Author(s)

#### Index Terms

#### Abstract

One of the most widely used techniques to look into transient behaviour of vibrating systems is the Krylov-Bogoliubov-Mitropolskii (KBM) method, which was developed for obtaining the periodic solutions of second order nonlinear differential systems of small nonlinearities. Later on, this method was studied and modified by numerous scholars to obtain solutions of higher order nonlinear systems. This article modified the method to study the solutions of semi-submerged sphere in a liquid which is floating owing to the gravitational force and the upward pressure of the liquid. This paper suggests that the results obtained for different sets of initial conditions by the modified KBM method correspond well with those obtained by the numerical method.

#### Cite This Paper

Shamima Aktar, M. Abul Kawser,"Asymptotic Solutions of a Semi-submerged Sphere in a Liquid under Oscillations", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.4, No.3, pp.66-79, 2018.DOI: 10.5815/ijmsc.2018.03.05

#### Reference

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

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

[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., 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.

[6]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.

[7]Sattar, M. A., An asymptotic Method for Second Order Critically Damped Nonlinear Equations, J. Frank. Inst., Vol. 321, pp. 109-113, 1986.

[8]Shamsul Alam, M., Asymptotic Methods for Second Order Over-damped and Critically Damped Nonlinear Systems, Soochow Journal of Math. Vol. 27, pp. 187-200, 2001.

[9]Mandelstam, L. and Papalexi N., Expose des Recherches Recentres sur les Oscillations Non-linaires, Journal of Technical physics, USSR, 1934.

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

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

[12]Murty, I. S. N., and Deekshatulu B. L., Method of Variation of Parameters for Over-Damped Nonlinear Systems, J. Control, Vol. 9, no. 3, pp. 259-266,1969.

[13]Bojadziev, G. N., On Asymptotic Solutions of Nonlinear Differential Equations with Time Lag, Delay and Functional Differential Equations and Their Applications (edited by K. Schmit), 299-307, New York and London: Academic Press, 1972.

[14]Osiniskii, Z., Vibration of a One Degree Freedom System with Nonlinear Internal Friction and Relaxation, Proceedings of International Symposium of Nonlinear Vibrations, Vol. 111, pp. 314-325, Kiev, Izadt, Akad, Nauk USSR, 1963.

[15]Mulholland, R. J., Nonlinear Oscillations of Third Order Differential Equation, Int. J. Nonlinear Mechanics, Vol. 6, pp. 279-294, 1971.

[16]Lardner, R. W. and Bojadziev G. N., Vibration of a Viscoelastic Rod with Small Nonlinearities, Meccanica, Vol. 12, pp. 249-256, 1977.

[17]Bojadziev, G. N., Damped Nonlinear Oscillations Modelled by a 3-dimensional Differential System, Acta Mechanica, Vol. 48, pp. 193-201, 1983.

[18]Bojadziev, G. N. and Hung C. K., Damped Oscillations Modelled by a 3-dimensional Time Dependent Differential Systems, Acta Mechanica, Vol. 53, pp. 101-114, 1984.

[19]Shamsul Alam, M., Ali Akbar M., Zahurul Islam M., A General Form of Krylov-Bogoliubov-Mitropolskii Method for Solving Non-linear Partial Differential Equations, Journal of Sound and Vibration, Vol. 285, pp. 173-185, 2005.

[20]Raymond P. Vito and Cabak G., The Effects of Internal Resonance on Impulsively Forced Nonlinear Systems with Two Degree of Freedom, Int. J. Nonlinear Mechanics, Vol. 14, pp. 93-99, 1979.

[21]Shamsul Alam, M., Perturbation Theory for n-th Order Nonlinear Systems with Large Damping, Indian J. pure appl. Math. Vol. 33, pp. 1677-1684, 2002.

[22]Shamsul Alam, M., Method of Solution to the n-th Order Over-damped Nonlinear Systems Under Some Special Conditions, Bull. Cal. Math. Soc., Vol. 94, pp. 437-440, 2002.

[23]Shamsul Alam M., Approximate Solutions of Non-oscillatory Systems, Mathematical Forum, Vol. 14, pp. 7-16, 2001-2002.

[24]Shamsul Alam M., Asymptotic Method for Certain Third-order Non-oscillatory Nonlinear Systems, J. Bangladesh Academy of Sciences, Vol. 27, pp. 141-148, 2003.

[25]Ali Akbar, M., Shamsul Alam M. and Sattar M. A., Asymptotic Method for Fourth Order Damped Nonlinear Systems, Ganit, J. Bangladesh Math. Soc. Vol. 23, pp. 41-49, 2003.

[26]Ali Akbar, M., 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.

[27]Ali Akbar, M., Shamsul Alam M. 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.

[28]Ali Akbar, M., M. Shamsul Alam, S. S. Shanta, M. Sharif Uddin and M. Samsuzzoha, "Perturbation Method for Fourth Order Nonlinear Systems with Large Dampingâ€ť, Bulletin of Calcutta Math. Soc., Vol. 100(1), pp. 85-92, 2008.

[29]Abul Kawser, M. and Ali Akbar M., an Asymptotic Solution for the Third Order Critically Damped Nonlinear System in the Case for Small Equal Eigenvalues, J. Math. Forum, Vol. XXII, pp. 52-68, 2010.